Question on the universal correctness of mathematics

[Deveno]

me personally? i am inclined to think not, but i am not 100% certain (P < 1?). if it was experimentally indicated that "odds" were observer-dependent, it wouldn't shake my world-view.

i believe that one of the possibilities is, that there actually is an "objective reality" that does not depend on being perceived to be. but i believe there are other possibilities, too. but i think that, to an extent, my personal beliefs are by and large, irrelevant, that communication with someone else dictates i adopt the convention that we share a common super-set (something which i am entirely unable to prove).

the assumption that there is something "out there" which follows "rules" is convenient, and useful. that doesn't make it true. we proceed on the assumption that our universe is consistent, and it's a reasonable assumption to make, but that does not constitute proof.

in fact, in actual (i use the word loosely, in light of the present subject matter) games of chance, there is always some deviation between predicted outcome, and actual outcome. roll a pair of dice 100 times, and tell me if 7's come up 16 or 17 times. the law of large numbers says only this will be the average for a large number of trials, in other words we are talking about a limit. well, with limits, we only have "equality" with an "infinite" number of trials, so there is always going to be some uncertainty. do you see, how already we are straying from certainty that P = 1/6? to make a statement we are sure of...we need to assume an "ideal world"...which certainly isn't this one.

and I'm ok with that, I'm perfectly happy to only be 99% confident of a statement. you seem not to be.

 

[Functor97]

"Quantities certainly existed before there were people. Two + two was four before anyone realized it, and the probability of drawing one of four aces from a deck of 52 was 4/52 before the first playing card was printed.

In fact, the fundamental notions of probability were discovered by Pascal in the 17th century while studying dice -- even though dice had been around for thousands of years. Do you think double 6's rolled up any more or less often after Pascal than before?"

Quoted this from yahoo. Just felt it might apply

This is interesting. How do you know quantities existed before people? The physicist John Wheeler did not believe anything can be said to have existed before we observe it. I agree with him. We are creating our own reality in our minds, time, space, all of it could be a human construction, or a conscious one. We are bits of data of the universe looking back at itself, there is no way to claim anything existed before we percieve it to.

It is absurd to think reality is using our laws of physics, i.e. that nature is using differential geometry to induce gravity or celestial mechanics to make the planets go "round and round", that is our model, that is how we explain it, and it is the best explanation possible. it is possibly a mistake to even think of nature "doing" anything.

 

[disregardthat]

I think it's vital to distinguish "observables" from objects of mathematics. And so the debate of whether physical objects can be said to have existed before they are observed is an entirely different one from the discussion as of now.

I personally think it's silly to think that physical objects "come into existence" at the moment of observation. Though it's form, i.e. appearance, may be created in our minds upon observation, the very existence of it is different, and in my opinion prior to observation.

But, when it comes to mathematical objects, such as numbers, polynomials and manifolds etc, these are not observables. These are inventions; formal structures for which we have rules of manipulation and such. Such rules are much less statements than rules, and must not be confused with statements of physics.

 

[Functor97]

Mind you i think mathematics is as real as any of our other perceptions. Mathematics and Physics study the same thing, via different methods, and that is the laws of reality. Physical objects are almost certainly packets of data, which are subject to quantum mechanical processess. Are you really touching your mouse? We know it is mostly "empty space" but it feels pretty solid in my hand! We have evolved to percieve these bits of data as "real" or "solid". When we do physics we are simply studying the relationship between these abstract entities, which surprise surprise is what we do in PURE MATHEMATICS! The two go hand in hand because they study the same thing from different "angles". Our experiments in physics are no more than advanced reductio ad absurdum statements. We can never understand "why" a physical law is true, for we observe it from nature and create postulates to match. Exactly like our mathematical axioms. It is wrong to think of mathematics as creating proofs from axioms, we create axioms for proofs just as often, there is an entire field called "reverse mathematics".

 

[Functor97]

I think it's vital to distinguish "observables" from objects of mathematics. And so the debate of whether physical objects can be said to have existed before they are observed is an entirely different one from the discussion as of now.

I personally think it's silly to think that physical objects "come into existence" at the moment of observation. Though it's form, i.e. appearance, may be created in our minds upon observation, the very existence of it is different, and in my opinion prior to observation.

But, when it comes to mathematical objects, such as numbers, polynomials and manifolds etc, these are not observables. These are inventions; formal structures for which we have rules of manipulation and such. Such rules are much less statements than rules, and must not be confused with statements of physics.

