no fundamental theorems please

[Paul Taylor <pt09@PaulTaylor.EU>]

I see that Takuo and I have still not overcome the "royalist" forces.

Claudio, Vaughan, Ross and others have mentioned various more
"sophisticated" versions of the Yoneda Lemma.

However, I contend that the more sophisticated the result is,
the LESS it deserves to be called "the fundamental theorem of
category theory".

I particularly like Euclid's algorithm for the highest common
factor as a historical and methodological example,   Without
meaning to dictate to number theorists how their subject should
be organised, let me suggest for the sake of argument that it
is a pretty good candidate for being called the "fundamental
theorem of number theory".

Gauss used the same idea to factorise polynomials, and no doubt
number theorists have many more "sophisticated" developments of it.

But the principal idea was Euclid's (or one of his colleagues),
not Gauss's, and definitely not that of any subsequent number
theorist!

Saunders Mac Lane said something about "the right" generality,
as opposed to the greatest generality.

As I say, I like Euclid's algorithm because the idea has
survived many many revolutions in the "official" foundations
of mathematics.

Theorems, like royalty, are rightly the victims of revolutions,
because the way in which we encapsulate a piece of theory as
a "theorem" depends as much on our current cultural prejudices
as it does on the real underlying mathematics.

For example, there is Cantor's "theorem" about a powerset being
strictly bigger than a set.  This belongs entirely to the dogma
of set theory.  When set theory is overturned, this miserable
and wholely misguided "theorem" will go in the dustbin of
mathematical history with it.

But Euclid's algorithm will live forever.

And the Yoneda Lemma will survive as long as Category Theory
does in a recognisable form.

But it still shouldn't be called the "fundamental theorem"!

Paul



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