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

[Aquamarine]

There is no absolute truth in mathematics and logics. There are many consistent mathematical systems, consistent in the sense that they are not breaking the basic rules. But there is no absolute way of knowing that the basic rules are true. And there are limitless basic rules that can be chosen. Different basic rules gives different systems, like the intuitionist logic,predicate logic or fuzzy logic.

So why then is mathematics/logic interesting and why are some systems studied instead of others? I would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.

 

[matt grime]

I wouldn't use the word true, and I would never use a comparitive form for it in this sense. It [the set of rules/things we use] is not "more true", but it is more reasonable, and better yet more *useful* than anything else we have. That is why we use the things we do in the models we make: it works. If there were a better one, we'd use that.

 

[Aquamarine]

I wouldn't use the word true, and I would never use a comparitive form for it in this sense. It [the set of rules/things we use] is not "more true", but it is more reasonable, and better yet more *useful* than anything else we have. That is why we use the things we do in the models we make: it works. If there were a better one, we'd use that.

I tend to agree, but then I see little difference between "truth" and "useful".

Since there is no known ultimate cause and therefore "truth", I think the best we can get is more "useful". In the sense that it allows us to better manipulate the world.

Regarding the current preferred system, predicate logic. Suppose physicists start making better predictions using some theories based on intuitionist logic. Would not this require that most of current mathematical proofs be examined again, and accepted or discarded depending on how they fare in this new logical system that seems better suited to the real world?

 

[CrankFan]

So truth in mathematics is ultimately derived from physics.

How do you derive from physics, the truth that $$\pi$$ is transcendental?

Regarding the current preferred system, predicate logic. Suppose physicists start making better predictions using some theories based on intuitionist logic.

Can you provide an example of what you have in mind? I seem to recall that intuitionistic theories are weaker than classical theories, any theorem of the former is a theorem of the later, but I may be wrong. An example of what you have in mind might make things clearer.

 

[Aquamarine]

How do you derive from physics, the truth that $$\pi$$ is transcendental?

Pi is transcendental according to the basic rules chosen. Physics (and in extension the real world) do not affect the theorems that can be proven when the basic rules have been chosen. The real world affects which basic rules to chose.

Can you provide an example of what you have in mind? I seem to recall that intuitionistic theories are weaker than classical theories, any theorem of the former is a theorem of the later, but I may be wrong. An example of what you have in mind might make things clearer.

We can look at an example higher up in hierarchy. Euclidean geometry was long considered the only possible system and that it also reflected the real world, restricting mathematical research to within this area. Then is was discovered that non-euclidean geometry could be constructed in logically consistent way. And also that physicists can make better predictions assuming non-Euclidean geometry. Since then, much research has been done in non-Euclidean geometry, probably much more than in Euclidean geometry.

Regarding intuitionist logic, let us assume that it is found that the most basic fundamental particles seems to follow the basic rules of intuitionist logic better than predicate logic. That when constructing theories and when using
Reductio_ad_absurdum one gets poor predictions. But if instead if using intuitionist logic one can construct simple theories that give good predictions. I think this would lead to a paradigm shift in mathematics, away from predicate to intuitionist logic.

 

[matt grime]

OK, next challenge - can we have one physical theory constructed using reductio ad absurdum?

Surely, since physics is attempting to explain what is there, as opposed to what one may show may exist, irrespective of its apparent use, or lack of, physics is reasonably intuitionist.

And, as someone has pointed out, anything that can be proved true in intuitionistic logic is true in predicate logic.

Fuzzy logic already is used in some of the applied science, as are other logic systems.

 

[HallsofIvy]

Why do you choose physics specifically? If calculus is used in economics would you say that it works because of physics?? As far as your "basic rules" are concerned, the whole point of mathematics is that all statements in mathematics are of the form "if A then B". One doesn't have to know IF A is true or not, only follow the consequences IF it were true.

 

[Aquamarine]

Why do you choose physics specifically? If calculus is used in economics would you say that it works because of physics??

I say physics since this since some people here question whether economics or psychology is science and I do not want to discuss that in this thread. However, with physics I mean empirical science. My point is that mathematics is also an empirical science, but one step further removed from the real world than for example physics. Another step away would be logic. And logic is not automatically true, many different kinds of logics can be constructed. For example:
http://en.wikipedia.org/wiki/Sequent_calculus

As far as your "basic rules" are concerned, the whole point of mathematics is that all statements in mathematics are of the form "if A then B". One doesn't have to know IF A is true or not, only follow the consequences IF it were true.

