Question on the universal correctness of mathematics

[disregardthat]

Platonism is the belief in perfect forms in a non-physical realm, forms which represent the objects in the world. Mathematical platonism is the belief in the existence of perfect mathematical objects in a different realm, as existing prior to our "discovery" of them. Which we incidentally does whenever we define something mathematical as the platonists want us to believe. And furthermore they want us to believe that we are exploring the properties of these mystical forms whenever we prove a theorem.

 

[Functor97]

Platonism is the belief in perfect forms in a non-physical realm, forms which represent the objects in the world. Mathematical platonism is the belief in the existence of perfect mathematical objects in a different realm, as existing prior to our "discovery" of them. Which we incidentally does whenever we define something mathematical as the platonists want us to believe. And furthermore they want us to believe that we are exploring the properties of these mystical forms whenever we prove a theorem.

Well my point is that you express the former view but reject the latter, of which the latter seems to be a subset of the first.

If we accept that there is an external reality, then it is very hard to ignore platonism. If there certainly was an external reality, which we model to a certain degree of success then our mathematics must do the same!

 

[Deveno]

To all interested in this subject, the following link is to the Stanford Encyclopedia of Philosophy. Provided you have the patience for a lengthy read, this particular article touches on the philosophy of mathematics. Very interesting material. I personally tend to align myself with mathematical Platonism...

that is a good article to gain some idea of what is at stake, among the various postions. i am a structuralist, a position that is not without its own epistemological concerns :). but i am by and large unconcerned with how "real" mathematics is, so the epistemological questions do not bother me.

 

[Functor97]

Some of you may have read or heard of the views of the cosmologist Max Tegmark.
Apparently our universe is not just described by math, but it IS mathematics:
http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf"
Apparently this answers all our philosophical and scientific questions. I like some aspects of it, but how does it stand up to the eternal question of Leiniz, "why something rather than nothing?". Do others here find the reversion to a pure mathematical explanation of everything disheartening or suitable?

Does the question of "what is mathematics" then become the most important? I mean why are all possible universes mathematical and not something else? This is the issue i have with the belief in external reality : ( Tegmark seems to advocate the asbolute "shut up and calculate" (his words not mine) approach. What is the point of science if all mathematics exists? Does it explain anything?
Seems like more questions than answers to me, the most irritating thing of all is i feel Tegmark is onto to something.

 

[Number Nine]

Apparently our universe is not just described by math, but it IS mathematics

Our universe is a discipline concerned with the study of quantity, structure, shape, and change?

The article you quoted does not suggest that the universe "is mathematics", it describes the universe as a mathematical structure, which is very different. The integers are a structure; they are not mathematics. A mathematical structure is just a collection of things with rules describing their behaviour.

 

[Functor97]

Our universe is a discipline concerned with the study of quantity, structure, shape, and change?

The article you quoted does not suggest that the universe "is mathematics", it describes the universe as a mathematical structure, which is very different. The integers are a structure; they are not mathematics. A mathematical structure is just a collection of things with rules describing their behaviour.

Maybe there is a break down in communication, but to say that our universe is a mathematical structure, is equivalent to saying it is a subset of mathematics. Tegmark claims that all our mathematical models (or at least the computable ones) are "physical" entities somewhere in the multiverse. The rules of a mathematical structure are mathematical, and thus i fail to see your point.

 

[Parmenides]

Functor97,

I would respond to such an intense question by suggesting that mathematics is perhaps only a half of what we would define as 'reality'. For example, certain current undertakings in modern cosmology attempt to describe a mathematical framework for ideas such as the possibility of a 'multiverse', an idea well beyond any hope experimental validation. I consider experimental and abstract reasoning as complementary methods for finding truth. Classical physics and its related mathematics could not account for energy radiation of a blackbody...something that could only be understood by observing that blackbody radiation is of course finite. The examples are endless; mathematics (our mathematics, at least) only seem to be valid as far as our observed experience.

I'm young and therefore subject to impulsive conceptions of things, but it seems to me that abstract, mathematical tools seem to only be considered by humans as useful if they are confirmed by observed experience and vice versa. As much as I wanted to be a pure rationalist, I think we must consider a union of rationalism and empiricism as our way of understanding reality.

Therefore, I find a purely mathematical description of reality a bit disheartening simply because we may not know what sort of conclusions math draws without subjecting it to observation. But I hold no real opinion on such a dense subject as a 20 year old : )

 

[Deveno]

Some of you may have read or heard of the views of the cosmologist Max Tegmark.
Apparently our universe is not just described by math, but it IS mathematics:
http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf"
Apparently this answers all our philosophical and scientific questions. I like some aspects of it, but how does it stand up to the eternal question of Leiniz, "why something rather than nothing?". Do others here find the reversion to a pure mathematical explanation of everything disheartening or suitable?

Does the question of "what is mathematics" then become the most important? I mean why are all possible universes mathematical and not something else? This is the issue i have with the belief in external reality : ( Tegmark seems to advocate the asbolute "shut up and calculate" (his words not mine) approach. What is the point of science if all mathematics exists? Does it explain anything?
Seems like more questions than answers to me, the most irritating thing of all is i feel Tegmark is onto to something.

the big question, here, really, is: does an objective reality exist? assuming one does is a crucial key step in Tegmark's argument, and he does not begin to address possible objections to that view.

