As a former mathematician, I can understand what he writes. Mathematically, it is basically a lot of ranting and raving over nothing. Set theory provides a foundation for mathematics. It is not the only foundation. Category theory is another. Pedagogically, his argument makes more sense. I think he is mostly arguing against using set theory in the teaching of math. True most mathematicians do not use set theory and mathematicians studied math for years without the concept of a set. Set theory is more used as a language and for that reason it is taught. In this sense, set theory is like turing machines and turing equivalence. Important concept, yes. Good to know, yes. Used by most in comp sci, no. Necessary, no.
...set theory is like turing machines and turing equivalence. Important concept, yes. Good to know, yes. Used by most in comp sci, no. Necessary, no.
I have to disagree with this position in regards to Turing machines. This is because the definition of Turing machines leads directly to the problem of P vs. NP. Within the last month, I came across a problem that turned out to be NP-complete, a problem which also involved a very large data set (coincidentally, the problem was set cover). Had I not been aware of Turing machines and the P vs. NP problem, I could have spent hours implementing an algorithm which turned out to be unusable on the large data set I was working with. Instead, I knew that it would be necessary to find a reasonable approximation to the solution using some sort of greedy algorithm, and the piece of software I wrote actually turned out to be useful to the project we were working on.
I think that is one thing that really appeals to me about Computer Science. Nearly every core course I have taken has helped me as a programmer and as a computational thinker. I think this is part of the original author's problem with Mathematics education, that so much is learned and discarded, or simply taken for granted without being understood. Everything I have learned thus far in my CS education has been conceptually understandable (admittedly with a little work at times) and has helped me in some way (well, except maybe for that one COBOL course I took).
and set theory leads to the fact that the axiom of choice is independent and unprovable. But most mathematicians accept that axiom of choice is true. The same is true for the continuum hypothesis with a smaller majority accepting that as true. Similarly most in comp sci accept that P does not equal NP. One does not need to understand turing machines to understand the concept of time and memory for computers. It is just a way of formalizing the problem. Nice to know that the problem can be formalized but not necessary to be taught. I never said that turing machines where not important. But I am equally sure that you did not formalize your algorithm into a turing machine to show that it was np-complete. The author's point is that the same is true for set theory although he likes to phrase it in a very provoking way that I wholly don't agree with. Mathematicians use the general concepts of set theory not the formal one. Comp sci/programing uses the general concepts of turing machines not the formalizations.
I hesitated to post this rant, as I'm far from a mathematician by training, and I'm certain to expose my ignorance by commenting further. That said, it touches on some of the stuff that bothered me about maths around the time I stopped seriously engaging it. That is, axioms as assumptions, and playing games with what appear to be arbitrary symbols (e.g. infinite sets). For me, this seems to stray from a sort of pure, meaningful mathematics I was initially attracted to, and those objections were never met with anything but bluster and appeals to authority.
I'm glad I read that article. Do any mathematicians here (or at least math majors) have any comment on it? I don't know much pure mathematics, myself, but the author makes some very interesting assertions and claims which have piqued my interest.
