[FOM] Alain Badiou
Vaughan Pratt
pratt at cs.stanford.edu
Fri Jul 24 01:35:32 EDT 2009
What one makes of Badiou is likely to depend to a great extent on one's
own take on foundations. The FOM readership has traditionally been
characterized by a distinct preference for set-theoretic foundations, a
point of view from which Badiou would appear to have a better handle on
how to relate mathematics and reality than would a category theorist,
given Badiou's acceptance of ZFC as a starting point for his own
(admittedly idiosyncratic) take on that relationship.
By the same token a category theorist is likely to find Badiou merely a
low-tech spokesperson for a point of view they have little sympathy for.
Personally I'm somewhat sympathetic to Badiou's general program, having
gotten interested myself in tie-ins between mathematics and ontology in
the past couple of years (but that's another story). I can also see how
ZFC is a sufficient starting point for that program, though I can't
agree at all that it a necessary one.
To me the essential difference between set theory and category theory is
their reversal of the roles of logic and algebra. Set theory is founded
on the logical notion of membership, a binary *relation*, which then
serves as a springboard from which to develop algebra.
Conversely category theory starts with the algebraic notion of an
associative composition of adjacent edges of a graph, a binary
*operation*, from which one can then develop notions of logic along
lines pioneered by Lawvere. (Categories are not algebraic in sets
however but in graphs: in CT language, Cat is monadic in Grph which in
turn is monadic in Set, but Cat is not monadic in Set, the basic
counterexample to composability of monads.)
To the extent that set theorists are wedded to logic as the core
framework of mathematics, their rejection of algebra as a possible
alternative core framework is understandable. This was not so clear
during the 19th century; logic did not really emerge as a preferred
basis for mathematics until Hilbert, Tarski, and Goedel lent their
prestige to the logical point of view favored by Frege, Peano, and
Russell over the more algebraic perspectives of Boole, Peirce, Dedekind,
and Schroeder. (I'd be interested in opinions as to whether Cantor as
the founder of set theory was more algebraic or logical in his outlook,
and likewise for De Morgan as the initiator of relation algebra, and for
Whitehead as the coauthor with Russell of the Principia.) However by
the time universal algebra became a subject in its own right, starting
say with Garrett Birkhoff's work in the mid-1930s, its founders had by
and large already accepted the inevitability of logic as the proper
framework on which to found mathematics.
The heresy that mathematics could *start* from algebra had its first and
arguably only serious revival in the 20th century (the two cylindric
algebra volumes, the Tarski-Givant book on set theory without variables,
and related works notwithstanding) with the mid-century emergence of
category theory, which I would characterize as equipping universal
algebra with tools powerful enough to disentangle itself from the
logical foundations it had fully accepted up to then. The Universal
Algebra and Category Theory conference, UACT, jointly organized by Ralph
McKenzie and Saunders Mac Lane and held at MSRI Berkeley in 1993,
brought together for the first time two communities that, when they met,
turned out to be living in dramatically different world views of
mathematics. When Bill Thurston, then Director of MSRI, opened the
conference with a confession to a sense of vertigo at the very thought
of the opposite of a category, the category theorists inhaled audibly in
perfect unison. Later that morning Fred Linton and Walter Taylor gave
back-to-back talks on the "right" definition of the notion of variety
(equational class) that were so utterly different that many of the
universal algebraists failed to realize the two talks even had a common
theme.
A century of investment in stabilizing the logical foundations of
mathematics has created a certain sense of inevitability. As I say on
my blog, "The self-evident is merely a hypothesis that is so convenient,
and that has been assumed for so long, that we can no longer imagine it
false." (To find my blog simply google for this statement.) Although
category theory has only been around half as long, this is nevertheless
long enough to have developed a culture of its own with at least two
generations who can't see the need for membership as a starting point
for the foundations of mathematics, and who prefer algebra over logic as
the place to start defining mathematics.
In matters of religion I prefer to be largely agnostic while allowing
that perhaps the number of gods is indeed a perfect square as most
people seem to believe. While I'd like to be just as agnostic on this
logic-vs-algebra schism, I have to confess a predilection for algebra
over logic as the proper starting point for mathematics. The logical
foundations of mathematics may well be more mature than its algebraic
counterpart, but I don't see how it follows that it's more effective.
Vaughan Pratt
Thomas Forster wrote:
> I imagine most listmembers who have heard of this chap share my
> suspicions. Does this chap know enough mathematics for his views
> on foundations to be of interest? Or is this just a case of fools
> rushing in where angels fear to tread? However I have recently had
> my ear bent by a very able former student of mine who says there is
> something in it and i should have a look. Take a squiz at this recent
> review of his:
>
> www.dpmms.cam.ac.uk/~tf/bat-on-badiou.pdf
>
> Does anyone have any useful thoughts on this matter..?
>
> tf
>
>
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