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:
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 im 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

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

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... http://www.physicsforums.com/showpos...&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

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/F...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 [itex]\dot{f}[/itex] or [itex]f\prime[/itex] instead of [itex]\dfrac{\mathrm{d}f}{\mathrm{d}x}[/itex]. I'm talking about a completely different sort of math.

lugita15

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

lugita15

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.)
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

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

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

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.

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.