FOM: arithmetic, geometry, projectibility, Frege

Robert Black Robert.Black at
Wed Jan 19 06:03:38 EST 2000

Mark Steiner:

>	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.

I agree that mathematical conjectures can have nondeductive support, as in
the Euler example discussed in your book. (Though I think there's a
difficulty about just how we should conceptualize this, which I'll come to
in a moment.) However, I don't think Frege would (or should) have been
happy with this, given (what I take to be) his views on inductive reasoning.

If you're in the rationalist-foundationalist tradition in epistemology,
then the most obvious way (perhaps the only way) to give an account of
inductive reasoning is that it's really *deductive* reasoning whose
conclusion is a statement about probabilities, of the general form:

it is probable to degree x that q.

Note here that the THEREFORE is logical entailment, and the number x is
part of the conclusion, not adverbial on the THEREFORE. The premises may
*or may not* involve probabilities. This was a fairly general view at the
time Frege was writing. I seem to remember there's a particularly clear
statement of it in Brentano, for example, and traces of it persist all the
way through to Carnap. I think that what Frege says at the end of section
10 of the _Grundlagen_ about induction being based on the theory of
probability (and his rejection of a Humean-empiricist account of induction)
is a pretty good indication that it was Frege's view.

Now if q in the schema above is a mathematical conjecture, and if you're a
logicist, then that conjecture is either a logical truth or a logical
falsehood, either it or its negation is entailed by the premises, and so
the only available values for 'x' in the above schema are 1 and 0. Thus the
rationalist picture of inductive reasoning combined with logicism rules out
correct but genuinely inductive arguments in mathematics.

It's worth adding here that even if you're neither a rationalist nor a
logicist there's still a remaining problem about how to conceptualize
inductive arguments in mathematics because at least in cases where we don't
take seriously the idea that a conjecture might be logically independent of
our axioms the usual idealization of probabilistic thinking which assigns
probability 1 to all logical truths runs straight up against the same

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

tel. 0115-951 5845
home tel. 0115-947 5468
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