What's Your Philosophy of Mathematics?

[lugita15]

Here's an explanation I wrote up a while back on Quora that details some major philosophies of mathematics:

Each major theory about the origin of mathematics has its own challenges to overcome. Any theory has to explain how mathematics is consistent, bountiful (meaning that there's always new things to discover), applicable to the physical world, and accessible to the human mind.

One philosophy is that math is just an invention of the human mind. This used to be associated with people on the fringe, called intuitionists or constructivists, who tried to establish a very narrow view of what mathematical techniques are allowable. But lately it's acquired more mainstream popularity because of George Lakoff's book Where Mathematics Comes From, which tries to explain math in terms of cognitive science and human psychology. The main problem with this view are that it doesn't explain how math is so self-consistent: most ideas we think up have all kinds of flaws and inconsistencies, so how has mathematics held up perfectly for so long? Also, why math is so useful in understanding the world around us?

The latter problem is most naturally addressed by physism, a philosophy originally proposed by Aristotle but which has come back into the limelight thanks to a series of books by Roland Omnes. Physism states that humans came up with math by observing the physical world. By studying the laws of physics, they were able to come up with mathematical rules which seem to govern how the world operates. The major problem with this philosophy is that mathematics is quite a expansive field, and it's not clear how much of it is grounded in actual physical phenomena. Sure, some things like calculus seem pretty well realized in the world, but can the same be said about more abstract branches like category theory? Probably not.

Formalism is yet another philosophy; it was all the rage a century ago, but now it's fallen out of favor. Formalists like David Hilbert believed that math is just a formal game we play using strict axioms and rules. But Godel's Incompleteness Theorems cast doubt on this: it turns out that mathematics is too expansive and bountiful (the technical term is "indefinitely extensible") to be captured by a single formal system. Also, it's hard to be absolutely sure that the system we're dealing with doesn't have some inconsistency lurking within. Finally, it seems too much of a coincidence that the universe behaves exactly according to the rules of a formal system we came up with millennia ago. (Unless you believe in computationalism, in which the universe really is just a big computer).

Last but not least, there is the most popular mathematical philosophy of all time, Platonism. Pretty much all mathematicians believe in this philosophy, which claims that there is an abstract realm called Platonic heaven where all mathematical structures reside. (In modern versions, we like to say that it's mathematical truths like 1+1=2 that are "out there", not actual objects like circles) It solves all of the problems listed above that plague the other major philosophies, but it has its own difficulty: how in the world are measly human beings able to discover truths about what goes on in Platonic heaven? If you're religious, the answer is obvious: we have souls, abstract nonphysical essences which can access Platonic heaven. But resorting to religion makes mathematics akin to theology, which seems unsettling to say the least.

In the interest of full disclosure, I'm somewhere close to logicism and/or platonism, not unlike the views of Gottlob Frege and Bertrand Russell, in that I believe that the truths of mathematics are objective and absolute, and I share their view that mathematics seems amenable to reason.

I'm really interested in the philosophy of math, so if you have any questions about it I'd be glad to help.

 

[AmPure]

Math is beautiful, i see it as a product of nature perhaps like the natural law of polarty's. i can compare it to our polarities as humans,, the Male is one polarity having straighter lines and features and more of a logical brain, and the other polarity Female having an intuitive "feeling" brain and character being the opposite of math (what ever that is?). essentially what I am saying is math is a beautiful form of logic, and it all comes together in nature. ~ just my ramblings hope you make something of it

 

[lugita15]

Math is beautiful, i see it as a product of nature perhaps like the natural law of polarty's.

Let me not comment on your male-female thing, but if you believe that mathematics arises from the properties of nature, then you would be an adherent of physism. But then how would you respond to the objection that there is so much mathematics that we do that is not directly grounded in our knowledge of the physical world?

essentially what I am saying is math is a beautiful form of logic

That would make you a logicist (which is pretty incompatible with physism).

and it all comes together in nature.

What do you mean "comes together in nature"?

 

[AmPure]

Perhaps properties of nature was a bad term, more that it arises from the properties of our Universe. (in my philosophy) As for pure mathematics that would be a pure form of the polarity (Logic) and where it "comes together in nature" is where you see Fibonacci sequences in plants, Physics, logic in situations, pretty much every use there is in our world here.
Perhaps i should have specified this was an "Other" personal philosophy, hopefully i cleared it up a bit for you

 

[apeiron]

In the interest of full disclosure, I'm somewhere close to logicism and/or platonism.

Physism would be closest for me, but then the debates begin.

I would start with the conventional point that all knowledge is modelling. So even mathematical truths are not pure knowledge - because even if the consequences of deduction are taken to be knowably true, the axioms from which those consequences are derived have to be assumptions (no matter how plausible).

A second point is that maths seems to divide between that which is possibly very true of the world - it describes the actual inevitable patterns of nature - and then a lot of elaboration which becomes just human invention. So even if the core of maths is physicist, then there could be a wandering off into intuitionist/formalist terrain.

A third point which I think is currently interesting in the philosophy of mathematics is the slogan "nature does not compute with infinite means". This is the claim that one of the ignored facts of (physicist) maths is that the natural patterns of real worlds are in fact restricted by material constraints.

