View Full Version : Why does math work in our reality?
Perspectives
Sep3-09, 02:02 AM
I知 reviewing my mathematics knowledge, except I知 looking for a different reason. I understand how it works you know, 1 plus 1 so on, I知 trying to understand why it works.
Pure and applied mathematicians and physicists have tied our understanding of reality with mathematics. They believe that if it computes it is real. This will do for a while, till we pose a question beyond our understanding.
But that痴 for a different time. Why does it work?
Any takers?
Hi there,
You have a different uderstanding of mathematics applications than mine. I don't see mathematics as an end solution to undrstand different real situations.
No matter what happens, with or without scientists, the world keeps on turning, and objects keep on falling when dropped. Therefore, I see science as a way to MODEL mathematically certain real life situations. The models can be right on the dot or far away from the truth, but the idea is to try to explain certain situation through mathematical models. Therefore, these models are made up to work in this world.
Take again falling objects. To my knowledge, it never happened that an object started floating when dropped. In addition, it is simple to see that the speed of the falling object increases. Therefore, scientists worked up a mathematical model that tries to explains this situation, anyways that tries to get as close as possible to reality.
Cheers
In 1960, the physicist Eugene Wigner published a beautiful essay titled
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
Here it is, enjoy the reading! :smile:
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
HallsofIvy
Sep3-09, 07:20 AM
This depends strongly on what you mean by "works". NO mathematics exactly fits a physical situation ("reality" if you like). All "reality" involves measurements that are approximate so the best we can hope for from a mathematical theory is that it work approximately.
All mathematical structures (theories) are "templates". Every mathematical structure involves "undefined terms", words that are defined using those undefined terms, axioms, and theorems proved from those axioms. To apply a mathematical structure to "reality", we have to assign meanings to those undefined terms. IF the axioms are true with those meanings assigned, then all theorems proved from those axioms are true and all methods of solving problems derived from those theorems will work.
Of course, the axioms [b]won't[b\] be perfectly true, only approximately true. The key to the "Unreasonable Effectiveness of Mathematics in the Natural Sciences" is having a large array of mathematical structures and choosing the one that best fits the specific application.
This depends strongly on what you mean by "works". NO mathematics exactly fits a physical situation ("reality" if you like). All "reality" involves measurements that are approximate so the best we can hope for from a mathematical theory is that it work approximately.
But does that mean that in the future the same can be said? That presently we just don't have a full understanding of mathematics.
I've been wondering this myself. It would've been a strange feeling for the first person who realized early on that mathematics actually had a use in real life. Its almost like the matrix, that behind the veil of reality its just numbers on paper.
Mentallic
Sep3-09, 09:00 AM
But does that mean that in the future the same can be said? That presently we just don't have a full understanding of mathematics.
It's not our understanding of mathematics that is flawed, it's the models we use to represent reality. Approximations will be made in our calculations of the real world for now, and a long time to come. There will never be exact solutions to reality, except approximations of desired accuracy (futuristic super computers taking all possible variables into consideration in its calculations).
Why does it work?
Hard work.
Over centuries mathematicians have worked hard to create a model of reality using symbols and logic. Its been refined and tweaked to help us predict the world. Millions of hours of work across multiple cultures, trial and error.
Might as well ask how can google map work, how can it show us how to get from Los Angeles to New York. Its impressive, for sure. Its just colors and lines on a computer screen. How does it know? It was designed to resemble the larger scale, in a very intentional and logical way.
There is no miracle or perfection to it. But of course, its been in the making quite a while, so for some it seems like magic. The pyramids weren't built in a day, or by one person, and they are an awesome site to behold. But people build them, one stone at a time, over many years.
WaveJumper
Sep3-09, 12:09 PM
Why does math work in our reality?
I'd say there was probably no other way to construct an orderly & comprehensible and consistent universe. What other way is there for a universe like ours to exist, in which it would not be reigned by total chaos and where particles would not have random values and no physical law would be possible?
How would there be laws of physics if there was no math? And how would there be you, if there were no laws of physics?
You want a world without math and laws of physics? You are not talking of a world, you are talking about an Idea.
The universe can be classically thought of as an aquarium, where we are the marine species that got smarter.
Tac-Tics
Sep3-09, 12:22 PM
Math probably "works" in reality because math always works. Math is nothing but what we can logically deduce about fundamental abstract ideas such as integers and sets. The space of logically-consistent ideas is much, much larger than what is real. The universe just one idea in an infinite sea of other things which are just as feasible.
Mathematics seems to correspond very well to reality, but that's only an evolutionary result. The best results all stem from the geometry of the world we find ourselves in. It's like in literature, how some people claim that all modern works are a retelling of something found in a Shakespearean play, all mathematics leads back in some form to ideas understood by Euler. Exponentiation, prime numbers, and bell curves are all relatively new ideas, but they all have some relation to the same circle understood 2,000 years ago by Pythagoras.
To echo Kant: How could it not work? Math is a product of our perception of the world and our ability to reason within it. Reason is shaped by our perception of reality, not the other way around. In other words, its not reality that is adapted to mathematics, but rather its our minds, and consequently mathematics, that are adapted to reality. Math is an inseparable part of the way we see and experience the world: 2 + 2 = 4 is not something we can conceive of as being false. More generally, we can't conceive of a world where math does not work.
"2 + 2 = 4 is not something we can conceive of as being false. "
Sure we can.
Argument, please?
junglebeast
Sep4-09, 12:58 PM
1+1=2 only because it was defined that way.
One could just as easily define a different set of symbols for numbers and operators so that 1+1 = 1. All of these symbolic choices were entirely arbitrary.
Everything in math results from a few base assumptions of countability that mirror the world
"2 + 2 = 4 is not something we can conceive of as being false. "
Sure we can.
Argument, please?
Really? Then I imagine you are quite comfortable with square circles and flat balls.
1+1=2 only because it was defined that way.
One could just as easily define a different set of symbols for numbers and operators so that 1+1 = 1. All of these symbolic choices were entirely arbitrary.
Everything in math results from a few base assumptions of countability that mirror the world
By 2 + 2 = 4, I mean the straightforward, everyday meaning of the statement: "Two and two objects put together, make four objects". If you begin to speak about a different "set of symbols" and "operators" you are in an arena far away from the intended context.
This is like objecting to "I am now writing on physicsforums" because the words "I", "am", "now" etc. have entirely arbitrary meanings and "one could just as easily define a different language so that this sentence is false". Yes, of course.
" I mean the straightforward, everyday meaning of the statement"
What relevance does this have to the validity of mathematical statements??
"2 + 2 = 4 is not something we can conceive of as being false. "
Sure we can.
Argument, please?
The case may be made clearer by considering a genuinely empirical generalization, such as 'All men are mortal.' It is plain that we believe this proposition, in the first place, because there is no known instance of men living beyond a certain age, and in the second place because there seem to be physiological grounds for thinking that an organism such as a man's body must sooner or later wear out. Neglecting the second ground, and considering merely our experience of men's mortality, it is plain that we should not be content with one quite clearly understood instance of a man dying, whereas, in the case of 'two and two are four', one instance does suffice, when carefully considered, to persuade us that the same must happen in any other instance. Also we can be forced to admit, on reflection, that there may be some doubt, however slight, as to whether all men are mortal. This may be made plain by the attempt to imagine two different worlds, in one of which there are men who are not mortal, while in the other two and two make five. When Swift invites us to consider the race of Struldbugs who never die, we are able to acquiesce in imagination. But a world where two and two make five seems quite on a different level. We feel that such a world, if there were one, would upset the whole fabric of our knowledge and reduce us to utter doubt.
...
A similar argument applies to any other a priori judgement. When we judge that two and two are four, we are not making a judgement about our thoughts, but about all actual or possible couples. The fact that our minds are so constituted as to believe that two and two are four, though it is true, is emphatically not what we assert when we assert that two and two are four. And no fact about the constitution of our minds could make it true that two and two are four. Thus our a priori knowledge, if it is not erroneous, is not merely knowledge about the constitution of our minds, but is applicable to whatever the world may contain, both what is mental and what is non-mental.
...
Let us revert to the proposition 'two and two are four'. It is fairly obvious, in view of what has been said, that this proposition states a relation between the universal 'two' and the universal 'four'. This suggests a proposition which we shall now endeavour to establish: namely, All a priori knowledge deals exclusively with the relations of universals. This proposition is of great importance, and goes a long way towards solving our previous difficulties concerning a priori knowledge.
...
In the special case of 'two and two are four', even when we interpret it as meaning 'any collection formed of two twos is a collection of four', it is plain that we can understand the proposition, i.e. we can see what it is that it asserts, as soon as we know what is meant by 'collection' and 'two' and 'four' . It is quite unnecessary to know all the couples in the world: if it were necessary, obviously we could never understand the proposition, since the couples are infinitely numerous and therefore cannot all be known to us.
http://www.ditext.com/russell/russell.html
See also the Kant / Math thread. I would be interested in seeing a counter-argument from a logician. One probably exists somewhere.
" I mean the straightforward, everyday meaning of the statement"
What relevance does this have to the validity of mathematical statements??
Explain. Of what type of validity are you talking about?
Mathematics is the study of structure, rythim, and patterns.
The Universe has structure, rythim, and patterns.
Therefore, math works in our reality.
What relevance does this have to the validity of mathematical statements??
See Frege's Theorem: http://plato.stanford.edu/entries/frege-logic/. Arithmetic follows from the validity of second-order logic and Hume's Principle. Are you denying the validity of second-order logic, or are you denying that "for any concepts F and G, the number of F-things is equal to the number G-things if and only if there is a one-to-one correspondence between the F-things and the G-things?" I suppose you could also be denying the validity of the proof itself...
junglebeast
Sep4-09, 02:26 PM
By 2 + 2 = 4, I mean the straightforward, everyday meaning of the statement: "Two and two objects put together, make four objects".
In that case, the reason that math works in real life can simply be stated as "conservation of energy" (from physics). Conservation of energy which is generally true at a macroscopic level allows things to be accumulated, and this allows us to define symbols for countable quantities, and everything else basically comes from that axiom.
In that case, the reason that math works in real life can simply be stated as "conservation of energy" (from physics). Conservation of energy which is generally true at a macroscopic level allows things to be accumulated, and this allows us to define symbols for countable quantities, and everything else basically comes from that axiom.
How do you explain the fact that a line a circle intersect at most at two points by "conservation of energy"?
junglebeast
Sep4-09, 03:55 PM
How do you explain the fact that a line a circle intersect at most at two points by "conservation of energy"?
Conservation of energy is why numbers have meaning in our reality. In order to talk about geometry you also need dimensions...and our reality has 3 spatial dimensions, so that gives meaning to 3 dimensional spaces.
A line intersects a circle in at most 2 points because we arbitrarily defined the concepts of "lines" and "circles" to behave that way: a circle is the set of points of constant radius from the centroid, a line is the set of points that can be represented as a linear combination of two points. To have a line intersect a circle in more than 2 points would contradict either the definition of a circle or the definition of a line, and therefore cannot be.
Because there is a real world analog for vector spaces of dimension less than or equal to 3 (namely, the space-like dimensions of our universe), the statement holds in the real world as well as the abstract made up world of math.
Of course, this is not coincidental -- everything in math was simply developed so that it could be used as a way to model reality. If there were another universe/reality in which conservation of energy did not hold, then the concept of a "number" wouldnt make sense at all...yet there might be other rules that governed their universe leading to a form of math that was completely absent of the concept of numbers
Conservation of energy is why numbers have meaning in our reality. In order to talk about geometry you also need dimensions...and our reality has 3 spatial dimensions, so that gives meaning to 3 dimensional spaces.
Junglebeast, the closest description to what you are saying that I can find in http://plato.stanford.edu/entries/philosophy-mathematics/ seems to be nominalist scientific reconstruction. This idea has largely been dismissed by philosophers, as discussed on that page. Is this correct, or would one of the other described theories better fit your view? Does your view fall outside of the usual classifications of foundational mathematical theories? Has this article missed a theory that you think should have been included?
In a nominalist reconstruction of mathematics, concrete entities will have to play the role that abstract entities play in platonistic accounts of mathematics. But here a problem arises. Already Hilbert observed that, given the discretization of nature in quantum mechanics, the natural sciences may in the end claim that there are only finitely many concrete entities (Hilbert 1925). Yet it seems that we would need infinitely many of them to play the role of the natural numbers — never mind the real numbers. Where does the nominalist find the required collection of concrete entities?
The argument presented above is similar to the one given by Russell that I previously quoted. Platonism and the four schools etc all hold math as existing or subsisting in some sort of idealized or abstract way, and not as being purely derivable from or existing in the physical realm. It is also possible, of course, that I am misreading these descriptions, in which case I would appreciate having that pointed out as well.
...mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories.
junglebeast
Sep4-09, 05:34 PM
kote,
I don't have the time or interest to read the past philosophical musings of every 19th century philosopher. It does not appear that what I'm saying can exactly be summarized by any of those categories. I also don't believe what I'm saying is a philosophical opinion at all, I think I'm just stating facts.
Math is simply a logical formalization of something -- anything. There are basic definitions and axioms, and on top of that everything else is derived from those fundamental axioms using logical proofs.
One can define ANY set of basic definitions and axioms and the resulting set of logical conclusions would be a body of mathematics. However, since we are interested in using mathematics to model reality, all the fields of mathematics are designed with basic assumptions that in some simplified way represent an aspect of reality.
The answer to the OPs question of "why does math work in reality" is really an ill-posed question because math doesn't actually work in reality. Math simply describes a space of conclusions that can be drawn from a set of assumptions. When you want to apply math to reality, you have to choose an appropriate set of assumptions that mimic the aspect of reality you are interested in modeling...and this usually implies choosing a physics model.
Even when you are talking about a question so basic as: "John has 1 apple. Beth gives John another apple. How many apples does John have?" which is translated into "1+1=2", this is still assuming a physics model, and that's why it makes sense in real life. You're assuming that there is conservation of energy, and conservation of momentum, among other things...because Apples aren't spontaneously appearing or disappearing. If there were a different alien reality where these principles did not hold, and you posed this question, they might say: You idiot! 1+1 = ?, because apples spontaneously appear and disappear at will!
One could define any laws of physics that they wanted, and as long as they don't contradict, you could build a field of math around those made up laws of physics. To the degree that the physics is accurate, the mathematical conclusions will be....although the word "degree" may be misleading in this context because obviously a small error can be magnified into a huge error in the prediction.
Mathematics works in describing the "physical world" because we constructed it in that way.
Ok, think about this in terms of this analogy: the world (the universe), a painter (scientists), and a painting of a landscape (applied mathematics).
WaveJumper
Sep6-09, 12:29 PM
Mathematics works in describing the "physical world" because we constructed it in that way.
Ok, think about this in terms of this analogy: the world (the universe), a painter (scientists), and a painting of a landscape (applied mathematics).
The analogy will hold, if the painter drew an animation and not a picture. That animation also has to be consistent for 14 billion years into the past, so it's not trivial.
The analogy will hold, if the painter drew an animation and not a picture. That animation also has to be consistent for 14 billion years into the past, so it's not trivial.
indeed, I oversimplified the matter.
but the reasoning as I see it correct: the "success" of mathematics is due to the fact that we "build" mathematics in such a way that it describes the universe.
This depends strongly on what you mean by "works". NO mathematics exactly fits a physical situation ("reality" if you like). All "reality" involves measurements that are approximate so the best we can hope for from a mathematical theory is that it work approximately.
All mathematical structures (theories) are "templates". Every mathematical structure involves "undefined terms", words that are defined using those undefined terms, axioms, and theorems proved from those axioms. To apply a mathematical structure to "reality", we have to assign meanings to those undefined terms. IF the axioms are true with those meanings assigned, then all theorems proved from those axioms are true and all methods of solving problems derived from those theorems will work.
Of course, the axioms [b]won't[b\] be perfectly true, only approximately true. The key to the "Unreasonable Effectiveness of Mathematics in the Natural Sciences" is having a large array of mathematical structures and choosing the one that best fits the specific application.
While I agree with your description of mathematics generally, I am not so sure that we can not have an ultimate mathematical/physical theory. Physicists differentiate between what they consider to be phenomenological theories and fundamental theories. For example the Shroedinger equation describes the spectrum of the hydrogen atom as a phenomenon but not in a fundamental way. this is because it takes coulomb forces as givens and does not explain them. But a theory like String theory attempts to explain everything fundamentally. Why could not a theory like this actually tell us everything exactly?
It is quite unnecessary to know all the couples in the world: if it were necessary, obviously we could never understand the proposition, since the couples are infinitely numerous and therefore cannot all be known to us.
The entire arguement is based on correspondence to observation. There is nothing special about math in this sense. It needs to be corroborated. 2 humans plus 2 humans can equal 5 if one gets pregnant. Therefore 2+2=5. It is just generally true, when you categorize things together in groups that 2+2=4.
Math is about generalizations, that is where its strength lies. But there is no math statement that stands on its own. Its correspondence to observation is what makes it a valid generalization. Observation is what math is built on.
27Thousand
Sep7-09, 11:51 PM
I知 reviewing my mathematics knowledge, except I知 looking for a different reason. I understand how it works you know, 1 plus 1 so on, I知 trying to understand why it works.
Pure and applied mathematicians and physicists have tied our understanding of reality with mathematics. They believe that if it computes it is real. This will do for a while, till we pose a question beyond our understanding.
But that痴 for a different time. Why does it work?
Any takers?
My view is that math doesn't explain, but rather describe how something works. E=MC^2 doesn't explain, but rather describes an observable principle. Same with Newton's inverse equation for gravitation. Then of course newer and better equations make up for situations it may not work.
So it doesn't explain, but rather describes/predicts how observable principles work.
My view is that math doesn't explain, but rather describe how something works. E=MC^2 doesn't explain, but rather describes an observable principle. Same with Newton's inverse equation for gravitation. Then of course newer and better equations make up for situations it may not work.
So it doesn't explain, but rather describes/predicts how observable principles work.
These are all scientific theories... Math is a different ball park. Yes, science uses math... no math is not science.
27Thousand
Sep8-09, 02:42 AM
These are all scientific theories... Math is a different ball park. Yes, science uses math... no math is not science.
Wait, so I'm confused here :confused: If math can "describe" observable principles, and make "predictions" for observations, how does it not work in our reality? It may not explain the way a scientific theory does, and may not actually be an observable principle itself (scientific law), but if it does a very good job of describing and predicting the observable, how is it not tied into some sort of reality? When I take a statistics class, math can accurately predict a range for probability in a bell curve, or describe what's already happened.
Math is pretty scary. It took us in directions we never imagined throughout human history, and has never been refuted. I view it as a very rigorous form of logic built from the ground up. It implies we live in a logical universe. That remains to be seen. but. still looks like a good bet. If the universe truly is illogical, I doubt we will ever comprehend it.
Hi there,
Wait, so I'm confused here :confused: If math can "describe" observable principles, and make "predictions" for observations, how does it not work in our reality?
I often give the following example to explain the usefulness of mathematics.
If you want to build a house, you will need a hammer (out of many other tools). The reasons a hammer works is very simple: a hammer is design and made to nail stuff in a wall.
You should mathematics in the same way. Mathematics are tools to help us model some theory that tries to explain something real. This tool is design and made to model these theories. Therefore, mathematics evolved over the years. When the known mathematics are not enough to explain this or that, scientist or mathematician will develop some more theories.
Cheers
There is a reverse way of looking at it. Our observations are not reality but merely reality as presented through our senses. Through science we start with our sense impressions and discover what reality is really like underneath. Mathematics is one tool for doing this.
Why can't we view science as a way of perceiving reality that is otherwise only partially known through the senses? Why are the senses so sacrosanct?
M Grandin
Sep9-09, 11:24 AM
For instance "HallsofIvy" and "Junglebeast" have already answered the OP question fully satisfactory, why I have not much to add. But expressed in own words, I could say
mathematics is just systematisized logics, where logics operating on logics may result in impressing formulas and mathematical complex using symbols resembling alien language to
common people. All resting on elementary building stones of logics and fundamental observations = axioms. These axioms may look like abstractions without regard to real world
- but may be more of physical observation than abstract thinking than people believe.
