[FOM] Mathematical Experiments.

[Bill Taylor]

Dan Goodman <dog at fcbob.demon.co.uk> wrote:

->The thing is that the 4CT proof couldn't "easily be humanly checked".

I guess the word "easily" is a problem here.  It is true that one human
COULD check it, taking a very long time - the work isn't so huge that it
would take more than a lifetime.  Only 40,000 cases or so have to be checked,
and each one is a simple routine of less than a minute. So in fact it could
all be done in just four months of doing that and nothing else. (We assume
he has computer printout ready to CHECK.)

And if we allow a team of several humans, a lot quicker.
 
->human checking the computer proof of the 4CT would be more likely to make a
->mistake checking it than the computer. 

Oh yes indeed!  I have already observed that humans can and do make mistakes
in proofs, and dismissed it after consideration.  But indeed, even computers
make mistakes, much though Pentium might like to deny it.

Humans are so prone to error, that the day will soon come when proofs of
certain types (and ultimately ALL types) will *have to be checked* by
computers to obtain publication.  The ideas behind this are frequently
discussed on this list.  But I don't think this has any effect on my theme.

->One can imagine a human checking it, but it's not actually possible. 

So I think in this case at least, it IS.  But I concede your point, that there
will be proofs that one human could never check in a lifetime.  Though he can
still check the computer's apparent correctness in various indirect ways.


->Is this significant? I think so, but I
->expect most people on this list would disagree.

It is sociologically, historically and academically significant; but not
(I think) logically or philosophically significant.

 
-> when you say the 4CT "could easily be humanly checked" you probably
->have something else in mind, e.g. there is nothing "in principle" stopping a
->human from checking it, no individual step in the calculation is
->unsurveyable by a person, only the calculation as a whole.

Well, I think the *proof* as a whole IS surveyable, and very easily - the
points that have to be *understood* are easy, and doable in just a few
minutes - the 4CT proof is pretty basic!  And also, each step is easily
doable.  There is a sense in which the whole proof is NOT "thinkable of"
all at the same time; but this applies to many other proofs as well I fancy.


-> I don't think Lakatos
->was trying to say that maths and science were the same, 

He did not say so directly, in that book.  But he HAS said so many other
times - it is an essential point of his whole philosophy of science.  It is
a view shared by many on sci.logic and some on this list.  I think it is
wrong, (though defensible), and I'm happy to note you think the same.

I have noticed one intriguing sociological effect - physicists seem to be
very prone to feeling that math and science are "essentially the same",
whereas mathematicians are more prone to think they're very different.
I wonder if a similar situation holds between chemists and physicists,
or between biologists and physi/chemists?

->he was just pointing
->out that the process of mathematical discovery is similar to that of
->scientific discovery.

So we all agree there, at least.

-> What makes it different is that what changes in the
->process of mathematical discovery is not the truth or otherwise of theorems,
->but the choice of theorem to look at.

Neatly summarized!

-> In "proofs and refutations" he traces
->the changing statement of Euler's theorem, not the change in the seeming
->truth value of some particular statement of it. Examples, essentially
->pictures, are the driving force for the changes, in his account.

Yes, this is how you or I would describe it.  But my impression is Lakatos
himself would *by no means* be prepared to leave it at that.  He would
probably say we were merely prevaricating and that the essential facts
to be understood had changed their truth-values as the process continued.
Or words to that effect.  I think this is *the* crucial difference between
the math-is-science folk and us math-isnt-science folk.  WE insist on
maintaining very fine distinctions; THEY insist that these are just 
covering up the real truth underneath.

Or so it seems to me.

------------------------------------------------------------------------------
         Bill Taylor                   W.Taylor at math.canterbury.ac.nz
------------------------------------------------------------------------------
         The chief difference between mathematics and physics is that
         in mathematics we have much more direct contact with reality.
------------------------------------------------------------------------------