FOM: arithmetic, geometry, projectibility, Frege
Mark Steiner
marksa at vms.huji.ac.il
Wed Jan 19 09:03:03 EST 2000
> 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:
>
> p_1
> p_2
> .............
> p_n
> THEREFORE
> 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
> problem.
>
Frege begins 10 by asking, "But might they [arithmetic truths]
be
inductive truths nevertheless? I take this to mean that he is
discussing the question of nondeductive reasoning in
mathematics--independently of Mill's view that arithmetic is empirical.
His answer is that induction cannot be performed where the domain is not
"uniform", which the natural numbers are not [he quotes Leibniz here],
particularly if you don't give them the kind of definition that Frege
wants to do. Once the definition is in, we can then start proving
things about the natural numbers from their definition. These facts can
influence the mathematician's degree of belief in as yet unproved
propositions. The common view (among philosophers, such as my teacher,
Carl Hempel) that nondeductive support of mathematical propositions is
impossible, using examples like Fermat's conjecture F(n), is fallacious
because the Fermat case seemed to involve the problem that Leibniz and
Frege were pointing to, namely the individual natures of the natural
numbers. Other examples, such as the Euler case in my book, do not
involve the individual natures of the natural numbers--and even where
Euler's arguments involved the natural numbers, such as when he
calculated to 100 decimal places the sum of 1/n^2 and also pi^2/6, the
individuality of the numbers does not come into question. And Harvey
has already mentioned the use of probabilistic "proofs" in mathematics
today. [Naturally, everybody realises that these are not proofs, but
the question here is whether you can get nondeductive support for
mathematical propositions falling short of proof.]
Obviously, to get an idealization of such nondeductive
reasoning, we
have (a) to distinguish between objective and subjective probability;
(b) lay down that the set of probability 1 propositions is not closed
under mathematical deduction. The present day models even for
subjective probability were not constructed to deal with mathematical
reasoning itself, but are applications to empirical questions, just as
you say. I believe that Prof. Haim Gaifman of Columbia has tried to
construct models of subjective probability that are not closed for the
very purposes of explaining the mode of rationality behind probabilistic
arguments.
I see no reason why a rationalist and a logicist could not
accept the
concept of nondeductive support alluded to here. A rationalist might
accept that, for example, the Riemann hypothesis is probably true, given
the different kinds of considerations that mathematicians have brought
forth to argue that it is, and that it is much more rational to try to
prove it rather than trying to refute it. And that this notion of
"rationality," is universal. Though most people would assign such a
view, that nondeductive support of a priori truths is possible, to the
empiricists. And in fact, the only philosopher I know who alluded to
the use of subjective probability arguments in a priori contexts was
David Hume (Treatise on Human Nature, Book I, Part IV, sec. 1)--he tries
to measure the probability of a PROVED mathematical statement, given
that we might have made a mistake in the proof! This is a common sort
of reasoning that has to be done by editors of mathematics journals, to
decide whether to publish a proof or not after it has been checked by
referees. Yet Hume did not succeed, and there is no model of
probability today to fit this kind of reasoning (even though it is
crucial to people's careers).
What Frege says at the end of 10 is again a piece of Mill
bashing--that
is, the idea that ALL arithmetic truth is COMPLETELY inductive is
circular, since inductive reasoning does not depend on "habit" (this is
Hume bashing) but on the theory of probability. And the theory of
probability is itself based on arithmetic.
What about the view which is still open, given even the truth of
logicism, and Rationalism, that inductive mathematical reasoning
(perhaps falling short of knowledge but perhaps sometimes not) is
possible? Frege says nothing directly about this, since his point is
polemical and negative. I must agree with Robert that (a) Frege
probably thought that all true arithmetic statements are provable in his
system (since it was inconsistent, they are); (b) Frege did not have a
mathematical model of probability which could assign a probability
neither zero nor one to a mathematical statement. Nobody did.
On the other hand, Frege was a mathematician (as Prof.
Benacerraf says,
we philosophers tend to forget this), and everybody in mathematics
engages in nondeductive reasoning, even if proof happens to be the
goal. Frege surely knew about Euler's results, and the mode of his
reasoning. He knew that you can rely pretty much on Euler's
nondeductive results (a mathematician told me who went through them that
they are almost never wrong). Frege also says in section 2, "laws like
the Associative Law of Addition are so amply established by the
countless applications made of them every day, that it may seem almost
ridiculous to try to bring them into dispute by demanding a proof of
them. But it is in the nature of mathematics always to prefer proof,
WHERE PROOF IS POSSIBLE, to any confirmation by induction." This seems
to concede the point that nondeductive (in this case, empirical, because
of the applications) reasoning in mathematics can confer very high
probability. The point of proving the associative law is not to
"establish" it as true (f.o.m. people take note): "The aim of proof is,
in fact, not merely to place the truth of a proposition beyond all
doubt, but also to afford us insight into the dependence of truths upon
one another." (I.e. Frege is doing Reverse Mathematics.) So I hope I'm
not pushing things too much, if I say that Frege could agree with
everything I say about Euler. I certainly got a lot of insights about
nondeductive reasoning in mathematics from these short comments Frege
makes in 10.
Thanks to Robert (and others) for enlightening comments. In
future
writings on Frege I'll try to take them into account. Thanks also to
the f.o.m. list and its organizers for creating a forum for friendly
(hopefully) cooperation among foundationalists, mathematicians, and
philosophers, because, to end again with a quote from Frege,
"Investigation of the conept of number is bound always to turn out
rather philosophical. It is a task which is common to mathematics and
philosophy. It may well be that the cooperation between these two
sciences, in spite of many demarches from both sides, is not so
flourishing as could be wished and would, for that matter, be possible."
Mark Steiner
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