So because these fundamental axioms and logics in fact are fundamental experienced "physics", using these systematically may also result in something matching real world. :approve:
For instance "HallsofIvy" and "Junglebeast" have already answered the OP question fully satisfactory, why I have not much to add. But expressed in own words, I could say
mathematics is just systematisized logics, where logics operating on logics may result in impressing formulas and mathematical complex using symbols resembling alien language to
common people. All resting on elementary building stones of logics and fundamental observations = axioms. These axioms may look like abstractions without regard to real world
- but may be more of physical observation than abstract thinking than people believe.
So because these fundamental axioms and logics in fact are fundamental experienced "physics", using these systematically may also result in something matching real world. :approve:
that is interesting - I experience mathematics not only as axioms and deductions but also as a branch of science. Mathematicians certainly do not think of themselves as mere logicians.
I think that there are mathematical objects of empirical study just as there are physical ones. There are mathematical theories just as there are biological or physical. Mathematical ideas require incredible imagination and are often derived from observation of mathematical objects and relationships - just as in any science.
The mathematics that is used for instance in General Relativity was first invented by mathematicians who were challenging our ideas of measurement and of intrinsic geometry. They came up with new theories which later - happened to have application in physics. A modern example is Chern-Simons invariants which were discovered during pure geometrical researches and later were found to have application in particle physics.
that is interesting - I experience mathematics not only as axioms and deductions but also as a branch of science. Mathematicians certainly do not think of themselves as mere logicians.
I think that there are mathematical objects of empirical study just as there are physical ones. There are mathematical theories just as there are biological or physical. Mathematical ideas require incredible imagination and are often derived from observation of mathematical objects and relationships - just as in any science.
The mathematics that is used for instance in General Relativity was first invented by mathematicians who were challenging our ideas of measurement and of intrinsic geometry. They came up with new theories which later - happened to have application in physics. A modern example is Chern-Simons invariants which were discovered during pure geometrical researches and later were found to have application in particle physics.
I'm not sure what the "mere" is for when you refer to logicians. If mathematics is the science of discovering real mathematical objects, the implication is that mathematical theorems, like in science, can be wrong, and proofs are not actually proofs but just hypotheses. Is this your stance?
If math is a science there is no such thing as mathematical proof and we should rewrite all of the textbooks. We should also allow for inconsistent mathematical theories, as is done in science, and not automatically accept "proofs" against the consistency of theories.
The application of math to the world is science, so the fact that math is used in scientific theories is irrelevant to the math itself. Euclidean space is just as valid as Minkowski space. Whether or not one gives a better model of reality is outside the realm of mathematics.
Math is either deduction from axioms or an inductive science. It can't be both.
I'm not sure what the "mere" is for when you refer to logicians. If mathematics is the science of discovering real mathematical objects, the implication is that mathematical theorems, like in science, can be wrong, and proofs are not actually proofs but just hypotheses. Is this your stance?
If math is a science there is no such thing as mathematical proof and we should rewrite all of the textbooks. We should also allow for inconsistent mathematical theories, as is done in science, and not automatically accept "proofs" against the consistency of theories.
The application math to the world is science, so the fact that math is used in scientific theories is irrelevant to the math itself. Euclidean space is just as valid as Minkowski space. Whether or not one gives a better model of reality is outside the realm of mathematics.
Math is either deduction from axioms or an inductive science. It can't be both.
My point is that mathematics is empirical and studies empirical objects just as any other science. Mathematicians even do experiments. They have the further more powerful tool that their theories can be substantiated by proof. If having this tool means that these other mathematical activities - including incredibly creative ideas - makes it not a science - then I think that is a definition - one that Gauss for instance did not agree with.
Math is both inductive and deductive. take this scenario. A person wants to know whether a certain geometrical property holds for a class of Riemannian manifolds. There are infinitely many such manifolds and they exist in all finite dimensions. Few examples are known and all of them are in low dimensions. So what does this person do? Does he try to deduce the answer? Maybe. But more likely he will start to look at examples. Based on these examples he will form hypotheses that he will check in other examples. If these hypotheses fail he will either modify them or look for new relationships and come up with new hypotheses and check them out again. Eventually he will find a property, a mathematical relationship, that reveals the truth or falseness of his original question. He then may consider the relationship that he has found ,though giving the answer, may not satisfactorily reveal how the property in question relates to broader questions of ongoing research. Thus he may revisit his investigation in search of other properties that allow this broader understanding. To me, this is science. the thought processes are the same.
the attitude that I have found is that mathematicians and physicists view the two as branches of the same subject. One PDE researcher said to me that his mathematical research though pure and not pointed at any scientific endeavor nevertheless examines certain geometrical minimization problems which he believes to relate to intrinsic features of the universe.
My point is that mathematics is empirical and studies empirical objects just as any other science. Mathematicians even do experiments. They have the further more powerful tool that their theories can be substantiated by proof. If having this tool means that these other mathematical activities - including incredibly creative ideas - makes it not a science - then I think that is a definition - one that Gauss for instance did not agree with.
Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.
If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.
Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.
If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.
I expanded my note to you. what is your reaction?
Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.
If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.
Einstein did not brush his teeth.
Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.
If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.
A subject that uses proof can still also use science. I do not believe that they are mutually exclusive.
I expanded my note to you. what is your reaction?
Sure, sociologically there is a method involved in how mathematicians decide what they want to prove. Whether a mathematician has a grant to investigate something and form a conjecture or whether the grant is to write a proof is not how I think we should draw the line. This is similar to how creatively coming up with new theories in science, theories for which no experiments have yet been done, is not part of anyone's definition of the scientific method. The science is in the experiments, not the conjecture, and not the sociological or psychological motivations.
I prefer to draw the line at inductive/deductive and idealized notions rather than looking at all of the sociological factors involved. This avoids problems such as the question of whether or not writing grant proposals is "math" or "science." It also avoids the fact that mathematicians are not infallible. How do we know that a proof is a proof and we haven't made a mistake? Well... we don't, but that's a fault of mathematicians and not of math.
Sure, sociologically there is a method involved in how mathematicians decide what they want to prove. Whether a mathematician has a grant to investigate something and form a conjecture or whether the grant is to write a proof is not how I think we should draw the line. This is similar to how creatively coming up with new theories in science, theories for which no experiments have yet been done, is not part of anyone's definition of the scientific method. The science is in the experiments, not the conjecture, and not the sociological or psychological motivations.
I prefer to draw the line at inductive/deductive and idealized notions rather than looking at all of the sociological factors involved. This avoids problems such as the question of whether or not writing grant proposals is "math" or "science." It also avoids the fact that mathematicians are not infallible. How do we know that a proof is a proof and we haven't made a mistake? Well... we don't, but that's a fault of mathematicians and not of math.
I do not think that the process of mathematical thought and investigation is sociological. It is a necessary aspect of mathematical ideation. the thought processes are fundamentally scientific. The experimenter does an experiment to verify a hypothesis or to examine a property of a physical system. A mathematician examines mathematical objects for the same reason, to verify a hypothesis or to examine a property. No difference.
While it is true that no proof can ever be know for sure - neither can the result of any experiment be know to be always repeatable. If experimenters did not believe that their evidence represents something immutable and invariant - they would never have a theory of anything. It is true that in science this is a belief - an act of faith perhaps- whereas in mathematics it is not.
Math is either deduction from axioms or an inductive science. It can't be both.
And where do axioms come from if not by induction? General ideas derived from particular impressions.
It is also obvious (since Godel at least) that all axioms demand an epistemic cut - the arbitrary insertion of an observer. At some point it is decided that all this is true, because all that is false. A crisp choice gets made. So even as generalities, axioms are always going to be subsets of the possible. A choice is made and stuff must get left behind. Or better yet, as good epistemic cuts are formally dichotomous, middles get excluded.
Have you read Robert Rosen or Howard Pattee? They have written good stuff on these matters.
And where do axioms come from if not by induction? General ideas derived from particular impressions.
It is also obvious (since Godel at least) that all axioms demand an epistemic cut - the arbitrary insertion of an observer. At some point it is decided that all this is true, because all that is false. A crisp choice gets made. So even as generalities, axioms are always going to be subsets of the possible. A choice is made and stuff must get left behind. Or better yet, as good epistemic cuts are formally dichotomous, middles get excluded.
Have you read Robert Rosen or Howard Pattee? They have written good stuff on these matters.
Well of course we choose axioms to give us systems that are useful in science etc, but the part about choosing axioms isn't math :smile:! I agree with what you said, I just see a choice of axioms as meta-math. Axioms are chosen on decidedly non-mathematical grounds like aesthetics or a presupposed scientific utility. Justification of consistent axioms is not itself mathematical.
I'll look into those authors.
Ok, so axioms go back in the philosophy bin!
All meta- level discussions are philosophical because that is the place for vague deliberations (as opposed to the crisply taken choices of maths and science).
We are talking about how we know the world. Philosophy is where the vague groping exploration of possibilities take place. Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements.
Actually "doing science" of course involves all three. We are in a modelling relation with reality (see Rosen). We start out with vague ideas and impressions and attempt to develop them into a crisp system of models and measurements.
Philosophy gets us started. Then we start to take the choices that swim into view.
Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality.
Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation. The apparatus of experiment and hypothesis, etc.
Maths works not because of some platonic magic but because reality is itself a collection of interactions that must settle into emergent patterns. There is a reduction of possibility that takes place "out there". And we are trying to do the same thing in our own minds.
I would argue that so far we have only really been doing half the job with the maths we've produced though.
We have a very well developed mathematics of atomism, a very poor mathematics of systems.
If you study hierarchy theory and other tentative examples of systems maths, they are indeed more "philosophical" - vaguely developed ideas rather than crisply taken choices.
But with chaos theory, tsallis entropy, fractal geometry, renormalisation group and scalefree networks, for example, systems math is starting to emerge in earnest.
The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or powerlaw realm.
So mathematics works because it is making crisp what was vaguely seen in philosophy. It works because reality is self-organising pattern. It works because it split off the model making issues from the measurement taking issus.
But it's job is far from complete. Atomism is well elaborated. But the field of systems mathematics is just in the process of being born.
Hippasos
Sep10-09, 03:12 AM
One amazing example observed physical behavior of quantum chaos conjectured by the pure mathematics area of number theory:
http://www.physorg.com/news142834558.html
Ok, so axioms go back in the philosophy bin!
All meta- level discussions are philosophical because that is the place for vague deliberations (as opposed to the crisply taken choices of maths and science).
We are talking about how we know the world. Philosophy is where the vague groping exploration of possibilities take place. Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements.
Actually "doing science" of course involves all three. We are in a modelling relation with reality (see Rosen). We start out with vague ideas and impressions and attempt to develop them into a crisp system of models and measurements.
Philosophy gets us started. Then we start to take the choices that swim into view.
Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality.
Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation. The apparatus of experiment and hypothesis, etc.
Maths works not because of some platonic magic but because reality is itself a collection of interactions that must settle into emergent patterns. There is a reduction of possibility that takes place "out there". And we are trying to do the same thing in our own minds.
I would argue that so far we have only really been doing half the job with the maths we've produced though.
We have a very well developed mathematics of atomism, a very poor mathematics of systems.
If you study hierarchy theory and other tentative examples of systems maths, they are indeed more "philosophical" - vaguely developed ideas rather than crisply taken choices.
But with chaos theory, tsallis entropy, fractal geometry, renormalisation group and scalefree networks, for example, systems math is starting to emerge in earnest.
The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or powerlaw realm.
So mathematics works because it is making crisp what was vaguely seen in philosophy. It works because reality is self-organising pattern. It works because it split off the model making issues from the measurement taking issus.
But it's job is far from complete. Atomism is well elaborated. But the field of systems mathematics is just in the process of being born.
How do you you know a prioi what reality is or isn't?
How do you know that reality is a "self collection of interactions" (whatever that means). Whatever that is supposed to measan, isn't that a model - not a very crisp one though - maybe a meaningless one.
Mathematics does not work because it is making crisp (what ever that means) what is vague in philosophy. You do not know why mathematics works. Nobody does.
"The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale." This is meaningless - and how do you know what the key is?
"Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation." What is this supposed to mean?
Science is not a system - it does not generalize observation - "generalizing observation" is an oxymoron.
Chaos theory and fractal math have little influence on scientific research. Both are fads. How then do you know then that these are the right direction of science?
"The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or power law realm." Ahat is this supposed to mean? Any scientist who heard you say this would smile politely then walk away. Why don't you go to a physics department and try it out on a mathematical physicist?
"Philosophy is where the vague groping exploration of possibilities take place." Not true. What is vague groping? Can you make that more crisp?
"Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements." That is wrong. Math is not for formalizing nor is science. Formalizing always occurs after the science and math have already been done. Axioms have little to do with scientific thinking. They are afterthoughts.
'Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality. " Causality is not what math studies.In fact ideas of causality have always been an impediment to physics and science has always tried to eliminate causality in order to make progress.
"
Math is not a way of creating definite models - models are formal devices - real mathematics is a way of discovering our ideas of space, geometry and number- empirical observation can guide these discoveries but it is not the only source of guidance. What empirical model of reality would you say the theory of Riemann surfaces represents? How about the theory of differentiable structures on manifolds? Which empirical data did the Riemann hypothesis model? - what observations did it make "crisp"?
How about Thom's theory of cobordism of differentiable manifolds? After you explain all of these to me, you can move on to Chern-Simons invariants and then rational homotopy theory/ Oh yeah and maybe you could help me out with which empirical data the theory of Bieberbach groups was designed to model.
Even the general theory of relativity did model model anything new - it was a reconceptualization of our ideas of space. Only after it was discovered was it found to predict certain new data that previous theories did not.
Science develops because people question or ideas of reality not because we model it. The Ptolememaic system was a great model of planetary motion. Yet it was questioned - not for empirical reasons but because people felt that it could not be consistent with the mind of God. When Gallileo said that the ball rolling on an inclined plane would rise to the same height he was discovering an idea of reality not explaining empirical data. In fact, people said to him that he was wrong because the ball did not rise exactly to the same height and the more it rolled back and forth the less it rose until it finally came to a stop. People said that on the contrary this confirmed Aristotle's patently accurate model of reality which was that an object in motion will come to its natural state of rest. the empirically correct model contradicted Gallileo's conclusion. His model was empirically false. Yet he said, 'If God wanted me to be wrong he would have made the ball miss by a mile not by an inch.'
Science is not a system - it does not generalize observation - "generalizing observation" is an oxymoron.
I'm confused here. Generalizing observation is exactly what science does. You take measurements and generate models or theories. Then, you check to make sure your theory fits all of the generalizations you intended it to fit by making additional observations.
Math is not for formalizing nor is science. Formalizing always occurs after the science and math have already been done. Axioms have little to do with scientific thinking. They are afterthoughts.
I suppose you'll have to tell that to Einstein regarding an assumption of invariance or QM regarding the Heisenberg Uncertainty Principle. Relativity theory is the formalization resulting from the axiom that the laws of physics are the same in any intertial reference frame.
In fact ideas of causality have always been an impediment to physics and science has always tried to eliminate causality in order to make progress.
Really? Science doesn't try to uncover the patterns between causes and effects? It doesn't try to tell you why certain things are observed? Are you saying that the less science has to do with physical cause and effect, the better it is? Physics is the study of physical causation.
In fact, people said to him that he was wrong because the ball did not rise exactly to the same height and the more it rolled back and forth the less it rose until it finally came to a stop. People said that on the contrary this confirmed Aristotle's patently accurate model of reality which was that an object in motion will come to its natural state of rest. the empirically correct model contradicted Gallileo's conclusion. His model was empirically false. Yet he said, 'If God wanted me to be wrong he would have made the ball miss by a mile not by an inch.'
Aristotle's model didn't make predictions. Gallileo's model was empirically more accurate. Being within an inch is closer than not even making a guess.
You posted a lot, so I'll have to apologize for only responding to parts.
I'm confused here. Generalizing observation is exactly what science does. You take measurements and generate models or theories. Then, you check to make sure your theory fits all of the generalizations you intended it to fit by making additional observations.
I suppose you'll have to tell that to Einstein regarding an assumption of invariance or QM regarding the Heisenberg Uncertainty Principle. Relativity theory is the formalization resulting from the axiom that the laws of physics are the same in any intertial reference frame.
Really? Science doesn't try to uncover the patterns between causes and effects? It doesn't try to tell you why certain things are observed? Are you saying that the less science has to do with physical cause and effect, the better it is? Physics is the study of physical causation.
Aristotle's model didn't make predictions. Gallileo's model was empirically more accurate. Being within an inch is closer than not even making a guess.
You posted a lot, so I'll have to apologize for only responding to parts.
Ok
Aristotle's model worked better
-science does not generalize observation-- observation is merely sense data.
Science does not fundamentally uncover patterns - pattern fitting is only a tool -what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.
- Einstein's theory of relativity originally did not,at first, explain new data - it discovered a new concept of space time - physicists believe that is was a totally unexpected and anomalous idea precisely because of this. They agree that physics could have gone along explaining known data just fine without it.
Most mathematical theories examine our ideas of space and quantity. I gave only a few examples. I think it would benefit the philosopher to do some real mathematics.
I guess underlying these thoughts is the idea that sense perception is merely a shadow of reality and its true nature needs to be discovered with this shadow as a guide. These is no doubt that science progressed on this assumption. The physicists and mathematicians who believed this are too numerous to list - e.g. Gauss, Riemann, Einstein, Kepler, Planck,most Quantum physicists, etc
There is a deep book on this subject written by Henri Poincare called Science and Hypothesis. In it he says that there are two schools of physics, the English school which denies the need for hypotheses about the nature of reality and believes that all science is just pattern fitting and the Continental school which says that underlying assumptions/hypotheses are necessary. the pattern fitters were notably Newton and Faraday both who explicitly claimed that they needed no hypotheses - maybe Maxwell also though I am not sure of his view on this. Curiously it seems that there were really two schools of physics centered around this controversy.
Science does not fundamentally uncover patterns - pattern fitting is only a tool -what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.
What is science if not a tool? Also, the Ptolemaic system is not "false," it is simply inelegant. Inelegance is not a mathematical or scientific criteria; it is aesthetics. And since when is a historical religious view the determining factor in scientific worth? Is acceptance by the church a requirement of good science?
Maybe it would help if you could explain what you think the goal of science is if it is not to formally generalize observations or to uncover physical causation.
I think it would benefit the philosopher to do some real mathematics.
Why do you assume we haven't done "real math?" What does performing calculations or derivations or constructing proofs have to do with the philosophical foundations of mathematics anyway?
What is science if not a tool? Also, the Ptolemaic system is not "false," it is simply inelegant. Inelegance is not a mathematical or scientific criteria, it is aesthetics. And since when is a historical religious view the determining factor in scientific worth? Is acceptance by the church a requirement of good science?
Maybe it would help if you could explain what you think the goal of science is if it is not to formally generalize observations or to uncover physical causation.
Why do you assume we haven't done "real math?" What does performing calculations or derivations or constructing proofs have to do with the philosophical foundations of mathematics anyway?
By doing mathematics - by which I mean coming up with new theories or extending existing theories in important ways - not just doing proofs and calculations - you will see my point.
If you have done this and don't agree with me then I am astounded and would like to discuss it further. Do you think the theory of Riemann surfaces was thought of as a calculation or a proof of something?
Science is a tool but is more than that. that is what I am trying to say. I am also saying that the idea of mathematics as merely a modelling tool is just plain wrong. If you are a mathematician, how is it that you believe this?
By the way, the Ptolemaic system - though a good pattern fit, a grand lovely model - was an impediment to the progress of science - yet it was able to predict and to model wonderfully.
By doing mathematics - by which I mean coming up with new theories or extending existing theories in important ways - not just doing proofs and calculations - you will see my point.
If you have done this and don't agree with me then I am astounded and would like to discuss it further. Do you think the theory of Riemann surfaces was thought of as a calculation or a proof of something?
Science is a tool but is more than that. that is what I am trying to say. I am also saying that the idea of mathematics as merely a modelling tool is just plain wrong. If you are a mathematician, how is it that you believe this?
It's because the idea that modeling deserves to be qualified by "mere" is a philosophical idea and not a mathematical one. Judgments about the value of modeling are not mathematical or scientific. I disagree that math is a science, so I see no problem in asserting that science is modeling while math is not. Math is a tool used by science to create models. Mathematical theories, in a way, can be said to be models themselves, they just aren't models of anything in particular without science attached. Math is the logical analytic extension of axioms or assumptions.
