I am trying to decide whether to study the foundations of mathematics i.e. formal logic, Godel's Incompleteness Theorem, set theory, etc.(adsbygoogle = window.adsbygoogle || []).push({});

The reason is that recently I have been taking advanced math courses such as topology, analysis, and abstract algebra and I am fascinated by this seemingly infallible system of logic that mathematicians have developed. There isalwaysa right answer in mathematics (at least for my homework problems). I love this concept of correctness and things being true or false.

As I get deeper into advanced math, I find that almost everything is constructed or proved or defined unlike in high school calculus for example where theorems are just given to you. So, I keep uncovering that there are actually reasons why the things I learned are true. So, now I have started to ask for the proof of everything I have learned and that has gotten me into the foundations of mathematics. I have realized that even though it appears that Rudin goes back and rigorously proves everything I learned in calculus, that is not really true since Rudin like most authors implicitly assumes tons of very subtle things about set theory that seem intuitive but really should come from somewhere in a system of formal logic. So, there is probably another textbook that goes back and rigorously proves everything Rudin assumes from even more basic principles. And so on.

But where does it end? That is what throttles my imagination! At some point there must a book that says, "We assume nothing. We lay down an ω-consistent recursive class κ of formulas and prove everythingformallywe need from our own axioms." I am debating whether I should actually try to understand what an "ω-consistent recursive class κ of formulas" and how everything I have learned in mathematics comes from one. The benefit would be that my appetite for logic would be satisfied i.e. I could classify much of my mathematical knowledge into theorems and axioms and I would find proofs of theorems that rely only on formal logic not at all on intuition (like when Munkres says "clearly these two curves are homotopic" or "we shall assume that what is meant by a set is intuitively clear").

One of the things that made me consider studying foundations was http://en.wikipedia.org/wiki/Mizar_system. I was shocked that a computer could actually understand abstract mathematical proofs and at first that drove me away from mathematics but eventually I came back wanting to understand things in the same way a computer does. It is hard to explain why I would want to do that, but having things be consistent in my brain is just something I like a lot.

So, anyway, I wanted someone who has studied foundations to tell me whether it has actually made them more or less impressed with mathematics and whether they think it is something worthwhile for me to do. Part of me thinks that it will be just a massive waste of time because that kind of math has no real applications and maybe foundations is just like the "dirty work" that some mathematician has to do but no one wants to because it is just so unelegant and tedious.

I don't know...I am really just making this thread for people to discuss their choice to study or to not study the foundations of mathematics.

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# Foundations of mathematics

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