# Is pure mathematics the basis for all thought?

1. Jul 5, 2011

### Functor97

I have been thinking much on the nature of pure mathematics. I believe this forum would make the best place to post over say the philosophy section, as i am more interested in the opinions of working mathematicians and physicists than philosophers.

In my opinion pure mathematics is the core of every academic discipline we humans have so far explored. I would extend that to state, that mathematics is in essence the art of thought it is the science of anything, and that all other disciplines may be reduced to applied mathematical problems. I understand this is quite a contentious statement, and i would gladly welcome your point of view.

The more i ponder this, the more i think that mathematics is all we have. We interact with the outside world via our senses, but these senses give us a warped view of our surroundings. This screen has no color, but our retina perceives color, due to the wavelength of electromagnetic radiation reflected, and our understanding of electromagnetic radiation is based upon physical formula which are simply applied forms of mathematical theorems. Quantum mechanics does not make sense to our intuition built up on the large scale, so we rely upon mathematics to understand the phenomenon. Many physics students object to quantum mechanics and claim that it is fundamentally wrong or not complete, because they cannot envision exactly where a particle is at a given instant, yet the mathematics still guides us. When people ask "why do like poles repel?", the answer can get quite complex leading to quantum field theory, and if an individual keeps asking "why?" each subsequent step is correct, it quickly devolves into mathematical reasoning not physical. This has led me to agree with max tegmark and a few others who conjecture that our universe is actually a mathematical structure, yet i do have a problem with this as to why conscious minds view only a certain form of mathematics. Does our pursuit of knowledge being grounded in mathematics, mean that external reality (if there is any?) is grounded in mathematics, or is that just our best guess, so to speak.

More generally, do you believe that mathematical inquiry, such as say number theory, complex analysis or functional analysis explains more about reality then quantum field theory for when we solve number theory problems, are we working with the very code of the cosmos? So our theories of physics are just the subset of our Matthematical theorems. Do many mathematicians hold this view? I guess this is some form of Neo Platonism, and as an atheist, i find the issue of an external mathematics problematic. Do you believe that pure mathematics should be given credence over particle physics, for if this conjecture is true, isn't the dependence of the Axiom of choice within a large cardinal system just as suitable a grand unified theory as M theory?

Finally if mathematics is just axioms, and we cannot prove an axiom to be true, and yet mathematics is the basis of all science, does this mean absolute truth is beyond us? (Not advocating the ends justifies the means)

Last edited: Jul 5, 2011
2. Jul 6, 2011

### dalcde

Physics isn't just the maths. It's more than that. In mathematics, we are allowed to define whatever we like (we can define the operation Ԅ to be $aԄb=ab^{\log_a b}$), but in physics, our rules have to agree with the real world.

I don't like to think axioms of something we take to be true. We define axioms to be true. We make certain definitions (axioms) and build upon them. A clear example (if you know some abstract algebra) is the axioms of groups. We don't take them to be true. They are rather things that constrain our system.

3. Jul 6, 2011

### Functor97

I understand that, i just have a problem with "defining" something to be true. It makes mathematics seem a game, where we make statements and draw those statements out to their logical conclusions. That very well maybe mathematics (there are certainly worse things to spend your time doing than playing such a beautiful game :tongue:) but i always thought of mathematics as the process by which we uncover the laws of god so to speak. (not saying there is a god, but i am sure you have heard the expression "God created the integers, all the rest is the work of men") The definition of axioms as things we state as true removes any chance at external truth, or some absolute system of thought. I by no means believe there to be one, but it does seem to be a common belief that mathematics is that system, i always thought of mathematics as striving towards perfect knowledge. If we decide upon truth, then how is mathematics different to some post modern literary theory, in which the english professor blows on about how everything (even time and space) are subjective?

Isn't physics just the mathematics which we apply in our universe. Would that not make physics a part of mathematics? Would mathematics then serve as an explanation for all physical theory? By explanation, i do not mean it to be final, rather each level of the onion "of truth" becomes more and more mathematical, till in essence we are just doing pure mathematics...

Last edited: Jul 6, 2011
4. Jul 6, 2011

### pwsnafu

That is my interpretation of mathematics. We call these things theorems. I don't know any mathematician who places more weight on the axioms themselves.

Specific example: is the axiom of choice true in your interpretation?