Mathematical theories often precede scientific theories. They are often surprising and paradoxical and cause us to think about things in new ways. Math is necessary for formal science and nearly all technology. Extending or applying mathematics can be a very creative process. None of that changes the fact that it involves only purely formal structures (or models), the application of which lies in the realm of science, and the value of which lies in the realm of ethics and aesthetics.
From Barry Mazur http://www.math.harvard.edu/~mazur/papers/plato4.pdf:
If we adopt the Platonic view that mathematics is discovered, we are suddenly in surprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one痴 faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets. ...
It seems to me that, in the hands of a mathematician who is a determined Platonist, proof could very well serve primarily this kind of rhetorical function洋aking sure that the description is on track預nd not (or at least: not necessarily) have the rigorous theory-building function it is often conceived as fulfilling.
Only math that involves "mere" abstract analytic constructions can talk about proof or universals. If math discusses anything other than models, it is reduced to mere inductive generalization, which cannot rationally be demonstrated as true.
Mazur also criticizes anti-platonistic views, but he focuses on the idea that math is socially constructed, which is not what we're talking about.
I also object to the view that what a mathematician says he thinks he is doing is valid evidence for what math actually is, unless truth is sociologically constructed. But if truth is sociological, math is already in trouble and needs to be knocked down a few levels.
None of that changes the fact that it involves only purely formal structures (or models),
that to me is just wrong unless I don't know what you mean by purley formal. Are all ideas purely formal? Is love purely formal?
Ideas are not formalisms to me. Is a Bach fugue purely formal if we think of it rather than listen to it. Is the painting in Van Gogh's mind purely formal before he paints it?
Mathematics is discovered. Mazur is right that this has potentially theological implications - but in my mind he is wrong that it is theistic. The reality of ideas is indisputable. Sense experience is always in doubt.
Mathematics is discovered. Mazur is right that this has potentially theological implications - but in my mind he is wrong that it is theistic. The reality of ideas is indisputable. Sense experience is always in doubt.
Ah, so you are a dualist :smile:! I almost agree with what you're saying here, except for one minor point.
You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.
If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.
If math is ideas that are discovered, then there exists an objective and eternal mind.
If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.
Ah, so you are a dualist :smile:! I almost agree with what you're saying here, except for one minor point.
You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.
If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.
If math is ideas that are discovered, then there exists an objective and eternal mind.
If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.
good explanation> I understand what you mean. I think our difference is largely semantic. A couple things though and this confuses me a bit. First of all why aren't relations just as objective as ideas?
Second, if we discover them did they come into existence at the moment of discovery or did we only become aware of them? If they come into existence then is the same relation in someone else's thoughts a different relation and if it is what unifies the two? In which mind does this unity exist and was that also created on the spot of discovery?
I like Riemann's idea that the universe itself has the intrinsic dynamics of a mind and that our minds partake in this and are part of it. He was I think a neo-Kantian and believed in intrinsic ideas. But it seems that he tried to extend this to explanation of physical phenomena as thoughts -e.g. particles. from this point of view there is a universal mind in a sense - but not a deity.
good explanation> I understand what you mean. I think our difference is largely semantic. A couple things though and this confuses me a bit. First of all why aren't relations just as objective as ideas?
It's been argued that all of philosophy is semantics :smile:. I'm inclined to agree. I would say that logical relations are objective. By objective here I mean not dependent on any particular point of view. I don't mean that they are objects or have existence.
Ideas are personal to the person who has them. They are about as subjective as you can get. Objectivity is trickier. It can be doubted whether or not there is such a thing at all. When it comes to logical or analytic truths, the best argument I can give for their objectivity (or universality) is that we can't conceive of them as being false or dependent on our point of view. Does it depend on my point of view that something can't both exist and not exist at the same time? Or that no unmarried men have wives?
Logic is the system in which we can make philosophical arguments, and, at least since Godel, we know that no logical system can prove its own validity. The fact that we can't rationally deny that 1=1, though, makes it as objective as may be possible in my opinion. This may also be a misuse of objective and subjective though, in that logic is supposed to come before either of those notions as a framework.I like Riemann's idea that the universe itself has the intrinsic dynamics of a mind and that our minds partake in this and are part of it. He was I think a neo-Kantian and believed in intrinsic ideas. But it seems that he tried to extend this to explanation of physical phenomena as thoughts -e.g. particles. from this point of view there is a universal mind in a sense - but not a deity.
I'm not familiar with Reimann's philosophy, but it might be relevant to point out that Kant did not believe in intrinsic ideas. From http://plato.stanford.edu/entries/kant-judgment/:
Kantian innateness is essentially a procedure-based innateness, consisting in an a priori active readiness of the mind for implementing rules of synthesis, as opposed to the content-based innateness of Cartesian and Leibnizian innate ideas, according to which an infinitely large supply of complete (e.g., mathematical) beliefs, propositions, or concepts themselves are either occurrently or dispositionally intrinsic to the mind. But as Locke pointed out, this implausibly overloads the human mind's limited storage capacities.
In other words, logic is innate, ideas are not. I like Bertrand Russell's treatment of the issue. Russell's laws of thought are somewhat similar to Kant's categories of perception. He talks about universals and relations in http://www.ditext.com/russell/russell.html chapters 7-10.Second, if we discover them did they come into existence at the moment of discovery or did we only become aware of them? If they come into existence then is the same relation in someone else's thoughts a different relation and if it is what unifies the two? In which mind does this unity exist and was that also created on the spot of discovery?
When we discover them, they come into existence as thoughts in our mind. From Russell (chap 9):
It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. We have here the same ambiguity as we noted in discussing Berkeley in Chapter IV. In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness.
As for unity with other people's thoughts, there is still trouble in overcoming solipsism. There's not even a guarantee that our own thoughts are true representations of logical relations. Proofs have certainly been shown to be wrong before. The best we can probably do is say that unless we accept solipsism, we must accept universals.
In geometry, for example, when we wish to prove something about all triangles, we draw a particular triangle and reason about it, taking care not to use any characteristic which it does not share with other triangles. The beginner, in order to avoid error, often finds it useful to draw several triangles, as unlike each other as possible, in order to make sure that his reasoning is equally applicable to all of them. But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal.
I wish I had better answers than I do, but if I had all the answers, this wouldn't be interesting :smile:. With post-modernism and social construction etc, people have been dissatisfied with analyticity all together and have simply denied that circles are necessarily round, so I suppose that's another option. There are some more serious semantic differences in that stance though, and I'm not a fan.
apeiron
Sep10-09, 06:32 PM
Math is not a way of creating definite models - models are formal devices - real mathematics is a way of discovering our ideas of space, geometry and number- empirical observation can guide these discoveries but it is not the only source of guidance. What empirical model of reality would you say the theory of Riemann surfaces represents? How about the theory of differentiable structures on manifolds? Which empirical data did the Riemann hypothesis model? - what observations did it make "crisp"?
How about Thom's theory of cobordism of differentiable manifolds? After you explain all of these to me, you can move on to Chern-Simons invariants and then rational homotopy theory/ Oh yeah and maybe you could help me out with which empirical data the theory of Bieberbach groups was designed to model.
'
I appreciate the fact you like a good argument but you have to at least make an attempt to understand what the other side is saying. Saying you find things meaningless, or you don't believe, is not a very interesting position to hold. So I'll limit my responses.
"real mathematics is a way of discovering our ideas of space, geometry and number"
You mean pattern and form. Self-consistent organisation.
"What empirical model of reality would you say the theory of represents?"
As you say, it all starts with observation of the world - modelling at its most natural level. The ancients noted the regularities of the world and generalised to create some structural truths about objects like triangles and morphisms like spirals.
Then mathematics has developed by even greater generalisation - as articulated by category theory for example. And demonstrated in the move from euclidean to non euclidean geometry, or simple to complex number. We find that we exist in a world that is highly constrained (3D and flat - and scaled) and then generalise by successively removing those constraints to discover if self-consistent regularity still exists.
And you can see what happens when you do this with quarternions and octonions. The regularity frays. Properties like division erode as you generalise the dimensionality. Instead of producing crisp algebraic answers, the meaning of the algebra becomes vague.
So the most general maths can cease to model crisp properties that were there in the original "empirical" view.
When this happens, many say maths has just wandered off into the wilderness - or a landscape in string theory's case.
But my philosophical approach is different. I am saying that generalisations lead back to vague potential. And the way to rescue the situation is by also building the global constraints - the selection rules that represent the idea of "self-consistency" - back into the maths explicitly. So maths with scale.
"Science develops because people question or ideas of reality not because we model it."
Ideas are models - ideas formalised.
apeiron
Sep10-09, 07:30 PM
Ok
-what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.
Epicycles did kind of fit with a harmony of the spheres. But we now think it ugly because it depended too much on construction - the addition of cycles - and not enough on global constraints (such as satisfying a universal law of gravitation).
At an instinctive level, we have long known that "good modelling" is about a natural balance of construction and constraint, local atoms and global laws. Now is the time to make this formally explicit in the form of an equilbration principle. Which is the main thing I've been working on with my interest in vagueness, dichotomies and hierarchies.
- Einstein's theory of relativity originally did not,at first, explain new data - it discovered a new concept of space time - physicists believe that is was a totally unexpected and anomalous idea precisely because of this. They agree that physics could have gone along explaining known data just fine without it.
Michelson邦orley? Mach and centrifugal force?
There is a deep book on this subject written by Henri Poincare called Science and Hypothesis. In it he says that there are two schools of physics, the English school which denies the need for hypotheses about the nature of reality and believes that all science is just pattern fitting and the Continental school which says that underlying assumptions/hypotheses are necessary. the pattern fitters were notably Newton and Faraday both who explicitly claimed that they needed no hypotheses - maybe Maxwell also though I am not sure of his view on this. Curiously it seems that there were really two schools of physics centered around this controversy.
One school worries about the information that must be discarded in modelling, the other doesn't.
Well actually the Newtons and the rest usually do wonder about the gap between reality in its fullness and their reduced descriptions that involve things like action at a distance.
But modern epistemology - Rosen's modelling relations being the best articulation I have come across - does away with this old hangover.
apeiron
Sep10-09, 07:48 PM
I disagree that math is a science, so I see no problem in asserting that science is modeling while math is not. Math is a tool used by science to create models. Mathematical theories, in a way, can be said to be models themselves, they just aren't models of anything in particular without science attached. Math is the logical analytic extension of axioms or assumptions.
The thread of prejudice running through your argument here is that knowledge is passive - it "exists". Whereas I am arguing from the opposite position that knowledge is active - it is about doing things, indeed getting things done. So that is why "modelling" is the chosen word. We do no represent reality or behold reality, instead we are seeking to have control over it - even if it is simply control over our perceptions at times.
Yes, you can talk about maths as people creating axioms and then investigating all the patterns that can flow from the axioms. This describes the day-to-day for many academics. It seems a very passive and interior exercise. And often is sterile. But the maths that gets sociologically rewarded is then the maths that turns out to be useful for control over the world, so betraying its true purpose.
So as I say - based on modelling relations epistemology - there is a natural divide into models and measurements. An observer needs the general of his ideas, the particulars of his impressions. And psychology tells us how these two develop from vague to crisp through their mutual interaction. The way a newborn baby learns to make sense of its world through active exploration.
This natural division is then repeated in our formalised disciplines. We have a method for constructing models, a method for making measurements. Maths is about fashioning tools for model construction. It may involve philosophy too in developing its crisp axioms.
apeiron
Sep10-09, 07:59 PM
Kantian innateness is essentially a procedure-based innateness, consisting in an a priori active readiness of the mind for implementing rules of synthesis, as opposed to the content-based innateness of Cartesian and Leibnizian innate ideas, according to which an infinitely large supply of complete (e.g., mathematical) beliefs, propositions, or concepts themselves are either occurrently or dispositionally intrinsic to the mind. But as Locke pointed out, this implausibly overloads the human mind's limited storage capacities.
In other words, logic is innate, ideas are not. I like Bertrand Russell's treatment of the issue. Russell's laws of thought are somewhat similar to Kant's categories of perception. He talks about universals and relations in http://www.ditext.com/russell/russell.html chapters 7-10.
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Yes, neurology tells us that the brain indeed has a "logic" - a way of arriving at a crisp local orientation to a global world. And that process is dichotomisation. Figure-ground, focus-fringe, attention-habit, conscious-preconscious, etc.
And this "real logic" has scale. There is always a local-global asymmetry involved. Global universals and their local particulars. Whereas modern symbolic logic has developed through the discard of scale - the reduction of asymmetry to (mere) symmetry. So the yes/no, on/off, binary and scaleless choices of information theory.
I appreciate the fact you like a good argument but you have to at least make an attempt to understand what the other side is saying. Saying you find things meaningless, or you don't believe, is not a very interesting position to hold. So I'll limit my responses.
"real mathematics is a way of discovering our ideas of space, geometry and number"
You mean pattern and form. Self-consistent organisation.
"What empirical model of reality would you say the theory of represents?"
As you say, it all starts with observation of the world - modelling at its most natural level. The ancients noted the regularities of the world and generalised to create some structural truths about objects like triangles and morphisms like spirals.
Then mathematics has developed by even greater generalisation - as articulated by category theory for example. And demonstrated in the move from euclidean to non euclidean geometry, or simple to complex number. We find that we exist in a world that is highly constrained (3D and flat - and scaled) and then generalise by successively removing those constraints to discover if self-consistent regularity still exists.
And you can see what happens when you do this with quarternions and octonions. The regularity frays. Properties like division erode as you generalise the dimensionality. Instead of producing crisp algebraic answers, the meaning of the algebra becomes vague.
So the most general maths can cease to model crisp properties that were there in the original "empirical" view.
When this happens, many say maths has just wandered off into the wilderness - or a landscape in string theory's case.
But my philosophical approach is different. I am saying that generalisations lead back to vague potential. And the way to rescue the situation is by also building the global constraints - the selection rules that represent the idea of "self-consistency" - back into the maths explicitly. So maths with scale.
"Science develops because people question or ideas of reality not because we model it."
Ideas are models - ideas formalised.
I apologize for saying things were meaningless - but your use of language I found inpenetrable -plus you took a lecturing tone. So I felt I was being lectured to with non-specific vague words. This was a sincere reaction and I felt very frustrated.
I still don't exactly know what your language means and that is why i gave up. I think clarity of expression is necessary and lack of clarity is a sign that the person does not know what they are talking about.
For instance you just told me what I mean - as if I don't know what I mean. That is condescending. I have no desire to fight with anyone and I am totally open minded. But you have not been clear as far as I am concerned and you have been lecturing. That is also not very interesting.
apeiron
Sep10-09, 11:04 PM
I think clarity of expression is necessary and lack of clarity is a sign that the person does not know what they are talking about.
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I accept I can be irritating. All I can say is that I was also feeling irritated.
Also I believe that any lack of clarity is due to the unfamiliarity of the ideas I am attempting to communicate rather than my alleged deficiencies as a communicator.
Yes, these ideas I am expressing do indeed come from a different community - a rather small band of systems thinkers such as Salthe and Pattee. And I understand how opaque they can seem. It took many years of discussion for me to come round to some of them. And we are also talking about work in progress - current research.
Anway, I have tried to create introductions to some of the key ideas like Vagueness - see this thread.
http://www.physicsforums.com/showthread.php?t=301514&highlight=vagueness
Ok So now that we realize that we are all sincere and serious here and are not trying to be dogmatic and contentious it would greatly interest me to understand how you view some of the examples that I suggested. Why not start with the examples of fields of mathematics that do not arise from attempts to explain empirical data. A simple one that would could all talk about without taking a math course first might be the discovery of hyperbolic plane geometry in the 18'th century.
My understanding is that people for centuries felt that Euclidean geometry was intrinsic to the idea of geometry itself - that the parallel postulate was indispensible to the idea of space.
I believe that Kant event thought that Euclidean geometry was an a priori synthetic idea meaning that it was not an empirical model but rather an intrinsic feature of our experience of spacial relations ships. But like logic which you seem to agree is intrinsic, Kant thought that Euclidean geometry was intrinsic.
Many others agreed with him and realized that if this were really true then the parallel postulate should be provable from the simple axioms of space that describe the way lines intersect and how they separate a plane. One axiom said that two points determine a line. Another said that a line separates a plane into two half planes. A third said that two lines in a plane can intersect in at most one point.
These guys already knew that parallels must exist - not empirically because that would be impossible to test - because they knew that two lines that intersect a third at right angles must be parallel. They just couldn't prove that they were unique. It was uniqueness that got them.
This lead them to question their intuition/picture of straight really meant. gauss finally came up with a model of plane geometry where lines were actually curves and where the parallel postulate failed. In his geometry there were infinitely many parallels through any point.
After that people thought that there were two possible intrinsic geometries of space and only after they realized this did they actually try to test it out - under the assumption of course that our picture of space that is derived from sense experience actually must obey geometrical laws. Gauss went out and measured large triangles on the Earth to see he he could detect angle defects away from 180 degrees.
So you need to take this Kantian or perhaps Platonic - you would know better than I - way of looking at things and tell me how it was only just discovering empirical relationships - generalizing observations - through models. This to me, and I know for sure for gauss and his colleagues - was an investigation into the intrinsic nature of our ideas of space. The empirical modelling part was not central to the investigations and came afterwards when Gauss realized that if one believed - by either philosophy or faith - that experience actually exhibits the laws of geometry that one should then be able to test for the two possibilities.
Let's make this the starting point and take this paragraph as a first step to get thing going.
apeiron
Sep11-09, 08:59 PM
My take here starts by saying it is a false dichotomy to think the situation would be EITHER empiricism OR platonism (or constructivism or intuitonalism, or however else we want to phrase this traditional divide between "looking out" and "looking in"). Instead - logically - it must always be BOTH. As the complementary extremes of "what can self-consistently be".
This is what happens because I chose asymmetric dichotomisation as the foundation of my logic. This is of course the unfamilar bit, even though it starts from ancient greek metaphysics (Anaximander, Aristotle), was messed about a bit by the likes of Hegel, and reappears in modern times with Peirce.
Now asymmetric dichotomisation says that any (vague) state of possibility or potential can only be (crisply) divided if that act of separation goes in two exactly "opposite" directions. And by opposite, this is not symmetric as in left/right or other kinds of symmetry breakings which have just a single scale. It must be an asymmetric breaking that is across scale and so results in completely unlike outcomes (as opposed to merely mirror reflections of the same thing).
If you are with me so far, then the classic examples of asymmetric dichotomies in metaphysics are local-global, substance-form, discrete-continuous, stasis-flux, chance-necessity, matter-mind, vague-crisp, subjective-objective, atom-void, space-time, location-momentum (and the list goes on, but these are among the "strong ones").
You can see that each is both the very opposite of the other, and yet also logically mutual or complementary. That is because each is defined actively as the exclusion of the other. Pure substance would be a stuff that has absolutely no form, and form is that which has absolutely no substance. (Even Plato had to have the BOTH of the forms and the chora).
So this is an emergentist and interactions-based logic or causality (a logic being a generalised model of causality in my book). You cannot have one side arise into being, into existence (or persistence) without also forming the other. As one arises (in thought or reality) by becoming everything that the other is not.
As I say, Anaximander was the first to articulate a vagueness => dichotomy => hierarchy approach to modelling causality, the logic of reality. Aristotle then polished it up (as in the law of the exclude middle). Today, you can see mathematical sketches of the idea in the symmetry breaking models of condensed matter physics, in hierarchy theory, and even in some basic stabs at maths notation.
Check out Louis Kauffman's musings on this.....
http://www.math.uic.edu/~kauffman/Peirce.pdf
The laws of form are another stab....
http://en.wikipedia.org/wiki/Laws_of_Form
A gateway to Peirce's writings (which are only a precursor to what I'm talking about)...
http://www.cspeirce.com/
And others currently treading some of the same ground (though I would have many criticisms of Kelso's actual approach)...
http://www.thecomplementarynature.com/
Anyway, I hope you can appreciate that this is like swapping in, swapping out, a complete computational architecture. There is standard logic based on atomism, mechanicalism, locality, and other good stuff which is like your classic sturdy von Neumann serial processing engine. It works, no question. Then over here in left field, there is an attempt to build an architecture of thought, a way of modelling, that is founded on very different basic computational principles. It is like the attempt to get neural networks off the ground. Some kind of global, holistic, hierarchical version of logic. And while it looks promising, it is still a long way from commercialisation.
But anyway, lets take these still developing ideas and apply them to the question you asked.
