Wednesday, September 08, 2010

Euler’s ‘2’ postscript

Since our reductionists (see previous post) assume that we can refer to abstract objects, let us see if any of the ways in which such reference might occur support their view of the referent of Euler’s ‘2’. It seems not; for suppose, for example, that reference to abstract objects is correctly described by Fictionalism. Then the view in question is like taking most twentieth century utterances of ‘Sherlock Holmes’ to refer to the character recently played by Benedict Cumberbatch on the BBC. Suppose instead that Gödelian platonism is true, so that we have something like a perception of abstract objects. Then the only choice we should make in our reference to them is the choice of their names. Between those two possibilities lies a Full-Blooded platonism, according to which all possible abstract objects exist. But that position is hardly available to those who don’t want—but don’t consider impossible—non-set-theoretic numbers. So in conclusion, it seems that our reductionists are quite eccentric after all.


Xamuel said...

The way I sleep at night is I push the bubble a little further down the wallpaper. The way I see it, I'm not studying objects at all, but proofs from a formal set of axioms, and rather than believing that 2*3=6, I believe that I've found a proof of 2*3=6. In other words, treat everything as syntax rather than semantics. If it turns out ZFC is inconsistent and sets as we know them don't exist, fine with me, the derivations from ZFC still exist (and in that case I'd sure love to be the first person to derive 1=0).

Of course, the bubble cannot be removed, the new questions are, what are formulas and what are proofs and what are derivations...

enigMan said...

But what does "two threes are six" mean? The importance of a proof of something usually lies in the meaning of that thing. And numbers have an obvious meaning and importance. If ZFC was inconsistent, then the derivations would still exist, but so would numbers. And surely mathematics is primarily concerned with numbers, not formal proofs. Surely it is an error to think otherwise; and what if ZFC is consistent but false of the numbers?

But it is interesting that even if we avoid the question of what numbers are, we have such questions as your last ones. I'm reminded of how we get proper classes even if we ignore the possibility of the natural numbers being collectively potential infinite.