FOM: arithemtic, geometry, projectibility, Frege
Mark Steiner
marksa at vms.huji.ac.il
Tue Jan 18 16:52:58 EST 2000
Robert Black wrote:
>
> 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.
>
Let me use the term "nondeductive" to mean induction of the All ravens
are black variety.
I understand Frege as separating the question of whether arithmetic is
empirical from the question whether nondeductive support is possible in
arithmetic, as he should. His conclusion is, that without his f.o.m.
type treatment of arithmetic, nondeductive ("inductive") confirmation
for arithmetic hypotheses is impossible because the numbers do not form
a "natural kind" without such treatment. This is true whether
arithmetic is empirical or not. Given the treatment of the Grundlagen,
however, which of course is supposed to be deductive and a priori, there
is no reason why mathematicians could not use nondeductive arguments
(like Euler's) to become persuaded that certain mathematical hypotheses
are correct, since the numbers, having been defined mathematically, then
become "natural kinds" (to some extent). Clearly, however, Frege
regards formal proof as the goal in mathematics, as he says explicitly.
Nondeductive arguments in mathematics can be regarded, by the way, as a
priori support, at least according to the meaning of a priori as "not
grounded in sense perception", since the "data" for such nondeductive
(or even probibalistic) arguments are not empirical--according to
Frege. (To complicate matters, though, I don't think that that's the
way Frege himself defined a priori.)
Separating the question of whether arithmetic is empirical from the
question of nondeductive support in mathematics, indeed allows him to
"bash" Mill, as you correctly point out.
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