Again, for me on the grounds of logic (all reality always works this way) I would come with the expectation that the story is going to be not either/or but instead both, and interactionist. So yes, strong dichotomies always emerge, and then the whole point is that they emerge because their existence is self-consistent in the wider view. They are mutually causal, or synergistic as asymmetric extremes.
Therefore it does seem that the creation of mathematics has this basic divide. There is either the pure development of ideas, or the discovery of ideas from observation. And my logic would force me to expect a mutually emergent story. The firming up of ideas inside a person's head allows them to make more detailed observations of the world, which in turn allow for more development of ideas inside their head. And these two parts of the action are driving each other ever further apart in scale. As the observations get ever smaller, ever finer, ever more particular, so the ideas get ever more general, ever more global and universal, ever more lacking in picky detail.
Now to take the specific example of non-euclidean geometry. The tale of the discovery follows this dichotomous logic. At first, forms got separated from substances in a way that divided the flat 3D world of immediate experience. Then as mathematicians realised that just three dimensions is a rather particular choice, and likewise just flat space was a rather particular choice, they could make a leap of generalisation to allow infinite dimensionality and any curvature. Their ideas became less particular, and so more general.
At the same time, this step in one direction brought with it a matching step in the ability to make ever finer "observations". It became possible to model some world with some particular curvature or number of dimensions. Maths could start exploring imaginary worlds of any crisply chosen design (and science could then use this new technology to test our actual world against the new variety of predicted designs).
So dichotomisation is the logic by which humans stepped back to see more. And then I would go further - from epistemology to ontology. Dichotomisation also is how the world probably actually emerges.
Taking non-euclidean geometry, we can see for example that "flat space" is precisely the average, the sum over histories, of curved space. If you have a dichotomous spectrum from purely locally hyberbolic space (disconnecting sea of points) to purely global hyperspheric space (curvature which makes a continuous or perfectly closed space) then flatness is the average, the equilibrium outcome, of these extremes "in interaction".
Of course this is still a hypothesis as I'm not sure how to go about constructing a mathematical proof of the idea. But I am just sketching the kind of answer I would expect to be the case if dichotomous logic is a valid logic.
There is another argument about why there would be just three spatial dimensions. But I can save that for some other time as it is even more left-field if Peircean semiotics is unfamiliar terrain.
To sum up, all my arguments stem from applying a different computational architecture. And it is not an arbitrary choice as - dichotomously - there would have to be exactly two deep models of logic/causality. Standard logic is one pole, and now I am working with people in developing the other pole. I see this as great news for good old fashioned atomistic logic as it cements its authority in place. It can be "right" because there is also the asymmetric view now making it "right" - that is, together they exclude the middle, all other possible approaches to logic.
So dichotomies rule. And the division over whether maths is derived from intuition or perception is a classic example of how both in interaction, creating a virtuous spiral of development, is the answer.
Then the logic of our minds is also the logic of reality itself. Dichotomies or symmetry breakings are also how things happen "out there" - how systems develop into being, complex hierarchies arising out of vaguer potentials because they are the self-consistent way a vagueness can be stabily, self-persistently, divided.
I am sure this is still indigestible. But just focus on some dichotomy and see for yourself if you can break it down differently.
Local-global is the most fundamental dichotomy I believe - pure scale. Though (dichotomously) it is then paired with an equally fundamental dichotomy vague-crisp. One talks about what exists, the other how what exists has developed.
But substance-form is the Athenian set-piece debate. Or you could back up a bit to consider the weaker dichotomies of stasis-flux or chance-necessity or atom-void.
I am digesting your words - thinking about them - will reply when I have something cogent to say.
apeiron
Sep15-09, 10:01 PM
these might also be useful....
A view on Rosen's modelling relation
http://www.osti.gov/bridge/servlets/purl/10460-5uGkyu/webviewable/10460.pdf
A review of his book
http://www.metanexus.net/magazine/ArticleDetail/tabid/68/id/2589/Default.aspx
systems biology - the project
http://www3.interscience.wiley.com/cgi-bin/fulltext/116833284/PDFSTART
The entire arguement is based on correspondence to observation. There is nothing special about math in this sense. It needs to be corroborated. 2 humans plus 2 humans can equal 5 if one gets pregnant. Therefore 2+2=5. It is just generally true, when you categorize things together in groups that 2+2=4.
Math is about generalizations, that is where its strength lies. But there is no math statement that stands on its own. Its correspondence to observation is what makes it a valid generalization. Observation is what math is built on.
If 2+2 = 5 because one gets pregnant and thus creating another human being, the equation would be 2+3 = 5; unless you don't consider the baby inside the womb to be a human being, but then why would you at the same time say that 2 humans plus another 2 humans equal 5?
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Basic arithmetic comes from our concept of units and quantity. 1 + 1 = 2 is simply 1 full unit of what we are considering, plus another full unit, equals 2 of those units. Other more abstract concepts can be deduced from previously deduced logical principles, we can't expect the Engilsh language to be capable of encompassing any idea there could ever be. Not all propositions refer to the most fundamental logical principles (but they could be deduced from other fundamental logical principles)
JoeDawg
Sep18-09, 02:51 PM
If 2+2 = 5 because one gets pregnant and thus creating another human being, the equation would be 2+3 = 5; unless you don't consider the baby inside the womb to be a human being, but then why would you at the same time say that 2 humans plus another 2 humans equal 5?
Why would you? well that is the point. The kind of system you create will depend on what you want to do with it. If you want to describe procreation as part of your system, then 1+1=3, and 2+2=5, etc...
Your math depends on correspondence. There are mathematical systems that don't include the concept of zero. There are systems that are based on one, a few, more than a few.
And this is why you literally have to create new math to describe experiene which is quite alien to what you normally experience, for example, quantum reality. Because different things are important to you.
Saying, why, and what is important, is just part of defining your system.
Why would you? well that is the point. The kind of system you create will depend on what you want to do with it. If you want to describe procreation as part of your system, then 1+1=3, and 2+2=5, etc...
Your math depends on correspondence. There are mathematical systems that don't include the concept of zero. There are systems that are based on one, a few, more than a few.
And this is why you literally have to create new math to describe experiene which is quite alien to what you normally experience, for example, quantum reality. Because different things are important to you.
Saying, why, and what is important, is just part of defining your system.
Actually I'm pretty sure it would look like this :
M⋃F → B(male union with female leads to baby or 1 union with 1 leads to 3)???
JoeDawg
Sep18-09, 06:23 PM
Actually I'm pretty sure it would look like this :
M⋃F → B(male union with female leads to baby or 1 union with 1 leads to 3)???
Except when it doesn't.
You could also describe it:
1+1= 2 1/2
Or you could add up the cells in each body.
Or mathematically represent the genetic information.
Its all about what you consider relevant. Abstract reasoning allows you to make whatever distinctions you like. This is why math can be very precise. The only limit is your assumptions, and what you consider relevant, that is, what the math corresponds to.
If, in your system, get 2+2=5 then what you are adding are not numbers! As a consequence of the properties of numbers, 2+2 always equal 4, and hence if 2+2=5 you are not dealing with numbers. This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics. If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.
JoeDawg
Sep19-09, 01:14 AM
If, in your system, get 2+2=5 then what you are adding are not numbers!
They are numbers, they just don't correspond to the same 'things'. Which is the point.
This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics.
Its not a flaw, its a different model.
If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.
And how do you confirm this?
They are numbers, they just don't correspond to the same 'things'. Which is the point.
No. As a consequence of the definitional properties of numbers 2+2=4 is always correct. 2+2=5 is false statement when dealing with numbers, and thus is the statement is correct, they is not numbers.
This is not a flaw in the mathematical system, but in your ability to properly model the situation with mathematics.
Its not a flaw, its a different model.
Well, a model with dissatisfactory accuracy, which was the point...
If this is a consequence in your system, then, clearly, the total amount of human beings cannot properly be computed by adding.
And how do you confirm this?
Obviously by observing that 2+2=5 is a result in the system, which is untrue for numbers.
JoeDawg
Sep19-09, 11:15 AM
Obviously by observing that 2+2=5 is a result in the system, which is untrue for numbers.
Sigh.
And where does the definition of 'numbers' come from?
Sigh.
And where does the definition of 'numbers' come from?
Try peanos axioms for example. And ZFC defines numbers based on the concept of sets. This is a rigorous construction of the natural numbers.
Math works in our reality because we define our physical concepts in mathematical terms. Mathematics is an extremely effective tool for describing physical theories for exactly this reason. Because physical laws seem to follow certain laws, we are naturally encouraged to apply our mathematical concepts too it, and with great accuracy.
JoeDawg
Sep19-09, 07:39 PM
Try peanos axioms for example.
And what is an axiom?
Its an assumption.
Why would we choose one axiom over another?
Because some axioms have a broader scope, they describe a wider range of experience.
Math is about generalizations, we draw these generalizations from observing consistency in the world, the greater the scope of the axiom, the more we can use it, and the more we do use it.
1+1=3 is useful, if you are talking about sex. But its not useful when you are discussing apples. 1+1=2 is useful for apples, as well as a large range of things we as humans consider relevant.
And what is an axiom?
Its an assumption.
Why would we choose one axiom over another?
Because some axioms have a broader scope, they describe a wider range of experience.
Math is about generalizations, we draw these generalizations from observing consistency in the world, the greater the scope of the axiom, the more we can use it, and the more we do use it.
1+1=3 is useful, if you are talking about sex. But its not useful when you are discussing apples. 1+1=2 is useful for apples, as well as a large range of things we as humans consider relevant.
An axiom is not an assumption as in the context of "taking it for granted". The concept of an axiom is a definition. When we are postulating an axiom, then we are defining whatever we are talking about. We are not talking about something we think we know something about, and then saying something we might think is true, and then take it for granted. An axiom is an assumption made in order to explore the consequences, and this is a critical point.
Math is about making generalizations, but it does not base itself upon empirical evidence, although it is inspired by it. Mathematics is (luckily) based upon rigorous definitions, which make silly statements like 1+1=3 meaningless if you are talking about numbers.
JoeDawg
Sep20-09, 12:02 AM
although it is inspired by it.
Inspired? What does that even mean?
which make silly statements like 1+1=3 meaningless if you are talking about numbers.
I think I see the problem here, you've decided that certain axioms of math have some sort of Platonic existence. But what 'numbers' are, is whatever they are defined to be. Now, some definitions are more useful.... empirically, and those are the ones we keep, use, modify, and refine. However, it is via observation that we decide which axioms are useful, and which are meaningless. You can't generalize from nothing, first you have to have instances, and then you develop rules based on those instances.... this is how logic and math work. Oh, and your patronizing tone is actually quite amusing. I don't disagree with most of what you said, I just don't think it means what you think it does.
Hehe, I think we have lost track of objectivity here. This debate has obviously come to a halt, I guess we have to agree to disagree. I won`t discuss anything but arguments.
However, I will say this: It doesn`t matter how we are choosing our axioms here, what is important is that we follow those we have chosen. In any reasonable definition of numbers, (read: axioms), 2+2=4.
I am patronizing?
Sigh.
And where does the definition of 'numbers' come from?
:smile:
JoeDawg
Sep21-09, 11:11 PM
In any reasonable definition of numbers, (read: axioms), 2+2=4.
And reasonable means... whatever Jarle agrees with.
Like I said... patronizing.
And reasonable means... whatever Jarle agrees with.
Like I said... patronizing.
You aren`t really discussing, are you?
apeiron
Sep22-09, 07:20 AM
And reasonable means... whatever Jarle agrees with.
Actually Jarle has been taking the more reasonable line here. Yes the idea of numbers may be a generalisation from experience, but it is also a maximally general one. As far as we can know. From the prime test, which is the self-consistency of the algebraic structures we find we can spin from the number system.
The "1+1=3 when we are talking about making babies" is a childish debating point. It is a description using numbers to talk about some specific kind of biological event. It is not a self-consistent consequence of the numbers themselves.
I certainly believe that we model reality. And also that our concept of number can be challenged. Axioms are always questionable.
But it becomes just silly to not understand that axioms are generalisations that can then have matchingly crisp or definite consequences. So you can't just try to assign your own private meanings to the objects of that system of morphisms as JoeDawg wants. The actual model consists of both its axioms and its consequences.
As ever, you have to keep your eye on the dichotomies at the centre of these things. o:) That is why maths moved on to category theory in its search for its fundamental ground. Structure-preserving change, patterns or symmetries that can persist.
JoeDawg
Sep22-09, 11:55 PM
The "1+1=3 when we are talking about making babies" is a childish debating point. It is a description using numbers to talk about some specific kind of biological event. It is not a self-consistent consequence of the numbers themselves.
I was never advocating that is was a replacement for 1+1=2, nor that we should throw out mathematics.
The point was simply that we derive our math from experience. The reason it is self-consistent is because we have specifically created it to be so, and moved from using numbers that only really work in specific cases, like with babies, to number systems that cover a wider amount of instances and with great precision. 1+1=3 may be descriptive, but only in a limited sense. 1+1=2 is also limited, however, integers are not as precise as decimals. And fractions also function differently. We have found ways to convert from one to the other, but each has problems and limitations.
And as we add more to math we specifically seek ways to make it consistent with what we already have. Its only self-consistent because that's the standard by which it is judged useful, and usefulness is about correspondence to reality.
slider142
Sep23-09, 06:40 AM
If you dislike the babies argument, consider a universe where Bose-Einstein condensates were more common than in ours. In that universe, the development of mathematics may have favored axioms that result in the theorem 1+1=1 versus the more exotic 1+1=2 branch (if anthropomorphic consciousness is even possible in such a universe).
I was never advocating that is was a replacement for 1+1=2, nor that we should throw out mathematics.
The point was simply that we derive our math from experience. The reason it is self-consistent is because we have specifically created it to be so, and moved from using numbers that only really work in specific cases, like with babies, to number systems that cover a wider amount of instances and with great precision. 1+1=3 may be descriptive, but only in a limited sense. 1+1=2 is also limited, however, integers are not as precise as decimals. And fractions also function differently. We have found ways to convert from one to the other, but each has problems and limitations.
And as we add more to math we specifically seek ways to make it consistent with what we already have. Its only self-consistent because that's the standard by which it is judged useful, and usefulness is about correspondence to reality.
Though I have been trying to stay out of this until I understand the philosophical points you are making my gut still objects to the idea that reality and mathematics are somehow disjoint. It seems to me that pure sensation has no intrinsic structure. So to say that you are generalizing from something without structure seems impossible. Controlled observation can only give us clues to the structure of reality. But reality is not something we directly experience or observe. Though a fusion of thought and observation we lift the veil of sense experience.
apeiron
Sep23-09, 07:41 AM
If you dislike the babies argument, consider a universe where Bose-Einstein condensates were more common than in ours.
If your reality were a Bose-Einstein condensate, where would the notion of a this one, as distinct from that one, derive? It would seem you would only feel like saying 1=1 at best? Experience would not yield a 1 plus a 1.
apeiron
Sep23-09, 08:32 AM
The reason it is self-consistent is because we have specifically created it to be so, and moved from using numbers that only really work in specific cases, like with babies, to number systems that cover a wider amount of instances and with great precision.
So you are arguing against yourself here. It seems there was some innate and inevitable trend to be discovered. A path that leads from the vaguely useful to the crisply useful, from the particular to the universal.
Of course, human civilisation did not actually start with a mathematics based on babies and then progress to something better.
Psychologically, the first and most natural dichotomy was probably the distinction between the one and the many. Or figure and ground, event and context, signal and noise. The idea of symmetry and then the symmetry breaking.
And anthropologically, if we want to focus on utility, the origins of maths probably had most to do with the cycles of the days and the seasons. Cycles of death and renewal. So more geometry than algebra. Though perhaps they did notch off sticks to count off cycles of the moon.
Counting became important in ancient agricultural civilisations with hierarchical ownership. Counting boards and tally sticks to keep track of the goats and sheafs of wheat. But I don't think even the Summerians recorded 1 goat + 1 goat as making 3. Or derive from that the further truth that if I have 3 goats and give you 1, then that must leave me also with only 1.
Again, mathematical systems must follow a certain path - the dichotomy defined by category theory. You must have the fully broken symmetry of the local and the global, the one and the many, the object and the morphism. Yes this is derived from experience - and also appears to be a truth about reality. Which is why maths works.
The mistake you keep making is then to just focus on one half of the dichotomy, of the broken symmetry. The number 1 does not stand alone. It is defined only in relation to its context. Which is why 1 has a stabilised meaning and cannot float free as something that could be defined anyway we choose.
Of course there is then a further epistemological wrinkle to all this. Out there in reality, symmetries are not truly "broken". Instead the breaking apart is merely approached in the limit. However in maths, as a modelling choice, we do treat symmetries as properly broken. So we treat the number 1 as not the limit of the act of separating the one from the many, but as actually - axiomatically - a thing which is separate, isolate, discrete. So maths is in fact unreal in this crucial regard. It appears to say something about reality which cannot in fact be.
JoeDawg
Sep25-09, 01:02 AM
It seems to me that pure sensation has no intrinsic structure.
In as far as its a function of biology, I would say it does, but I'm not sure what you mean here. The human mind instintively separates experiences into events and objects.
So to say that you are generalizing from something without structure seems impossible. Controlled observation can only give us clues to the structure of reality. But reality is not something we directly experience or observe. Though a fusion of thought and observation we lift the veil of sense experience.
Well, I would say reality is what we experience, both with regards to thinking and observation. The source of reality is the mystery.
JoeDawg
Sep25-09, 01:30 AM
It seems there was some innate and inevitable trend to be discovered. A path that leads from the vaguely useful to the crisply useful, from the particular to the universal.
Utility is relative. Consistency in experience gives us the foundation. The direction and scope are up to us. Narrowing the scope when needed just gives you a better picture of what you want to see... like a fractal.
Of course, human civilisation did not actually start with a mathematics based on babies and then progress to something better.
It would have been more basic than that, but the arbitrariness of the example is also important to my point. I think many people want there to be some ultimate math, like a ToE, but math is what it is used for.
And anthropologically, if we want to focus on utility, the origins of maths probably had most to do with the cycles of the days and the seasons. Cycles of death and renewal. So more geometry than algebra. Though perhaps they did notch off sticks to count off cycles of the moon.
The ancient Egyptians invented geometry to deal with the problem created by the Nile flooding the land. It was good for the land, but it made it difficult to allot farmland. The flooding destroyed all landmarks. Geometry solved this problem. It was also useful with regards to astronomy, and the building of tombs. But no perfect circles exist.
Again, mathematical systems must follow a certain path - the dichotomy defined by category theory.
One doesn't need math to have categories or dichotomies. Math just formalizes what our minds and bodies already do.
The mistake you keep making is then to just focus on one half of the dichotomy, of the broken symmetry. The number 1 does not stand alone. It is defined only in relation to its context. Which is why 1 has a stabilised meaning and cannot float free as something that could be defined anyway we choose.
I agree there has to be context, which is why I used procreation as context. It's simplistic and not broadly useful, but it makes the point. Without context all math is just squiggles on a page.
Of course there is then a further epistemological wrinkle to all this. Out there in reality, symmetries are not truly "broken". Instead the breaking apart is merely approached in the limit. However in maths, as a modelling choice, we do treat symmetries as properly broken. So we treat the number 1 as not the limit of the act of separating the one from the many, but as actually - axiomatically - a thing which is separate, isolate, discrete. So maths is in fact unreal in this crucial regard. It appears to say something about reality which cannot in fact be.
The map is not the territory.
In as far as its a function of biology, I would say it does, but I'm not sure what you mean here. The human mind instintively separates experiences into events and objects.
Well, I would say reality is what we experience, both with regards to thinking and observation. The source of reality is the mystery.
biology is an explanation of experience - there may be other explanations - these are not what experience is - if you are saying that all ideas generalize experience then from this point of view you are just making a statement about biological theory.
Different people attach different intellectual constructs to experience of the outside world. Does this mean that they have different instincts? Does this mean that there are multiple realities?
The Impressionist era artists, particularly, Monet and Cezanne, tried to eliminate intellectual constructs from their images. They rejected the idea that such things as perspective and the theory of light actually are part of experience. They viewed these things as intellectual overlays. Their goal was to record experience at the moment of sensation just before the division into objects occurs. This is what Cezanne meant when he said that when he paints he tries to learn from nature. This is what I meant by unstructured experience. This is why their pictures often appear flat and are not supported by a geometrical skeleton as in classical art. At that time, people believed that pure experience was individual and even racial. from your point of view, they would say that there are as many realities as there are individuals. And I get the impression that this is what you are saying.
