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From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <1355.9102231618@s4.sys.uea.ac.uk>
Date: 23 Feb 91 16:18:47 GMT
Sender: daemon@ucbvax.BERKELEY.EDU
Lines: 92

News glitches have tangled the thread somewhat.  Recent messages are:

1.  I sent a message dated 17 Feb.
2.  I reposted (1) on 20 Feb, having wrongly suspected it didnt get out
the first time.
3.  In message <1991Feb20.215853.22149@sics.se>, Torkel Franzen replied
to (1).
4.  In a message of 21 Feb I replied to (3).
5.  Finally, in message <1991Feb21.202449.22562@sics.se>, Torkel Franzen
replied a second time to (1), suspecting that message (3) didnt get out.

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 (to quote you more precisely: "I don't believe any very
good answers have been arrived at").  I am reminded of a notorious
non-solution to the theological problem of evil, which consists in saying
that indeed God is good, but good in some mysterious sense that no-one
understands.

Now going back the question of the coherence of physical theory vs. that
of mathematics.  I am dissatisfied with my previous attempts to address
this.  As usual, if the following conflicts with my previously expressed
views, it supersedes them.

One has a reason for expecting the sun to rise again, other than that it
has done so previously: its rise is also predicted on the basis of a
great deal of other background beliefs about the world.  However, if
those background beliefs are brought into the foreground, and one asks
why the entire body of belief relevant to sunrises should continue to
apply, then one is in the same position as one is with respect to the
consistency of the entire theory of arithmetic.  There is no longer any
separate background to provide extra support for a particular prediction.

So why do I believe in the predictions of physical theory?  I don't know. 
Why do I believe in the consistency of arithmetic?  I don't know that
either.  Which brings me to the gist of message (5):

>So, as far as the evidence for consistency is concerned, my conclusion
>is that your grounds for claiming that arithmetic is consistent are no
>less arbitrary, peculiar, or subjective, and no more empirical in any
>obvious sense, than just those appeals to intuition which you so sternly
>rejected in your original postings.

As in my message (4), I agree, as far as that goes; however, I would
claim at the same time that your grounds for expecting the next sunrise
have the same character.  If empirical evidence is, as you claim,
worthless in mathematics, it is worthless in the physical sciences also.

You can believe the axioms of arithmetic, and therefore believe in their
consistency and the truth of arithmetic theorems; or you can apply
various degrees of doubt to the various subtheories, which will carry
over to their theorems.  Likewise, you can believe in physical theory,
and believe in its various predictions, such as that the sun will rise
again; or be variously doubtful about its different parts and
correspondingly doubtful about their predictions.

Now when the theorems are found to be true in particular instances, and
their study to have unearthed no inconsistency, what does this tell us
about the mathematical theory or its subtheories?  When the predictions
are found to be satisfied in particular instances, what does this tell us
about the physical theory or its subtheories?  The two problems are, it
seems to me, the same problem, the problem of what evidence is evidence
of, and no good answer has been given to it.

In practice, one must make decisions about how much of a physical or
mathematical theory to believe (or at least to adopt as a working
hypothesis and stop worrying about pro tem), without having any solution
to the problem of how such decisions can be made.  In such matters,
people ascribe their choices to something they call "intuition", which
only gives the problem a name, but does not solve it.  And for me at
least, such intuitive understanding of and confidence in a mathematical
theory comes after and arises from working with it and seeing it work,
not from contemplating its axioms and seeing that they are true.  The
latter provides me at best with prima facie evidence that the axioms may
have succeeded in describing the perhaps vague concepts that they were
intended to, and may be worth working with further - no more.

>There exists a considerable body of thought,
>not necessarily found in Usenet articles

No kidding!  :-)

--
Richard Kennaway          SYS, University of East Anglia, Norwich, U.K.
Internet:  jrk@sys.uea.ac.uk            uucp:  ...mcsun!ukc!uea-sys!jrk

"But...surely you believe that tables and chairs exist?  What's this, if it
isn't a table?" he said, rapping on the table.  "A working hypothesis," I
replied.  Then Vimalakirti's doorkeeper disappeared again.