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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <1991Feb26.122804.26512@sics.se>
Date: 26 Feb 91 12:28:04 GMT
Article-I.D.: sics.1991Feb26.122804.26512
References: <1355.9102231618@s4.sys.uea.ac.uk>
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In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 23 Feb 91 16:18:47 GMT

In article <1355.9102231618@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

>Notwithstanding your preference for a reply to (5) rather than (3), I
>must first comment on one more point from (3): "obscurantism".
>This is the pot calling the kettle black with a vengeance, coming from
>someone who believes that there is excellent evidence for the consistency
>of arithmetic, and not only cannot say what it is, but believes no-one
>else can either.

  What I was talking about in referring to the lack of "good answers"
was of course a philosophical or theoretical analysis of mathematical
evidence. As for saying what the evidence for the consistency of
arithmetic is, I thought I had done so: it's no different from the
evidence for mathematical theorems in general: we have proved it,
using valid axioms and principles of reasoning.

  So how do we know that the axioms are true and the principles of
reasoning valid? This is where it is difficult to say anything very
interesting - which is why these are classical questions in the philosophy
of mathematics. As I put it, concerning the induction principle: given
that the natural numbers are s(0), s(s(0)), etc, it follows that
anything that is true of 0 and is true of s(x) whenever it is true of
x is true of all natural numbers. This is not mysterious - it's just
that there is nothing to say to make this more convincing than it
already is.  What should we call the evidence for this assertion?
Reflection on the meaning of our concepts? Seeing into mathematical
reality? Understanding the rules of language? Visualizing hypothetical
entities? These alternatives, and others, have been pursued in the
philosophy of mathematics. The induction principle itself is, however,
far better understood and less problematic than any of the answers to
the philosophical questions.

  Other basic mathematical principles, such as the axiom of choice or
the existence of the power set of N are of course far less
transparent, and in many cases the question of what we do or do not
have evidence for is to a considerable extent a matter of subjective
judgment. It sometimes appears that you assume that mathematical
evidence, when invoked, must be thought apodictic, incontrovertible,
unshakeable.  This has been no part of my argument.  I have claimed
that your proposed empirical evidence for the truth of a particular
mathematical theorem ("arithmetic is consistent") is worthless, and
that there is such a thing as mathematical evidence.  Constructivists
and intuitionistists of various schools have a different idea of what
is good mathematical evidence than I have, and as I pointed out in an
earlier article, I have no quarrel with their views in this particular
context.

>If empirical evidence is, as you claim, worthless in mathematics, it is
>worthless in the physical sciences also.

  What empirical evidence means in physics and the other sciences is a
large question which I won't try to enter into. Of course the fact
that the sun has risen so far is not good empirical evidence that it
will rise forever - on the contrary, we believe that it will
eventually rise no more. What I claim is that the empirical fact that
no inconsistency has been found in arithmetic so far is worthless as
evidence for the assertion that there exists no inconsistency in
arithmetic. Since you presented no other empirical evidence for the
consistency of arithmetic, I make no claims regarding empirical
evidence in general - nothing having been said as to what would in general be
meant by empirical evidence for the truth of mathematical theorems.

  Your remaining remarks seem to me to dilute the issues to a point
where I see little to argue against. You admit to an "intuitive
understanding of and confidence in a mathematical theory", but seem
reluctant to say that this yields any kind of evidence for the truth
of mathematical theorems. I find it hard to take this seriously. There
is no need to suppose that this intuitive understanding and confidence
results from some kind of pure contemplation, nor that it is
infallible or unchangeable. But to declare (to return to the very
special case of the consistency of arithmetic) that the ordinary mathematical
evidence for the truth of a theorem - i.e. its proof - is no evidence
at all, that is what I call obscurantism.

  So, to sum up. I am not concerned with arguing the philosophy or
theory of mathematical evidence here. I do claim that the empirical
evidence for the consistency of arithmetic (consisting in the
empirical fact that no inconsistency has been deduced) is very poor,
and with this you seem, this time around, to agree. (Or so it would
appear, in view of your low opinion of appeals to "seeing", which you
concede to be no worse than your appeal to the empirical.) I also
claim that there is nothing special about consistency theorems when
one considers questions of mathematical evidence.  (This was earlier
formulated in polemical terms as a rejection of the "rituals"
surrounding talk of consistency.) This you don't seem to deny.  And
finally, I claim that it is obscurantism to pretend that there is no
other evidence for the truth of mathematical theorems than the
"empirical". Here you apparently disagree, but I don't see how to take
the matter further with the material now at hand.