

Set Theory: Should You Believe? - dood
http://web.maths.unsw.edu.au/~norman/views2.htm

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arvid
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.

~~~
BrandonM
_...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).

~~~
arvid
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.

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BrandonM
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.

