Why does math work in our reality?

[Perspectives]

I’m reviewing my mathematics knowledge, except I’m looking for a different reason. I understand how it works you know, 1 plus 1 so on, I’m 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’s for a different time. Why does it work?

Any takers?

 

[fatra2]

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

 

[arildno]

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!
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

 

[HallsofIvy]

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 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.

 

[Blenton]

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]

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).

 

[JoeDawg]

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]

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]

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.

 

[Werg22]

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.

 

[arildno]

"2 + 2 = 4 is not something we can conceive of as being false. "

Sure we can.

Argument, please?

 

[junglebeast]

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

 

[Werg22]

"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.

 

[arildno]

" I mean the straightforward, everyday meaning of the statement"

What relevance does this have to the validity of mathematical statements??

 

[kote]

"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.

 

[Werg22]

" 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?

 

[Pinu7]

Mathematics is the study of structure, rythim, and patterns.

The Universe has structure, rythim, and patterns.
Therefore, math works in our reality.

 

[kote]

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]

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.

 

[Werg22]

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]

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" wouldn't 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

 

[kote]

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]

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.

 

[tauon]

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]

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.

 

[tauon]

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.

 

[wofsy]

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 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?

 

[JoeDawg]

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 argument 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]

I’m reviewing my mathematics knowledge, except I’m looking for a different reason. I understand how it works you know, 1 plus 1 so on, I’m 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’s 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.

 

[Sorry!]

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]

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 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.

 

[Chronos]

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.

 

[fatra2]

Hi there,

Wait, so I'm confused here 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

 

[wofsy]

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]

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.

 

[wofsy]

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.

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.

 

[kote]

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.

 

[wofsy]

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.

 

[kote]

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.

 

[wofsy]

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?

 

[wofsy]

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.

 

[wofsy]

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.

 

[kote]

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.

 

[wofsy]

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.

 

[apeiron]

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.

 

[kote]

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 ! 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.

 

[apeiron]

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]

One amazing example observed physical behavior of quantum chaos conjectured by the pure mathematics area of number theory:

http://www.physorg.com/news142834558.html

 

[wofsy]

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.'

 

[kote]

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.