FOM: arithemtic, geometry, projectibility, Frege

[Robert Black]

Mark Steiner:

>>E.g., don't you think that "addition on the integers is
>> commutative" is projectible? Or is it simply a temporary view of 20th
>> century thinkers that is subject to radical change in the future?
>
>	Believe it or not Frege discusses related questions.  He seems to argue
>that his version of f.o.m. renders this a projectible hypothesis, or
>more exactly, that without f.o.m. (meaning in his case the logicist
>program) it is not projectible, since each number is unique.  This is
>different, he says, from the situation in geometry, where the points of
>space are indistinguishable.

This sems to me - at least as most naturally interpreted - to be a pretty
odd reading of Frege. I take it you're talking about section 10 of the
_Grundlagen_, which is in the context of an attack on Mill's idea that
general arithmetical truths are known by induction.

Mill-bashing was an extremely popular sport in 19th-century German academic
philosophy, which prided itself on opposition to empiricism (often
asociated with materialism). [One suspects that the attacks on Mill are
often really disguised attacks on homegrown German empiricists or
materialists (particularly Buechner) who were regarded as too despicable to
be even named.] So Frege can be confident that his readers will agree with
him here anyway: arithmetic is to be a priori, so induction (and any modern
associated talk about projection) is going to be irrelevant.

As one of his arguments (not the only one) Frege uses the point that since
every number is different from every other, inductive conclusions about all
numbers won't be strongly supported. Frege quotes Leibniz but this also
harks back to Kant's view that since every number is different from every
other, arithmetic, unlike geometry, can't be based on general axioms. And
Frege does indeed make the point that 'every position in space and every
point in time is as good in itself as any other'. But he's here talking
about empirical induction of the 'all ravens, whenever and wherever, are
black' variety, certainly not about geometry. At least at the time of the
_Grundlagen_ Frege agreed with Kant that the truths of geometry aren't
known inductively either, but are synthetic a priori.



Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

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