Yes. But if you follow the the sequence backward you will find axioms that have been picked among many possible other axioms. There is no ultimate principle from which everything can be derived. And again, my point is that is the real world that determines which axioms to pick. Those that help makes better predictions are chosen. Chosen in the sense that mathematicians concentrate most of their efforts on those mathematics.

OK, next challenge - can we have one physical theory constructed using reductio ad absurdum?

Surely, since physics is attempting to explain what is there, as opposed to what one may show may exist, irrespective of its apparent use, or lack of, physics is reasonably intuitionist.

And, as someone has pointed out, anything that can be proved true in intuitionistic logic is true in predicate logic.

Fuzzy logic already is used in some of the applied science, as are other logic systems.
12-05-2004 05:37 PM

One could maybe argue that predicate logic have already failed in physics. That wave-particle duality would be an example.

I do not know enough physics or logic to know if QM is consistent with predicate logic. But if not, another logic would seems to be better choice since QM seems to make the best current predictions in its field of explanation.

Another interesting idea. What if the it turns out that that real world is discrete, that there is no infinity or continuity in nature. Would this mean that the current concept of limit and differentiation would have to be reworked?

 

[matt grime]

What has wave particle duality to do with predicate logic? Apart from your opinion that anything that *ought* to be dichotomic is the same as predicate calculus. The answer by the way is nothing.

As was pointed out in another thread, discrete models of space currently don't work, and for the umpteenth time limits and derivatives are mathematics, they are not physical objects! The nature of the real world would not invalidate it, in fact it is quite obviously independent of the nature of the physical world in some sense.

QM in case you didnt' notice has not stopped Newton's Laws of Motion from still being taught and used.

 

[HallsofIvy]

Pi is transcendental according to the basic rules chosen. Physics (and in extension the real world) do not affect the theorems that can be proven when the basic rules have been chosen. The real world affects which basic rules to chose.

Then I think the problem is that you have only studied mathematics that was designed for applications (and you seem to be using the word "physics" in a way I find peculiar). I know many forms of mathematics in which the axioms (what you call the "basic rules") have nothing to do with "the real world", for example the various finite geometries. Of course, those are not then used in applications.

I think you are looking through the "wrong end of the telescope". Mathematics, in its truest sense, does not derive axioms from "the real world". Of course, when you want to apply mathematics to "the real world", you choose the particular types of mathematics whose axioms do, in fact, appear to correspond to the real world.
Sometimes, of course, it happens that newly discovered properties of the "real world" turn out to correspond to axioms of forms of mathematics that had been developed previously without reference to the real world. I'm thinking in particular of the application to General Relativity of Riemannian spaces which were develope before the "basic rules" of General Relativity were known.

 

[Aquamarine]

What has wave particle duality to do with predicate logic? Apart from your opinion that anything that *ought* to be dichotomic is the same as predicate calculus. The answer by the way is nothing.

Actually there seem to have been a quite interesting discussion about this:
http://en.wikipedia.org/wiki/Is_logic_empirical?
http://en.wikipedia.org/wiki/Quantum_logic

As was pointed out in another thread, discrete models of space currently don't work, and for the umpteenth time limits and derivatives are mathematics, they are not physical objects! The nature of the real world would not invalidate it, in fact it is quite obviously independent of the nature of the physical world in some sense. QM in case you didnt' notice has not stopped Newton's Laws of Motion from still being taught and used.

I am not saying that the universe is discrete or continuous, in my limited understanding that is one point where QM and GR disagree.

It is true that the current limits and derivatives would not be logically falsified if the universe proved discrete. They are consistent according to the basic rules chosen. My point was that if physicists could make better predictions with limits and derivatives defined somewhat differently, they and mathematicians would abandon the current limits and derivatives. Hypothetically, a discrete universe could simulate a continuous universe at the macro level while being discrete at the micro level. The interesting predictions would be in the borderland, where the discrete and continuous mix. If the new limits and derivatives would make better predictions here than the old which ignore the discrete, they would be better suited for the real world.

