## Is pure mathematics the basis for all thought?

 Quote by SteveL27 That idea is easily refuted by the existence of non-Euclidean geometries. Through a point not on a given line, you can assume there are zero, one, or more than one parallels to the given line. Each choice gives a logically consistent geometry. But these three choices can not all be true of the world we live in. Math is not physics. Even within math, you can play the same game. Given the Zermelo-Fraenkel axioms, you can assume the Axiom of Choice (AC) or its negation. Either way you get a consistent set theory. Same with the Continuum Hypothesis (CH); and there are also a number of less well-known axioms with the same property of being independent of ZF, with no "real world" way of knowing whether the axiom or its negation should be accepted into mainstream math.
I do not see how that follows. Non Euclidean Geometry is a generalisation of Euclidean geometry, it extends it, it does not contradict it.
I have yet to take a course in advanced mathematical logic, but to me it seems that our axioms are so basic that we cannot come up with a different form of mathematics, it is intwined within our way of thought. For example, could you change a basic axiom of mathematics and come up with a system that seems interesting and beautiful but is totally distinct from our current research areas of mathematics? I do not know, like i said, i am unfamilar with this work as an undergraduate, but if it can be done, why has it not been done?

Maybe i am looking at this from the wrong angle, for some of the arguments on this page make me think of mathematics as a subset of physics. We take our physical intuition and generalise it. That makes sense from a biological/evolutionary stand point, what would be the advantage of us accessing the "source code" of the universe, if it were distinct from our need to survive. Would anyone agree with that? I think many pure mathematicians would object, as they often take pride in "useless research" as Hardy said.

I can find one pure mathematician who agrees with this standpoint. Vladimir Arnold, the key protagonist of the anti bourbaki tradition, said "Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." Thoughts?

 Quote by Functor97 I do not see how that follows. Non Euclidean Geometry is a generalisation of Euclidean geometry, it extends it, it does not contradict it.
You can have a self-contained, consistent system of geometry that is Euclidean; and a self contained, consistent system (lots of different ones, in fact) that are non-Euclidean.

The physical universe can not be both. It must be one or the other. In this case math is a tool for describing universes. It does not discriminate between the hypothetical ones and the real one.

 Quote by Functor97 I have yet to take a course in advanced mathematical logic, but to me it seems that our axioms are so basic that we cannot come up with a different form of mathematics, it is intwined within our way of thought. For example, could you change a basic axiom of mathematics and come up with a system that seems interesting and beautiful but is totally distinct from our current research areas of mathematics?
The most well-known example is the Axiom of Choice (AC). It says, innocently enough, that you can simultaneously choose an element from each one of a collection of nonempy sets. AC turns out to be independent of the other standard axioms of Zermelo-Fraenkel (ZF) set theory.

So, you can do math with or without AC. If you use AC then you can prove many standard theorems that mathematicians (and physicists) use daily. But you also get unavoidable anomalies such as the famous Banach-Tarski paradox, which says you can cut a solid in 3-space into a finite number of pieces; rearrange the pieces using rigid rotations and translations; and end up with TWO copies of the original solid. This result is disturbing to many people.

On the other hand if you reject AC, you get a perfectly good, logically consistent theory (well, ZFC is consistent if ZF was consistent in the first place -- which is another story!). But in this choiceless theory, you have a vector space without a basis; a ring without a maximal ideal; the product of compact topological spaces might not be compact; and a lot of standard theorems can't be proved.

So the overwhelming majority of mainstream mathematicians freely use AC. Not because it's "true" in any conceivable meaning of the word -- I mean, who the heck really knows whether the real numbers can be well-ordered, which is one of the equivalents of AC -- but because it's convenient. It let's you prove more theorems, so mathematicians use it.

In other words, no pun intended: It's a matter of choice :-)

There are other examples but this is the most famous one.

http://en.wikipedia.org/wiki/Axiom_of_choice

 Quote by Functor97 I do not know, like i said, i am unfamilar with this work as an undergraduate, but if it can be done, why has it not been done?
It's done every day. Set theorists, logicians, and computer scientists deal with axioms and provability every day. Why computer scientists? They're interested in what you can do with finite strings of symbols, which they call programs. Logicians are interested in what you can do with finite strings of symbols, which they call proofs. It's the same subject. Godel, Church, Turing in the 30's, very active area of research ever since. The set theorists have a long list of wild axioms that they study. Each axiom gives you a different set of properties for the real numbers. Of particular interest are the large cardinal axioms. Large cardinals are sets so large that their existence can't be proven from ZFC. But some of them are starting to work their way into standard mathematics. A large cardinal is implicitly used in Wiles's proof of Fermat's Last Theorem. Foundations are always in a state of flux.

