Is the Empirical Nature of Mathematics and Logic the Key to Understanding Truth?

[Aquamarine]

It might interest you to know that Euclidean geometry plays an essential role in the very definition of differential geometry (the geometry used by GR).

I am not a physicist or mathematician but I quess you mean that a manifold is like Euclidean space when near a point. But this does not mean that the Euclidean manifold gives better predictions than the pseudo-Riemannian manifold. Which would you say is a more true theory of the real world?

 

[Hurkyl]

Yes. In fact, the very definition of a manifold is that it is "locally Euclidean". Without Euclidean space, you do not have manifolds.


It is a somewhat unusual usage of the word "truth" to suggest that a theory can be "more true" than one of the components upon which it depends in an essential way.

And as another interesting bit of trivia, all of differential geometry can be done entirely within Euclidean space. In a very real sense, differential geometry is not a new theory, it's just a new way of looking at an old theory. This also puts your usage of the word "truth" in an awkward position.



It's clear that you intend "true" to refer, in some sense, as a sort of rating about how well a mathematical theory models the "real world", but I think you'll be very hard pressed to give it a precise meaning that is consistent with your usage. (let alone the "typical" usage of the term)

 

[jcsd]

What Aquamarine calls 'truer' systems, he means stronger systems in the sense of Goedel's second incomplteness theorum. Clearly that a system is stronnger than another syas nothing about their relative 'truth'.

 

[Aquamarine]

Yes. In fact, the very definition of a manifold is that it is "locally Euclidean". Without Euclidean space, you do not have manifolds.


It is a somewhat unusual usage of the word "truth" to suggest that a theory can be "more true" than one of the components upon which it depends in an essential way.

And as another interesting bit of trivia, all of differential geometry can be done entirely within Euclidean space. In a very real sense, differential geometry is not a new theory, it's just a new way of looking at an old theory. This also puts your usage of the word "truth" in an awkward position.



It's clear that you intend "true" to refer, in some sense, as a sort of rating about how well a mathematical theory models the "real world", but I think you'll be very hard pressed to give it a precise meaning that is consistent with your usage. (let alone the "typical" usage of the term)

Are you saying that it is not possible to tell if Euclidean or pseudo-Riemannian mathematical model gives better predictions in the real world? That you cannot tell which model is more true of the real world?

 

[Hurkyl]

Are you saying that it is not possible to tell if Euclidean or pseudo-Riemannian mathematical model

I am saying that anything that can be modeled by differentiable manifolds can be modeled equally well in Euclidean space.

This fact comes about because every n-dimensional differentiable manifold is homeomorphic to an n-dimensional surface in some higher dimensional Euclidean space.

Informally speaking, this means any statement about a differentiable manifold is true if and only if it is also true about the corresponding surface in Euclidean space.

Furthermore, if the manifold is Riemann, the surface can be chosen so the Riemann metric coincides with the Euclidean metric. For pseudo-Riemann manifolds, I think there's a corresponding theorem for Minowski space (which is just Euclidean space with a different dot product)

 

[Aquamarine]

I am saying that anything that can be modeled by differentiable manifolds can be modeled equally well in Euclidean space.

This fact comes about because every n-dimensional differentiable manifold is homeomorphic to an n-dimensional surface in some higher dimensional Euclidean space.

Informally speaking, this means any statement about a differentiable manifold is true if and only if it is also true about the corresponding surface in Euclidean space.

Furthermore, if the manifold is Riemann, the surface can be chosen so the Riemann metric coincides with the Euclidean metric. For pseudo-Riemann manifolds, I think there's a corresponding theorem for Minowski space (which is just Euclidean space with a different dot product)

You did not answer my questions.

 

[Hurkyl]

anything that can be modeled by differentiable manifolds can be modeled equally well in Euclidean space.

This doesn't?

 

[Aquamarine]

So you are saying that the both models are equally good for the real world?

 

[Nereid]

IF the 'real world' is one in which GR is 'true', then yes. The same holds for a many areas in physics ... the maths used is partly for convenience; the systems are 'formally equivalent'.

 

[Aquamarine]

Just to be certain. The real world is both Euclidean and non-Euclidean at the same time and both models are equally useful when making predictions about the real world.

 

[Hurkyl]

Yes.

A lower dimensional example of the embedding theorem might help for understanding.


The surface of a sphere is a non-Euclidean geometry, right? However, the sphere is embedded in Euclidean 3-space. Thus, any statement about the surface of the sphere is simultaneously a statement about Euclidean 3-space.


Conversely, Euclidean space is a differentiable manifold.



Addendum: IIRC, for the topologies that physicists consider physically reasonable for the universe, you don't even need to appeal to a higher dimensional space!