I talked about this development here... https://www.physicsforums.com/showpost.php?p=3791415&postcount=244

So I don't hold with Platonism. But I would argue that the forms that nature can take are materially limited. For a reality to exist, there are constraints that will emerge. So the Platonic notion of a realm of forms is in this way an objective fact. But instead of the forms existing dualistically in some detached and placeless heaven, they are what emerge by a process of actualisation. They "exist" in the way that definite limits exist - by in fact marking where all further possibility ceases.

And then - here the argument turns physicist - we have developed a language to describe these "eternal forms", these constraints of nature, in pragmatic fashion.

 

[lmoh]

That would make you a logicist (which is pretty incompatible with physism).

Before I give my answer, I must ask, why are they incompatible? My own view seems to be that mathematics is just a variation of logic and reason, so both can be questioned in the same way, but I also believe that it was originally based upon our observations of the real world, so my current view seems to fall under both (but mainly leaning towards logicism).

BTW, You did not describe logicism and fictionalism in your initial post.

 

[Whovian]

I'm not exactly a Logicist, but that option's close enough, so I selected it.

Simple. I don't view mathematics as needing some species to invent it, or anything of the sort. I've thought of it as, ah, just existing independent of a Universe to give it practical applications. The fact that most branches of Mathematics somehow tie in with the Universe's behaviour is just a very informed decision on the Universe's part to keep Physicists working as Mathematicians. (Sorry, love Douglas Adams's writing style and couldn't help but mimic it.) And if you're wondering what concepts are central to algebraic mathematics, my opinion is that they are the integral, the derivative, and the limit.

So, basically, I think of the Universe as being based around Mathematics, not Mathematics being based around the Universe.

A similar thread on a different forum (note I said similar, not the same): http://www.artofproblemsolving.com/Forum/viewtopic.php?f=138&t=446895

(I'm known as bdejean there)
I'm still wondering if this "ultimate framework" exists. And I'm not talking about just using different symbols or different notation, like using $\dot{f}$ or $f\prime$ instead of $\dfrac{\mathrm{d}f}{\mathrm{d}x}$. I'm talking about a completely different sort of math.

 

[lugita15]

So even mathematical truths are not pure knowledge - because even if the consequences of deduction are taken to be knowably true, the axioms from which those consequences are derived have to be assumptions (no matter how plausible).

Even if the theorems of mathematics are derived from axioms that just have to be assumed (which is debatable, see Frege's work in logicism), isn't it still true that the fact that the axioms do logically imply the theorems is logically true? (The only way you could dispute that if you do something like this.) So in that sense, don't the chains of deductions, used in mathematical proofs, constitute pure logical knowledge? Like the fact that the Pythagorean theorem is derivable from Euclid's axioms.

A second point is that maths seems to divide between that which is possibly very true of the world - it describes the actual inevitable patterns of nature - and then a lot of elaboration which becomes just human invention. So even if the core of maths is physicist, then there could be a wandering off into intuitionist/formalist terrain.

I think you would be surprised how little mathematics is actually directly grounded in human knowledge of the physical world. See Hartry Field's work in mathematical fictionalism. Field, set out to formulate all known laws of physics using as little mathematics as possible, and he found that he could do it with almost no mathematics at all! That's right, no real numbers, no natural numbers, none of the things that we regularly use in physics! So he concluded that almost all of the mathematics we use is NOT grounded in our knowledge of physics.

So then then what is the rest of math based on? You mentioned that you think it could be intuitionism or formalism. Either way, how is it that the rest of matematics is able to stay consistent? Is it just a marvelous coincidence that we happened to be using a consistent system of mathematics, or do you allow the possibility that it's inconsistent and we haven't discovered it yet? Also, concerning formalism, how do you get around the Godel's theorem objection?

So I don't hold with Platonism. But I would argue that the forms that nature can take are materially limited. For a reality to exist, there are constraints that will emerge. So the Platonic notion of a realm of forms is in this way an objective fact. But instead of the forms existing dualistically in some detached and placeless heaven, they are what emerge by a process of actualisation. They "exist" in the way that definite limits exist - by in fact marking where all further possibility ceases.

And then - here the argument turns physicist - we have developed a language to describe these "eternal forms", these constraints of nature, in pragmatic fashion.

But where do these constraints, that natural law must conform with, come from? You say that the constraints are necessary for reality to exist, but where does necessity itself come from? Why is logical necessity not universe-dependent? Or do you admit that there is such a thing as prexisting logical truth, that is universe-independent?

 

[Whovian]

I think you would be surprised how little mathematics is actually directly grounded in human knowledge of the physical world. See Hartry Field's work in mathematical fictionalism. Field, set out to formulate all known laws of physics using as little mathematics as possible, and he found that he could do it with almost no mathematics at all! That's right, no real numbers, no natural numbers, none of the things that we regularly use in physics! So he concluded that almost all of the mathematics we use is NOT grounded in our knowledge of physics.

I've been waiting for something like this. But ... a copy on Amazon is $469.78 ... They're probably overcharging, but still.

 

[lugita15]

Before I give my answer, I must ask, why are they incompatible? My own view seems to be that mathematics is just a variation of logic and reason, so both can be questioned in the same way, but I also believe that it was originally based upon our observations of the real world, so my current view seems to fall under both (but mainly leaning towards logicism).

