FOM: bad set membership

Harvey Friedman friedman at
Fri Jan 23 18:13:02 EST 1998

[NOTE: The title was inspired by what appears after the # sign below.]

This is a response to McLarty 11:15AM 1/23/98. By the way, we still don't
have a concession from you that you need much more than a topos with
natural number object to naturally do undergraduate real analysis. How
about it? Also, we don't have a concession from you that there is no
coherent conception of the mathematical universe that underlies categorical
foundations in your sense. You should make it more clear that you are very
proud of the fact that there is no such coherent conception of the
mathematical universe on which you base your approach so that the rest of
us can stop thinking that you are even trying to do foundations of

You would have greater credibility if you would say, e.g., that you are
doing o.o.m. instead of f.o.m. O.o.m. is organization of mathematics. Even
here, you have a lot of convincing to do to make it stick that it is all
that illuminating. Simply wrapping yourself up in Grothendieck is
insufficient, according to my mainstream mathematical sources, who are
perfectly satisfied with the usual set theoretic foundations of

>Categorical foundations suggest a wider range of topics. (Simpson and I
>agree on this fact, but he feels it invalidates the claim to foundationality).

If you call anything and everything "foundations" that would also suggest
even more topics - namely, all topics!

>Categorical foundations are closer in style and methods to mainstream

Actual real anaysis, actual algebraic topology, etcetera, are even closer
in style than category theory and topos theory to mainstream mathematics,
since they are actually mainstream mathematics. Let's call them foundations
of mathematics too!!

>(Friedman considers mainstream mathematics intellectually >corrupt.)

You like to summarize complex positions of mine by means of loaded
adjectives like "corrupt"? I think this is silly and counterproductive.
Show me where I said "corrupt."

#>Categorical foundations emphasizes functions, and considers it meaningless
>to talk about "membership" between arbitrary sets--e.g. it is meaningless to
>ask whether the set of integers is a member of the set of symmetries of the

So you are, after all, struggling to make sense of what you are doing by
saying that you are banning certain kinds of propositions! This is very
useful to fom. Just when we thought that you weren't going to give us a
clue!! At least you are now trying to say something of a definite
conceptual nature, in order to help us see what this categorical
"foundations" is about. Please, please, please, please, please expand on
this. Keep talking!!

So when I show a 3 year old three cards on a table and get him to identify
it as a group, I have to spank him if he says that the slice of bread on
the table is not in that group?? Child abuse!! By the way, in case you
couldn't figure it out, that slice of bread is definitely not an element of
that set of cards.

I seem to know something else you don't: no set of integers is a member of
the set of symmetries of the plane. Every set of integers is at most
countable, yet every symmetry of the plane is uncountable.

Some of us try to use philosophically coherent thinking, whereas I'm not
sure this is all that high among your priorities; you would rather
misappropriate some technical spadework of some core mathematicians for
very definite mathematical purposes as "foundations," thereby gutting the
whole idea of genuine foundations, with its origins in the contemplations
of great philosophers.

Keep giving us more clues as to what this is all about. It's exciting!

>(Friedman feels this violates the expectations of small children.)

How cute. Kids at 3 grasp three cards on a table as a unit, and later the
idea that any of those cards could even be a unit. You can generate just
about everything in set theory from that. Your kids at 3 grasped - well I
don't know what. They must be smarter at 3 than most of the adult

We're still waiting to see those beautiful fully formal axioms of topos
theory with natural number object and well pointedness and choice to
compare side by side with my axioms in 12:54AM 1/23/98. I'm looking forward
to this.

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