Aatu Koskensilta wrote: > Lee Rudolph wrote: [...]
>> 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.
I consulted the web site ``Earliest Known Uses of Some of the Words of Mathematics", for which I can't find the author.