Certainly much of mathematics was originally discovered based on physical observations, like if you put one rock next to another rock you get two rocks, so 1+1=2. But the question is not how humans happened to come across mathematics, but rather what is the nature of mathematics itself? Is mathematical truth dependent on the properties of the physical universe? Suppose we lived in a universe in which whenever you put one rock next to another rock you somehow get three rocks. Would that mean that 1+1 would equal 3 in that universe, or would it still equal 2? (Of course, in that universe we might have chosen to give the name "addition" to a completely different mathematical operation, one that makes 1 and 1 yield 3. But the question is not about the names we happen to give to mathematical notions, but the mathematical notions themselves.)

BTW, You did not describe logicism and fictionalism in your initial post.

You're right, I didn't. Logicism is the belief that the concepts of mathematics can be reduced to purely logical notions, and that once you translate mathematical statements to purely logical statements, they can be shown to be tautologies. Logicism originated with German philosopher Gottlob Frege, who tried to start off by showing that arithmetic (meaning the study of natural numbers) can be reduced to logic. He wrote a groundbreaking logical analysis of the concept of Number (meaning reducing the concept of number to logic) in his short book The Foundations of Arithmetic (which I highly recommend reading). After that, he wanted to rigorously derive all the laws of arithmetic (like commutativity of addition) from pure logic, which he tried to do in his meticulous and complex symbolic treatise The Basic Laws of Arithmetic.

Unfortunately, Bertrand Russell discovered that the formal system Frege had been using for this purpose had an inconistency in it, so Russell and Whitehead wrote their three-volume magnum opus the Principia Mathematica, a symbolic treatise that tried to fix the inconsistency in Frege's system and to derive even more of mathematics that Frege had attempted from pure logic. Unfortunately, Russell's effort was also unsuccessful, not because it was inconsistent but merely because it used one axiom that was not purely logical, the Axiom of Reducibility. So then for most of the twentieth century the logicist project was pretty much abandoned, until recently when a Crispin Wright, Bob Hale, and others found that much of Frege's original work could be salvaged. They call their attempt neologicism, and although it has some issues to iron out it looks promising. You can read more about Frege's logicism and the neologicists in this excellent article.

Concerning fictionalism, the idea is pretty simple. Works of fiction have their own internal systems of truth and falsity. For instance, in the works of Arthur Conan Doyle, "Sherlock Holmes lived on Baker Street" is a true statement, and "Sherlock Holmes lived on Main Street" is a false statement. Yet in reality, both of those statements are wrong, because Sherlock Holmes didn't live anywhere. Philosopher Hartry Field proposed that mathematics is also similarly a fictional "story", and that when we say "for every prime number there is a bigger prime number", we don't (or shouldn't) really mean that there actually such things as numbers but rather that within the fictional story of mathematics, there are numbers and it is true that for every number there is a bigger number. In Field's view, mathematics is just a convenient story that we find useful to think in terms of when dealing with certain problems, but that it is not actually necessary for any purposes. To demonstrate this, he wrote a book "Science without Numbers", in which he found that he could formulate the known laws of physics without using the notion of numbers at all! That is a serious challenge to the philosophy of physism, which claims that mathematics is grounded in physics.

I hope that helps.

 

[lugita15]

Simple. I don't view mathematics as needing some species to invent it, or anything of the sort. I've thought of it as, ah, just existing independent of a Universe to give it practical applications

What you're articulating is just the viewpoint known as Platonism. Pretty much all logicists are Platonists, but most Platonists are not logicists. Logicists specifically believe that not only is mathematical truth absolute and universe-independent, but it is also reducible to logical truth.

 

[lmoh]

Thanks for the info. I guess I am going to stick my my original gut instinct and opt for logicism. My position on this is mainly as a response to something like Wigner's "Unreasonable effectiveness of Mathematics". My response has always been that such effectiveness was based upon the regularity of the universe, which mathematics describes. I imagine most of substantial mathematics is also based upon such regularities (I am referring to logic here) as well. So just so long as both strictly adhere to a certain set of rules, then I don''t see a reason why a connection between the two cannot be made. It does not go much further than that in my opinion, so mathematical truths probably just equate to logical truths, and the latter doesn't seem at all suprising, which is why I am not fond of platonism.

 

[lugita15]

Thanks for the info. I guess I am going to stick my my original gut instinct and opt for logicism. My position on this is mainly as a response to something like Wigner's "Unreasonable effectiveness of Mathematics", which I always thought was based upon the regularity of the universe, through which I imagine most of substantial mathematics is also based upon (I am referring to logic here). It does not go much further than that in my opinion, so mathematical truths probably just equate to logical truths, and the latter doesn't seem at all suprising, which is why I am not fond of platonism.

Logicism is not at all incompatible with platonism. In fact, virtually all logicists are platonists. Platonist believe that mathematics is about something objectively real, and logicists believe that that something is just logic.

And as far as logicism being obvious, what makes you think that? If it was so obvious, why would the reduction of mathematics to logic require geniuses like Frege and Russell?

 

[ilhan8]

intuitionism.
on every field I use intuitive methods. I am right brained.

 

[Pythagorean]

I think both platonism and physism; a combination of invention and discovery.

 

[lmoh]

Logicism is not at all incompatible with platonism. In fact, virtually all logicists are platonists. Platonist believe that mathematics is about something objectively real, and logicists believe that that something is just logic.

I don't know if that would apply to my position, but of course, I am just an amateur on the issue as I said. Platonism to me seems to imply that mathematics is special in its own right (with a separate realm for mathematical truths), which to me strikes me as being a little mystical and unnecessary. My own take is that most mathematical developments following basic mathematics are derived from logic and reasoning*, so it would not be suprising that there are such mathematical truths. For the most part, I am probably just reducing the issue of mathematics to one about logic.

*When I was saying that mathematics is based upon logic, I was mainly referring to its development in comparison to that of the sciences. This is mainly as a response to the Wigner paper noted earlier, which is why I don't think that my position is well grounded. I don't think I agree that it is purely based upon logic, so right now, I don't think the logicist heading completely applies here. I may have to look at it some more.

And as far as logicism being obvious, what makes you think that? If it was so obvious, why would the reduction of mathematics to logic require geniuses like Frege and Russell?

Did I say that logicism was obvious? At best I am only saying that the universe being logical is not something that most people would disagree with.

 

[John Creighto]

-I think it depends on the area of math. 1+1=2 is synthetic (see Kant), in that we are assigning definitions. (So Formalism)

-The natural numbers are a generalization of this synthetic knowledge extended to an infinate set.(Same as above)

-Geometry on the other hand represents the physical world. Primitively it depends on the physical world -- in that a triangle on a plane differs from a triangle on the surface of a sphere.

-Logic I think has a psychological origin but one which arose from evolution. Therefore while this is constructivist it is still is indirectly physism.

-Given all math is reducible to logic and logic is inductively learned though evolution then all math evolves through a process which learns something about the physical world.

-Now where does learning come from? Why are learning processes like evolution fundamental to the world?

-Finally consider that Godel's Incompleteness Theorems says we cannot prove a system of mathematics from the rules within the system, so how is it then that we come to learn about math because in order to learn we need some principle by which to evaluate the truth of what we learn. If this principle is induction then our enter justification for mathamatics is tautological.

 

[apeiron]

Even if the theorems of mathematics are derived from axioms that just have to be assumed (which is debatable, see Frege's work in logicism), isn't it still true that the fact that the axioms do logically imply the theorems is logically true?

But the rules of logic are just as axiomatic at base. They depend on certain critical assumptions, like the law of the excluded middle, which by definition may be true within the realm of our modelling, but not necessarily true of the world.

So he concluded that almost all of the mathematics we use is NOT grounded in our knowledge of physics.

Fictionalism just seems to be making the modelling point to me. Whether the concepts we employ seem more concrete, or more abstract, they are still in the end all just concepts - general ideas derived by inference from experience.

And I don't really take physism to be referring to physics - especially something so particular as Newtonian mechanics - but rather a codification of metaphysical concepts. So maths/logic is based on our fundamental categorisation of nature - general sharp ontological distinctions such as discrete~continuous, chance~necessity, substance~form, stasis~flux, etc.

So then then what is the rest of math based on? You mentioned that you think it could be intuitionism or formalism. Either way, how is it that the rest of matematics is able to stay consistent? Is it just a marvelous coincidence that we happened to be using a consistent system of mathematics, or do you allow the possibility that it's inconsistent and we haven't discovered it yet?

My point there was that maths has an intrinsic freedom which means it can be used to talk about real things, but also to talk about imaginary things. Just as language is free to talk about horses and unicorns.

So the same rules of syntax can carry the semantics from the realm of the real to the realm of the imaginary.

Mathematicians set up a machinery to generate patterns. Then they get busy discovering every pattern that can exist as a result of this machinery. Wolfram's exhaustive cataloguing of cellular automata is a good illustration here. Then some of these patterns are discovered to model reality in a useful way. And we feel tempted to believe this is because reality works in this way - although we can never in truth leap that epistemic divide.

Also, concerning formalism, how do you get around the Godel's theorem objection? But where do these constraints, that natural law must conform with, come from? You say that the constraints are necessary for reality to exist, but where does necessity itself come from?

The constraints of nature would be self-organised limits on nature's inherent dynamism. So whatever is stable in nature is emergent. The necessity of reality lies in its developmental history.

But the epistemic trick of maths/logic is to jump to the end of the story - to presume that what emerges is simply existent. So stasis is taken for granted. Limits actually "are" - whereas in nature, limits are precisely what are not. Limits mark the edge of reality.

It is the old debate about infinity. A naturalistic perspective - one that insists on descriptions that are material - will argue that infinity is a limit that can only be approached. But maths - inventing its Platonic realm of forms - just drops the requirement of a material means and takes the limit as something that exists.

Why is logical necessity not universe-dependent? Or do you admit that there is such a thing as prexisting logical truth, that is universe-independent?

What mathematicians call logical necessity, the universe would call historical inevitability.

The difference is that mathematicians presume they are unlimited in their pattern spinning - any possible pattern is also (within Platonia) an actual pattern. Whereas reality (as a mix of material and formal cause) erases the possible in developing into something actual.

The tricky thing at the centre of all this is that maths works so well in describing the patterns of reality because it does chop away the material limits of reality - Plato's chora. So the world is modeled in terms of forms, and the material aspects of the world are left unformalised as the separate business of making the measurements which might animate the models.