To me, whether experience is instinctively organized or not - and I am not sure how anyone knows this - that does not mean that experience is not separable into is cognitive constructs and sensory elements. To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.
S.Vasojevic
Sep25-09, 08:52 AM
Well I think it is because our world is about relations between things. Why object that has mass of 1 kg has twice smaller mass then object of 2 kg? If you say it has four times smaller mass, what would you say comparing it to object that has mass of 4 kg? Nature of course does not care about numbers, they are symbols made by us in our modest attempt to explain our existence through relations between entities.
JoeDawg
Sep29-09, 01:30 AM
biology is an explanation of experience - there may be other explanations - these are not what experience is - if you are saying that all ideas generalize experience then from this point of view you are just making a statement about biological theory.
It seems to be the case that Ideas are distinct from Experiences.
And it also seems to be the case that certain Ideas are linked to certain Experiences.
Experiences however, seem to have more detail and specificity. Ideas seem more vague.
If you want to accept the radical empiricism of Berkeley and state that matter doesn't exist and such... yes, that is another 'explanation'... but accepting the conceit of the common understanding of modern science and physicalism, yes, all ideas generalize experience.
Different people attach different intellectual constructs to experience of the outside world. Does this mean that they have different instincts? Does this mean that there are multiple realities?
Defining reality can be confusing. Often people equate reality with existence, which is fine, but it ignores the fact that experience sometimes contradicts existence. At which point you have two 'realities'. Dreams are a good example of this. Last night I flew through an alien city... does the alien city exist the way my computer does... well no, but it does, or did, exist.
So yes, in that sense everyone has their own 'reality' following them around. And those realities all seem to come from the same source.... existence.
As to instincts... there are biological instincts and learned instincts, so yes different people can have different instincts. An example of the latter instinct would be something you are trained to do, and therefore do automatically without thinking.
At that time, people believed that pure experience was individual and even racial. from your point of view, they would say that there are as many realities as there are individuals. And I get the impression that this is what you are saying.
In terms of experience sure, but not ontologically. Its true we can't know for certain whether 'its all just a dream', but this is where being reasonable comes in. I see no reason to doubt that there is a common source for experience.
To me, whether experience is instinctively organized or not - and I am not sure how anyone knows this - that does not mean that experience is not separable into is cognitive constructs and sensory elements. To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.
Problem there is 'true sensory experience' is not self-reflective. Think of how animals and babies seem to live in the moment. Any analysis of experience, and certainly taking the time to paint it on canvas, would entail cognitive constructs. Even talking or thinking about an experience... puts it within a constructed framework. Meaning is cognitive. Experience just is, and then it's gone.
apeiron
Sep29-09, 05:20 AM
To me the intellectual constructs are those things which are not experienced and the sensory data is what is given - that which is directly experienced.
The idea that experience is given - the ineffability of qualia - is not something that would be supported by psychology and neuroscience. Experience is also constructed.
The terms I prefer to use here are "ideas" and "impressions" as it helps preserve the constuctedness of both aspects of awareness. One is not being favoured over the other in term of veridity (or lack of it).
Now what is the nature of the actual divide (I mean dichotomy) that you are sensing here? The instinctive separation you want to make?
It is between the general and the particular, the model and the measurements. Or as in Grossberg's neural nets, the long term and the short term memories.
So in all these ways of saying the same thing, we have something that acts as the longer lived context - the idea that constrains. And then we also have the moment to moment impressions, the fleeting train of events, that constructs some particular state of experience.
And the two scales of mental activity are in interaction. They are not separate processes but instead separate levels of process.
So ideas are the established habits of memory, anticipation and thought which serve to give shape to impressions. They make sense and organise each moment. Awareness is created top-down.
But equally, impressions over time build up the ideas. The brain learns by generalising from what it thinks happened (the experiences it constructed) and so builds broader, sturdier, habits of interpretation/perception.
So all this activity is subjective. There is no direct objective access to reality. But, being a systematic approach, the subjective collection of ideas~impressions does come to model reality very well for our purposes.
And then maths/science/philosophy are activities that try to repeat this basic cognitive formula on a still broader social scale. Societies have purposes and evolved models of reality that serves them.
The idea that experience is given - the ineffability of qualia - is not something that would be supported by psychology and neuroscience. Experience is also constructed.
While this is arguably true, its also more complex than that...
Experiences and Ideas exist on the level of consciousness. They are, in a sense, immediate.
Neuroscience and psychology work on the level of explanation.
So you can indeed describe experience as unconstructed. Its only after experience has been assessed and compared that we get the sciences, and then we work backwards for an explanation. It is then that we can view experience as constructed.
Which is not to say that neuroscience and psychology don't offer good explanations, but they are heavily dependent on that 'ineffable qualia', which is our primary mode of being.
Ideas are more obviously constructed, since they involve internal (mental) processes, not external sources.
So you can indeed describe experience as unconstructed. Its only after experience has been assessed and compared that we get the sciences, and then we work backwards for an explanation. It is then that we can view experience as constructed.
I understand what you mean but this is what I call the introspectionist fallacy.
In fact paying attention to your experiences is a highly artificial and learnt skill. Animals and babies can't do it. And it takes a lot of practice and scaffolding for even modern Western adults.
You can argue that there are degrees of construction. So the naked sensation of redness is perhaps less obviously mediated than your perception that a ship on the horizon is a large object a long way away.
Yet still the very act of stopping and contemplating "redness" is a highly constructed - and constructing - action. Your brain has to suppress attention to much else to manage to make it seem like the redness of something red is filling your awareness.
Or to use a better example, think of the tricks that an impressionist painter goes through to see the distant hills as purple not brown or green. Cut out a little square in white card and hold it up to physically block out the contextual information that is fooling your appreciation of the pure actual colour.
Yes, some things may be less mediated, less apparently constructed. But in the end, all experience is the result of some act of mediation, some constructive effort and not about naked witnessing.
In fact paying attention to your experiences is a highly artificial and learnt skill.
Well, sure. But that is not really what I'm referring to.
Animals and babies can't do it.
And this would be the example of 'experience'. Quite a lot of our adult life is constructed, but thats because we build ideas around 'sensations'. This is why differentiating between ideas and experience is important. Pleasure and pain, for instance, are immediate. They don't even need to be localized in time or space, within our minds, although quite often they are... and in that case part of the experience is constructed.
Yet still the very act of stopping and contemplating "redness" is a highly constructed
Contemplating yes, but experiencing no. Obviously if you are calling it red, you're attaching an idea to it. But there are lots of times, when we see something for the first time, we don't place it... within a framework, at least not immediately.
Impressionist painters are trying to simulate raw experience.... on canvas. Not something I think you can really do successfully, but they try.
Yes, some things may be less mediated, less apparently constructed. But in the end, all experience is the result of some act of mediation, some constructive effort and not about naked witnessing.
Like I said, the fact biology 'constructs' a sensation in the mind is, I think, quite a different thing. Biology is an explanation. We as adults may reflect on sensations, almost immediately, and certainly our brains seem to want to categorize everything. But before we learn to do this, and on occasion when something intense or unexpected happens, we do have a kind of raw experience. And that is the sort of thing I am talking about. Adrenaline junkies crave this, and so do people who meditate.
A person's first orgasm, for instance, can completely shatter their reality. It's only after, when they organize thoughts around it, that it becomes 'constructed'.
You are arguing here from personal prejudice rather than psychological or neurological fact. Which makes for an unproductive conversation as usual.
To ground your ideas, why not simply tell me at which point as a photon strikes a retinal receptor you feel that there is this supposed transition from raw input to mediated experience - constructed in the sense that the processing has begun in earnest.
Or if you prefer to focus on pain, then again, where after the finger is pricked with a pin does the percept swim into view. We know the neurology of the pain pathway. Where is the location where the magic of qualiahood achieved and there is an experience ready to be contemplated?
Yes, agreed there are degrees of mediation. And Sperling's iconic memory experiments would be a good line of evidence for you to be arguing here I would have thought.
But I fear you will never get the essential point that I am arguing. Which is that all experience is processing - mental construction - and ideas and impressions are then two extremes of this one process. They are not two different kinds of thing.
But if you insist on being dualist, taking the position that qualia are primal - naked conscious facts - then you will have to accept all the mystical and unscientific baggage with comes with such a philosophy.
If experience is constructed then it must have been constructed from something. that something was either not experienced or it was non-constructed experience. If it is not constructed, then it is given.
Experience has two fundamentally different ingredients - that which is unchanging, such as the idea of space and that which is changing such as the perception of a color. One is certain the other unpredictable.
An attempt to isolate these two through e.g. through introspection does not invalidate the difference just because the attempt is contrived.
Apeiron I don't know why you are getting insulting with me again. I feel like you are telling me that I am so stupid and biased that I don't have the right to be part of this discussion.
If that it true why not just ignore my posts?
I choose to be part of this discussion whether you like it or not.
I don't think you don't understand the basic fact of philosophy. That is that empirical theories of experience do not explain it they merely describe data that in the past has been detected in some experiment. You have no proof that these same outcomes will occur in the next experiment. These are merely constructs. The existential nature of experience - the true subject of philosophy (as opposed to science) - does not deal with empirical constructs. Your references to experiments and photons and whatever illustrate this that you do not agree with this. You think it is all mystical. The fact that you refute me by referring to neurological and psychological "fact" shows you believe this. A fact for you is some testable result of an experiment.
I think that you confuse philosophy with the theory of knowledge. Empiricists and positivists and others argue that statements about experience can only refer to testable results and can mean nothing more than the outcomes that they predict. This is a theory of meaning not a theory of the existential nature of experience.
If i say 'This is a piece of chalk', you say well what does that mean? What testable results does that imply? If I say experience is given, you say what experiments can I do that give that statement meaning?
I personally think that the idea that all meaning is really a collection of testable empirical outcomes is a definition that ignores the fundamental existential nature of experience. While it is valid for Science it begs the questions of Philosophy.
Pythagorean
Oct1-09, 03:13 PM
to the OP (I haven't read this thread, I've participated in a couple like it):
Because we designed it to.
Apeiron I don't know why you are getting insulting with me again. I feel like you are telling me that I am so stupid and biased that I don't have the right to be part of this discussion.
Ha, no I was insulting JoeDawg this time round o:). Sorry if that wasn't clear in the quoting.
The existential nature of experience - the true subject of philosophy (as opposed to science) - does not deal with empirical constructs.
I would be surprised if everyone agreed this was what philosophy was about. But maybe that is because my interests are clearly meta-physics and epistemology. Get these right and the rest follows I believe - even ethics and aesthetics.
As to my use of scientific facts, I follow Rosen's modelling relations approach. It is all about the interaction between ideas and impressions, models and measurements. So the "facts" inform the opinion, just as much as the opinon informs (or prejudices) the facts. You tend to see what you believe, and that is a meta-fact that our attempts to understand the world must deal with systematically.
This makes me impatient both with those who haven't worked sufficiently on forming their opinions (doing the philosophy) and noting the facts (doing the science).
I personally think that the idea that all meaning is really a collection of testable empirical outcomes is a definition that ignores the fundamental existential nature of experience. While it is valid for Science it begs the questions of Philosophy.
I was a psychology/biology student in the 1970s so felt all the frustrations of science's failure to tackle the issue of mind. And science is still generally failing to do its job here. It is the proper scientific (and mathematical) generalisation of the idea of mind which is my major life project. And I just don't see any real division between the philosophical and scientific aspects of this quest.
But if you insist on being dualist, taking the position that qualia are primal...
Consciousness is primary to epistemology, ontology is something entirely different.
I never said anything about dualism.
You're nothing but a cranky troll.
Math Is Hard
Oct2-09, 01:07 AM
In fact paying attention to your experiences is a highly artificial and learnt skill. Animals and babies can't do it. And it takes a lot of practice and scaffolding for even modern Western adults.
That appears completely untrue. Even simple sea slugs can learn from experience. If they did not pay attention to an experience, it seems unlikely they would be able to pair it with an outcome and consistently modify their behavior (protectively) when a dangerous event repeated.
I suppose, they couldn't "mull it over" after the fact, but there was a sensory register (attention paying) at some point.
That appears completely untrue. Even simple sea slugs can learn from experience. If they did not pay attention to an experience, it seems unlikely they would be able to pair it with an outcome and consistently modify their behavior (protectively) when a dangerous event repeated.
OK, this is getting ludicrous. Provide me with citations that Aplysia "pays attention" in Kandel's classic habituation experiments.
You in fact have this example exactly about front. Aplysia is genetically wired to respond to a prod and habituation is "learning" to ignore what is not actually dangerous. Or rather a simple tiring of the circuitry via the simplest feedback. No anticipation involved.
As to babies and chimps, I would refer you to Mead and Vygotsky. Feel free to debate the actual science.
Math Is Hard
Oct2-09, 05:43 PM
Then I might be unclear on the definition of "attention". The way I see it, it can be something that is controlled:
http://www.mybrilliantkidz.com/wp-content/uploads/studying-child.jpg
Or it can be something that is uncontrolled:
http://i2.photobucket.com/albums/y31/Fredcat/Cats%2003/Cats-CatsWatchingBirds.jpg
Controlled and uncontrolled would be a woolly distinction. And also irrelevant to my statement - "In fact paying attention to your experiences is a highly artificial and learnt skill."
It is selectively attending to the "contents of awarenesss" - introspection - that I was talking about. Extrospection is what brains are designed for. Introspection is a skill humans cultivate (and is difficult because it is essentially unnatural).
Math Is Hard
Oct3-09, 12:35 AM
Controlled and uncontrolled would be a woolly distinction. And also irrelevant to my statement - "In fact paying attention to your experiences is a highly artificial and learnt skill."
It is selectively attending to the "contents of awarenesss" - introspection - that I was talking about. Extrospection is what brains are designed for. Introspection is a skill humans cultivate (and is difficult because it is essentially unnatural).
OK, I think I understand you better now. Thanks for clarifying.
So, it is your belief that we are not biologically wired for metacognition (thoughts about our thoughts), but that we can learn it, and that non-human animals are incapable of this?
So, it is your belief that we are not biologically wired for metacognition (thoughts about our thoughts), but that we can learn it, and that non-human animals are incapable of this?
Correct.
I知 reviewing my mathematics knowledge, except I知 looking for a different reason. I understand how it works you know, 1 plus 1 so on, I知 trying to understand why it works.
Pure and applied mathematicians and physicists have tied our understanding of reality with mathematics. They believe that if it computes it is real. This will do for a while, till we pose a question beyond our understanding.
But that痴 for a different time. Why does it work?
Any takers?
Even though a calculation computes, the result may not reflect reality. To give you an example, lets assume a computer monitor has an area of 93.5 square inches. It's width is 2.5 inches larger then its length. What is the length and width of the computer monitor?
The area of the computer monitor is computed as A=LW.
We know the area so 93.5 = LW.
We also know the width is 2.5 inches larger then the length so 93.5 = x(x+2.5).
After we distribute the x, we arrive with 93.5 = x^2 + 2.5x.
After we set the equation to 0, we have 0 = x^2 + 2.5x - 93.5
Since we are too lazy to factor, we use the quadratic equation:
http://physicsforums.com/attachment.php?attachmentid=21039&stc=1&d=1255143899
The b value is 2.5.
The a value is 1.
the c value is 93.5.
After we plug in the values and do some calculating, we arrive with two solutions.
The roots of the quadratic are X = 8.5 and x = -11.
So we test our results with the area formula A=LW.
The area a will be 93.5.
The length will be 8.5.
The Width will be (8.5+2.5) or 11.
We plug in the values 93.5 = (8.5)(8.5+2.5).
We do the addition (8.5+2.5) = (11) first.
Then we times 8.5 with 11 to get 93.5.
The equation now appears as 93.5 = 93.5, and it its a true equation.
Thus, our first solution of 8.5 works, and we can tell the length is 8.5 inches and the width is 11 inches.
The next solution was -11, and we do the same thing to test the results.
After filling in the values, 93.5 = (-11)(-11 + 2.5).
We again arrive with 93.5 = 93.5, which is a true equation; however, the length of the monitor would be -11 inches, and the width would be -8.5 inches.
Can a monitor have a negative length and width? As far as mathematics is concerned, the answer is yes; however, we are presented with a physical limitation. So we disregard the negative result in favor of the positive result.
While I agree with your description of mathematics generally, I am not so sure that we can not have an ultimate mathematical/physical theory. Physicists differentiate between what they consider to be phenomenological theories and fundamental theories. For example the Shroedinger equation describes the spectrum of the hydrogen atom as a phenomenon but not in a fundamental way. this is because it takes coulomb forces as givens and does not explain them. But a theory like String theory attempts to explain everything fundamentally. Why could not a theory like this actually tell us everything exactly?
The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.
To illiterate the problem, I will create a very simple system.
Simple System:
In the United States, there is only one person named Joe who works as a professional landscaper. Joe mows lawns for a living. All inhabitants of the United States either mows their own lawn, or Joe to mows their lawn for them.
Limitation: If Joe does not mow his own lawn, then who does?
According to the system, if joe does not mow his own lawn, then Joe mows his own lawn.
See the problem?
This is why a TOE cannot be created. No matter how many axioms you add, you wind up with this same limitation. You can add axioms all day long with countless pages of complex details of the system, but you will eventually wind up with the Joe problem. If an attempt to force the Joe variable is made, the entire system becomes inconsistent.
SixNein
Oct10-09, 01:24 AM
Ah, so you are a dualist :smile:! I almost agree with what you're saying here, except for one minor point.
You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.
If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.
If math is ideas that are discovered, then there exists an objective and eternal mind.
If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.
People should drop the human element from the entire discussion because the human element complicates the problem. I think it would be best to form the question outside of the human mind completely; instead, people should assign the question to a computer. If a computer finds the solution to p=np, did the computer discover the solution or invent it?
Mattara
Oct10-09, 02:24 PM
The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
There are places where this incompleteness theory is not applicable.
The problem is due to Godel's theorem of incompleteness. The theorem is very important because it shines a light on a fundamental limitation on systems. The limitation occurs when an attempt is made to explore properties of a system with the system. In a basic nutshell, the attempt cannot be complete and consistent at the same time. I personally think this manor of wording is very misleading to people, so allow me to reword it. In a basic nutshell, you cannot create enough axioms in order to have consistency and completeness. Since you do not have enough axioms, your system is incomplete. If an attempt to force completeness despite the lack of axioms is made, then the system will be inconsistent.
To illiterate the problem, I will create a very simple system.
Simple System:
In the United States, there is only one person named Joe who works as a professional landscaper. Joe mows lawns for a living. All inhabitants of the United States either mows their own lawn, or Joe to mows their lawn for them.
Limitation: If Joe does not mow his own lawn, then who does?
According to the system, if joe does not mow his own lawn, then Joe mows his own lawn.
See the problem?
This is why a TOE cannot be created. No matter how many axioms you add, you wind up with this same limitation. You can add axioms all day long with countless pages of complex details of the system, but you will eventually wind up with the Joe problem. If an attempt to force the Joe variable is made, the entire system becomes inconsistent.
I don't see how Godel's theorem applies here. I thought it applied to specific models of axiomatic systems.
SixNein
Oct12-09, 11:43 AM
I don't see how Godel's theorem applies here. I thought it applied to specific models of axiomatic systems.
Formal, Finite, and Self Referencing are the requirements.
Are physical theories formal? Yes.
Are physical theories Finite? Yes.
Are physical theories self referencing? Yes.
*poof*
vectorcube
Oct16-09, 04:11 AM
Last time i was in this forum, there was this stupid guy that say "the laws of nature is derived from logic". Obviously, the guy is misguided, but i often feel that common people don ` t see the difference between math and physics. Math is a tool used to describe physical laws, and the tool to tease out the consequence of the laws.
Last time i was in this forum, there was this stupid guy that say "the laws of nature is derived from logic". Obviously, the guy is misguided, but i often feel that common people don ` t see the difference between math and physics. Math is a tool used to describe physical laws, and the tool to tease out the consequence of the laws.
Physical laws are mathematical in my opinion. So math is not a tool only. It is the law.
vectorcube
Oct17-09, 03:23 AM
Physical laws are mathematical in my opinion. So math is not a tool only. It is the law.
