FOM: replies to Hersh on logicism and certainty
M. Randall Holmes
M.R.Holmes at dpmms.cam.ac.uk
Mon Oct 5 05:27:32 EDT 1998
You have made a solid case. Logicism lives.
Who's arguing? When did I say it was dead?
I said it had been unable to achieve its original goals.
That's not saying it's dead.
If you read my recent posts, you will realize that I admit no such thing.
I describe precisely how I think that mathematics can successfully be
founded on logic, and I support the proposition that properly proved
mathematical statements are certainly true.
I proposed and advocated a different kind of philosophy,
based on mathematical practise as it really is, not as
it's supposed to be.
That's proposing a competitor to logicism. It's not
saying logicism is dead.
No such competing viewpoint is necessary, as mathematics is really
as it is supposed to be (as well as we can manage with our admittedly
erratic faculties :-) )
We agree. Logicism lives.
We agree on the truth of that statement, yes.
(Hersh replies further (re indubitability))
You say one cannot have a usable or worthwhile notion of partial
or incomplete rigor or proof without having in advance a notion
of perfect rigor or certainty.
I don't think this necessity has ever been demonstrated.
I think this is obvious, and attested in the actual behavior
of mathematicians (probably including yourself in unreflective moments).
Certainly it works the other way--*if* one could have a meaningful,
valid criterion or notion or test of absolute certainty or perfect
proof or rigor, then indeed that would be useful in dealing with
imperfect or incomplete proofs.
Imperfect or incomplete what, exactly? The imperfect falls short of ...
the perfect :-)
However, it may be that the search for perfect or absolute certainty or
proof or rigor is actually made by successively refining and making more
satisfactory our standards of incomplete proof.
History does show successive refinements and strengthening of proof
(betweenness in geometry, uniform convergence in analysis, and others
in set theory which you are better qualified than I to tell about.
You can say that those advances were made possible on the basis of
an existing notion of absolute certainty. I would say that our
notion of absolute certainty was developed in the course of more
careful and critical incomplete proofs.
How can you tell that the standards of proof are refined or strengthened
if you do not know what an adequate proof is? The notion of the absolute
certainty of mathematics is as old as Pythagoras and Euclid; it was not
a brainstorm of 19th century mathematicians.
I emphasize that "incomplete proofs" means virtually all proofs
as presented to seminars or colloquia or printed in journals and treatises
or broadcast on the world wide web. Harvey recently gave examples
of typical ways in which gaps are consciously left in published
proofs. Complete proofs, for most problems
and theorems of any substance, would be far too long and tedious to be
read or published.
I emphasize this point very strongly myself. How can you recognize
a gap if you don't know what a complete proof would look like? The
experience of the Automath project, by the way, suggests that the
increase in size of completely formal proofs expressed in a suitable
notation is by a constant factor and not perhaps by such a large factor
as one might suppose (though the tedium of such proofs is undeniable).
I think the recent experience of Mizar supports this (is anyone in the
Mizar group on this list?)
(I close (there is mercy in the world!))
Sincerely, M. Randall Holmes
holmes at math.idbsu.edu or mrh29 at dpmms.cam.ac.uk
Boise State University and the University of Cambridge
must be held harmless for any silly thing I may say.
"And God posted an angel with a flaming sword at the gates
of Cantor's paradise, that the slow-witted
and the deliberately obtuse might not glimpse
the wonders therein." (Holmes, with apologies to Hilbert)
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