FOM: Toposy-turvey
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Mon Jan 19 07:42:11 EST 1998
It seems that the Harvey Friedman's critics will
not make a big problem for Colin McLarty, because
the latter has silently distanced his "course"
from the categorical mainstream.
Indeed he does not start his "course" with
a prayer like
"there are no Categories except for Categories
and the Topos is their Prophet".
He starts and continues with the Bourbaki set
theory. (He did not specify this clearly, indeed.)
The Bourbaki set theory is known to have just
two parts: a foundational part and a Prayer.
The foundational part is a reasonable set theory,
perhaps equal to ZFC (if not speaking about Regularity,
and perhaps Choice and Replacement, I do not remember
the details well), at least it grounds the real analysis
pretty much the same way as ZFC does.
The Prayer is: (*) functions are not sets.
Since an axiomatic taboo to define ordered pairs and
graphs would be very ridiculous, the Prayer quickly
leads to the conclusion:
that a function is foundationally different from its graph.
This would be allright with the philosophical part
of fom subscribers, who even might see here some reflection
of Kantian ideas. This would be perhaps allright with
set theorists as well because they easily see, following
Harvey Friedman, that what is right in this approach is
just a "slavish copy" from usual set theory, while all the
rest is a "generalized nonsense", so there is no reason
to follow this up further.
However this will not be that allright with students,
because eventually a smart student asks:
|-- Dear professor, you told us that functions are
different from their graphs. We are not meaningless kids
here, in particular we know that the computer on the next
table is one and the same thing whichever way you call it,
computer, or Rechner by German, or accordingly in Japan or
other language -- and remains for us one and the same
thing as long as it works one and the same way.
Functions and their graphs seem to work, mathematically,
one and the same way, because for a function to _work_
means to _get values from arguments_.
So would you tell us now shall we
regard the (*) above just as a prayer or you can now
ground it for us in a short, clear, and mathematically
sound way, if possibly omitting prayer-like references
to particular personal ideas of your greater colleagues. --|
Vladimir Kanovei
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