We are free to choose which axioms we want, and we look at their conclusions. Whether our choice of axioms is "extensionally true" is irrelevant. Mathematics is nothing more than applied logic. That is: mathematics is limited by human thought.

Just because PA has a statement X which is true but not provable in PA doesn't mean much. I can strengthen my axioms and then study the problem. There may exist an Axiom B such that X is true in PA+B and but false in PA-B. Well that's interesting in itself. And certainly not subjective.

5. Jul 6, 2011

### chiro

Hello Functor97 and welcome to the forums.

One reason why math is so powerful (and why it is applied everywhere) is due to the properties of numbers, sets, functions, graphs, and all the other objects out there in math.

Numbers can represent anything. They encode information about anything. You can use the same kind of argument for functions and sets in similar ways.

The axiomatic way also matches much of human behavior when it comes to things that are probably not considered mathematical.

For example consider a trainee just hired by an employer. The employer has years and years of experience of this job and has to quickly train up the employee. The employer spends about 10-15 minutes outlining a "compressed form" of training to the employee. It doesn't cover every situation, but it allows the employee to get a picture of how to handle things that aren't said and how to use that knowledge to handle a specific situation.

This looks a lot like what scientists do: they come up with principles that are both minimal in description and complexity and maximal in descriptive capability for the domain being described.

You can probably find these kind of analogues in everyday life if you look hard enough.

The big difference is though, that in many human situations rigor is not something you really need. In maths though you do need it. This kind of formality can be a pain in the neck, but it also increases confidence in what you are doing.

6. Jul 6, 2011

### Functor97

So you would view mathematics as a tool rather then the process? I guess this is very subjective, but it is often said that Mathematics is the queen of the sciences, and i was hoping there would be some bootstrap which would reduce physical reasoning to mathematics. Is physics the mathematics of our universe? or is Mathematics, Physics we primates have evolved to abstract for evolutionary gain? Both maybe?

7. Jul 6, 2011

### chiro

It's a language and a way of thinking and analyzing.

Human beings are primarily constrained by their ability to describe, classify, and analyze. This constraint is language. Before you wish to do anything, you need to figure out what it is that you want to do: you need to describe what you want to achieve.

Math is a language that suits a specific purpose and provides a framework for a certain way of thinking. Like any language it is constantly evolving: things become more general, isolated things merge together to form new things, and completely new things are discovered and added to make sense of a previously unknown phenomena.

Like any language it is optimal for its purpose. You don't write a hundred page book in English to describe something if you can write it in a formula that covers a page. At the same time it might not be enough to write just a few mathematical laws to communicate what you need to. Use a particular language for its strengths and if one isn't strong in enough ways, use another or create one that fits the purpose.

If there comes a better way to describe the physical universe, it will most likely be embedded into mathematics. This is my opinion, but the evidence is there based on what has happened historically.

8. Jul 6, 2011

### Functor97

But it is a language in which certain rules remain constant. Take Hardy for instance, he was a strict platonist and believed in an objective mathematical reality. The number 23 is prime not because we want it to be, but because it fits our definition of a prime. If mathematics was purely a language i would challenge you to create a prime that does not fit out previous definition. Of course you cannot, that is a contradiction...I will agree mathematics is a language, but it is the language of the universe, the language of reason. You cannot shape this language as much as you want to, it shapes your view, in that mathematics is very much like physics. We may draw the lines, dots and squigels, but the "Background" reason is independent of us, or at least our ability to influence it. As mathematics is the ultimate language for the physical world, would that not mean it explains the physical world? I would claim It is the "deepest" part of the physical world we can access...
I think that writing mathematics off as a language, does not do it justice, maybe call it the only language...that may suffice?

9. Jul 6, 2011

### Functor97

You just need to look at modern particle physics. The deeper we go, the more mathematical reality becomes. In high school we could all draw pictures of what we thought was going on, when you reached quantum mechanics that vanished. Now we must use mathematics, but how does this differ from pure mathematics itself. Einstein showed that the bending of spacetime is gravity, and the bending of spacetime is just geometry, and how do we understand geometry? enter pure mathematics. Sure a physicist working in solid state or optics is not going to notice this trend as much as a high energy specialist, but the deeper into reality we go, the more and more we must embrace pure mathematics. I do not see this ending, that is why i believe that pure mathematics is the foundation of all physical understanding...

10. Jul 6, 2011

### chiro

Languages are used to describe and classify things and mathematics does exactly that.