Again, I am not saying this will be. Just an example of how mathematics could change which axioms to study, similar to the shift from Euclidean to non-Euclidean before.

 

[Aquamarine]

Then I think the problem is that you have only studied mathematics that was designed for applications (and you seem to be using the word "physics" in a way I find peculiar). I know many forms of mathematics in which the axioms (what you call the "basic rules") have nothing to do with "the real world", for example the various finite geometries. Of course, those are not then used in applications.

I think you are looking through the "wrong end of the telescope". Mathematics, in its truest sense, does not derive axioms from "the real world". Of course, when you want to apply mathematics to "the real world", you choose the particular types of mathematics whose axioms do, in fact, appear to correspond to the real world.
Sometimes, of course, it happens that newly discovered properties of the "real world" turn out to correspond to axioms of forms of mathematics that had been developed previously without reference to the real world. I'm thinking in particular of the application to General Relativity of Riemannian spaces which were develope before the "basic rules" of General Relativity were known.

Yes, there are areas of mathematics that seem to have no connection with the real world today But most mathematics are used and studied because it is useful in the real world. If it was not useful, why should society spend resources on it? Then society could as well spend the resources on teaching dead languages. So if mathematicians want to keep their status and pay, they have to choose to study systems built from axioms suited to the real world.

Regarding the second paragraph: Sometimes theory is before experiments. Sometimes experiments are before theory. Still, in the end both experiments and theory must fit together.

 

[matt grime]

Well, that's an interesting opinion, but factually flawed from my experience of being employed by mathematics departments (I would estimate in my current department less than 1/2 do maths that is applied to the real world, and it's predominantly an applied maths department too; this is what engineering departments are for). There is nothing to suggest all maths must eventually translate to the physical world (or it'll have it's funding cut). Intellectual rigour need not have a practical direct use.

Sometimes theory never meets experiment, thank goodness.

Interestingly I appear to be an impossible object in your opinion - a mathematician who's paid, by the state, to do mathematics that isn't practically applicable. At least it's nice when people tell me what I "have" to be.

Good to see you're so against all the higher aspects of humanity at least - utility should be one of the last things we think about in lots of cases. As the old saying goes, it'll be a great day when the schools are fully funded by the state and the military has to hold a bring and buy say to pay for its aircraft carriers.

 

[selfAdjoint]

matt, I want to bring up another sense of empirical. Take a mathematician who sets out to create something new - that is what math research is. The guy has in his mind a conspectus of prior mathematics in his area and he considers the relationships he knows about. Sometimes ther is a little "mix-and-match" novelty he can do, sometimes it's "monkey see monkey do" analogy with other math, and sometimes by building out a train of thought with these humble techniques he can come to a point where insight supervenes and he makes a discovery. (I want to apologize here to all the excellent women mathematicians for my choice of pronouns).

Now isn't a creation path like this "empirical" in its relationship to the world of existing mathematics, which to a mathematician is as real and present as the sensual world?

 

[matt grime]

I missed the post replying to me quoting back my post.

Lots of quantum things in general (such as quantum groups, cohomology etc) use quantum in a certain mathematical sense.

I still stand by my assertion that wave particle duality and predicate logic are not necessarily linked, but that is a philosophical position about the nature of logic (and mathematics) not being physical entitites. I would take it to say that the ideas of QM have led to a new way of thinking about things by introducing measurements of failure to commute (deformations), rather than it being the other way round, but again that is a personal opinion about philosophy.

However it is all hypothetical isn't it?

Mathematicians are obtuse people, they will study what they want to. Some will make connections to the real world some won't.

The development of mathematics and physics has to varying degrees been linked throughout its course. It diverged strongly in the 40's, but they are starting to come together now, it appears.

I don't think calculus wll die (something I'm a little sad about...)

It may interest you to know that there is a non-analytic concept of derivative; we algebraists have been using it for years. It uses something called the space of Dual Numbers.


Self Adjoint: I haven't actually used the word empirical directly, or addressed what may or may not be meant by it.

 

[CrankFan]

I do not know enough physics or logic to know if QM is consistent with predicate logic.

What does it mean for an arbitrary physical theory to be "consistent" with an arbitrary mathematical theory?

