Re: Set Theory: Should you believe?

Norman Wildberger wrote:
I have posted an article at http://web.maths.unsw.edu.au/~norman/views.htm
that has caused a bit of discussion in some logic circles.

My claims in short: 1) most of `elementary mathematics' is not sufficiently well understood by the mathematical establishment, leading to weaknesses in K12 and college curriculum, 2) the current theory of `real numbers' is a joke, and sidesteps the crucial issue of understanding the computational specification of the continuum, and 3) `infinite sets' are a metaphysical concept, and unnecessary for correct mathematics.

Analysts and set theorists are welcome to send me reasoned responses.

Assoc Prof N J Wildberger
School of Maths
UNSW

After casually reading his notes, I think that he is saying this. The axioms of modern set theory are too burdensome for actual mathematicians, who in practice take a somewhat Platonic view of their subject. But he is unsatisfied with the pragmatic Platonist approach that we take, particular in the manner in which mathematics is taught at pre-college levels. He thinks we need to roll up our sleeves and rethink foundations so that we get something that really is usuable, so that we can finally truly rid ourselves of Platonism, superstition, instinct, gut reaction, religion, etc, etc.

Personally I really like the Platonic approach, using set theory as a highly convenient crutch. I'm not going to preclude the possibility that one day a set of foundations for mathematics will be found, that will greatly simplify and advance the extent that we will be able to think about mathematics, but I think it will take a great genius, and also some crisis of cicumstance (perhaps the discovery of some horribly unresolvable contradiction).

But on the whole, even if his tone was not exactly politic, I liked a lot of what he said. But I don't think it is going to change the world.

Stephen
.

Relevant Pages

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