 

[Nereid]

I haven't the faintest idea what 'the real world' is! And I would argue that 'the real world' cannot be known through science (for some good discussions on this, and whether 'the real world' can be known through something other than science, see several active threads elsewhere in PF Philosophy).

The strongest statement one can make about successful physics (and science in general) theories is something like this: "within its domain of applicability, all results from good experiments and observations are consistent with {the theory}". Sometimes the domain of applicability is very large (e.g. the universe, in the case of GR); sometimes the testing is fairly shallow (e.g. only the 'weak field' and 'static' parts of GR have been tested); sometimes the degrees of accuracy of the tests are astonishing (e.g. QED has been tested to 12 (16?) decimal places); sometimes there is a 'huge' gap in what can be tested (e.g. no physical theories have been directly tested on distance scales smaller than ~10^-18m, and even indirectly only very weakly).

But there is no claim - in the theories themselves - that they are the real world.

Just in case ... the very good accuracy of (most) successful scientific theories means that you can post things on PF, buy useful things cheaply in your local shop that were made in factories on the other side of the world, have the choice to live to a 'ripe old age' (probabilistically), etc.

 

[Aquamarine]

I haven't the faintest idea what 'the real world' is! And I would argue that 'the real world' cannot be known through science (for some good discussions on this, and whether 'the real world' can be known through something other than science, see several active threads elsewhere in PF Philosophy).

The strongest statement one can make about successful physics (and science in general) theories is something like this: "within its domain of applicability, all results from good experiments and observations are consistent with {the theory}". Sometimes the domain of applicability is very large (e.g. the universe, in the case of GR); sometimes the testing is fairly shallow (e.g. only the 'weak field' and 'static' parts of GR have been tested); sometimes the degrees of accuracy of the tests are astonishing (e.g. QED has been tested to 12 (16?) decimal places); sometimes there is a 'huge' gap in what can be tested (e.g. no physical theories have been directly tested on distance scales smaller than ~10^-18m, and even indirectly only very weakly).

But there is no claim - in the theories themselves - that they are the real world.

Just in case ... the very good accuracy of (most) successful scientific theories means that you can post things on PF, buy useful things cheaply in your local shop that were made in factories on the other side of the world, have the choice to live to a 'ripe old age' (probabilistically), etc.

Would you agree that a theory that is more consistent with experiments is more true?

 

[Hurkyl]

I would not agree.

My primary objection is the vagueness of the term "true". I always strive to say what I really mean than invoke vague and often contentious terms.

Of course, if you defined "truth" as a measure of consistency with experiment, then I would certainly agree that a theory more consistent with experiment is more true, by your definition.


My next objection is that it does not make sense to ask if a mathematical theory is consistent with experiment. It does, however, make sense to ask if the interpretation of physical phenomena within a mathematical theory agrees with experiment.

 

[Aquamarine]

I would not agree.

My primary objection is the vagueness of the term "true". I always strive to say what I really mean than invoke vague and often contentious terms.

Of course, if you defined "truth" as a measure of consistency with experiment, then I would certainly agree that a theory more consistent with experiment is more true, by your definition.


My next objection is that it does not make sense to ask if a mathematical theory is consistent with experiment. It does, however, make sense to ask if the interpretation of physical phenomena within a mathematical theory agrees with experiment.

I can agree with much of what you say. Truth has been difficult to define in philosophy. It seems that it is not possible to decide semantic truth within in a formal language, but only using an outside language. That would seem to lead to the old infinite regress argument.
http://en.wikipedia.org/wiki/Semantic_theory_of_truth

But the title of my thread was "Is mathematics empirical?". And I have seen nothing to contradict that. In the sense that what is studied is mainly decided by what can be of use in the real world, in the sense that it helps make better predictions about the real world. Those mathematics that are of no help in making predicions and will not help in making predictions in the future are frankly not any different from playing an interesting but useless game.

 

[matt grime]

You are misquoting my post.

You asked if some theories were more true than others without defining "true" and it's comparative usage here. Since most of us think something is true or it isn't, how is asking for some claification about this, and to get the debate away from seemingly pointless ideas, misquoting? Quoting out of context perhaps...


As Hurkyl points out, you're using true in some vague way that really isn't clear to us.

Every differential manifold of dimension N can be embedded isometrically in euclidean 2N-1 space if I recall my manifolds correctly, though I wouldn't put money on that.

Note, that lots of mathematical physicists don't even use manifolds, they use algebraic varieities.

 

[Aquamarine]

You asked if some theories were more true than others without defining "true" and it's comparative usage here. Since most of us think something is true or it isn't, how is asking for some claification about this, and to get the debate away from seemingly pointless ideas, misquoting? Quoting out of context perhaps...