Another interesting idea. What if the it turns out that that real world is discrete, that there is no infinity or continuity in nature. Would this mean that the current concept of limit and differentiation would have to be reworked?

No.

 

[Aquamarine]

Well, that's an interesting opinion, but factually flawed from my experience of being employed by mathematics departments (I would estimate in my current department less than 1/2 do maths that is applied to the real world, and it's predominantly an applied maths department too; this is what engineering departments are for). There is nothing to suggest all maths must eventually translate to the physical world (or it'll have it's funding cut). Intellectual rigour need not have a practical direct use.

Sometimes theory never meets experiment, thank goodness.

Interestingly I appear to be an impossible object in your opinion - a mathematician who's paid, by the state, to do mathematics that isn't practically applicable. At least it's nice when people tell me what I "have" to be.

Good to see you're so against all the higher aspects of humanity at least - utility should be one of the last things we think about in lots of cases. As the old saying goes, it'll be a great day when the schools are fully funded by the state and the military has to hold a bring and buy say to pay for its aircraft carriers.

You may not have noticed but we live in a world of limited resources. These should be allocated where they are most useful. One must always think of the alternative use, for example saving lives in health-care with more money and intelligent people.

That many must work so that a few can spend their time in useless games seems absurd. And if theoretical mathematics is of no use in the real world, then it is no better than spending time with a complex computer game. Interesting maybe for the individual, but not something others should be forced to support. Regarding the state, is is noted for its inefficiency in all areas due to lack of competition.

But of course theoretical mathematics with no apparent utility today is much better than a computer game. Exactly because it may be of use in the future. So that is one of the better ways the state spends its money, compared to many others which often even have a negative effect. But note that since it is possible to construct and study countless useless mathematics which could consume all resources in the world, some calculation regarding future use should be made when giving money to today useless mathematics.

Regarding the use of mathematics, it is of course used today in most of society. And most of those using mathematics are certainly not using axioms with no connection to the real world.

 

[Aquamarine]

What does it mean for an arbitrary physical theory to be "consistent" with an arbitrary mathematical theory?

Two arbitrarily chosen theories are mostly useless without some connection to the real world, for example data from experiments.

If we have data, then we can construct and test a physical theory using a mathematical theory. Note that the mathematical theory will restrict which physical theories are possible to make or affect how good the predictions are or how complex the physical theory must be to fit the data.

It would be quite difficult to construct a good physical theory that fits the data if one were forced to work only within the mathematical theory of Euclidean geometry.

 

[CrankFan]

Two arbitrarily chosen theories are mostly useless without some connection to the real world, for example data from experiments.

What makes you think THE theory of predicate calculus is related to QM in some obvious way? As far as I can tell, substituting "theory of recursive functions" or "theory of formal languages" in place of "theory of predicate calculus" is an arbitrary choice.

If we have data, then we can construct and test a physical theory using a mathematical theory. Note that the mathematical theory will restrict which physical theories are possible to make or affect how good the predictions are or how complex the physical theory must be to fit the data.

We all know that mathematics can be used as a tool to build physical theories. What I'm wondering is; what exactly is the process you used to come to the determination that predicate calculus is false based on the empirical evidence that suggests that light has a dual nature?

 

[matt grime]

There is no absolute truth in mathematics and logics.

would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.

Can we get back to the original post? It seems that this isn't a discussion about mathematics or anything in particular of that nature. It is purely a debate about what people mean by "true". And in particular comparitive ideas.

 

[Aquamarine]

What makes you think THE theory of predicate calculus is related to QM in some obvious way? As far as I can tell, substituting "theory of recursive functions" or "theory of formal languages" in place of "theory of predicate calculus" is an arbitrary choice.

We all know that mathematics can be used as a tool to build physical theories. What I'm wondering is; what exactly is the process you used to come to the determination that predicate calculus is false based on the empirical evidence that suggests that light has a dual nature?

That question has been asked and answered earlier. And I did not claim that QM has falsified predicate logic.

 

[Aquamarine]

Can we get back to the original post? It seems that this isn't a discussion about mathematics or anything in particular of that nature. It is purely a debate about what people mean by "true". And in particular comparitive ideas.

You are misquoting my post.

 

[Nereid]

Reading this thread for the first time, in one sitting, it seems surreal ... an awful lot of people talking past each other ...

