Why does math work in our reality?

[SixNein]

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]

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]

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".

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]

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.

 

[wofsy]

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]

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]

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]

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?

 

[wofsy]

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]

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]

... 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]

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".

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]

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]

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.

 

[wofsy]

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]

I am interested in the notion that a consistent structure must underlie all possible worlds.
I quote from post 138, in which vectorcube wrote:

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]

I am interested in the notion that a consistent structure must underlie all possible worlds.
I quote from post 138, in which vectorcube wrote:

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]

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]

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]

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]

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]

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]

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]

no
i don't

 

[arithmetix]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.

 

[wofsy]

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]

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]

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]

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.