Physical laws does not have to be mathematical, and math is a tool, because there are infinite many mathematical results that do not have any applications to physical world.
Pythagorean
Oct17-09, 03:31 AM
Physical laws are mathematical in my opinion. So math is not a tool only. It is the law.
Your argument is rather meaningless:
premise 1: physical laws are mathematical
premise 2: (not stated, but implied... not sure I know what it is)
conclusion: It (presumably math) is the law.
In every case, your argument is vague. Premis 1 doesn't really say anything:
"Physical laws are mathematical". In what sense? Surely not in every sense, or it would just be mathematics! Physics is conceptual too, it has quality, not just quantity.
Premise 2: Your still have to tell us your second premise. You have to tell us why you think that Premise 1 leads to your conclusion.
Conclusion. "It is the law" This is meaningless. Is the law of what? Of the whole universe? That's quite a claim.
Come back with a more descriptive argument that can actually be argued.
Physical laws does not have to be mathematical, and math is a tool, because there are infinite many mathematical results that do not have any applications to physical world.
I agree that math is a tool but to say that there is more math than physical laws and more than one mathematical description does not change what i am saying. Kepler thought of physics as a process of ever better approximation to truth. Intermediate theories reflect and guide us towards this truth even though they may be incomplete and non-unique for the set of phenomena that they predict.
To think that mathematics is somehow separate from reality seems to me to be an arbitrary hypothesis and also seems to contradict all of experience.
WaveJumper
Oct17-09, 11:23 AM
Your argument is rather meaningless:
premise 1: physical laws are mathematical
premise 2: (not stated, but implied... not sure I know what it is)
conclusion: It (presumably math) is the law.
In every case, your argument is vague. Premis 1 doesn't really say anything:
"Physical laws are mathematical". In what sense? Surely not in every sense, or it would just be mathematics! Physics is conceptual too, it has quality, not just quantity.
Premise 2: Your still have to tell us your second premise. You have to tell us why you think that Premise 1 leads to your conclusion.
Conclusion. "It is the law" This is meaningless. Is the law of what? Of the whole universe? That's quite a claim.
Come back with a more descriptive argument that can actually be argued.
He probably read somewhere about the mathematical universe. While i am not a proponent of the theory, it's been a deep conceptual problem in physics to identify and conceptualise the basic constituents of matter in a non-mathematical way. All efforts so far have proved contradictory and incomplete. The basic building blocks of the material universe(electrons, quarks, quanta in general) cannot be unambiguously described without resorting to maths. They are not 'material' in any of the traditional ways that we are accustomed to dealing with. What is an electron? What is a quark? As i am certain you are aware, those are not easy questions to answer. At all.
It takes a leap of faith to jump from "Matter can only be described through mathematics" to "Matter is pure mathematics", though.
vectorcube
Oct17-09, 02:52 PM
I agree that math is a tool but to say that there is more math than physical laws and more than one mathematical description does not change what i am saying. Kepler thought of physics as a process of ever better approximation to truth. Intermediate theories reflect and guide us towards this truth even though they may be incomplete and non-unique for the set of phenomena that they predict.
To think that mathematics is somehow separate from reality seems to me to be an arbitrary hypothesis and also seems to contradict all of experience.
Contradict all experience my ***. Math is a internally consistent thing don` t need the physical world to make it`s statements true. Physics is about the world! There is a unlimited number of physical realities we can imagine up, and describe with math. Why do we live in a world without jelly monsters? We just don` t! There is no mathematically reasons why things are the way they are. They just are. Case closed.
Contradict all experience my ***. Math is a internally consistent thing don` t need the physical world to make it`s statements true. Physics is about the world! There is a unlimited number of physical realities we can imagine up, and describe with math. Why do we live in a world without jelly monsters? We just don` t! There is no mathematically reasons why things are the way they are. They just are. Case closed.
I understood your point before you responded. I am not saying something superficial as you seem to think.
Theorem proving makes math internally consistent but that just means that there is something internally consistent about the universe. If it did not mean that then you would not even be able to talk.
Pythagorean
Oct17-09, 03:26 PM
He probably read somewhere about the mathematical universe. While i am not a proponent of the theory, it's been a deep conceptual problem in physics to identify and conceptualise the basic constituents of matter in a non-mathematical way. All efforts so far have proved contradictory and incomplete. The basic building blocks of the material universe(electrons, quarks, quanta in general) cannot be unambiguously described without resorting to maths. They are not 'material' in any of the traditional ways that we are accustomed to dealing with. What is an electron? What is a quark? As i am certain you are aware, those are not easy questions to answer. At all.
It takes a leap of faith to jump from "Matter can only be described through mathematics" to "Matter is pure mathematics", though.
Quite. That leap of faith would be his premise 2. I'm not sure if you could even prove premise 1 in the absolute sense. Physics is mathematical in some sense, but it's not 100% mathematical. Even in the case of leptons (electrons) and quarks. I do agree that these aren't easy questions to answer, but we have gained both a conceptual and mathematical understanding of them as a community. If the unnamed premise 2 were graspable, why would one pick mathematical over conceptual?
To inject my obvious opinion:
I think people have a blind respect and fear for mathematics. Mathematics is like dynamite: it's definitely a tool to respect for its power... but it's still a tool.
Quite. That leap of faith would be his premise 2. I'm not sure if you could even prove premise 1 in the absolute sense. Physics is mathematical in some sense, but it's not 100% mathematical. Even in the case of leptons (electrons) and quarks. I do agree that these aren't easy questions to answer, but we have gained both a conceptual and mathematical understanding of them as a community. If the unnamed premise 2 were graspable, why would one pick mathematical over conceptual?
To inject my obvious opinion:
I think people have a blind respect and fear for mathematics. Mathematics is like dynamite: it's definitely a tool to respect for its power... but it's still a tool.
I never said the universe is only a mathematical theorem. I said physical law is mathematical and thereby is intrinsic to the universe. If you think mathematics is only a tool, I would say that that is a leap of blind faith. why do you use it at all?
Pythagorean
Oct17-09, 03:42 PM
I never said the universe is only a mathematical theorem. I said physical law is mathematical and thereby is intrinsic to the universe. If you think mathematics is only a tool, I would say that that is a leap of blind faith. why do you use it at all?
This argument is still not sound.
"Why do we use it at all" seems to be your argument for it being more than a tool.
We could apply the same logic to a hammer and the fact that we us a hammer it all is not a sufficient argument for it having some deep, universal meaning.
Mathematics is a much bigger, diverse tool, and I'm not arguing that it's as simple as a hammer, it's not even the same kind of tool. I'm just pointing out that you still haven't made an argument. I invite you to think about it more and develop a better argument, though. I'm willing to listen.
This argument is still not sound.
"Why do we use it at all" seems to be your argument for it being more than a tool.
We could apply the same logic to a hammer and the fact that we us a hammer it all is not a sufficient argument for it having some deep, universal meaning.
Mathematics is a much bigger, diverse tool, and I'm not arguing that it's as simple as a hammer, it's not even the same kind of tool. I'm just pointing out that you still haven't made an argument. I invite you to think about it more and develop a better argument, though. I'm willing to listen.
In fact the ability to predict the outcome of the use of a hammer does have some deep universal meaning.
SixNein
Oct18-09, 12:21 AM
He probably read somewhere about the mathematical universe. While i am not a proponent of the theory, it's been a deep conceptual problem in physics to identify and conceptualise the basic constituents of matter in a non-mathematical way. All efforts so far have proved contradictory and incomplete. The basic building blocks of the material universe(electrons, quarks, quanta in general) cannot be unambiguously described without resorting to maths. They are not 'material' in any of the traditional ways that we are accustomed to dealing with. What is an electron? What is a quark? As i am certain you are aware, those are not easy questions to answer. At all.
It takes a leap of faith to jump from "Matter can only be described through mathematics" to "Matter is pure mathematics", though.
I think people misunderstand the role of mathematics or physics in general. So I'm going to make a chart to explain it better...
======Formal System=====|====Physical System====
------- Mathematics -----<Observation>----- Stars --------
------- Physical Theories -<Observation>---- Planets -------
===========================================
---- Formal Description <Observation> Event
Physical theories are very formal. They do not have the ability to describe the universe; instead, they can only model it using mathematics. The goal of physics is to place limitations on mathematical equations that are validated through observation. Mathematics does the same thing in a sense, but it works without physical imitations.
I think; therefore, I am.
The above statement is the only thing people can know for sure.
vectorcube
Oct18-09, 01:26 AM
I understood your point before you responded. I am not saying something superficial as you seem to think.
Theorem proving makes math internally consistent but that just means that there is something internally consistent about the universe. If it did not mean that then you would not even be able to talk.
Wrong in some many levels.
1: The notion that you could make a math theorm true by proving it is a very constructivistic approach to math. It is also a deeply misguided one. Why? Intuitively, A theorm is true independent of anyone. fermat ` s theorm would be true even if that some guy did not prove it.
2. Because math is an internally consistent system. This follows that it is also independent from contingent nature of the world.
__________________________________________________ ___________
The world is contingent. That is, the world, being made of constitutes, and laws that governs those constitutes need not be the way they are for any mathematical reasons. As such, there could be different physical realities with different constitutes, and dynamical laws. This is as oppose to math propositions which needs to be the case in all possible worlds.
So:
For P, such that P is a law of nature, then there is a possible W, such that -P hold in W.
For P, such that P is a matheamatical proposition, then P is ture in all possible worlds.
Wrong in some many levels.
1: The notion that you could make a math theorm true by proving it is a very constructivistic approach to math. It is also a deeply misguided one. Why? Intuitively, A theorm is true independent of anyone. fermat ` s theorm would be true even if that some guy did not prove it.
2. Because math is an internally consistent system. This follows that it is also independent from contingent nature of the world.
__________________________________________________ ___________
The world is contingent. That is, the world, being made of constitutes, and laws that governs those constitutes need not be the way they are for any mathematical reasons. As such, there could be different physical realities with different constitutes, and dynamical laws. This is as oppose to math propositions which needs to be the case in all possible worlds.
So:
For P, such that P is a law of nature, then there is a possible W, such that -P hold in W.
For P, such that P is a matheamatical proposition, then P is ture in all possible worlds.
you also did not get what I was saying.
math is not separate from the world and the world is not contingent. Only knowlege from sense experience is contingent.
I am not sure what you mean by mathematical reasons or why reasons at all have anything to do with math or physics.
arithmetix
Oct18-09, 09:28 AM
My answer:
Maths is a language in which true things can be said, so that the consequences of these truths are implied in the grammar of the statement. I declare that any such careful grammar partakes of whatever spirit is in formally recognised mathematics, and that for the purposes of your question, all statements purporting to be absolute truth are mathematics-like statements.
For me even such simple statements as "it is there" and "it is not there" comprise mathematics-like truths.
Then although I know this answer is scarcely an answer at all, maths works because it is a formalisation of our practical, day-to-day experience.
Is this answer to your point?
My answer:
Maths is a language in which true things can be said, so that the consequences of these truths are implied in the grammar of the statement. I declare that any such careful grammar partakes of whatever spirit is in formally recognised mathematics, and that for the purposes of your question, all statements purporting to be absolute truth are mathematics-like statements.
For me even such simple statements as "it is there" and "it is not there" comprise mathematics-like truths.
Then although I know this answer is scarcely an answer at all, maths works because it is a formalisation of our practical, day-to-day experience.
Is this answer to your point?
yes - but I do not agree that math is a language - i agree that we can talk about it and think about it - to say that a 4 dimensional projective space is just just a word in a language with some grammar is just like saying that a tree is just part of a language.
Mathematical objects are perceived just as trees or cars.
vectorcube
Oct18-09, 02:28 PM
you also did not get what I was saying.
math is not separate from the world and the world is not contingent. Only knowlege from sense experience is contingent.
I am not sure what you mean by mathematical reasons or why reasons at all have anything to do with math or physics.
Wrong. I do get what you are saying, and i can tell you that you are no different that what the ancient greeks, kelper, rationalist and common people think about the relationship between math& physics. Namely, there is no difference whatsoever. The Ideas that one could make some equations up because it is beautiful, and that it would apply to the real world is extremaly misguided.
In reality, people go on in the real world, makes observations, and described the regularities they see using math, and tease out the consequence of the mathematical description.
math is not separate from the world and the world is not contingent.
Tell me how? Give me a single example, and i will shut the hell up. What you are saying here is no different as saying that the laws of physics is logically necessary, but that is the same as saying it is logically impossible to have a world in which E=Mc^3.
There are infinite many ways which reality could be different. It is entire possible that there is a world govern by a 2-d cellular automata. I suppose the people in such a world could easily figure out the symmetries, and laws. Still, it would be a world.
Wrong. I do get what you are saying, and i can tell you that you are no different that what the ancient greeks, kelper, rationalist and common people think about the relationship between math& physics. Namely, there is no difference whatsoever. The Ideas that one could make some equations up because it is beautiful, and that it would apply to the real world is extremaly misguided.
In reality, people go on in the real world, makes observations, and described the regularities they see using math, and tease out the consequence of the mathematical description.
math is not separate from the world and the world is not contingent.
Tell me how? Give me a single example, and i will shut the hell up. What you are saying here is no different as saying that the laws of physics is logically necessary, but that is the same as saying it is logically impossible to have a world in which E=Mc^3.
There are infinite many ways which reality could be different. It is entire possible that there is a world govern by a 2-d cellular automata. I suppose the people in such a world could easily figure out the symmetries, and laws. Still, it would be a world.
I do not think that math and physics are identical. But to say that math is just a tool is a blind assumption. I do not view ideas and sense data as existing in two different worlds as say Plato did. That is the whole point that I am making.
You say there are many possible universes. Well that is not a contingent statement and I don't think that you know that.
That there are many possibilities that we can conceive of does not mean that there are actually many possibilities. And even if there were, they would all be distinguished in their intrinsic mathematical laws.
Further the many possibilities would have to be connected through a universal law.
vectorcube
Oct18-09, 03:31 PM
You say there are many possible universes. Well that is not a contingent statement and I don't think that you know that.
I am saying that there infinite many ways things could be for the universe, and that is what makes it contingent. There could be a universe described by cellular automata, but we do not happen to live in such a world. Mathematical propositions are true in all possible worlds, and that is what makes it necessary.
That there are many possibilities that we can conceive of does not mean that there are actually many possibilities.
I do believe some form of modal realism. You are correct that possibilities only exist if something actually exist. My point that is that there on logical reasons for excluding these worlds based on logic, and logic alone.
And even if there were, they would all be distinguished in their intrinsic mathematical laws.
Something like every possible world corresponds to a fundamental equation of some specific form. I am sure if you open yourself, you see that the world of "harry potter" is logically possible, but there is no governing dymanical law.
Further the many possibilities would have to be connected through a universal law.
Not so. Suppose for a contradiction that such a law exist that govern the entire ensemble of universes. Say law U. But U and -U is also logically possible. Thus, -U would govern it `s own possible ensenble. contradiction.
Here is the thing you need to know. For a law of nature L, -L is a logically possibility.
For a mathematical proposition P, -P is logically impossible.
You benefit greatly by reading Nozick ` s principle of fecundity.
arithmetix
Oct18-09, 05:25 PM
Some few hours ago, from a distant time zone, I stated that math is a language and met with disagreement.
I have found that until a student understands that 'divided by' (maths) translates to 'per' or 'for every' (english) he will make no progress, but that to show a student that equivalence is to set his feet on the road of understanding the process of division.
Given that the four arithmetical operations are the foundations of the most popular mathematical truths, and that they are explicable in English, it seems to me that the mathematical representation of those truths is no different to the English exposition of them.
Hence mathematics is a language.
Pythagorean
Oct18-09, 05:42 PM
yes - but I do not agree that math is a language - i agree that we can talk about it and think about it - to say that a 4 dimensional projective space is just just a word in a language with some grammar is just like saying that a tree is just part of a language.
Mathematical objects are perceived just as trees or cars.
Perhaps you underestimate language. Check out linguistic relativity. The wiki on it has some notes on the empirical research involved.
Also, out of curiosity, what is your academic standing in mathematics?
Perhaps you underestimate language. Check out linguistic relativity. The wiki on it has some notes on the empirical research involved.
Also, out of curiosity, what is your academic standing in mathematics?
no academic standing - been reading on my own and sitting in on classes. Great fun.
Pythagorean
Oct18-09, 05:57 PM
no academic standing
I'm probably not the first to tell you, but as someone who does have an academic standing in physics, I can tell you that mathematics doesn't perfectly describe things in physics. It describes things much better than traditional language does, it's more descriptive in terms of quantification and it's more complex, allowing it to be used to discuss a lot of different situations, but it's still very much a language.
The real universe, however, is very stochastic, and we generally take advantage of the convenience of approximations and where we can, waving our arms about and saying "this mathematical relationship is only good in this situation and only to this accuracy."
Even in quantum mechanics, after the initial groundwork is laid down... it's approximation after approximation after approximation to get to a model of real world applications.
I'm probably not the first to tell you, but as someone who does have an academic standing in physics, I can tell you that mathematics doesn't perfectly describe things in physics. It describes things much better than traditional language does, it's more descriptive in terms of quantification and it's more complex, allowing it to be used to discuss a lot of different situations, but it's still very much a language.
The real universe, however, is very stochastic, and we generally take advantage of the convenience of approximations and where we can, waving our arms about and saying "this mathematical relationship is only good in this situation and only to this accuracy."
Even in quantum mechanics, after the initial groundwork is laid down... it's approximation after approximation after approximation to get to a model of real world applications.
Approximations do not mean that there isn't a mathematical underpinning to the universe. Quite the contrary. By the way, pulling academic rank is in my view inappropriate to this conversation. Do you just want me to stop thinking and take your word for it?
A physicist once told me that rather than physics and mathematics being approximations to observed phenomena, the opposite is true. Observed phenomena and their explanations are approximations to the correct physics/reality.
Pythagorean
Oct18-09, 06:11 PM
Approximations do not mean that there isn't a mathematical underpinning to the universe. Quite the contrary. By the way, pulling academic rank is in my view inappropriate to this conversation. Do you just want me to stop thinking and take your word for it?
A physicist once told me that rather than physics and mathematics being approximations to observed phenomena, the opposite is true. Observed phenomena and their explanations are approximations to the correct physics/reality.
no, I want you to take some classes and think about it for yourself after you've had some exposure to it.
Your last sentence in your last paragraph, I agree with, by the way. But in my case, it contributes to my point. I don't see it as an "opposite".
no, I want you to take some classes and think about it for yourself after you've had some exposure to it.
Your last sentence in your last paragraph, I agree with, by the way. But in my case, it contributes to my point. I don't see it as an "opposite".
What classes do you suggest. I have taken a course in Quantum Mechanics, General realtivity, have read Feynmann's Lectures on Physics.
BTW On a Riemannian manifold with a potential function the metric can be modified so that the paths of particles in the presence of the potential are geodesics. Why can't this be done with the gravitational potential and give another way to do GR?
arithmetix
Oct18-09, 06:53 PM
I reply to this message:
[yes - but I do not agree that math is a language - i agree that we can talk about it and think about it - to say that a 4 dimensional projective space is just just a word in a language with some grammar is just like saying that a tree is just part of a language.
Mathematical objects are perceived just as trees or cars.]
My answer:
I think that the 4-dimensional space is an object that may be described in a language about which everything is, in principle, known... whereas the tree is an object about which we know nothing except what we have discovered by experiment.
The 4d object is part of a mental map we have found to be congruent with 'reality', and while the universe of mathematical truths is like the 'real world' in that we may discover things we did not previously know in it, it is not the same as the real world... the map is not the territory.
I reply to this message:
[yes - but I do not agree that math is a language - i agree that we can talk about it and think about it - to say that a 4 dimensional projective space is just just a word in a language with some grammar is just like saying that a tree is just part of a language.
Mathematical objects are perceived just as trees or cars.]
My answer:
I think that the 4-dimensional space is an object that may be described in a language about which everything is, in principle, known... whereas the tree is an object about which we know nothing except what we have discovered by experiment.
The 4d object is part of a mental map we have found to be congruent with 'reality', and while the universe of mathematical truths is like the 'real world' in that we may discover things we did not previously know in it, it is not the same as the real world... the map is not the territory.