As Hurkyl points out, you're using true in some vague way that really isn't clear to us.

Every differential manifold of dimension N can be embedded isometrically in euclidean 2N-1 space if I recall my manifolds correctly, though I wouldn't put money on that.

Note, that lots of mathematical physicists don't even use manifolds, they use algebraic varieities.

Let us abandon truth since there is no consensus of what it is and it seems not possible to define semantic truth within a formal language. Or at least discuss that in a different thread. Let us go back to the question that you edited out:

So why then is mathematics/logic interesting and why are some systems studied instead of others?

 

[Hurkyl]

But the title of my thread was "Is mathematics empirical?". And I have seen nothing to contradict that. In the sense that what is studied is mainly decided by what can be of use in the real world, in the sense that it helps make better predictions about the real world. Those mathematics that are of no help in making predicions and will not help in making predictions in the future are frankly not any different from playing an interesting but useless game.

Well, the sense in which you mean "empirical" differs from the sense in which I understand "empirical". One definition listed by Google (the others are similar) is:

Derived from experience or experiment.

There's a substantial difference between being motivated by experimental concerns, and actually derived from experiment.

 

[Aquamarine]

Well, the sense in which you mean "empirical" differs from the sense in which I understand "empirical". One definition listed by Google (the others are similar) is:



There's a substantial difference between being motivated by experimental concerns, and actually derived from experiment.

I have repeatedly said that the real world cannot say anything about the consistency of mathematical proofs, once the axioms have been chosen. My point is that most of those who do mathematical research are using axioms chosen because of usability in the real world.

 

[selfAdjoint]

My point is that most of those who do mathematical research are using axioms chosen because of usability in the real world.

And I continue to ask you to validate that statement by showing it to be true for an accessible data set of mathematical papers, e.g. the math section of the arxiv over thepast year. BTW, you won't often find a set of axioms in a math paper, what you find is references to previous papers.

 

[matt grime]

I would like to echo selfAdjoint there. The foundations of maths, whatever they may be, are of very little interest to mathematicians anymore. We do what we do. The vast majority of mathematics (including mathematical physics) papers have no reference to the world or experiment. In some sense that's what makes it hard to follow but it is also a necessity in order to make any progress at all. Are the underlying rules, rules that we make little reference to, chosen for their applicability to the real world? To be honest who knows, or cares - it's all a matter of interpretation anyway. For instance, do the axioms of ZF encapsulate the real world of "sets"? Many will say no. Indeed as soon as we start to do maths we create far more objects (used in a reasonably accurate sense for the category theorist) than exist, so the rules we need to manipulate them will not have any reflection in the real world: the set of all sets is purely a theoretical issue. There are many ways to pass beyond finite collections of things, and none of them can really be said to encapsulate the real world since there are only a finite number of objects in it, so how can we say the (very necessary) rules we choose there reflect anything 'real'?

If we take the axiom of choice as false then there are vector spaces without well defined bases. If we accept it then there are unmeasurable sets, and there is the Banach Tarski paradox to deal with.

Here is another discussion about such things:

http://www.maa.org/devlin/devlin%5F6%5F01.html

 

[Aquamarine]

I will repeat my previous post:

Maybe. But much mathematical research is not there. And probably almost all of that is applied. Much is never reveled. Like cryptography. The NSA is the world's largest employer of mathematicians with PhDs and has budget larger than the CIA. Or mathematics in military research. Or mathematics considered trade secrets by corporations. In finance mathematics is a central part of forecasting prices, often using exotic techniques.

And very importantly, everything involving programming computers is mathematical research into boolean algebra. All the steps from designing the CPU to high-level programming is an enormously complex application of boolean algebra. Including the research into efficient algorithms for database searches, data compression or graphical dispaly of 3D objects.

 

[Aquamarine]

I would like to echo selfAdjoint there. The foundations of maths, whatever they may be, are of very little interest to mathematicians anymore. We do what we do. The vast majority of mathematics (including mathematical physics) papers have no reference to the world or experiment. In some sense that's what makes it hard to follow but it is also a necessity in order to make any progress at all. Are the underlying rules, rules that we make little reference to, chosen for their applicability to the real world? To be honest who knows, or cares - it's all a matter of interpretation anyway. For instance, do the axioms of ZF encapsulate the real world of "sets"? Many will say no. Indeed as soon as we start to do maths we create far more objects (used in a reasonably accurate sense for the category theorist) than exist, so the rules we need to manipulate them will not have any reflection in the real world: the set of all sets is purely a theoretical issue. There are many ways to pass beyond finite collections of things, and none of them can really be said to encapsulate the real world since there are only a finite number of objects in it, so how can we say the (very necessary) rules we choose there reflect anything 'real'?

