Philosophy of Mathematics

Swapnil

I was wondering, what kinds of question does one asks in philosophy of mathematics (by mathematics, I mean like algebra, calculus, arithmetic, geometry, not logic though).

These are the only one I can think of:
1) "Why does math describe the physical world so elegently? Is there an intricate connection between nature and mathematics?"
2) "Do numbers exist?"
3) "Is math invented or discovered?"

Can you guys think of any other interesting questions?

Swapnil

Thanks but I am not particularly looking for resposne to the questions I asked. I just wanna know what types of questions are raised in philosophy of mathematics and get a jist of what philosophy of mathematics is all about.

selfAdjoint

Another question philosophers ask about mathematics is "In what sense are mathematical results true" aka "What is mathematical truth?"

Also note that your questions 2 and 3 are so closely related that some might consider them equivalent. If numbers exist independently of human minds, then so do facts about them, and these facts would have to be discovered, not invented. Going the other way, things that are discovered, not invented have an existence independently of human minds (No?) therefore if mathematical idea are d not E then they are independently existing, and in particular then numbers are independently existing.

neurocomp2003

what is computable. what is a language/grammar

NickJ

More technical philosophers of mathematics -- those who address "foundational issues" -- also debate things like how to understand the set-theoretic hierarchy (is it iterative?).

And another general question: what entitles one to accept mathematical axioms?

Also, among those who accept the existence of numbers, there are questions about the nature of that existence: are numbers just abstract structures or are they more like Platonic forms?

And, in an overlap with philosophers of science, some debate whether mathematics is dispensible to the practice of science (are there any results in science that we could not obtain IN PRINCIPLE without using mathematics, or is the use of mathematics just a convenient shortcut). (Indispensibility would be a good reason to think that numbers exist.)

Swapnil

So is there a specific branch of philosophy that focuses more on nature of different types of number. For example, complex numbers, infinitesimals, infinity, negative numbers, fractions etc. I would really like to know how philosophers justify the existence of these numbers.

selfAdjoint

Somebody in the nineteenth century (Kummer?) said, "God made the whole numbers; all else is the handiwork of humanity".

Then the set theorists of the early twentieth century derived the whole numbers from the axioms of set theory.

All the different kinds of numbers you cite are easy to model starting from pairs of whole numbers for the rationals, going on to simple matrices and taking limits. There is no great philosophical issue about them as such. Of course the enterprise to describe arithmetic by set theory runs into Goedel's great critique which is a mightly subject in philosophy.

HallsofIvy

That just muddies the water- any philosopher of mathematics or everyday mathematician know that one does not "accept mathematical axioms". Every statement in mathematics is of the form "If these axioms are true, then ....". The "acceptance" of axioms is purely hypothetical.

Pythagorean

I don't have another question, but I think for number 3) the answer is a topic of debate. I think the periodic table implies that math was discovered.

selfAdjoint

O. Lismahago

“A mother tells her infant that two and two make four, the child remembers the proposition, and is able to count four to all the purposes of life, till the course of his education brings him among philosophers, who fright him from his former knowledge by telling him that four is a certain aggregate of unites; that all numbers being only the repetition of an unite, which, though not a number itself, is the parent, root, or original of all number, four is the denomination assigned to a certain number of such repetitions. The only danger is, lest, when he first hears theses dreadful sounds, the pupil should run away; if he has but the courage to stay till the conclusion, he will find that, when speculation has done its worst, two and two still make four.”

- Samuel Johnson, The Idler, No. 36. Saturday, 23 December 1758.

moving finger

yes, I have another question.

If a natural number is defined as a number which is generated by adding 1 to itself a finite number of times (as a natural number is indeed defined in number theory), how can the cardinality of the set of natural numbers be infinite?

Best Regards

Hurkyl

The relevant proof by contradiction is a rather trivial exercise. (assume the cardinality is finite. Then there is largest natural number. Add 1 to it) So I think the more pertinent question is "why do you think it shouldn't be infinite?"

moving finger

It is not that I think it should not be infinite - it is that I think an infinite set of natural numbers is inconsistent with the notion that all natural numbers are finite.

This "proof by contradiction" shows that the cardinality of the set must be infinite - but it does not follow from this that such a cardinality is consistent with the definition of a natural number as "add 1 to itself a finite number of times".

We can arrange the complete set of natural numbers in ascending order, from 1 upwards.
If every natural number is finite then the total number of numbers in our set (the number of members of the set) must also be a finite number (why? because if the set is complete then the number of members of the set will be numerically equal to the largest member of the set).
How can the cardinality of the set be infinite if it contains a finite number of members?

Best Regards

Pythagorean

sorry, not the periodic table itself, but the way elements aline so well. Our fundamental building blocks are linear. We have an element with with one electron/proton, we have one with two, three, four, etc, etc.

When I first learned that in chemistry in high school, I was very amazed at the order.

I don't know if you can ever really answer the question "was math discovered or invented" without opining, so I'm not trying to prove that it was discovered, just explaining why I have my opinion.