FOM: 'constructivism' as 'minimalistic platonism'
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Wed Jun 7 22:59:51 EDT 2000
I said (in my reply to Schuster):
>> The working mathematician, like normal people, in using "A or ~A"
actually
>> means "either A is true or ~A is true, but I don't know which".
and Fred Richman replied:
>Lots of people on both sides of the aisle try to explain what working
>mathematicians mean by "A or B". Constructivists like to say that they
>mean "not( not A and not B ). From my experience, that seems
>ridiculous. On the other hand, I can't believe that they mean "but I
>don't know which". Such a thought would never occur to them.
I agree - I put it very badly. (However, this "philosophical" thought *does*
regularly occur to me, now that I'm interested in the debate between
classicists and intuitionists). What I should have said is this. Consider a
mathematician who is happy with classical reasoning. I suspect that most
mathematicians are, but perhaps that's either highly debatable, or
irrelevant, or a measure of their epistemological naivety (but I don't know
which!). Suppose they use LEM ("A or ~A") in a proof. Then I'd claim that
they *just* mean "either A is true or ~A is true", but they *don't*
necessarily intend that assertion to carry any further implication that they
are *already* justified in accepting either A, or justified in accepting ~A.
So perhaps I should have put it like this:
(i) The realist thinks the statement "Either A is true or ~A is true, but
I/we/humanity/etc., might never find out
which!" is a coherent and sensible thing to say (and perhaps even a priori
true),
whereas
(ii) The constructivist/intuitionist thinks the statement "Either A is true
or ~A is true, but I/we/humanity/etc., might never find out which" is
somehow (conceptually) incoherent (for suitable choice of A).
Is that the right way to state the difference between the realist use of
"or" and the constructivist use of "or"?
If so, this goes back to the point about the two disparate notions of truth
in play. For a realist, mathematical truth is correspondence (independently
of epistemic matters), while for a constructivist, mathematical truth must
be epistemically accessible. The realist wants to separate truth from
epistemic accessibility -- so that there can (in principle) be facts beyond
our ken. The constructivist (as I understand it) is motivated by the idea
that facts (in this case, mathematical facts) must be epistemically
accessible, so that there cannot (in principle) be facts beyond our ken. I
find the idea of making sense of a realm of epistemically-accessible truths
very important - so that's an important epistemological project that
constructivist approaches can serve. However, I find it equally important to
recognize the coherence and possibility of a realm of
epistemically-inaccessible truths/facts (c.f., the Chaitin book I mentioned
before, and general considerations about the incompleteness/undecidability
theorems). I see no good a priori reasons for restricting the structure of
mathematical reality to fit our abilities to find out about it. So, realists
want to keep facts and knowability of facts separate.
(If I had to explain further I would add the following: the physical
universe might exhibit all kinds of rich mathematical structure which our
puny human brains are incapable of grasping; we certainly shouldn't place
strong constructive limitations on our mathematical methods when our goals
are to understand the world. In particular, if non-constructive mathematics
is indispensable to theoretical physics, then radical constructivism seems
doomed. That's a big "if", I know. Let's see).
>How do mathematicians use such statements? At some point in a proof
>they might have an integer n > 1. This will not in general be a
>specific integer, like 88001, about which they might conceivably say,
>"88001 is either prime or composite, but I don't know which". They
>might say, "either n is a prime, or n = ab for integers a,b > 1", then
>proceed to argue in each of those two cases. The idea that they do not
>know which alternative holds would make no sense to them in this
>situation. It's not like the case of 88001. Here they don't even know
>what n is, so how would they know if it were prime or composite?
I think they're entitled to say "whatever this n is, it must be either prime
or composite". There isn't any genuine possibility that it might be neither!
I suspect that they're entitled to add "perhaps I shall never know, of this
n---whatever it is---whether it is one or the other". Have I understood the
point?
I agree that there's some kind of intelligibility problem here, connected to
the scope difference between statements like "I know that there is an n such
that A(n)" and "There is an n such that I know, of n, that it is A". I think
there's a big philosophical literature about this kind of thing, about
"knowing what something is" (e.g., "knowing who the prime minister is",
etc.).
The real question seems to be: do you need to know what this integer n is
*before* you're entitled to assert that either it is A or it is ~A?
I said:
>> The intuitionist rejects the introduction (during a proof) of "A or B"
>> until there is already either a proof of A or a proof of B. But I do
>> not see why we need to make this *extremely restrictive* assumption.