If we take the axiom of choice as false then there are vector spaces without well defined bases. If we accept it then there are unmeasurable sets, and there is the Banach Tarski paradox to deal with.

Here is another discussion about such things:

http://www.maa.org/devlin/devlin%5F6%5F01.html

And why then should society support those mathematics, if it is of no usefulness in the real world? Why not spend the money on teaching dead languages? Or on those prefering to play Everquest all day? Or on better health care? Or on reducing poverty? Why should taxpayer's money support those who want to play an useless game?

 

[master_coda]

And why then should society support those mathematics, if it is of no usefulness in the real world? Why not spend the money on teaching dead languages? Or on those prefering to play Everquest all day? Or on better health care? Or on reducing poverty? Why should taxpayer's money support those who want to play an useless game?

Money is spent on mathematics because there are people who are interested in its results. Just like any other type of research.

 

[Aquamarine]

Money is spent on mathematics because there are people who are interested in its results. Just like any other type of research.

Yes, but why should the state do that, using taxpayer's money? For those mathematics of no use in the real world? I am not arguing against indviduals spending their own money.

 

[HallsofIvy]

"The state" appears to believe that knowledge is a good thing in and of itself. It is also true,as I pointed out before, that often applications for pure mathematics are found AFTER the pure mathematics is done for no other reason than to "see how it works"- Riemannian Geometry was defined and researched long Einstein realized it could be used to describe general relativity.

By the way, the "state" doesn't support a heck of a lot of pure mathematics research. That typically is done by university professors who are paid to teach. The research is pretty much "on the side" (though an important side: it makes the university look better!). Wiles, for example, wasn't given any government support for his research into Fermat's Last Theorem (no doubt part of his salary was through government funding but it wasn't for the purpose of proving Fermat's Last Theorem)- which, by the way, is a good example of "non-empirical" mathematics.

Your original statement, by the way, was "truth in mathematics is ultimately derived from physics." Have you given up on that?

 

[CrankFan]

And why then should society support those mathematics, if it is of no usefulness in the real world?

Great idea!

Take non-Euclidean geometry. A perfect example of mathematics which had no obvious use in the "real world" at the time of its development. It's madness that anyone wasted their time with such fantasies when their time could have been equally well spent playing EQ!

And digital cryptography. More garbage. It just seems like many of the useful theorems of cryptography, discovered many centuries ago or earlier, have obvious application. The enlightened know that this is just an illusion; any result that follows from investigations into a branch of mathematics with no obvious application is necessarily useless in the "real world".

Just think about how much better off science would be if mathematicians never wasted any time on pure mathematics.

Oh wait a second...

 

[Aquamarine]

"The state" appears to believe that knowledge is a good thing in and of itself. It is also true,as I pointed out before, that often applications for pure mathematics are found AFTER the pure mathematics is done for no other reason than to "see how it works"- Riemannian Geometry was defined and researched long Einstein realized it could be used to describe general relativity.

By the way, the "state" doesn't support a heck of a lot of pure mathematics research. That typically is done by university professors who are paid to teach. The research is pretty much "on the side" (though an important side: it makes the university look better!). Wiles, for example, wasn't given any government support for his research into Fermat's Last Theorem (no doubt part of his salary was through government funding but it wasn't for the purpose of proving Fermat's Last Theorem)- which, by the way, is a good example of "non-empirical" mathematics.

Your original statement, by the way, was "truth in mathematics is ultimately derived from physics." Have you given up on that?

Yes, I have given upp a consensus opinion of truth, especially since it is not possible to define semantic truth within a formal language. But the title and essence still stands. Is mathematics empirical?

You do seem to agree that the value of apparently useless mathematics above playing a computer game is that they may be useful in the future. And my point is that this is what decides mathematical research, other people are not interested in supporting useless games played of mathematicians. In return for working for the mathematicians, they want something in return. So most mathematical research is forced into that useful in and agreeing with the real world. In that sense it is empirical. And do not argue for beauty or being like poetry, almost all people except of the mathematicians themselves do not understand what they are doing or producing.

If the state do not support much pure mathematics, then that is support for my view.

 

[master_coda]

Yes, but why should the state do that, using taxpayer's money? For those mathematics of no use in the real world? I am not arguing against indviduals spending their own money.

Because not very many people want the state to only spend money on things "of use in the real world".

 

[Aquamarine]

Because not very many people want the state to only spend money on things "of use in the real world".

You think so? That they do not want the state to concentrate its limited resources on things useful, like health care or reducing poverty? And it they do not want that, why should mathematics be supported instead of playing Everquest 24/7/52.