 

[apeiron]

-I think it depends on the area of math. 1+1=2 is synthetic (see Kant), in that we are assigning definitions. (So Formalism)

-The natural numbers are a generalization of this synthetic knowledge extended to an infinate set.(Same as above)

-Geometry on the other hand represents the physical world. Primitively it depends on the physical world -- in that a triangle on a plane differs from a triangle on the surface of a sphere.

-Logic I think has a psychological origin but one which arose from evolution. Therefore while this is constructivist it is still is indirectly physism.

-Given all math is reducible to logic and logic is inductively learned though evolution then all math evolves through a process which learns something about the physical world.

-Now where does learning come from? Why are learning processes like evolution fundamental to the world?

I agree that every -ism seems to apply to some degree. And I would argue that this is because each attempts to mark some definite philosophical boundary on our modelling of the world. We want to be either "completely this" or "completely that", when actually being so extreme is not possible. We must always remain within the boundaries that we can define.

So the actual task would be to narrow down the -ism spinning to its simplest division.

Platonism (our experiencing of form, reason, computation) certainly appears to be one of these limiting extremes. And then our particular material experience of the world seems to be the other.

Maths tries to divorce itself as much as possible from the material and the particular (so as to maximise its abstract generality). But then in doing so, it is defining itself just as much by what it is moving away from as what it is moving towards.

And thus all these -isms, all these attempts to say that maths is founded monadically on "one thing", seem to carry some truth. But look closer, and it is always going to involve this kind of epistemic manouevre. Thesis and antithesis. To become one thing, you have to also become not the other thing. And so every self must include its other.

Finally consider that Godel's Incompleteness Theorems says we cannot prove a system of mathematics from the rules within the system, so how is it then that we come to learn about math because in order to learn we need some principle by which to evaluate the truth of what we learn. If this principle is induction then our enter justification for mathamatics is tautological.

CS Peirce tried to fix this by arguing that abduction paves the way for induction and so, in turn, deduction.

So all that is actually needed to start the ball rolling is some kind of creative fluctuation, some random or spontaneous leap. A guess is good enough.

Although Peirce also pointed out that humans seem to make unreasonably good guesses. And so our actual starting point for reasoned thinking looks already highly evolved. Brains are natural generalising engines, and the formalised machinery of induction and deduction could emerge quite easily once humans developed the necessary syntactical machinery of speech.

 

[disregardthat]

Formalism/constructivism.

The other alternatives are ridiculous! First of all, mathematics is not reducible to logic, something which should be obvious for anyone who know euclidean geometry. Platonism doesn't make any sense, what would it mean if it was so? Physicsm is ambiguous, "based" in what way? I don't understand fictionalism, in what way does it contradict the others?

 

[apeiron]

I'm somewhere close to logicism and/or platonism, not unlike the views of Gottlob Frege and Bertrand Russell, in that I believe that the truths of mathematics are objective and absolute...

So what are your grounds for believing maths is objective rather than subjective?

As I say, I take the modelling approach where all knowledge is subjectively derived even if rationally structured. Reality may be "mathematical" and so our impressions of it will come to match if we observe closely enough, but there is always going to be an epistemic gap that means our knowledge is never actually objective.

And this stance in turn seems more consistent with our actual beliefs about triangles and other Platonic forms.

The essential attribute of a Platonic form is that it is perfect, absolute, eternal. And Platonists agree that the material world we are modelling is always imperfect. An actual triangle can never exactly match the ideal.

A Platonist responds by saying, well, if my ideal does not exist out there in the real world, then it must exist in some other dualistic realm - that happens to be objectively accessible to the human mind (in some way that doesn't get explained, although divine souls are historically invoked).

So that answer lacks commonsense.

On the other hand, the modelling approach would say a Platonic form is our idea of a material limit. A triangle is a model of perfection which stands as an absolute boundary on what can actually be.

And hey what do you know, out in the real world, material reality is giving the same answer. A triangle is a limit on what it can achieve. It is - as I mentioned - the very place that reality cannot arrive at. Perfection is exactly what lies that infinitesimal step outside what can exist.

So this view now seems like commonsense.

We model the world in terms of its limits - what cannot actually objectively be the case. And the world indeed does not have perfect triangles or anything else in the Platonic bestiary of ideal mathematical objects.

Platonic forms "don't exist" in our heads (even though we can treat limits as conceptual objects and give them names). And they don't exist out there either. So no metaphysical difficulties are raised.

 

[chiro]

I don't know if mathematics is a 'random' invention of the human mind, but what I do think is that it is going to most probable being realized given our linguistic abilities.

To me mathematics is like any language although it's focus and application is different.

Lots of people think that language and analysis are two separate things but they are not. When you define something very clearly, you have taken the necessary steps to analyze something and hopefully you are representing something in an optimal way.

The use of the spoken and written words help us do exactly the same things that we often do using mathematics, but the exception is the nature of the language as mathematics is not only broader in its scope, but also extremely precise and these two seemingly contradictory properties create something that is extroadinarily powerful.

To me mathematics as a whole field endeavors to do a few things: it tries to generalize the representation and thus extend the language, it tries to gauge some level of internal consistency within the language to bring clarity to its descriptive capacity, and it tries to create a way of looking at transformations in a general way so that one can build multiple perspectives on an otherwise single thing.