Why isn't everything in principle known about the tree? Or why isn't there anything about the tree which is in principle knowable? Or if there is nothing in principal that is knowable about the tree then how can we form a theory about it? Or isn't any theory that successfully predicts new experiments just a theorem and in principal knowable?
vectorcube
Oct18-09, 07:40 PM
Approximations do not mean that there isn't a mathematical underpinning to the universe. Quite the contrary. By the way, pulling academic rank is in my view inappropriate to this conversation. Do you just want me to stop thinking and take your word for it?
A physicist once told me that rather than physics and mathematics being approximations to observed phenomena, the opposite is true. Observed phenomena and their explanations are approximations to the correct physics/reality.
You should reply to me in what i said on post 138.
vectorcube
Oct18-09, 07:47 PM
Perhaps you underestimate language. Check out linguistic relativity. The wiki on it has some notes on the empirical research involved.
Also, out of curiosity, what is your academic standing in mathematics?
It is not necessary falses. Mathematical platonism do see mathematical proposition as being descriptive. There is nothing incoherent about this idea.
vectorcube
Oct18-09, 07:52 PM
My answer:
I think that the 4-dimensional space is an object that may be described in a language about which everything is, in principle, known... whereas the tree is an object about which we know nothing except what we have discovered by experiment.
The 4d object is part of a mental map we have found to be congruent with 'reality', and while the universe of mathematical truths is like the 'real world' in that we may discover things we did not previously know in it, it is not the same as the real world... the map is not the territory.
This makes no sense to me. If math is a map, then it is a composite of many maps of which one correponds to reality. That is to say, our universe is a mathematical structure describle captured by some axioms.
SixNein
Oct18-09, 07:52 PM
I'm probably not the first to tell you, but as someone who does have an academic standing in physics, I can tell you that mathematics doesn't perfectly describe things in physics. It describes things much better than traditional language does, it's more descriptive in terms of quantification and it's more complex, allowing it to be used to discuss a lot of different situations, but it's still very much a language.
The real universe, however, is very stochastic, and we generally take advantage of the convenience of approximations and where we can, waving our arms about and saying "this mathematical relationship is only good in this situation and only to this accuracy."
Even in quantum mechanics, after the initial groundwork is laid down... it's approximation after approximation after approximation to get to a model of real world applications.
I would like to point out that mathematics is not exact. There is approximations in mathematics and even uncertainty.
Alas, I would ask what is physics? Is physics not the mathematical relationships found in our universe?
vectorcube
Oct18-09, 08:01 PM
Alas, I would ask what is physics? Is physics not the mathematical relationships found in our universe?
Physics is certainly not some random relationship conjured up because it is "beautiful".
apeiron
Oct18-09, 08:17 PM
I think that the 4-dimensional space is an object that may be described in a language about which everything is, in principle, known... whereas the tree is an object about which we know nothing except what we have discovered by experiment.
The 4d object is part of a mental map we have found to be congruent with 'reality', and while the universe of mathematical truths is like the 'real world' in that we may discover things we did not previously know in it, it is not the same as the real world... the map is not the territory.
The place to start this debate would be epistemology - acceptance that all knowledge is modelling. Knowledge is always a map (and so embeds a human purpose, representing where we want to go).
Then the question becomes what kind of knowledge of reality is maths?
Clearly it is knowledge of the most general or universal kind. The most general or universal that we find useful *as a map*.
So we can know a tree at many levels of modelling, from memories of particular trees in my garden to what we get taught about plant life in general in botany class. But what kind of mathematical level generalisations can we make about trees?
The fractal nature of tree branching would be one "deep insight". It connects trees to many other phenomenon like river systems and other dissipative structures.
Dissipative structure theory is of course a physical theory about energy flows and entropy degraders. And fractals are the product of mathematical equations. So we can see how there is a path from what some like to call qualitative-to-quantitative description.
A tree is a highly qualitative experience as we know "so many things" about it. But in a stuck together, constructed, componential sort of way. The idea of "tree-ness" is multi-dimensional. Then dissipative structure theory is a much more general description that is also much more constrained in its application. It has qualitative aspects (like energy flow) but also offers "things that must be measured" - such as quantities which get conserved. Then fractals are completely general, so general they no longer appear to refer to any real life instances. There are no qualitative aspects, just the quantitative - variable plugged into equations.
So what I am arguing is that all knowledge is modelling. Qualitative or quantitative. Then modelling does follow a hierarchy of generalisation. You start with "raw experience" (or the kind of natural world modelling that animal brains evolved to do, which also embed purposes of course). Then move away from raw experiences of trees and fish and ponds to increasingly more general, and thus reduced (stripped of qualifying specifics, trimmed of unneccessary phenomenal dimensions) levels of modelling. Physics is our word for the limit of science, the limit of description for what is real. Then maths is the step beyond, into generalities that are not real, that are pure quantity - but which can have qualities like energy or inertia plugged back into their frameworks and so become a tool for doing physics or other reality modelling.
Well, I say maths is pure quantity, but of course the axioms of maths are the vestiges of qualitative description. We boil reality right down to the last irreducibly necessary concepts - like assumptions about continuity~discreteness, stasis~change, chance~necessity.
Maths itself is of course a mix of disciplines.
You have algebra~geometry (the discrete vs the continuous descriptions). Algebra and geometry are in effect the exploration of the world of all possible general objects or general structures.
Then you have logic, which is the generalisation of causality, the generalisation of reality's rules.
And within maths there are levels of generalisation, as made explicit in category theory. So topology is more general than geometry. Arguably, by taking away the quantitative aspects of geometry - distance and angle and curvature - topology becomes a more qualitative level of description. Yes, it does reduce geometry towards the axiomatic nub, the ideas like continuous~discrete dimensionality that are its founding assumptions.
So category theory ends up with a ur-reduced, ur-general, map of reality. Objects and morphisms. Like a map which is a blank sheet of paper with an arrow saying "you are here" and a second saying "everything else is somewhere else". :smile:
To sum up, all knowledge is modelling. All modelling is shaped by purpose. Maps have reasons. Maps also want to be as simple as possible - particulars are reduced to leave generalisations.
Then human knowledge starts up where animal knowledge left off. We start off with a highly subjective or qualitative view of reality and work towards an objective or quantitative view. Maths is an almost purely general level of map making, but even here some founding qualitative axioms are required.
SixNein
Oct18-09, 08:51 PM
Physics is certainly not some random relationship conjured up because it is "beautiful".
I think most people misunderstand the beauty of mathematics. People have to think deeper then symbols and equations in order to see it. The beauty of mathematics is the understanding it imparts to the mathematician. In some cases, a mathematician may be the first human to set foot on a new world, and he maps it so that physicists and engineers may find their way. The world the mathematician sees is described as beautiful.
I am saying that there infinite many ways things could be for the universe, and that is what makes it contingent. There could be a universe described by cellular automata, but we do not happen to live in such a world. Mathematical propositions are true in all possible worlds, and that is what makes it necessary.
I do believe some form of modal realism. You are correct that possibilities only exist if something actually exist. My point that is that there on logical reasons for excluding these worlds based on logic, and logic alone.
Something like every possible world corresponds to a fundamental equation of some specific form. I am sure if you open yourself, you see that the world of "harry potter" is logically possible, but there is no governing dymanical law.
Not so. Suppose for a contradiction that such a law exist that govern the entire ensemble of universes. Say law U. But U and -U is also logically possible. Thus, -U would govern it `s own possible ensenble. contradiction.
Here is the thing you need to know. For a law of nature L, -L is a logically possibility.
For a mathematical proposition P, -P is logically impossible.
You benefit greatly by reading Nozick ` s principle of fecundity.
Logical possibility has nothing to do with what I am saying. You seem to think that I do not understand that one can never prove that the universe must be a certain way. I am saying that what is knowable - must contain mathematical stuctures - and that what is knowable is what is actually real - we can not speak about what is not knowable - to me this is not a question of logic, or formalisms - but a question of what is knowable.
vectorcube
Oct18-09, 09:33 PM
I think most people misunderstand the beauty of mathematics. People have to think deeper then symbols and equations in order to see it. The beauty of mathematics is the understanding it imparts to the mathematician. In some cases, a mathematician may be the first human to set foot on a new world, and he maps it so that physicists and engineers may find their way. The world the mathematician sees is described as beautiful.
I guess you fail to see the point. The point is that there is difference between math and physics, and most people ignore that difference. People don` t make up equations in physics unless there is a physical motivation, and constrint imposed by physical reality.
vectorcube
Oct18-09, 09:36 PM
I am saying that what is knowable - must contain mathematical stuctures - and that what is knowable is what is actually real - we can not speak about what is not knowable - to me this is not a question of logic, or formalisms - but a question of what is knowable.
That is still wrong. Take the harry potter universe. This universe cannot be described by math, yet, it is logically possible, and knowable.
Pythagorean
Oct19-09, 12:20 AM
What classes do you suggest. I have taken a course in Quantum Mechanics, General realtivity, have read Feynmann's Lectures on Physics.
BTW On a Riemannian manifold with a potential function the metric can be modified so that the paths of particles in the presence of the potential are geodesics. Why can't this be done with the gravitational potential and give another way to do GR?
Just out of curiousity:
Did your QM course involve linear algebra with eigenspinors using the pauli matrice and complex operators as observables on a wave function in the schroedinger equations? Did your general relativity class have you solving tensor equations?
I haven't studied general relativity nor Riemannian manifolds. If you really understand the mathematics behind that question, why don't you pose the question mathematically and find out where it breaks down? Is that question even relevant to our discussion?
Just out of curiousity:
Did your QM course involve linear algebra with eigenspinors using the pauli matrice and complex operators as observables on a wave function in the schroedinger equations? Did your general relativity class have you solving tensor equations?
I haven't studied general relativity nor Riemannian manifolds. If you really understand the mathematics behind that question, why don't you pose the question mathematically and find out where it breaks down? Is that question even relevant to our discussion?
the course was a standard first course given by Brian Greene at Columbia University. I also correspond with a Physics Professor on QM. Once a month a group of friends get together to discuss the measurement problem. Currently we are studying Bohm's deterministic theory of QM.
I asked this same question of my GR ( grad course) prof and he referred me to some papers and thought that there are in fact alternatives to GR based on this concept. The mathematics of the question is simple differential geometry and the students in the class were familiar with it. If you would like to learn about this I would be glad to write a different thread for you to read - say in the GR section of PF. I asked the question to you only just out of curiosity thinking that you might have some thoughts on it since you said you have academic credentials.
arithmetix
Oct19-09, 07:49 AM
well it sounds to me as if what the original questioner, who has taken the name 'perspectives', wants is certainty about knowledge. Absolute, unquestionable certainty, for me, comes out of meditat
ion, and studying Tao, and prayer. Such deep certainty is intransmissible and indescribable, but is very simple to apprehend if you get quiet enough.
I don't personally believe that any absolute truths can be clearly stated in any language. I do think that this whole thread has been contributed to by philosophers however, and I suggest that philosophy would be a good thing to study along with the maths.
please see next post before responding!!
arithmetix
Oct19-09, 08:33 AM
... a bit of a think later, and having reviewed the rules:
I now withdraw from further discussion on some of what I have said, since I have opened an area of this debate which, far from physics, treats of religion. My views are my own and I have the right to them and the right to state what they are, but this is a dangerous area to get into since it may easily lead to heated discussion.
Anyone wishing to respond to the religious aspect of my post is welcome to respond to me personally, and if they have anything interesting to say we could find another venue.
Thank you.
vectorcube
Oct19-09, 12:58 PM
The place to start this debate would be epistemology - acceptance that all knowledge is modelling. Knowledge is always a map (and so embeds a human purpose, representing where we want to go).
Then the question becomes what kind of knowledge of reality is maths?
Clearly it is knowledge of the most general or universal kind. The most general or universal that we find useful *as a map*.
So we can know a tree at many levels of modelling, from memories of particular trees in my garden to what we get taught about plant life in general in botany class. But what kind of mathematical level generalisations can we make about trees?
The fractal nature of tree branching would be one "deep insight". It connects trees to many other phenomenon like river systems and other dissipative structures.
Dissipative structure theory is of course a physical theory about energy flows and entropy degraders. And fractals are the product of mathematical equations. So we can see how there is a path from what some like to call qualitative-to-quantitative description.
A tree is a highly qualitative experience as we know "so many things" about it. But in a stuck together, constructed, componential sort of way. The idea of "tree-ness" is multi-dimensional. Then dissipative structure theory is a much more general description that is also much more constrained in its application. It has qualitative aspects (like energy flow) but also offers "things that must be measured" - such as quantities which get conserved. Then fractals are completely general, so general they no longer appear to refer to any real life instances. There are no qualitative aspects, just the quantitative - variable plugged into equations.
So what I am arguing is that all knowledge is modelling. Qualitative or quantitative. Then modelling does follow a hierarchy of generalisation. You start with "raw experience" (or the kind of natural world modelling that animal brains evolved to do, which also embed purposes of course). Then move away from raw experiences of trees and fish and ponds to increasingly more general, and thus reduced (stripped of qualifying specifics, trimmed of unneccessary phenomenal dimensions) levels of modelling. Physics is our word for the limit of science, the limit of description for what is real. Then maths is the step beyond, into generalities that are not real, that are pure quantity - but which can have qualities like energy or inertia plugged back into their frameworks and so become a tool for doing physics or other reality modelling.
Well, I say maths is pure quantity, but of course the axioms of maths are the vestiges of qualitative description. We boil reality right down to the last irreducibly necessary concepts - like assumptions about continuity~discreteness, stasis~change, chance~necessity.
Maths itself is of course a mix of disciplines.
You have algebra~geometry (the discrete vs the continuous descriptions). Algebra and geometry are in effect the exploration of the world of all possible general objects or general structures.
Then you have logic, which is the generalisation of causality, the generalisation of reality's rules.
And within maths there are levels of generalisation, as made explicit in category theory. So topology is more general than geometry. Arguably, by taking away the quantitative aspects of geometry - distance and angle and curvature - topology becomes a more qualitative level of description. Yes, it does reduce geometry towards the axiomatic nub, the ideas like continuous~discrete dimensionality that are its founding assumptions.
So category theory ends up with a ur-reduced, ur-general, map of reality. Objects and morphisms. Like a map which is a blank sheet of paper with an arrow saying "you are here" and a second saying "everything else is somewhere else". :smile:
To sum up, all knowledge is modelling. All modelling is shaped by purpose. Maps have reasons. Maps also want to be as simple as possible - particulars are reduced to leave generalisations.
Then human knowledge starts up where animal knowledge left off. We start off with a highly subjective or qualitative view of reality and work towards an objective or quantitative view. Maths is an almost purely general level of map making, but even here some founding qualitative axioms are required.
weird. So math is a "map". Why? because people evolved from monkeys?
vectorcube
Oct19-09, 05:03 PM
Math is :
1. about sturctures( possible or actual).
Physics is:
1. Finding relationships between different quentities( observer, or not).
I think it is very easy to see the similarities/differences between the two.
Pythagorean
Oct19-09, 05:15 PM
the course was a standard first course given by Brian Greene at Columbia University. I also correspond with a Physics Professor on QM. Once a month a group of friends get together to discuss the measurement problem. Currently we are studying Bohm's deterministic theory of QM.
I asked this same question of my GR ( grad course) prof and he referred me to some papers and thought that there are in fact alternatives to GR based on this concept. The mathematics of the question is simple differential geometry and the students in the class were familiar with it. If you would like to learn about this I would be glad to write a different thread for you to read - say in the GR section of PF. I asked the question to you only just out of curiosity thinking that you might have some thoughts on it since you said you have academic credentials.
Our modern physics class was a two-semester class that involved QM, Nuclear, and a choice between GR or nonlinear dynamics at the end. We unanimously voted for nonlinear dynamics. I have never been interested in GR, personally. Nonlinear dynamics (to me) is more applicable and diverse in terms of the world we experience on a day-to-day basis.
I've read part of a book by Brian Greene, "The Fabric of the Cosmos" (which I own). And I loved his discussion of Newton's bucket in the beginning of it, but I'm fairly turned-off by string theory, so the book didn't hold my interest enough to finish after he got into that.
Anyway, to put us back on topic, my opinion on the matter of mathematics is that it's a core way to quantify human thinking in terms of traditional logic. Not every mathematical abstraction we can think of necessarily pertains to the real world, but every mathematical abstraction we can think of does necessarily pertain to the computational methods of the human brain. In other words, I suspect any mathematical theory you can come up with has a good chance of telling you how the human brain makes abstractions and uses them to make (and check) constant predictions of the world around it, though any particular math theory you come up with won't actually be useful for making predictions about the world.
This is a lot like farting and watching porn. We don't have any evolutionary purpose for farting or watching porn, but we do have evolutionary purposes that farting and watching porn are a byproduct of.
Our modern physics class was a two-semester class that involved QM, Nuclear, and a choice between GR or nonlinear dynamics at the end. We unanimously voted for nonlinear dynamics. I have never been interested in GR, personally. Nonlinear dynamics (to me) is more applicable and diverse in terms of the world we experience on a day-to-day basis.
I've read part of a book by Brian Greene, "The Fabric of the Cosmos" (which I own). And I loved his discussion of Newton's bucket in the beginning of it, but I'm fairly turned-off by string theory, so the book didn't hold my interest enough to finish after he got into that.
Anyway, to put us back on topic, my opinion on the matter of mathematics is that it's a core way to quantify human thinking in terms of traditional logic. Not every mathematical abstraction we can think of necessarily pertains to the real world, but every mathematical abstraction we can think of does necessarily pertain to the computational methods of the human brain. In other words, I suspect any mathematical theory you can come up with has a good chance of telling you how the human brain makes abstractions and uses them to make (and check) constant predictions of the world around it, though any particular math theory you come up with won't actually be useful for making predictions about the world.
This is a lot like farting and watching porn. We don't have any evolutionary purpose for farting or watching porn, but we do have evolutionary purposes that farting and watching porn are a byproduct of.
Interesting idea about the brain. Do you think that if viewed as a physical object rather than experiential (if that is a word) the formation of mathematical ideas in it means anything about Nature?
An interesting historical aside is that Riemann seems to think that idea formation was the true model of the universe and tried to compare finite objects to ideas and continuous space to the mind as a whole. His model for the universe was that it had the same processes as the mind.
I think this philosophical attitude inspired his invention of shocks in non-linear wave equations. The shock is like a "new idea" that lawfully arises in the "mind" to resolve an apparent contradiction, in this case a multi-valued signal. The law is preserved - but the mind/Nature if you will - creates a new object in order to preserve it and also changes the meaning of what a solution to the equation is.
He came up with this idea rather than introducing diffusion terms (changing the law) to prevent a multiple signal. His view I guess was that Nature does not change its laws but rather creates new things if it has to to preserve the laws.
More generally I think that the attitude that unchanging truth is to be discovered behind inexact observation is a guiding koan of modern physics. It can be seen in Einstein,Lorenz, Gallileo, Riemann. This is one reason that I can not accept the view that math is mere modeling. If that were Kepler and Newton's view then the Ptolemaic system would never have been rejected. It was an incredibly accurate model and could be adjusted periodically to preserve its accuracy. It was rejected precisely because it did not present universal law - in fact it contradicted the idea of universal law since it had to be modified from time to time - sort of like putting diffusion terms into the wave equation.
This thread because of its empiricist/positivist bias doesn't even care about this and doesn't care what thoughts inspired the progress of science. In fact I am sure that someone in this thread will write that the Ptolemaic system was perfectly fine and Gallileo and Kepler and Newton and Riemann and Einstein and all of the others were jerks - they just didn't understand anything.
arithmetix
Oct19-09, 06:23 PM
I am interested in the notion that a consistent structure must underlie all possible worlds.
I quote from post 138, in which vectorcube wrote:
[QUOTE=vectorcube;2399004]I am saying that there infinite many ways things could be for the universe, and that is what makes it contingent. There could be a universe described by cellular automata, but we do not happen to live in such a world. Mathematical propositions are true in all possible worlds, and that is what makes it necessary.
Also quote from vectorcube in the same post:
"I do believe some form of modal realism. You are correct that possibilities only exist if something actually exist. My point that is that there on logical reasons for excluding these worlds based on logic, and logic alone."
I ask: is it really possible to determine, using logic, whether a universe must be logical? Or even whether this universe is logical? (I have noticed that logic fails us on some kinds of question.)
For instance imagine that a prime cause exists, and imagine that we have a project to dtermine the nature of that cause. Then should one of us stumble on the exact description of that cause there would be no way for him to prove his discovery out, because logic demands that a statement be proven by an alternative analysis which must be shown to lead to the same result.
