Lee Rudolph wrote: > Aatu Koskensilta <email@example.com> writes: > >> My remarks are based on somewhat more mundane considerations; in group >> theory one often says things like "let A be an Abelian group such that >> ..." but in number theory one does not say "let <W,s,f,g> be a structure >> in which all the arithmetical consequences of ZFC hold" or "let >> <W,s,f,g> be a natural number structure in which the twin prime >> conjecture holds". > > I agree (based on observations of number theorists) that "one does > not say" that sort of thing, and I am open to being persuaded (indeed, > I am predisposed to be persuaded) that if a school of mathematicians > got into the habit of saying that sort of thing (while continuing to > do mathematics) then we (and possibly they) might want to say that > what they were doing (though still mathematics, and possibly very > fine mathematics) was no longer "number theory" (or, weaker, no > longer *just* "number theory"): but I don't see such a response > is self-evidently right.
That certainly would be the natural reaction. In fact, there already is a discipline of mathematics where one can expect to hear such things. No one calls it "number theory". It seems highly unlikely that there ever could be a school of mathematics in which studying e.g. structures in which the arithmetical consequences of ZFC hold was called "number theory". That sort of a terminological shift would require a major upheaval in the way people think about natural numbers.
> In other words (I guess), is there anything more than historical > chance and prejudice behind the feeling (which I certainly share, > but don't feel particularly justified in sharing) that "natural > numbers should be *categorical*, dammit"?
It's a basic property of our conception of the natural numbers that they don't bifurcate into a multitude of non-isomorphic structures. There are some people of ultra-intuitionist and ultra-finitist peruasion - Esenin-Volpin and Edward Nelson come to mind - who do think that there are many different natural number lines and reject the ordinary conception of the natural numbers as incoherent or unjustified. This just goes to show, once again, that it is to a large extent a matter of personal preference and inclination what one finds convincing and coherent, and that even in mathematics there probably is no principle someone competent hasn't doubted or rejected.
-- Aatu Koskensilta (firstname.lastname@example.org)
"Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus