Re: terminology

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 5/23/2010 8:39 AM, Colin McLarty wrote:
> It is an interesting impulse in higher category theory to avoid
> identity in favor of isomorphism on the level of objects, and to avoid
> isomorphism in favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.

Is identity even definable?  I thought it was a kind of received wisdom,
like the natural numbers.  All of us seem to be working with the same
notions of = and N, but what are they, exactly?

That was only intended as a rhetorical question, btw.  We can readily
agree on some properties of = and N, which logicians of various stripes
have gone to the trouble of spelling out, and which number theorists
both analytic and algebraic have expanded on.

Moreover most of us would agree that the proposition "the prime factors
of M = 7^7^7^7 + 5^5^5^5 + 1 (7#4 + 5#4 + 1 where m#n denotes an
exponential stack of n m's) are all greater than 2 billion and there are
more than a thousand distinct such" not only makes perfect sense but is
either true or false.  However fewer might be willing to join me in
insisting that it is certainly true.

Those who question excluded middle for this proposition may have
received different wisdom about N than the rest of us, though if I'm
right then there's a constructive proof of the proposition that can be
checked on any laptop in under an hour, which should then oblige the
intuitionistic objectors to stand down.

(No, I don't currently know a single prime factor of M and I don't
believe anyone else does either.  I do however know the least prime
factors of both M + 1 and M + 958; leaving the former as an exercise,
the latter is 1,985,781,901.  M in decimal is 1755522...1375469 where
the number of omitted digits when itself written in decimal has 695,975
digits, so although M in binary wouldn't fit in the universe let alone a
laptop's random-access memory its length in binary would easily fit in
the latter.  The requisite calculations for all these observations take
only minutes on an ordinary laptop.)

Without exponentiation in the language, M would not be known to us: with
only the polynomial operations the requisite expression 7*7*...*7 +
5*5*...*5 + 1 would stretch beyond the farthest known galaxies.  This
question about M, which is a question about N, could therefore not have
arisen.  With it, the question becomes part of our understanding, or
lack thereof, of N.

The same can be said of identity.  The richer the language, the more
tools we have to probe our understanding of identity, and the clearer
our lack of complete understanding of it becomes.

Vaughan Pratt

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