Aquamarine has, IMHO, posed some very interesting questions, and certainly in the right place in PF.

First though, some personal clarifications - economics is just as much a 'science' as physics is (the scientific method is applied just as rigourously). Whatever 'reality' the latest and best physics or economics theories suggest, the maths used to describe those theories has its own, independent 'reality' (and if a successful theory can be described using several different mathematical frameworks - insert your favourite examples here - isn't this a practical demonstration of what I just said?). Throughout the history of maths and science (go back as far as you like ... even pre-Greece), there has been a fertile interchange between the two; for a great many of those who have contributed to our current body of theory - directly or indirectly - it would be hard to make a clear distinction; for others, their working lives have had only the most tenuous connection with the other domain (tho a detailed study of how mathematicians have worked may show a greater influence of 'external' ideas than is popularly believed).

So, to what I see as the core of Aquamarine's post: of the truly vast potential 'space' mathematicians could explore (workers in the 'truth mines' - anyone read the SF novel "Diaspora"?), the regions most heavily explored have tended to be those with an apparent connection to 'models of the physical universe'. This is NOT to say that all (or nearly all) mathematicians work in areas that are seen to be (potentially) 'useful', nor that there aren't wonderful results far from the beaten track (e.g. Cantor's work?).

Perhaps we could use the scientific method to examine this idea? I read somewhere that there are ~250,000 new 'theorems' published every year. Perhaps we could take the collective works of mathematicians over the last 50 (500?) years and analyse them in some way?

Finally, there is Aquamarine's 'utility' question (the economics of mathematics? not to be confused with 'usefulness' - utility is a term in economics with a specific and precise meaning): the allocation of scarce resources (salaries for people to do math). I suspect that this is somewhat beyond the farthest shores of economics today ... for example, innovation has only recently begun to be decently addressed (economically, so to speak), and I suspect 'multi-generational' returns are a grey area. This matters not one bit at the individual level (there will always be those who are independently wealthy - free to pursue whatever interests them, and free to fund whomever they like, for example) - but it may be of great interest in the broad.

 

[selfAdjoint]

So, to what I see as the core of Aquamarine's post: of the truly vast potential 'space' mathematicians could explore (workers in the 'truth mines' - anyone read the SF novel "Diaspora"?), the regions most heavily explored have tended to be those with an apparent connection to 'models of the physical universe'. This is NOT to say that all (or nearly all) mathematicians work in areas that are seen to be (potentially) 'useful', nor that there aren't wonderful results far from the beaten track (e.g. Cantor's work?).

Perhaps we could use the scientific method to examine this idea? I read somewhere that there are ~250,000 new 'theorems' published every year. Perhaps we could take the collective works of mathematicians over the last 50 (500?) years and analyse them in some way?

A cheap and easy substitute for this would just to peruse the mathematical section of the arxiv. I'll bet a lot that you wouldn't find a majority of the papers there motivated by describing physical reality.

 

[jcsd]

It is an emprirical process, you learn about maths from the empirical observation of textbooks.

 

[CrankFan]

That question has been asked and answered earlier. And I did not claim that QM has falsified predicate logic.

When you wrote:

"There are some logical and mathematical systems that are more true than others"

What examples did you have in mind?

 

[Aquamarine]

A cheap and easy substitute for this would just to peruse the mathematical section of the arxiv. I'll bet a lot that you wouldn't find a majority of the papers there motivated by describing physical reality.

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]

When you wrote:

"There are some logical and mathematical systems that are more true than others"

What examples did you have in mind?

I would consider the non-Euclidean geometry behind GR more true than Euclidean geometry. Or predicate logic more true than the logic of Aristotoles. Based on that they have enabled theories with better accuracy in the real world. This does not mean that Euclidean geometry is not consistent using its own axioms. But that it is a less true theory regarding the real world.

 

[Hurkyl]

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).

 

[CrankFan]

It is an emprirical process, you learn about maths from the empirical observation of textbooks.

Learning from texts is one way to learn about mathematics.

Is it impossible to develop a theorem, in your head, independent of any of the 5 senses?

I guess a lot of this depends on what is meant by "empirical" and "empirical process". If we are using a definition of empirical in which every thought process is an empirical process, then what do we gain by referring to a certain process (carried out by humans) as an empirical process?

 

[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)