 Quote by Functor97 Maybe i am looking at this from the wrong angle, for some of the arguments on this page make me think of mathematics as a subset of physics. We take our physical intuition and generalise it. That makes sense from a biological/evolutionary stand point, what would be the advantage of us accessing the "source code" of the universe, if it were distinct from our need to survive. Would anyone agree with that? I think many pure mathematicians would object, as they often take pride in "useless research" as Hardy said.
Art is inspired by our experience of the real world. But art far transcends the real world. Same with math. Or, what does a symphony or a pop tune have to do with our need to survive?

 Quote by Functor97 I can find one pure mathematician who agrees with this standpoint. Vladimir Arnold, the key protagonist of the anti bourbaki tradition, said "Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." Thoughts?
That's really interesting. You know, I only heard about Bourbaki as these French guys who wrote the textbooks that are the standard for the way all the graduate students are trained to think about math these days. But I have never heard of what it means to be "anti-Bourbaki." Can you tell me more about that? What is it they don't like?

As far as that quote, of course math is a toolkit for physics. It just happens to be a lot more. But this is an old debate. I wouldn't pretend to be qualified to speak for mathematics. I'll just let xkcd have the last word ...

http://xkcd.com/435/

 Recognitions: Science Advisor The current wikipedia article on "Philosophy of mathematics" lists several varieties of beliefs about mathematics. I haven't bothered to understand the distinctions among them but it's interesting how many posters one encounters on this forum who advocate some version of "everything is math" or "math is a reality that exists outside the axioms that people create for it", etc. I would call this "mathematical Platonism". The wikipedia article suggests that my classification system doesn't have enough species. To me, the most interesting aspect of thought, mathematical or otherwise, is to consider what we know about it - which is practically nothing. It appears to be conducted by some sort of self-modifying biological network that doesn't work very well (at physics or math) when it is first created. As it ages and gains experience, it (in its own opinon) begins to grasp things that it considers to be truths. It becomes very impatient with other biological networks that express any contradiction to them. I suppose it's useful as a self-motivational tool to believe that our brains are touching some important, eternal verities. Yet if someone makes a mathematical claim and then presents an incoherent incomprehensible justification for it, we don't admit that he has provided a proof. So, since we don't know how our brain works, why should we be so trusting of its conclusions?
 Were to even start with your original post?? "nature of pure mathematics" Well the ideas that math is it the real world and man has observed it and found and used it is one view. The other opposing is that man has made all of this up. There exist others that fall in between. It goes with out the speculation to Man, Memory, Quantization, also size and amount of Quantity as an abstract symbol as numbers, The list above is a crude development of child and of mankind from hunter gather: the process of development. Other questions to ask are How does mathematics so easily fall into use with the physical world. Are the relations of math so easily suited to the situation, or again is it the thinking of man and how we develop quantity of measurement? Again no real answer, yet if math was so "real" then using it to communicate with other species on this earth would be easy, it is not. Even with in man exist places were man looked at quantity as: Enough for today. Any more was ignored. Communication with math would have been futile to try and communicate with the group when it was found. Of course time and economy as well as contact with modern man is and has changed this. So far the only real answer to this is Man, memory, quantization, Individual symbols for quantity, study of such relations. It is in my opinion that for man to have managed to string together all we have and be able to "seem to get it right" is a achievement unto itself.
 This is a most interesting thread and I'm glad to see how well tempered the discussion has been. So I offer a few comments of my own, although others here are far more advanced pure mathematicians than myself. The original post seems to ask the question "is (pure) mathematics all we have?" I take the point about maths being 'internal thought' , developed 'in the dark independent of outside influence' and offer the comment that the Trachenberg system of arithmetic is a prime example of this. One point about this system of internal thought is that it includes many systems that do not exist in 'reality'. Another point is whether you accept that logic is part of pure maths or a separate discipline and what you even mean by logic. To me Physics differs from Maths in that Physics is about measurement and quantities. Graphs in Physics have units, graphs in maths have axes which are indistinguishable. Finally my stomach would disagree with the premise 'maths is all we have' since my method of cooking is largely non mathematical. go well