Of course this is impossible for a sole, prime cause... therefore it is not possible to address this kind of question, using logic.
vectorcube
Oct19-09, 07:03 PM
I am interested in the notion that a consistent structure must underlie all possible worlds.
I quote from post 138, in which vectorcube wrote:
[QUOTE=vectorcube;2399004]I am saying that there infinite many ways things could be for the universe, and that is what makes it contingent. There could be a universe described by cellular automata, but we do not happen to live in such a world. Mathematical propositions are true in all possible worlds, and that is what makes it necessary.
Also quote from vectorcube in the same post:
"I do believe some form of modal realism. You are correct that possibilities only exist if something actually exist. My point that is that there on logical reasons for excluding these worlds based on logic, and logic alone."
I ask: is it really possible to determine, using logic, whether a universe must be logical? Or even whether this universe is logical? (I have noticed that logic fails us on some kinds of question.)
For instance imagine that a prime cause exists, and imagine that we have a project to dtermine the nature of that cause. Then should one of us stumble on the exact description of that cause there would be no way for him to prove his discovery out, because logic demands that a statement be proven by an alternative analysis which must be shown to lead to the same result.
Of course this is impossible for a sole, prime cause... therefore it is not possible to address this kind of question, using logic.
I don `t understand your example. Logic by itself cannot tell us anything at all. What it tells use is that for a proposition p, p&-p is impossible.
arithmetix
Oct19-09, 09:53 PM
Quoting myself:
"... For instance imagine that a prime cause exists, and imagine that we have a project to determine the nature of that cause. Then should one of us stumble on the exact description of that cause there would be no way for him to prove his discovery out, because logic demands that a statement be proven by an alternative analysis which must be shown to lead to the same result.
Of course this is impossible for a sole, prime cause... therefore it is not possible to address this kind of question, using logic. "
Because prime cause contains all results, every result must lead back to prime cause, but because each result does arise in the same way (same cause) no independent demonstration of cause is possible. Hence prime cause cannot be proven to be prime cause, even if we somehow know what prime cause is.
If that helps... (?)
vectorcube
Oct19-09, 10:33 PM
Quoting myself:
"... For instance imagine that a prime cause exists, and imagine that we have a project to determine the nature of that cause. Then should one of us stumble on the exact description of that cause there would be no way for him to prove his discovery out, because logic demands that a statement be proven by an alternative analysis which must be shown to lead to the same result.
Of course this is impossible for a sole, prime cause... therefore it is not possible to address this kind of question, using logic. "
Because prime cause contains all results, every result must lead back to prime cause, but because each result does arise in the same way (same cause) no independent demonstration of cause is possible. Hence prime cause cannot be proven to be prime cause, even if we somehow know what prime cause is.
If that helps... (?)
I am uncertain what this has to do with logically possibilities.
apeiron
Oct19-09, 10:42 PM
Quoting myself:
"... For instance imagine that a prime cause exists, and imagine that we have a project to determine the nature of that cause. Then should one of us stumble on the exact description of that cause there would be no way for him to prove his discovery out, because logic demands that a statement be proven by an alternative analysis which must be shown to lead to the same result.
Of course this is impossible for a sole, prime cause... therefore it is not possible to address this kind of question, using logic. "
Because prime cause contains all results, every result must lead back to prime cause, but because each result does arise in the same way (same cause) no independent demonstration of cause is possible. Hence prime cause cannot be proven to be prime cause, even if we somehow know what prime cause is.
If that helps... (?)
Alternatively, thinking in terms of prime causes could be where you go wrong - it is not actually "logical".
The other way to look at it is teleological. Perhaps only some certain outcome is self-consistent. So start from any kind of initial conditions and the system will develop to arrive at the same old place.
Kind of like attractors in dynamics. And that would be how maths developed - the sub-set of patterns that are self-consistent over the total space of potential patterns. Realities would be the same, and thus "mathematical" - or at least ameniable to modelling as patterns.
Syllogistic reasoning is a tool. But it is also useful to understand Aristotle's wider story on causality - his four causes. Purpose or teleology is something we need to bring back into logic modelling. In systems approaches, it is what is called global constraints.
vectorcube
Oct19-09, 11:18 PM
Alternatively, thinking in terms of prime causes could be where you go wrong - it is not actually "logical".
The other way to look at it is teleological. Perhaps only some certain outcome is self-consistent. So start from any kind of initial conditions and the system will develop to arrive at the same old place.
Kind of like attractors in dynamics. And that would be how maths developed - the sub-set of patterns that are self-consistent over the total space of potential patterns. Realities would be the same, and thus "mathematical" - or at least ameniable to modelling as patterns.
Syllogistic reasoning is a tool. But it is also useful to understand Aristotle's wider story on causality - his four causes. Purpose or teleology is something we need to bring back into logic modelling. In systems approaches, it is what is called global constraints.
what do you mean here?
arithmetix
Oct20-09, 12:58 AM
In post 170, vectorcube agrees that we are unable to determine whether we are in a logical universe, and offers the notion of a teleological model for consideration.
I think that vectorcube is right (if that is indeed what he means to say) and I wonder whether this will help us with the ostensible subject of our thread. If we are unable to determine whether the universe is or is not logical, we are unable for the same reasons to determine "why" math works, and the question is (regrettably) answered.
vectorcube
Oct20-09, 01:05 AM
In post 170, vectorcube agrees that we are unable to determine whether we are in a logical universe, and offers the notion of a teleological model for consideration.
I think that vectorcube is right (if that is indeed what he means to say) and I wonder whether this will help us with the ostensible subject of our thread. If we are unable to determine whether the universe is or is not logical, we are unable for the same reasons to determine "why" math works, and the question is (regrettably) answered.
Hold on. I say no such thing!
Look, answer me this:
Do you think there is true contradiction?
arithmetix
Oct20-09, 02:48 AM
no
i don't
arithmetix
Oct20-09, 02:53 AM
sorry about that very short post. After I've done the cooking I'll come back and work out exactly what I do think.
vectorcube
Oct20-09, 03:17 AM
sorry about that very short post. After I've done the cooking I'll come back and work out exactly what I do think.
If there are no true contradiction, then what can follow is that there is no possible world with true contradiction. Since our world is a possible world, then it follows that there is no true contradiction in our world. Thus, Our world is logically possible.
apeiron
Oct20-09, 07:20 PM
If there are no true contradiction, then what can follow is that there is no possible world with true contradiction. Since our world is a possible world, then it follows that there is no true contradiction in our world. Thus, Our world is logically possible.
Alternatively, only "logical contradictions" make reality logically possible.
So we must have both substance and form, chance and necessity, change and stasis, discrete and continuous, atom and void, matter and mind, etc, etc. Each arises as the negation of the other - thesis and antithesis. And then in interaction they create (or in modelling recreate) our reality.
At a superficial level, contradiction seems a bad thing logically. But at a deep level, it is what philosophically and mathematically we have always found.
vectorcube
Oct21-09, 01:22 AM
Alternatively, only "logical contradictions" make reality logically possible.
I don` t see how our world is depended on a logical contradiction.
So we must have both substance and form, chance and necessity, change and stasis, discrete and continuous, atom and void, matter and mind, etc, etc. Each arises as the negation of the other - thesis and antithesis. And then in interaction they create (or in modelling recreate) our reality.
At a superficial level, contradiction seems a bad thing logically. But at a deep level, it is what philosophically and mathematically we have always found.
This is rather confusing. What i mean be a "logically possible world" should be interpreted as an semantic for modal logic.
Pythagorean
Oct23-09, 01:04 AM
Interesting idea about the brain. Do you think that if viewed as a physical object rather than experiential (if that is a word) the formation of mathematical ideas in it means anything about Nature?
I think, that as we've already generally accepted, it shows us that we can only form a close approximation of nature. Even with our empirical (observation) and mathematical (theory) cleverness, nature is just outside of our perception in the context of science and logic. Traditionally, science and logic required that things were deterministic. But part of accepting that we can't really know things exactly as they makes us give up "confidence" in our deterministic principles, so we have stochastic systems, which is a way to make up for the flaws in understanding that cripple determinism. I would venture to say that stochastic studies are still very much deterministic in the sense that you're still trying to determine things in a logical, rational fashion. You've only relieved yourself of the accountability of being wrong by using words like "confidence" and "probability".
An interesting historical aside is that Riemann seems to think that idea formation was the true model of the universe and tried to compare finite objects to ideas and continuous space to the mind as a whole. His model for the universe was that it had the same processes as the mind.
I thought Einstein had used Riemann Geometry anyway, but I've mostly only heard about GR "in the halls".
I think this philosophical attitude inspired his invention of shocks in non-linear wave equations. The shock is like a "new idea" that lawfully arises in the "mind" to resolve an apparent contradiction, in this case a multi-valued signal. The law is preserved - but the mind/Nature if you will - creates a new object in order to preserve it and also changes the meaning of what a solution to the equation is.
He came up with this idea rather than introducing diffusion terms (changing the law) to prevent a multiple signal. His view I guess was that Nature does not change its laws but rather creates new things if it has to to preserve the laws.
More generally I think that the attitude that unchanging truth is to be discovered behind inexact observation is a guiding koan of modern physics. It can be seen in Einstein,Lorenz, Gallileo, Riemann. This is one reason that I can not accept the view that math is mere modeling. If that were Kepler and Newton's view then the Ptolemaic system would never have been rejected. It was an incredibly accurate model and could be adjusted periodically to preserve its accuracy. It was rejected precisely because it did not present universal law - in fact it contradicted the idea of universal law since it had to be modified from time to time - sort of like putting diffusion terms into the wave equation.
But to me, this says that physics is not mere modeling. If math was all that mattered, than the Ptolemaic system would have been fine. Observations matter too, though. There's something just weird about planets doing little loop-de-loops out of nowhere. That would be unsettling to must of us who were genuinely curious and studied the night sky.
Just like, for Einstein, it would have been unsettling if gravity was instantaneous and that.. if the Sun disappeared, we'd still stay forever bathed in it's light as the gravity instantly let us go, but the light took time traveling the speed C to us.
Math is not the one to go to to say "this is weird... there's something not right about this", intuition is. We then go and break apart our question into mathematics to answer the question. That's what math does for us: makes the information manageable.
Realize that this is also a matter of logistics. Nobody wants to have to adjust their calendar, it's inconvenient. And then there's the whole egocentric part about being at the center of the universe....
This thread because of its empiricist/positivist bias doesn't even care about this and doesn't care what thoughts inspired the progress of science. In fact I am sure that someone in this thread will write that the Ptolemaic system was perfectly fine and Gallileo and Kepler and Newton and Riemann and Einstein and all of the others were jerks - they just didn't understand anything.
I disagree. Thoughts that inspired the progress of science are very fascinating to most of us, especially when you realize how often mistakes and random guessing have helped to discover nature. We love seeing where intuition helps and where it hinders. No rational individual is going to say Ptolemaic system was fine simply because there's no reason to believe that we'd be the center of the universe and there's a simpler explanation.
SixNein
Oct23-09, 02:55 AM
[QUOTE=Pythagorean;2406426
But to me, this says that physics is not mere modeling.
[/QUOTE]
But it is....
Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.
I don't see why people think physics is something more grand or special.
Pythagorean
Oct23-09, 03:07 AM
But it is....
Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.
I don't see why people think physics is something more grand or special.
I'm not making the argument that it's grand or special. But it's more than "mere modeling". It's effective and functional modeling! It's not completely arbitrary and incorrect. It works, and it works well. It helps make predictions about reality. Of course, this couldn't be done accurately without the language of mathematics.
vectorcube
Oct23-09, 03:11 AM
But it is....
Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.
I don't see why people think physics is something more grand or special.
Agreed.
I know all the arguments for the independent existence of abstract objects, and i still find weird. It is so weird.
vectorcube
Oct23-09, 03:14 AM
I'm not making the argument that it's grand or special. But it's more than "mere modeling". It's effective and functional modeling! It's not completely arbitrary and incorrect. It works, and it works well. It helps make predictions about reality. Of course, this couldn't be done accurately without the language of mathematics.
Very vague. "Funcational modeling" don` t give me any insight.
The role of math is in the formulation of physical laws, and the role it plays to carry out the implications of those laws. I find laws of nature to be more useful than the language used to describe it.
Pythagorean
Oct23-09, 03:46 AM
Very vague. "Funcational modeling" don` t give me any insight.
The role of math is in the formulation of physical laws, and the role it plays to carry out the implications of those laws. I find laws of nature to be more useful than the language used to describe it.
Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.
Geometry itself was originally very empirical.
Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.
So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"
Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).
We can make up any function we chose in mathematics. It doesn't have to represent any physical observations to be mathematics.
vectorcube
Oct23-09, 04:02 AM
Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.
Would the postulates of quantum mechanics be functional modeling?
Geometry itself was originally very empirical.
Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.
So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"
fine.
Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).
It seems to me that you do think there is an initial assignment between the mathematical symbols( 1, 2 ,3 ..), and physical world. You were talking about numbers being assingned to different people. When one person die, we use subtraction, right?
Even in this basic assignment of numbers with people, we are created a semantic map between the math symbols and the real world( people). Notice that we can say:
1) the initial assignment formulated in terms of mathematical symbols. Each person corresponds to a number, etc.
2) The consequence of the mathematical manipulation( ex 2+3=5) makes predictions about the real world( there are 5 people).
This is exactly what i am saying. math give us a precise formulation of laws( in 1), and help us tease out the consequence of those laws( in 2).
Pythagorean
Oct23-09, 04:17 AM
It seems to me that you do think there is an initial assignment between the mathematical symbols( 1, 2 ,3 ..), and physical world. You were talking about numbers being assingned to different people. When one person die, we use subtraction, right?
Even in this basic assignment of numbers with people, we are created a semantic map between the math symbols and the real world( people). Notice that we can say:
1) the initial assignment formulated in terms of mathematical symbols. Each person corresponds to a number, etc.
2) The consequence of the mathematical manipulation( ex 2+3=5) makes predictions about the real world( there are 5 people).
This is exactly what i am saying. math give us a precise formulation of laws( in 1), and help us tease out the consequence of those laws( in 2).
Yeah, I don't think we're in disagreement here at all. I was just elaborating in my last post. But as for 2), my point was more focused on the fact that we developed the rules for (a+b=x) based on observation. The mathematics is the language we used to further define the operations we were doing (the + and the =) as well as the numbering system.
My point is that the mathematics transcends reality. The number system goes to negative numbers in mathematics. -4 enemies doesn't make sense in the context of our model. So mathematics makes unreasonable predictions if we don't constrain it based on more observations. In this way, we invent math as we go along to fit our observations.
But math isn't one thing. I'll bet everybody has their own idea of what mathematics is. To me, it's just an academic branch. The "deeper thing" going on in both mathematics and physics is logic.
vectorcube
Oct23-09, 04:37 AM
Yeah, I don't think we're in disagreement here at all. I was just elaborating in my last post. But as for 2), my point was more focused on the fact that we developed the rules for (a+b=x) based on observation. The mathematics is the language we used to further define the operations we were doing (the + and the =) as well as the numbering system.
I will not make any comment about why 1+2=3, because it will go outside the topic. I will say the assignment in coming up with a math model is the assignment between math symbols with physical quantities.
Ex:
M stands for mass
C stands for light
E stands for energy
so that the model MC^2=E tell us something about the world.
My point is that the mathematics transcends reality. The number system goes to negative numbers in mathematics. -4 enemies doesn't make sense in the context of our model. So mathematics makes unreasonable predictions if we don't constrain it based on more observations. In this way, we invent math as we go along to fit our observations.
I am not going to taking about math transcending physical reality because we can count to -4. Outside the topic.
I would say we use the math to describe our observations. We come to a model by the assigment of math symbols to physical quantities, and we find relationships between the math symbols from generalization in empirical experiments.
The "deeper thing" going on in both mathematics and physics is logic.
I would not say physics is logic. Physics is about trying to know how the world works.
Logic is the study of formal arguments such that true premises lead to true conclusion. They are apples and oranges.
SixNein
Oct23-09, 03:09 PM
Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.
Geometry itself was originally very empirical.
Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.
So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"
Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).
We can make up any function we chose in mathematics. It doesn't have to represent any physical observations to be mathematics.
Integers is a modern concept. Integers feel natural now, but they were hard to get accepted at first. There is no real observation of negative distance, negative apples, or anything of the kind; however, integers really revolutionized money and more importantly debt. Even the number zero took time. Why would a farmer need to count zero sheep? After he or she lost all sheep, he or she was out of business anyway.
Mathematics still works by and large from observations. Many mathematicians reword problems into another form that can be observed or visualized. A good example would be p=np. If you could solve the traveling salesman problem, then it would automatically answer the p=np question.
Pythagorean
Oct24-09, 04:45 PM
Integers is a modern concept. Integers feel natural now, but they were hard to get accepted at first. There is no real observation of negative distance, negative apples, or anything of the kind; however, integers really revolutionized money and more importantly debt. Even the number zero took time. Why would a farmer need to count zero sheep? After he or she lost all sheep, he or she was out of business anyway.
I think this is a mathematical perspective. From an observational perspective, integers were an easy concept. Babies and monkeys alike can tell when something they want is missing, they can tell the difference between 2 and 3 cookies. It's very natural, I think, for us to take account of these things, starting with the emotional framework, "I want more of what I like, rather than less of what I like". From this more/less concept arises counting in our later development.
Studying integers as mathematical objects, I agree, is a more modern concept.
A speculative citation: The first known use of numbers, 30,000 BC, was tally marks, an integer number system:
http://en.wikipedia.org/wiki/Number#History_of_integers
SixNein
Oct24-09, 05:17 PM
I think this is a mathematical perspective. From an observational perspective, integers were an easy concept. Babies and monkeys alike can tell when something they want is missing, they can tell the difference between 2 and 3 cookies. It's very natural, I think, for us to take account of these things, starting with the emotional framework, "I want more of what I like, rather than less of what I like". From this more/less concept arises counting in our later development.
Studying integers as mathematical objects, I agree, is a more modern concept.
A speculative citation: The first known use of numbers, 30,000 BC, was tally marks, an integer number system:
http://en.wikipedia.org/wiki/Number#History_of_integers
I would agree that natural or whole numbers can be observed, but integers are a different story. Can a monkey tell that he's got negative apples? No apples and negative apples are different concepts.
I'm inclined to believe that natural numbers are older than mankind. I would even venture a wild theory that dinosaurs could do some kind of counting. The concept of integers, however, separates us from the wild.
Aristotle's Law of Inertia said that a body comes to its natural state of rest unless acted upon by an outside force. By the Renaissance no counter example had ever been observed. Yet Gallileo said that a body in motion will stay in motion unless acted upon by an outside force. His evidence was the inclinded plane experiments. But as he was dersively told by his comtemporaries, the balls eventually come to a state of rest and in fact do not rise to the same height. So there was another ingredient other than just modeling observation in Gallileo's thinking and in fact in his day his theory contradicted observation while Aristotle's did not.
Pythagorean
Oct24-09, 08:12 PM
I would agree that natural or whole numbers can be observed, but integers are a different story. Can a monkey tell that he's got negative apples? No apples and negative apples are different concepts.
I'm inclined to believe that natural numbers are older than mankind. I would even venture a wild theory that dinosaurs could do some kind of counting. The concept of integers, however, separates us from the wild.
Oh, of course I've been using the wrong word! I meant natural/whole numbers from the beginning. I've always had a tendency to call natural numbers integers because we use the word commonly in physics for n = 1, 2, 3, etc. I now see that you mean to exclude 0 and negative numbers (and probably infinity too?) which I would agree with. Of course, I'm sure zero was observed early on, but it wasn't something that was discussed or recognized, as you may have implied earlier.
vectorcube
Oct24-09, 11:32 PM
Mathematics still works by and large from observations.
Can you show me how for vector spaces, and group theory?
What about algebraical notions like fields, rings etc?
vectorcube
Oct24-09, 11:39 PM
His evidence was the inclinded plane experiments. But as he was dersively told by his comtemporaries, the balls eventually come to a state of rest and in fact do not rise to the same height. So there was another ingredient other than just modeling observation in Gallileo's thinking and in fact in his day his theory contradicted observation while Aristotle's did not.
How is that? His`s only claim is that acceration ( or g) is the same for all large, and smell things. I don ` t see why this is NOT a observation generalized law.
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