FOM: arbitrary objects

William Tait wwtx at
Thu Jan 31 14:41:43 EST 2002

In a paper ``Finitism'' (Journal of Philosophy 1981) I took the
notion of an arbitrary number as basic for understanding what is
finite in a function F : N arrow N or in a proof of a general proposition
such as forall x, y [x+y = y+x].  I hesitate to mention this now,
since I haven't time to get into a discussion of the merits of my
analysis. (There is a bit more in a paper ``Remarks on finitism''
soon to appear in the collection of essays _Reflections_. But I think
there is still more to be said.)

Arnon Avron  wrote

>I should say that I find the discussion about "arbitrary objects"
>(or "arbitrary numbers") rather embarrassing, especially that
>it is made by logicans. When I read it I got the feeling that
>Gentzen (and his analysis of Natural Deduction) and Tarski
>(with his semantical analysis of formulas, using structures
>and assignments) had never existed, and that
>variables and their correct use are still a mystery...

Maybe one should be embarrassed by talk of arbitrary objects only if 
one takes that notion to explain the meaning of variables. For 
example, that the notion of an indeterminate (and the pre-Cohen 
notion of a generic object) may be understood in terms of free 
variables does not mean that there may not be another way to 
understand them that throws a different kind of light. (On the other 
hand, I don't know Kit Fine's work on this subject.)

Bill Tait

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