In addition to this, it ends up with these goals to create a language that is remarkable in terms of what can be encoded in only a few set of symbols: again this relates the idea of the breadth of what can described as well as the low information content that needs to describe something that is so broad in its definition.

Like any language, it has its exploratoy aspects where people explore mathematics and create problems and new language within the language just like poets and writers create poems, stories, and plays that create whole new themes, conflicts, and ways of expression and thought. I think it was Hardy that said that he was like a poet and in many ways I agree with him if that was the case.

Langauge is used to analyze, to create, to abstract, to solve problems, to express oneself and to communicate amongst other things and I think it does all of these better than most other languages.

 

[alt]

Is there an alternative missing ? Ie, 'Mathematics is a language'

I have often seen it stated here, even by long term, well respected mentors, that

'Mathematics is a language - like the French language for example'

I didn't vote because of this missing alternative which I think is quite a compelling one, but if I had to, would go for Physism.

 

[lugita15]

Is there an alternative missing ? Ie, 'Mathematics is a language'

I have often seen it stated here, even by long term, well respected mentors, that

'Mathematics is a language - like the French language for example'

That is just formalism. It is the view that mathematics is just a symbolic language made up by humans, and that it has no underlying significance or truth to it, except perhaps truth concerning the properties of the language itself. In my write-up in the OP, I discuss some problems with this philosophy; perhaps the most significant issue is Godel's theorem.

 

[Whovian]

Is there an alternative missing ? Ie, 'Mathematics is a language'

I have often seen it stated here, even by long term, well respected mentors, that

'Mathematics is a language - like the French language for example'

I didn't vote because of this missing alternative which I think is quite a compelling one, but if I had to, would go for Physism.

And, similarly, one might come up with a completely different way to express mathematics, which would be considered yet another language. I think the question here is about the concepts in mathematics, not how we express them.

 

[Pythagorean]

Well firstly, I agree that mathematics is a language (but a very accurate one that allows for things natural languages don't!)

but I agree with Whovian; we could ask the same kind of questions about natural language and the fact that "they're a language" isn't really an answer. The question is really whether our conceptual perspective of nature is "realistic".

In the case of mathematics though, a special case, I think it's a question of whether the axioms would still be true if it weren't for humans.

Our natural languages aren't explicitly axiomatic (but then again, I'm not sure if mathematics as a field is really explicitly axiomatic or if that is just an ideal or only applied to particular subfields, or what)

But... I think logical truth statements in philosophy are the comparable natural language version of axioms, though I'd think the subjectivity of natural language contaminates the axioms a bit.

 

[fuzzyfelt]

Nice thoughts in this thread.

 

[Paulibus]

As I write, this thread has 27 posts. I guess others will follow. This poll has the makings of a long story that, so far, illustrates very well a point I’ve mentioned in the recent “Ultimate Question...” thread: the discussions of philosophers don’t seem take much account of the progress of biology and palaeontology over the last hundred and fifty years or so.

To recapitulate briefly: It is now accepted knowledge, especially from evidence gathered over the last few decades, that we are one of several species of great apes that evolved in Africa over the last few million years. We are the species which, driven by the forcing hand of evolution, somehow acquired the ability and compulsion to invent and communicate with rapidly evolving languages. In the latest few evolutionary instants our drive to talk has led first to the invention of numbers and from this beginning to the evolution of mathematics. And by stimulating technology our ability to count and quantify has helped to elevate our numbers to our six or seven billion chattering individuals that now infest this planet. Think Facebook.

Acknowledging this success should play a part in selecting among the seven ‘isms’ offered in this poll. We now understand better than earlier disadvantaged folk what we are, and should take this into account when discussing what we do and why.

 

[lugita15]

1+1=2 is synthetic (see Kant), in that we are assigning definitions.

Frege wrote his famous book, The Foundations of Arithmetic, to refute Kant's view that arithmetic is synthetic. I find his arguments quite persuasive.

Geometry on the other hand represents the physical world. Primitively it depends on the physical world -- in that a triangle on a plane differs from a triangle on the surface of a sphere.

But you can define the different systems of geometry, Euclidean and non-Euclidean, axiomatically, without reference to the physical world.

Logic I think has a psychological origin but one which arose from evolution. Therefore while this is constructivist it is still is indirectly physism.

You can say that human use of logic, like human use of physics has a psychological origin. But do you really think the basic laws of logic are contingent upon human psychology or the laws of physics? For instance, what about the law that says a statement always implies itself? For instance, "If logic has a psychological origin, then logic has a psychological origin." Do you really think that this law could possibly be wrong, regardless of the universe in which we live or the nature of our psyche? Because if you're skeptical of that, you're questioning the very basis of argument itself.

Finally consider that Godel's Incompleteness Theorems says we cannot prove a system of mathematics from the rules within the system, so how is it then that we come to learn about math because in order to learn we need some principle by which to evaluate the truth of what we learn.

This is not an accurate characterization of Godel. Roughly speaking Godel's 1st theorem says that any sufficiently strong consistent formal system must be incomplete, and his 2nd theorem says that any sufficiently strong consistent formal system cannot prove its own consistency.

