Is pure mathematics the basis for all thought?

In summary, the conversation discusses the role of pure mathematics in understanding the world and its relationship with physics. The participant believes that mathematics is the core of all academic disciplines and that all other disciplines can be reduced to applied mathematical problems. They also discuss the idea that our universe may be a mathematical structure and wonder why we, as conscious beings, perceive only a certain form of mathematics. The conversation also touches on the concept of axioms and their role in mathematics, and whether absolute truth is attainable through mathematics. Finally, there is a discussion about the relationship between physics and mathematics and whether mathematics serves as an explanation for all physical theories.
  • #1
Functor97
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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)
 
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  • #2
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 [itex]aԄb=ab^{\log_a b}[/itex]), but in physics, our rules have to agree with the real world.

Functor97 said:
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)

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
dalcde said:
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 [itex]aԄb=ab^{\log_a b}[/itex]), 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.

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...
 
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  • #4
Functor97 said:
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 is my interpretation of mathematics. We call these things theorems. I don't know any mathematician who places more weight on the axioms themselves.

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.

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

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?

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
Functor97 said:
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)

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
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
Functor97 said:
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?

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
chiro said:
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.

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
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
Functor97 said:
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?

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
Functor97 said:
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?

The art of mathematics is to make good definitions and axioms, and build great work (theorems) upon them. You can define addition to be
[tex]a+b=0,[/tex]
but it won't do much good.
 
  • #12
chiro said:
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.

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
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 isn't 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.
 
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  • #14
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
dalcde said:
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 [itex]aԄb=ab^{\log_a b}[/itex]), 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.

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.
 
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  • #16
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.
 
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  • #17
Functor97 said:
... i always thought of mathematics as the process by which we uncover the laws of god so to speak.

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
SteveL27 said:
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?
 
  • #19
Functor97 said:
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.
Functor97 said:
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.

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

Functor97 said:
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.
Functor97 said:
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?

Functor97 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?

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
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?
 
  • #21
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 without 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.
 
  • #22
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
 
  • #23
Studiot said:
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)??
 
  • #24
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.
 
  • #25
Just thought I'd like you all know that I have the answer: Nope.
 
  • #26
KingNothing said:
Just thought I'd like you all know that I have the answer: Nope.

LOL
That was fun.
 
  • #27
Studiot said:
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.
 
  • #28
KingNothing said:
Just thought I'd like you all know that I have the answer: Nope.
Thank you -- I appreciate that.

Let's all go home.
 
  • #29
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?
 
  • #30
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.
 
  • #31
So what about my culinery example?
 
  • #32
Studiot said:
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 that's my view.
 
  • #33
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?
 
Last edited:
  • #34
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.

https://www.physicsforums.com/showthread.php?t=511119

go well
 
  • #35
Studiot said:
Do you mean we use physical length, width etc to draw the graph on paper?

Even when you do it in your mind..

An old philosopher joke
If you conceive nothing: Do you still exist?

I have had to change my view of mathematics to a form of : studies of quantity and the relations there of.

I went through what archeologist have found, and some of what anthropologist have found.

The development of quantity by man has gone through changes to what is and is not accepted. It even took the old philosophers games and rules of argument and mathematics and others to formalism it into logic of argument. Even today there are different forms of such that allow different trains of thought to what is and is not accepted as proof, not just in mathematics.

I have had to discard the "purity" idea of such. If so it could easily be used to communicate with other species of our world. Some times it is difficult to use it to communicate with it to other people. I would say it is not so 'pure".

That I right or wrong, Who is to say. Not me.
Yet varied questions keep pooping up by folks thinking about careers, questions of why math unto its self, works for other science etc... The ideas of both science and mathematics having the potential not to be so exact, etc.

Back to the original subject. :

I have also found across the years people searching for the ideas of a set system, to go into fields such as science an math, and others.

I have encountered folks that leave physics when they hit the uncertainty principle.

Or math due to the original posters statement of
"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?"

More often than not: (The idea the world around us is not so ridged, that it is not an exact) seems to be the root of the conflict and confusion.
Of course the root of the question is way deeper than just the math or science. It seems to bring up questions in the root of the person them selves. It creates an internal crisis for the individual, and the lack of "truth", "Purity", and the result of the idea of "control".

To be honest I have not found a good way to talk to and or see a person, who would be good and or great in these fields: to continue past the self conflict encountered with these questions and problems. To many times I see people just stopping and going different routes due not
to the questions asked here, but the internal conflict they create in them selves from the questions. I have no doubt that a great many talents to math and science, are lost in this manner.
 
<h2>1. Is pure mathematics the only basis for all thought?</h2><p>No, while pure mathematics is a fundamental aspect of thought and understanding, it is not the only basis for all thought. Other factors such as personal experiences, cultural influences, and emotional responses also play a significant role in shaping our thoughts and perceptions.</p><h2>2. How does pure mathematics contribute to our thought processes?</h2><p>Pure mathematics provides a systematic and logical approach to problem-solving and critical thinking. It helps us to analyze and understand complex concepts, make connections between seemingly unrelated ideas, and develop abstract reasoning skills.</p><h2>3. Can pure mathematics be applied to real-world situations?</h2><p>Yes, pure mathematics can be applied to real-world situations in fields such as physics, engineering, economics, and computer science. It provides a framework for understanding and modeling natural phenomena, making predictions, and developing practical solutions to problems.</p><h2>4. Is pure mathematics necessary for all fields of study?</h2><p>No, while pure mathematics is essential in certain fields, such as physics and engineering, it is not necessary for all fields of study. Other disciplines, such as literature and history, rely more heavily on language, interpretation, and analysis of human experiences.</p><h2>5. How does the study of pure mathematics benefit society?</h2><p>The study of pure mathematics has numerous benefits for society. It has led to advancements in technology, medicine, and other fields, improving our quality of life. It also fosters critical thinking skills, which are essential for making informed decisions and solving complex problems. Additionally, pure mathematics has a beauty and elegance that can inspire creativity and appreciation for the world around us.</p>

1. Is pure mathematics the only basis for all thought?

No, while pure mathematics is a fundamental aspect of thought and understanding, it is not the only basis for all thought. Other factors such as personal experiences, cultural influences, and emotional responses also play a significant role in shaping our thoughts and perceptions.

2. How does pure mathematics contribute to our thought processes?

Pure mathematics provides a systematic and logical approach to problem-solving and critical thinking. It helps us to analyze and understand complex concepts, make connections between seemingly unrelated ideas, and develop abstract reasoning skills.

3. Can pure mathematics be applied to real-world situations?

Yes, pure mathematics can be applied to real-world situations in fields such as physics, engineering, economics, and computer science. It provides a framework for understanding and modeling natural phenomena, making predictions, and developing practical solutions to problems.

4. Is pure mathematics necessary for all fields of study?

No, while pure mathematics is essential in certain fields, such as physics and engineering, it is not necessary for all fields of study. Other disciplines, such as literature and history, rely more heavily on language, interpretation, and analysis of human experiences.

5. How does the study of pure mathematics benefit society?

The study of pure mathematics has numerous benefits for society. It has led to advancements in technology, medicine, and other fields, improving our quality of life. It also fosters critical thinking skills, which are essential for making informed decisions and solving complex problems. Additionally, pure mathematics has a beauty and elegance that can inspire creativity and appreciation for the world around us.

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