 

[Functor97]

Functor97,

I would respond to such an intense question by suggesting that mathematics is perhaps only a half of what we would define as 'reality'. For example, certain current undertakings in modern cosmology attempt to describe a mathematical framework for ideas such as the possibility of a 'multiverse', an idea well beyond any hope experimental validation. I consider experimental and abstract reasoning as complementary methods for finding truth. Classical physics and its related mathematics could not account for energy radiation of a blackbody...something that could only be understood by observing that blackbody radiation is of course finite. The examples are endless; mathematics (our mathematics, at least) only seem to be valid as far as our observed experience.

I'm young and therefore subject to impulsive conceptions of things, but it seems to me that abstract, mathematical tools seem to only be considered by humans as useful if they are confirmed by observed experience and vice versa. As much as I wanted to be a pure rationalist, I think we must consider a union of rationalism and empiricism as our way of understanding reality.

Therefore, I find a purely mathematical description of reality a bit disheartening simply because we may not know what sort of conclusions math draws without subjecting it to observation. But I hold no real opinion on such a dense subject as a 20 year old : )

well you are older than me so we can feel brash together.
The problem with concluding that there is more to reality then math is problematic because we cannot begin to define something that we cannot define. It seems regressive to claim that we are more than the sum of our parts.

 

[1mmorta1]

Is there any purpose in this debate? Can there ever be an answer? This seems about as productive as Christians arguing with Muslims.

I, and others like myself, believe that the universe has always followed the same rules, that we discover those rules and learn more and more about their implications. I don't necessarily believe that, in some corner of a higher dimension, there are 5 euclidian solids floating around with the axioms and lemmas of all discovered and yet to be discovered mathematics. I just feel that reality exhibits certain properties, and that math is a way of taking the purest, most raw properties and using them.

Others feel that math is a human creation, and that(apparently) when we create math, the universe behaves accordingly.

Neither can be proven with science, though naturally I feel my own beliefs make the most sense.

 

[Number Nine]

Maybe there is a break down in communication, but to say that our universe is a mathematical structure, is equivalent to saying it is a subset of mathematics. Tegmark claims that all our mathematical models (or at least the computable ones) are "physical" entities somewhere in the multiverse. The rules of a mathematical structure are mathematical, and thus i fail to see your point.

Mathematics is a discipline, not a "thing". A neuron is not "neurobiology", a government is not "political science", a collection of things with rules governing their behaviour is not "mathematics".

 

[Deveno]

Is there any purpose in this debate? Can there ever be an answer? This seems about as productive as Christians arguing with Muslims.

I, and others like myself, believe that the universe has always followed the same rules, that we discover those rules and learn more and more about their implications. I don't necessarily believe that, in some corner of a higher dimension, there are 5 euclidian solids floating around with the axioms and lemmas of all discovered and yet to be discovered mathematics. I just feel that reality exhibits certain properties, and that math is a way of taking the purest, most raw properties and using them.

Others feel that math is a human creation, and that(apparently) when we create math, the universe behaves accordingly.

Neither can be proven with science, though naturally I feel my own beliefs make the most sense.

one possible purpose of a discussion such as this, is to clarfy to yourself and others what it is you believe. the meanings of words are not precisely fixed, and when we understand more clearly the ways in which we each see things, it helps to communicate our ideas, we have a basis for deciding which analogies to make, for example.

the belief that the world acts in accordance with some fixed principles, which behave consistently, is not a new one, and i daresay held by a great many people. many of those people believe that mathematics is the purest and most unambiguous way to communicate this perceived behavior.

but even deterministic systems can exhibit behavior which is indistinguishable from non-determistic behavior. and some people see a conflict between a universe that "runs like clockwork", and the notion of personal choice and freedom, or even randomness.

furthermore, our best efforts to show that mathematics is itself free from contradiction (and for those people who believe the universe itself is free from contradiction, this is a necessary thing, since using an inconsistent mathematics to describe a consistent world is self-defeating) have met with some serious set-backs in terms of some justifiably famous incompleteness theorems (consider the analogy with the heisenburg uncertainty principle, which posits a similar incompleteness in our abilty to measure, and thus know the world's behavior).

the tl,dr; version: some people believe Mathematics is Truth, but they can't prove it.

 

[disregardthat]

Well my point is that you express the former view but reject the latter, of which the latter seems to be a subset of the first.

If we accept that there is an external reality, then it is very hard to ignore platonism. If there certainly was an external reality, which we model to a certain degree of success then our mathematics must do the same!

Don't you see the difference between believing in platonic perfect forms in a non-physical realm, and physical objects in the physical world existing even though you haven't observed them?

The first one is almost a religious thing, and the second, in many regards just common sense.

 

[cmb]

This seems about as productive as Christians arguing with Muslims.

That's an outrageous analogy!

This is a bona fide point of 'natural philosophy', which intrinsically means there is 'no answer', so providing each party listens to each other's arguments, the learning and the 'philosophical act' is in the process of the debate itself. This is the fundamental inverse of a debate of religious zealots, who seek to destroy the argument for the sake of arriving at forcing an unprovable proposition and disallowing discussion.

So don't turn the thread into looking like a religious argument, please! Your're trying to close down the act of discussing [the nature of mathematics], exactly like a religious zealot would try to do.

 

[Hootenanny]

We are veering wildly from philosophy to utter nonsense here. I'm going to lock the thread, pending moderation.