 Quote by Studiot This is a most interesting thread and I'm glad to see how well tempered the discussion has been. So I offer a few comments of my own, although others here are far more advanced pure mathematicians than myself. The original post seems to ask the question "is (pure) mathematics all we have?" I take the point about maths being 'internal thought' , developed 'in the dark independent of outside influence' and offer the comment that the Trachenberg system of arithmetic is a prime example of this. One point about this system of internal thought is that it includes many systems that do not exist in 'reality'. Another point is whether you accept that logic is part of pure maths or a separate discipline and what you even mean by logic. To me Physics differs from Maths in that Physics is about measurement and quantities. Graphs in Physics have units, graphs in maths have axes which are indistinguishable. Finally my stomach would disagree with the premise 'maths is all we have' since my method of cooking is largely non mathematical. go well
"graphs in maths have axes which are indistinguishable."

Can you show me how these axis do not have measurement, and quantity?
Can you show me a graph in math that does not show the relation, and or relations; in a mechanical form (the graph)??

 Graphs in Physics have units such as kilogrammes, centipoises or whatever along the axes. The graphs are only valid within the bounds of physical laws. Graphs in mathematics have pure numbers only. Any graph I can draw in maths will be 'valid'. However I cannot draw a valid graph in Physics extending say the value of youngs modulus v temperature to negative temperatures. I could certainly draw a line in the negative half plane as a valid mathematical act, it would just have no meaning in Physics.
 Just thought I'd like you all know that I have the answer: Nope.

 Quote by KingNothing Just thought I'd like you all know that I have the answer: Nope.
LOL
That was fun.

 Quote by Studiot Graphs in Physics have units such as kilogrammes, centipoises or whatever along the axes. The graphs are only valid within the bounds of physical laws. Graphs in mathematics have pure numbers only. Any graph I can draw in maths will be 'valid'. However I cannot draw a valid graph in Physics extending say the value of youngs modulus v temperature to negative temperatures. I could certainly draw a line in the negative half plane as a valid mathematical act, it would just have no meaning in Physics.
Well the ideas that "Pure" numbers that have no designation as an item or dimension is correct. Yet when we make a graph, we use length width etc to make it. Even in the basic counting we place value on such numbers as quantity but no specific name to the value as time, energy, space, orange etc...

We try and keep the quantity, but remove the qualifier for the ideas of mathematics.

 Quote by KingNothing Just thought I'd like you all know that I have the answer: Nope.
Thank you -- I appreciate that.

Let's all go home.

 Yet when we make a graph, we use length width etc to make it.
Do you mean we use physical length, width etc to draw the graph on paper?

 I was not claiming that math is all there is, but mathematics is all we can know, its as good as it gets. As our physics becomes more and more "advanced", in an attempt to answer more physical questions, it has become more and more mathematical. So much so that many pure mathematicans worked and work in the physical based string theory. The intuition of quantum mechanics and subsequent theories have been mathematical. Not rigorous certainly, but mathematical none the less. We do not ask why mass bends spacetime, but rather why certain gauge theories work, and the answer to the latter is mathematical. I think the whole quest for understanding is mathematical, without axioms in the end, we will never have a theory of everything. What happens if string theory/ loop quantum gravity or some other theory describe our universe perfectly? We are left to ask, why these equations, why does this constant not take another value? It just keeps on going, and like mathematics it is boundless. Our senses evolved to deal with short time spans, short distances and newtonian intuition. How can we delve deaper into reality without changing our senses? will mathematics elvolve our senses? I do not claim to have an answer. But i am very interested in other people's.
 So what about my culinery example?

 Quote by Studiot So what about my culinery example?
Lets just say it gives me food for thought ; )

Seriously though, I think mathematical skills have helped you in the kitchen. Its just so applied you do not really think about it. Well thats my view.

 Recognitions: Science Advisor I think when people are asking this, they are asking the wrong question. Perhaps what motivates the question is the same that motivates the following one: is logic the basis for all thought,-or,-is logical laws laws of thought? That is a much more interesting question and worthwhile to answer. Another questions might be which is also touched upon in this thread: are the laws of euclidean geometry laws of geometric intuition?

 I think mathematical skills have helped you in the kitchen.
'helped' is a long way from 'forms the basis of'.

I did ask if you considered logic part of pure maths and the question has again be raised by
disregardthat.

I agree the thread would benefit from firming up on definitions.

A similar question is also current here and the references in post 21 and 31 are qite relevant.