FOM: Toposy-turvey

Kanovei kanovei at
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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