Lets say you have the word "cat" and with a Venn diagram you draw a 2D shape that corresponds to all written, spoken, graphical, etc definitions that pertain to cat. That set will contain a boundary. At some classification point there is a definitive point where things are no longer "cats". The union of "cats" and "not cats" is anything that can be described within the limits of the language.

Just like your cat example, all your math definitions are exactly the same, in the context of the language that is math. The fact that you called something prime and described it means you artificially created your "prime" subset and with your definition have declared a boundary that separates what is "prime" and what "isn't prime". Both are completely disjoint.

I agree that math is the best language we have because of its domain, its clarity, its generality, and its ability to describe so much more than any other language we have: no arguments there.

One example that comes to mind is a tribe (I forget the actual geographical vicinity) that did not have a complete system for numbers. They had words that corresponded to zero, one, two and three, but anything more than three was considered just one word (kind of like our infinity with the fact that finite numbers were also included): it was like our definition of "many".

With something like english I can talk about a chair. What exactly is a chair? Well its something you sit on. "But I can sit on a table you say", but then you say "but that's not a chair". Eventually you might get something that is a good definition.

With math its a lot simpler. We can add constraint after constraint by treating sensory input as a mathematical signal and then using these constraints to get ridiculously close to a very specific definition of chair.

So in short I agree that math is the "super" language or at least the best language we have.

11. Jul 6, 2011

### dalcde

The art of mathematics is to make good definitions and axioms, and build great work (theorems) upon them. You can define addition to be
$$a+b=0,$$
but it won't do much good.

12. Jul 6, 2011

### Functor97

Even if we change the constraints the method of thought remains the same, and that is what i would claim is the essence of mathematics. It is certain. Or we may say, it is as certain as we want it to be. As to external reality, who knows what that is, does it even exist? As soon as we examine it, does it change? For all intents and purposes we must accept mathematics as external reality, it very well maybe just the way our brains are structured, but i don't see how we can examine the cosmos without our minds.

I think many pure mathematicians would object to dalcde's claim that mathematics should be built around utility. Yet i do see the point, many pure mathematicians do work within systems that are derived from very physically intuitive concepts, and then extended. Would any pure mathematicians care to expound on what drives you in your research?

13. Jul 6, 2011

### Functor97

All of our physical laws are mathematical statements. Thus i see mathematics as the underpining of all thought. I don't think this is possible to change, it just is. Look at some of the papers from leading particle physicists, take edward witten for instance http://arxiv.org/find/all/1/au:+witten_edward/0/1/0/all/0/1" he isnt sitting around in a patent office constructing thought experiments, he is exploring pure mathematical structures, in the hope that these will explain the nature of our reality.

Last edited by a moderator: Apr 26, 2017
14. Jul 6, 2011

### kramer733

Ok this is going to be off topic but why does functor97 keep on writing a new text below his previous text? JUST EDIT YOUR PREVIOUS POST! Jesus man! I've seen you do that like 4-5 times or something. To me it's very annoying.

As for my input on this, math is honestly just a tool. Like it teaches you how to think but i more or less see it as a tool now.

15. Jul 6, 2011

### SteveL27

The group axioms aren't true of the integers under addition? Who knew!

To be clear: I disagree. The group axioms are true of any group. What do you say to that?

And by the way, axioms are not definitions. I don't know why so many people are confused about that.

Last edited: Jul 6, 2011
16. Jul 6, 2011

### dx

Scientific thinking is a free play with concepts (words) whose justification lies in the measure of survey over the experience of the senses which we are able to achieve with its aid. All knowledge is originally represented within a conceptual framework adapted to account for previous experience, and any such frame may prove too narrow to comprehend new experiences. Mathematics is essentially an extension of our ordinary language developed for the logical representation of relations between experience, supplementing it with appropriate tools for representing relations for the communication of which ordinary language in not sufficient, and also with its well-defined abstractions allows the representation of the harmonious relationships of theoretical physics where at each stage appropriate widening of the conceptual framework to grasp new experience brings greater unity and harmony to the whole description.

Last edited: Jul 6, 2011
17. Jul 6, 2011

### 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.

18. Jul 6, 2011

### 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.
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?

19. Jul 7, 2011

### SteveL27

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.

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.

http://en.wikipedia.org/wiki/Banach–Tarski_paradox

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

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.

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?

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/

20. Jul 9, 2011

### Stephen Tashi

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?

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