THEY ARE NOT INVENTIONS! Well at least in the sense that the laws of physics are not inventions. It goes against the entire scientific process to claim that something has exists before we observe it. We assume that the universe is homogenous and that is all we need to do science. We do not even need to state that there is an "external reality" because by definition it is untestable. Now I could be wrong, being the stupid ape that i am But I do not see how we can believe in an external reality, before getting mystical and relying upon some bogus spiritual or religious hypothesis.

 

[disregardthat]

There is nothing religious about believing in the existence of stars we have not directly observed yet.

And no, the study of mathematics is nothing like the study of physical objects. Mathematical objects are just not physical, they are concepts. As such, they are definitely inventions. Platonism is outdated, bogus, and much more mystical and religious than you suggest unobserved objects are.

 

[Functor97]

There is nothing religious about believing in the existence of stars we have not directly observed yet.

And no, the study of mathematics is nothing like the study of physical objects. Mathematical objects are just not physical, they are concepts. As such, they are definitely inventions. Platonism is outdated, bogus, and much more mystical and religious than you suggest unobserved objects are.

But how do you know that these stars exist? You assume that there are more stars then we are aware of because you accept the universe is homogenous, and you have seen lots of stars here, so there must be more over there. If you could explore the entire universe, would you still assume that there were more stars?
The study of physics and mathematics are isomorphic. They appear different, but the more physics you study the more mathematical it becomes. Quantum mechanics showed us what happens when the belief that physical and abstract entities are exclusive is upheld.
You say that platonism is outdated, but your beliefs suscribe to the platonic trend.

 

[disregardthat]

You say that platonism is outdated, but your beliefs suscribe to the platonic trend.

Not at all, it doesn't seem like you know what platonism is.

 

[Deveno]

But how do you know that these stars exist? You assume that there are more stars then we are aware of because you accept the universe is homogenous, and you have seen lots of stars here, so there must be more over there. If you could explore the entire universe, would you still assume that there were more stars?
The study of physics and mathematics are isomorphic. They appear different, but the more physics you study the more mathematical it becomes. Quantum mechanics showed us what happens when the belief that physical and abstract entities are exclusive is upheld.
You say that platonism is outdated, but your beliefs suscribe to the platonic trend.

i have to say i disagree that physics and mathematics are isomorphic. i think there is a subset of physics isomorphic to a subset of mathematics, and that's as far as I'm willing to go.

there are assumptions about a physical world that a mathematical system is not obligated to conform to, and vice versa. mathematics is not validated by physics, nor does mathematics prove physical theories. in fact, i find the argument that science justifies mathematics in some way particularly repugnant, but that's just a personal bias on my part.

 

[Functor97]

Not at all, it doesn't seem like you know what platonism is.

Sure you deny the existence of mathematical forms, but platonism is slightly broader then that. You believe in the existence of forms which our physical theories model, but not perfectly. Sounds very much like what Plato said in the "Republic" with the shadows dancing on the wall of the cave.

If i have misunderstood your views, feel free to enlighten me : )

 

[1mmorta1]

This conversation seems to bring out a religious and mystical side of mathematicians...interesting

 

[Functor97]

i have to say i disagree that physics and mathematics are isomorphic. i think there is a subset of physics isomorphic to a subset of mathematics, and that's as far as I'm willing to go.

there are assumptions about a physical world that a mathematical system is not obligated to conform to, and vice versa. mathematics is not validated by physics, nor does mathematics prove physical theories. in fact, i find the argument that science justifies mathematics in some way particularly repugnant, but that's just a personal bias on my part.

I think physics is a subset of mathematics. All our laws of physics are mathematical statements. Of course the methods are different, but i would question how different they really are. I think physics is a lot more mathematical then a lot of physicists are willing to admit. However i believe that mathematics is a lot more empirical (at least quasi empirical) then a lot of mathematicians are willing to admit.
Its a middle-ground we need, between the hardcore platonists and the hardcore empiricalists, at least in scientific philosophy. None of the things we are discussing really changes the process of either the natural sciences or mathematics.

 

[Deveno]

This conversation seems to bring out a religious and mystical side of mathematicians...interesting

it is, isn't it? in general i would say we feel this gap between "what we know" and "what there is to be known". it is hard to estimate a path across this gap, without having a belief in something, even if it is the dry faith of rationality.

 

[Parmenides]

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...

 

[Parmenides]

http://plato.stanford.edu/entries/philosophy-mathematics/

Sorry : )

 

[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.