Bourbaki and Foundations
Harvey Friedman
hmflogic at gmail.com
Sat May 14 20:24:05 EDT 2022
I want to bring in some facts that might better direct this discussion.
The first is from the very famous and very influential book
[1] Saunders Mac Lane, Categories for the Working Mathematician
I quote from page 10:
A category (as distinguished from a metacategory) will mean any
interpretation of the category axioms within set theory.
Here are the details. A graph (often called a diagram scheme) is a set O of
objects, a set A of arrows, and two functions ...
In this graph, the set of composable pairs of arrows is the set A = [an
expression with { ... }]
A category is a graph with two additional functions ...
So it is clear that Mac Lane explicitly uses set theory as the foundation
for category theory without hesitation, right from the beginning.
How can this be explained?
Well I think that Mac Lane realized that any attempt to remove set theory
from the picture, and have notions like category as primitive, is likely
doomed not only in terms of its complexity, but also in terms of its
Philosophical Coherence. So Mac Lane didn't even attempt to do such a
thing, at least in this book.
>From a purely practical mathematical point of view, there really is no need
to have any kind of philosophical coherence (except possibly for certain
teaching purposes). One can give some standard presentation of category
theory without having to address any issue of philosophical coherence, and
get right into the mathematics itself.
Or you can adopt Mac Lane's approach in [1] and have a page or two with
precise definitions to start things out sitting on top of set theory.
For getting the real mathematics going in the sense of mathematical
culture, it doesn't make any difference which of these two. If
philosophical coherence is desired or at issue, the second way is best. At
least until there is a philosophically coherent version without sitting on
set theory.
The second is illustrated through the notion of interpretation (and
stronger notions such as synonymy). Basically, any version of category
theory proposed, sitting on or not sitting on set theory, is equivaelnt to
set theory at least in the sense that there is an interpretation between
set theory and the category theory going both ways. I.e., there are ways of
talking about each in terms of the other so that any proof in one framework
is automatically converted to a corresponding proof in the other framework,
and vice versa. So there is an automatic grand reconciliation.
This automatic grand reconciliation is actually much stronger than one
might expect in certain crucially important ways. It is commonly agreed
that there are common ways of talking about very basic objects of
mathematics. All approaches pretty much treat the ordered ring of integers
in equivalent ways. This also extends further, in treating the ordered
field of real numbers with distinguished part Z in equivalent ways. One can
argue that one can take these equivalences even further.
The mutual interpretability of set theory and category theory of various
kinds is very strong in that they preserve the truth of statements about
the ring of integers and even the ordered field of real numbers with
distinguished part Z.
So basically, the point is that for reasonably well grounded mathematics of
a standard kind, the mathematics one has is completely interchangeable -
and automatically.
Harvey Friedman
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