If this principle is induction then our enter justification for mathamatics is tautological

I have no idea what you're talking about here.

 

[apeiron]

For instance, "If logic has a psychological origin, then logic has a psychological origin." Do you really think that this law could possibly be wrong, regardless of the universe in which we live or the nature of our psyche?

I'd say John Creighto is correct that logic arises out of cognitive evolution. But then, as I say, the modern mind does its little trick of "taking the limit".

So the animal mind is quite happy to induce a general idea, such as a bell means the expectation of food, but the idea would not have the absolutism that we demand of logic.

So an animal would be thinking the equivalent of "the bell is as reliable a signal as possible". While the logician would be thinking either the bell is true or false.

The animal's expectation remains semantically bounded - so it is realistic. The logician switches the game to talking in terms of those bounds, so any subsequent utterances are now applies unreal labels to the world.

If you want to talk about the foundations of maths, this epistemic cut that maths/logic/semiotics makes is critical. It is the trick that moves you across the line into a world that is formally unlimited (where thought, logic, induction, whatever, is no longer materially bounded in the same way).

The grounds of maths/logic is untruth . It pretends the world is full of definite things. And that proves to be a very useful new cognitive trick.

 

[ThomasT]

I think it starts with the ability to perceive the presence of a definite, bounded something and the absence of that something, and the ability to perceive congruence and incongruence.

It's an emergent phenomenon of an emergent phenomenon ...

I voted for physism.

 

[bohm2]

I think there may be a specific innate capacity for acquiring mathematical knowledge but unlike intuitionism/constructivism I don't believe it's an arbitrary invention of the human mind/brain. Unless I'm misunderstanding constructivism.

 

[John Creighto]

But the epistemic trick of maths/logic is to jump to the end of the story - to presume that what emerges is simply existent. So stasis is taken for granted. Limits actually "are" - whereas in nature, limits are precisely what are not. Limits mark the edge of reality.

It is the old debate about infinity. A naturalistic perspective - one that insists on descriptions that are material - will argue that infinity is a limit that can only be approached. But maths - inventing its Platonic realm of forms - just drops the requirement of a material means and takes the limit as something that exists.

With regards to taking the limit, in section II of the introduction of Critique of Pure Reason (Translated by F. Max Muller) there is the following relevant quote.

"II.

We are in possession of certain Cognitions a priori,
and even the ordinary understanding is never without them.

All depends here on a criterion, by which we may safely distinguish between pure and empirical knowledge. Now experience teaches us, no doubt that something is so or so, but not that it cannot be different. First, then, if we have a proposition, which is thought, together with its necessity, we have a judgment a prior; and if, besides, it is not derived from any proposition, except such as is itself again considered as necessary, we have an absolutely a priori judgment. Secondly, experience never imparts to its judgments true or strict, but only assumed or relative universality (by means of induction), so that we ought always to say, so far as we have observed hitherto, there is no exception to this or that rule. If, therefore, a judgment is thought with strict universality, so that no exception is admitted as possible, it is not derived from experience, but valid absolutely a priori. Empirical universality, therefore, is only an arbitrary extension of a validity which applies to most cases, to one that applies to all:"

pg 25-26 of Basic writings of Kant, Edited and with an Introduction by Allen W. Wood, copyright 2001, ISBN: 0-375-75733-3

This quote is not found in the Gutenberg version which is available free online.
http://www.gutenberg.org/ebooks/4280

I warn anyone that Kant's writings are quite difficult to read and consequently I would not suggest him for an introduction to philosophy. If anyone wants to learn about the basic concepts of how we obtain knowledge about the world, I would suggest either Bertrand Russell’s, "The Problems of Philosophy" of Aristotle's "Metaphysics" as an Introduction. As an aside I here Hume is quite difficult as well.

 

[apeiron]

Empirical universality, therefore, is only an arbitrary extension of a validity which applies to most cases, to one that applies to all.

Yes, but here Kant is surely making the contrast with pure a priori universality? So the view he ends up taking is both related and subtly different.

My point here was that everyone recognises the underlying dichotomies at work - such as Kant's synthetic~analytic, or constitutive~regulative, distinctions. And people keep trying to force an either/or answer as to which is fundamental, instead of recognising how the answer is both/together.

This poll was set up as another prime example of that reductionist trope. Either maths has to be real or invented, rational or empirical, objective or subjective, etc.

Kant's answer on maths - that it is synthetic a priori - is in fact a powerful insight here.

The way I would describe it is that humans generalised their way to some ultimate abstractions such as the natural numbers and their fundamental operations. This was knowledge derived from experience of the world, and thus not a priori. But then there is that final step, that epistemic cut, which shifts us from an "imperfect" material world to the immaterial world of our imagination where we grant the unreal - the "in the limit" - a (Platonically) concrete reality. So now we are indeed dealing with analytic truth - what we deem to be just self-evident (having apparently "completely" eliminated the need for material foundations).

Then the genius bit. We start to synthesise with this "immaterial material" we have created. We can get going on constructing mathematical objects using numbers and their operations (or more broadly, structures and their morphisms). So truths become synthetic a prior - true by principles of constitutive judgement.

I say genius, but this semiotic trick was already discovered by nature. Genes and words are also symbolic means by which to construct states of regulative constraint. Logic, maths, computation, information theory, etc, are just taking this habit of nature to a higher level of abstraction and thus applicability.

