guenther vonKnakspot wrote: > Stephen Montgomery-Smith wrote: > >>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 > > > Hi, without any intention of antagonizing you, let me ask you this > question: What of what he said did you like and for what reasons? > I would like to understand what the appeal of such a rant as > Wildberger's is. In my view, it is evidently wrong in many parts, wrong > under light scrutiny in many others and falacious for the most part of > the rest. The only statement I can think of which is undebatable is the > fact that education in mathematics is ever more deficient. But this is > also strongly suggested by a lot of other factors, as the ever growing > number of crackpots, the ever more present wrong notion that > mathematics is dependent on computability and the expanding belief that > mathematics is somehow subjected to the constrains of physical reality. > Thanks in advance for any effort you may put into your answer. > Regards. >
Well let me start by saying that I am only stating an opinion so there is no need to get upset or mad about it. I do admit that I didn't deeply scrutinize his writings, but I didn't feel that he said anything mathematically false. So it really all boils down to feelings about the current state of mathematics.
Next, while he was definitely ranting, and perhaps venting a certain anger he also feels, I think it is important to listen to the message rather than the messenger or the style in which the message is communicated.
There is a clear distinction between someone who expresses discomfort with the present way we do mathematics which is at worst an unpopular view, and saying that Cantor's diagonal argument is logically flawed which at best is a crackpot position to hold. This web site is clearly in the former category.
The view that mathematics should try to constrain itself to physical reality is, in my opinion, not a crackpot position. I think this is what Morris Kline was getting at in his books "Mathematics: Loss of Certainty." Attempts to place axiomatic number theory and axiomatic set theory on firmer ground by proving its consistency starting with a much weaker system are doomed to failure by Goedel's Theorem, and thus the possibility that the whole thing is built on a house of cards becomes ever more reasonable. I agree with Kline that the strongest evidence for the consistency of these theories, or at least some weaker version which would still contain most of our important discoveries, is just how very well it seems to work in practice.
Let me use an example. The mathematicians Wiener and Banach lived at about the same time. Banach chose a much more abstract direction to take his mathematics, looking at general properties of infinite dimensional normed vector spaces beyond Hilbert spaces and spaces of continuous or measurable function. Wiener looked more to real life applications, and building on the work of Einstein put the phenominum of Brownian motion onto a rigourous mathematical foundation. Many very talented mathematicians have studied Banach spaces, but it has proven to be a remarkably barren field. On the other hand Wiener's approach has led to the rich theory of stochastic calculus, which has proved extremely useful in studing the heat equation, harmonic functions, and even economics.