Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties.

For example, in group theory you deal with groups; in topology, open sets; in analysis, the set of reals, etc.

Is there something more general which englobes all maths? Is this a too reductionist view?

Maybe I should replace set theory with category theory, but the same question holds.

share|improve this question

closed as primarily opinion-based by Michael Hoppe, Claude Leibovici, Yiorgos S. Smyrlis, Lucian, Your Ad Here Mar 4 at 11:14

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

2  
Set theory + logic + various definitions, maybe. –  naslundx Mar 4 at 9:28
    
See also this post –  Mauro ALLEGRANZA Mar 4 at 10:07

2 Answers 2

I disagree that most branches of mathematics are just an application of set theory and logic. The fact that most areas of mathematics use set related notions and employ logic does not mean they are applications of these areas.

For instance, would you say that English Literature = English Words + English Grammar? After all, every piece of English literature uses English words and follows (more or less) the English grammar.

There are many views on what mathematics is. One school of thought on the subject is called formalism and it (more or less) asserts that mathematics is devoid of any meaning and everything is just formal manipulations on a piece of paper. There is no doubt that this is a factually true statement, in the sense that all our proofs are formal manipulations on a piece of paper that follow some very strict rules. But, there is (to me at least) no doubt that such a point-of-view is devoid of any relevance to the working mathematician. A mathematician never wakes up in the morning and says "let's choose some axioms at random and see what I can prove from them". Instead, a mathematician has a plan. A program. Something to achieve. That something transcends the formalism. There is a reason why we study the algebraic structures we do and not the endless other examples of algebraic systems that follows any set of arbitrary axioms. There is a reason why we study certain objects like $\mathbb R ^n$ or $\ell _p$ and not a trillion (and more) other arbitrary sets that can be concocted in a myriad of nonsensical ways. What gives sense to the mathematics we care about can't be attributed to set theory and logic. Neither set theory nor logic tell us how to find interesting axioms to study.

The answer does not change if you replace set theory by category theory.

share|improve this answer
2  
Very well written! –  Martin Brandenburg Mar 4 at 16:16

No.

For example, if we spend the time to formalize group theory in a set theory (take ZFC as an example), then by the end of this long and arduous exercise, we probably haven't learned a whole lot of new concepts in group theory. We've just worked out how to implement the old concepts in ZFC. Furthermore, we're probably no better at discovering new group-theoretic ideas than we were when we started the whole enterprise. In fact, we might be worse, since now we've (potentially) allowed ourselves to get stuck in the ZFC way of doing things.

The same critique can be directed at any approach to founding mathematics; since its not the only way of doing things, we have to be careful not to let our creativity be stifled by just focusing on that one particular approach. This does not mean we should avoid foundations; rather it means we should always be open to new foundational ideas.

By the way, the main application of set theory to (basic) group theory is that you can actually prove the existence of entities that we take for granted, like Cartesian products of groups. Their existence follows from the existence of Cartesian products of sets, which is a basic result of set theory. (Actually, some approaches take the existence of a Cartesian product of sets as an axiom).

Anyway, the point is this. Just because we can formalize a lot of mathematics using set theory + logic, this does not mean that mathematics $=$ set theory + logic.

share|improve this answer
1  
(+1) but I have to disagree with the claim that the main application of set theory to group theory (basic or not) is, e.g., the existence of products and the like. These are trivialities. An example of an application of set theory to basic group theory would be the existence of certain maximal subgroups by employing Zorn's Lemma. An application to advanced group theory would be any of a multitude of results that construct a certain counter example to a group theory claim by means of, e.g., large cardinals. –  Ittay Weiss Mar 4 at 11:05

Not the answer you're looking for? Browse other questions tagged or ask your own question.