So it is complicated. The material world creates material states of global regulative constraint via emergence. Then humans create immaterial descriptions of these global states. And from there, we use this mental material to construct immaterial worlds of our unlimited imagining. Then to complete the loop, we can measure our constructed worlds - our mathematical models - against the actual behaviour of the material world again.

So for instance, we give names to numbers, names to operations. A global concept like "many" is reduced to some particular actual Platonic thing, like 122,988.0879. These atomistic entities can then be combined by fixed rules such as "add" or "subtract". Then we can compare the behaviour of the model back to events in the world to show it is all "true" - that the trip into the realm of the rational, though the land of the analytic and synthetic a priori, maintained the empirical correspondence we ultimately must value (unless we are idealists or Platonists, I guess ).

Kant was concerned with further issues, like where our judgements on time and space came from - whether to force them into basket of the empirical or rational. These were a problem at the time because they were clearly general ideas, but ones that seemed to arise right down at the level of basic perception rather than loftier a-perception.

We now know enough about the evolution of the brain to see how those concepts are the result of earlier pre-human rounds of semiosis. They are biologically evolved abstractions written into the brain's architecture, whereas maths is a subsequent culturally evolved abstraction that gets learnt.

Abstraction is indeed about "taking the limit" - crossing the line from material emergence to immaterial reification. And the evolutionary view can show how this has been happening in steps, with the biggest jump being enabled by the human invention of syntactic language.

 

[lugita15]

Formalism/constructivism.

How would you respond to the criticisms of each of those philosophies I give in the OP?

First of all, mathematics is not reducible to logic, something which should be obvious for anyone who know euclidean geometry.

What bearing does Euclidean geometry have on the question of whether mathematics is reducible to logic? Perhaps you mean that it is a matter of physical observation what geometry our universe conforms to. That may be true, but both Euclidean and non-Euclidean geometry constitute internally valid systems. We may of course choose to work with either system, but both can be modeled perfectly using the real number system, which is based on rational numbers, which are based on natural numbers, which are based on ... (see Frege's short book The Foundations of Arithmetic for the rest).

Platonism doesn't make any sense, what would it mean if it was so?

What do you mean what would it mean? In the Platonist view, humans discover, not invent mathematical truth. So when we find out that there are infinitely many prime numbers, we are finding something out about something objective real, just as when we find out how many moons Jupiter has. In the traditional form, Platonism says that there is an abstract realm called Platonic heaven in which all the mathematical objects reside, like the number 9 and the perfect circle, and that through reason, intuition, or both (depending on your flavor of Platonism), humans are capable of finding out truth about the properties of Platonic heaven and what exactly is there. A more modern version of the philosophy says that although mathematical truth concerns a reality every bit as real as physical reality, there are no actual squares and triangles bouncing around in Platonic heaven. Thus in the modern view, it is the statements of mathematics, not mathematical objects, that correspond to the properties of mathematical reality.

Physicsm is ambiguous, "based" in what way?

Physism is the belief that mathematics consists of the patterns the physical world possesses, at least those patterns which humans have noticed so far. To put it another way, in this view mathematics is just the set of recurring properties of the world of that underlies the laws of physics. For instance, if humans see that particle motion conforms to Newton's 2nd law of motion, which is a second-order differential equation, then at the very least humans conclude, or perhaps hypothesize, that calculus, and thus the real number system, is part of whatever makes the universe tick. According to physism, if we lived in a physical universe that behaved differently, then mathematics would be different in that universe. Consequently, if our knowledge of the physical universe changes, the philosophy requires us to re-examine our supposed knowledge of mathematics. (Unlike Platonism, where mathematical truth is independent of our sensory experience of the world.)

This view conveniently solves the indispensability problem: why is it that an abstract subject like mathematics is so often applicable to the study of the physical universe, and why is mathematics so necessary to the formulation of the laws of physics? The response of physism is that this is so definitionally, because mathematics is defined to be that which is necessary to account for the behavior of the physical world.

The main problem with this is that most of mathematics doesn't seem to be grounded in our knowledge of physics. Numbers seem fairly well-grounded: you can count how many stars there are, or how tall a tree is. But what does category theory or Ramsey theory have to do with the physical world?

I don't understand fictionalism, in what way does it contradict the others?

In the context of the works of Arthur Conan Doyle, consider the statements "Sherlock Holmes lived on Baker Street" and "Sherlock Holmes lived on Sesame Street". In reality both statements are wrong, because Sherlock Holmes is nonexistent. But can't we still say that within the fictional world dreamt up by Arthur Conan Doyle, the first statement is right and the second is wrong? In the same way, Hartry field said that mathematics is just a fiction, albeit one that is often convenient. In his philosophy known as fictionalism, there are no such things as prime numbers, but we can still say that within the fictional world of mathematics, there are infinitely many prime numbers. It is in most direct contradiction with physism. Whereas physism says that mathematics is that which is indispensable to physics, Field said that NO mathematics is indispensable to physics, and that it is just a useful fiction we can discard at any time. He wrote a book, Science Without Numbers, which attempts to show how all the known laws of physics can be formulated without any mathematics whatsoever, in particular without real numbers or even natural numbers!