Stephen Montgomery-Smith wrote: > Gerry Myerson wrote: > > In article <firstname.lastname@example.org>, > > "Gene Ward Smith" <email@example.com> wrote: > > > > > >>Gerry Myerson wrote: > >> > >> > >>>I personally don't put set theory in the same category as astrology > >>>or creation science. Maybe Norm does. I don't know. > >> > >>Norm apparently puts number theory in that category. > > > > > > For what it's worth, Norm has written papers in number theory, > > e.g., > > > > MR1314396 (96a:11029) Wildberger, N. J. Row-reduction and invariants of > > Diophantine equations. Proc. Indian Acad. Sci. Math. Sci. 104 (1994), > > no. 3, 549--555. > > > > Whether this was before he came to his current views on set theory, > > I do not know. > > This affirms what I felt about the OP, that while his views are > unpopular, he is not a crank. (I also just checked him out on > MathSciNet, and it is obvious that he is an active research > mathematician.) In my opinion, his remarks about the natural numbers > are spot on. The only way I disagree with him is that I don't think > that now is the time to work on these problems. > > This whole thread has left a bad taste in my mouth because of the speed > at which people were willing to spew abuse on him. Just because he has > a superficial resemblence to crankpot anti-Cantorians doesn't make him > one. His views are controversial, but that doesn't mean we respond > rudely, rather it gives us an opportunity to engage in civilized > conversation. Even if we end up still disagreeing, we might have > learned something, or at least honed our arguments better. > > Now, one of the respondents said that Norm had rejected the axiom of > infinity, and so how is he supposed to do number theory.
It is quite easy to do number theory without the axiom of infinity. If you replace the axiom of infinity with its negation, you get a system that is bi-interpretable with PA. I don't know whether Norman would be happy with PA as a foundation. (I've been rather frustrated that Norman doesn't seem to be willing to specify which systems he would accept as good foundations).
> But nowhere in > his article has he rejected the axiom of infinity. He has rejected the > whole notion that axioms are the way to go. I would paraphrase what he > said slightly differently - axioms (might) describe the natural numbers, > but they don't define them. There is clearly a problem that there are > numbers between 1 and googolplex that we can never write down or > meaningfully describe. What makes us think that they are really there? > The evidence is at best empirical. > > The responses have led me to think that some people here believe in the > modern axiomatic system like a religion. > > Stephen