FOM: "crises" and crises
Lou van den Dries
vddries at math.uiuc.edu
Wed Oct 15 19:44:51 EDT 1997
Let me try to respond to some of the matters raised by Harvey's email
containing a kind of questionnaire for me.
Most of the questions Harvey asks are not sufficiently clear cut for me
to give a clear cut answer. Also, I do not presume to be familiar enough
with the interests of all (or even most) "really important active core
mathematicians", or of "the general scientific public", etc. to have
simple answers to some of the questions. (And I would probably mistrust
simple answers that someone else would provide.) And I just got the
"Math into the 21st Century" out of the library, so it's too early to
respond to the question Harvey asked me about it.
Harvey writes (in connection with the Russell paradox): It is compelling
that there should be a property that holds of exactly the properties
that obey a given well defined condition.
Hmm, I never found this compelling in the least, from the first time
I heard the Russell paradox (as a schoolboy). My reaction to it was from
the beginning that one should be quite clear what one means by "properties"
here, and what it means for one property to be the same as another
(I mean, a second) property. For example, I would have accepted such
a "law of thought" if it concerned well defined properties of natural
numbers, and properties were taken in the extensional sense. And even then
it remains to specify in some way "welldefined property" and well defined
condition". Anyway, when one insists on certain obvious requirements
of precision (which seem to me also "laws of thought" the Russell
paradox disappears. Of course, this line of thought leads naturally
to Zermelo's comprehension axiom in set theory. (Or to the theory of
types.)
Harvey asks: Do you hold one or both of these viewpoints:
1) that the program I outline will never be carried out;
2) if carried out, will not create a crisis of the kind I indicate.
If I am not mistaken "the program" refers to creating "natural"
statements about (large) finite sets of a combinatorial nature
that can only be proved by going far beyond ZFC. ("finite sets"
should perhaps be replaced here by "finite combinatorial objects";
and of course, they should be provable from axioms addedto ZFC that
are in some way plausible.)
Well, it would be foolish to assert 1). But yes, i would tend to go
for 2). Combinatorics of large finite objects is certainly an interesting
field in itself, and, for example, the Ramsey theorem and its variants
play an interesting role outside that field also. But that a revolution
in that field (if it occurs) would create a major crisis in the
foundations of mathematics to be compared with the greatest
intellectual events of all time (as Harvey wants it to be), seems to
me very unlikely. (I could be wrong of course. Perhaps, if "probability"
considerations would come into play, there might be more to it than
I think; even so, it seems to me to create more of a "local crisis"
than one which implicates mathematics in general.)
I do not particularly care for statements like
"Goedel's work in the 1930's represent the most impressive
mathematical achievements of the 20th century".
Impressive and original as Goedel's work is, I don't see the point of
this kind of adoration. Who are the all knowing intellectuals who say
this? (Sorry, "achievements" should be "achievement".)
Best regards,
Lou van den Dries
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