Fred replied:
>Only if you make this assumption do your theorems acquire
>computational interpretations. The intuitionist views a proof as a
>program. The statement "A or B" is typically used as a branch point in
>that program. If the program can't figure out which branch to take,
>it's not going to be able to proceed.
I agree entirely. The realist and the constructivist surely have different
views as to what counts as a justification for a theorem. The realist is
just more liberal in their view of what counts as evidence. The
constructivist must insist that a non-constructive proof, in lacking a
computational interpretation, lacks a sufficient justification for us to
consider the theorem "proved" (deduced classically) as having been "properly
justified".
I think that the realist must reply something like this. The theorem proved
*is justified*: i.e., if our initial assumptions are true, then the theorem
must be true (because classical deductions preserve truth); and we are
surely justified in our initial assumptions (say the axioms of PA or
whatever). Consequently, we must be justified in accepting the theorem. The
realist seems to have a broader, more liberal concept of (deductive)
justification---even non-constructive evidence is evidence.
The whole issue turns on having a computational interpretation, which is
related to some notion of epistemic accessibility---non-constructive
theorems lie at a more remote "epistemic" distance from human computable
accessibility. But what's *wrong* with a theorem that doesn't have a
computational interpretation?
(Aside from the very fact that it doesn't have one).
Is there a motivating philosophical assumption within constructivism that
there cannot be mathematical facts which resist constructive proof (or,
removing some of the double negations: that all mathematical facts are
constructively provable)? Perhaps that is the main claim of what Hellman
calls "radical constructivism" (c.f., Dummett). Then the question is: why
should mathematical reality be built like that? I mean, to suit human minds?
The only way to justify that supposition is by saying that we "create" or
"construct" mathematical reality as we go along, and that brings us back to
the Simpson-Frank debate about subjectivism as the central philosophical
motivation for constructivism.
Perhaps "subjectivism" is the wrong word--perhaps "epistemically accessible
via computation" is a better description for Bishop-style constructivism.
But a realist can still say: why should all mathematical facts be
epistemically accessible via computation? A liberal constructivist
(Hellman's terminology) can presumably say "Maybe there are such facts, but
I'm just *more interested* in the
epistemically-accessible-via-computation-ones". A radical constructivist
will presumably say "It is incoherent even to suppose that there could be
such epistemically-inaccessible mathematical facts--the classical
mathematician is engaged in theology". (This is (roughly) the sort of thing
that my radical constructivist friends do say!)
As a further note on the use of "or", I think I have an example of a famous
mathematician (well - 2 famous mathematicians in fact!) claiming "A or B",
even though he didn't already have a proof of either A or B. The example is
nice, because it connects up with some other interesting philosophical
topics in f.o.m.
In Goedel's Gibbs Lecture (delivered at Providence on December 26th 1951),
he presents the following dilemma (I'm quoting from Hao Wang's "From
Mathematics to Philosophy" (1974, Routledge)), which Wang describes as
being, in Goedel's opinion, one of "the two most interesting rigorously
proved results about minds and machines". I quote:
------------
2. The second result is the following disjunction: Either the human mind
surpasses all machines (to be more precise: it can decide more
number-theoretical questions than any machine) or else there exist number
theoretical questions undecidable for the human mind. (Wang 1974, p. 324).
-----------
So, Goedel (or at least Goedel-Wang) seemed to be asserting the disjunction
"A or B", and thought that this disjunction is a "rigorously proved result".
Of course, Goedel also believed A (minds are better than machines), but he
didn't first prove A and then deduce "A or B". Rather, he first proved "A or
B" and then argued that B is false, thereby justifying A (by disjunctive
syllogism). My point is that he did not justify his assertion of "A or B" by
a prior proof of either A or a proof of B. Rather, he proved "A or B" by
reflecting on the Incompleteness Theorems, and then argued for A on the
basis of his argument that B is false.
Perhaps I'm wrong, but this seems to be a case of a justified disjunction
where we're very unsure of the justification of either disjunct.
(Or perhaps we shouldn't apply intuitionist logic to extra-mathematical
claims like this).
(Or even perhaps, the formulation of this claim as a disjunction is
misleading, for there is precise constructive content here: namely, a
conditional roughly of the form C --> A: any justification/proof of the
claim that all number theoretical questions are mentally decidable can be
converted to a justification/proof of the claim that minds are better than
machines).
Regards - Jeff
Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk
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