Alain Badiou's Mathematics of the Transcendental

José Manuel Rodríguez Caballero josephcmac at
Tue May 24 02:06:34 EDT 2022

Harvey Friedman wrote:

> I'm not attacking category theory at all, nor am I attacking PDE, nor
> am I attacking algebraic geometry, nor am I attacking analytic or
> algebraic number theory.

I'm attacking the idea that category theory serves as any kind of
> philosophically coherent foundation for mathematics -- as can readily
> be tested by the general intellectual community.

I agree that, assuming fixed foundations of mathematics, we can give
category theory the status of a branch of mathematics such as PDE,
algebraic geometry, and number theory. If I well understood, to give
category theory the status of a valid approach to foundations of
mathematics it is required to show that it is philosophically coherent.
According to

Marquis, Jean-Pierre. "Category theory and the foundations of mathematics:
philosophical excavations." Synthese 103.3 (1995): 421-447.

the notion that S is a foundation for T can be instantiated as follows:

(i) LogFound(S,T): S is a (relative) logical foundation for T.
> (ii) CogFound(S, T): S is a (relative) cognitive foundation for T.
> (iii) EpiFound(S, T): S is an (relative) epistemological foundation forT.
> (iv) SemFound(S, T): S is a (relative) semantical foundation for T.
> (v) OntFound(S, T): S is an (relative) ontological foundation for T.
> (vi) MetFound(S, T): S is a (relative) methodological or pragmatic
> foundation for

If I am not mistaken, the philosophical coherence is EpiFound(S, T) and
OntFound(S, T).

In the same paper, we read that set theory can be used as a foundation of
mathematics if we accept that:

(1) mathematics is truly the science of the realm of sets;
> (2) set theory is part of logic, the latter being the universal science
> upon
>  which every other science is based; Set theory is just, in a sense,
>  applied logic to mathematical concepts;
> (3) set theory captures the fundamental, i.e. the most general, cognitive
>  operations upon which the whole of mathematical knowledge is based;
> (4) the axioms of set theory possess an epistemological property, e.g.
>  self-evidence, truth, indubitability, which gives them a priviledged
>  status;
> (5) a set theory is indispensable for doing mathematics, if only to provide
>  a uniform and good control on questions of size, but mostly for
>  definitions, constructions and techniques of proofs. Thus a set theory
>  is heuristically and methodologically inescapable.

and category theory is an alternative foundation of mathematics in the
following sense:

(1) category theory is heuristically fundamental;
> (2) the theory of the category of all categories is the ontologico-logical
>  foundation for mathematics;
> (3) category theory provides a methodological foundation for mathematics.
> (4) toposes provide an adequate logical foundation for 'ordinary'
> mathematics
>  via their internal language; some have suggested that the free
>  topos should be taken as the foundation for mathematics, e.g. Lambek
>  and Scott, whereas others that a theory of well-pointed toposes with
>  choice should be investigated, e.g. Mac Lane;
> (5) topos theory provides the appropriate framework for the investigation
>  of 'local' logical foundations, that is foundations for specific parts
>  of mathematics, e.g. differential geometry or algebraic geometry;
>  moreover, the axioms for a topos constitute the foundational invariants
>  of the logical foundations of mathematics

I wonder if Alain Badiou was able to show the philosophical coherence of
category theory, that is EpiFound(S, T) and OntFound(S, T), in his book:

Badiou, Alain. *Mathematics of the Transcendental*. Bloomsbury Publishing,

In Mathematics of the Transcendental, Alain Badiou painstakingly works
> through the pertinent aspects of category theory, demonstrating their
> internal logic and veracity, their derivation and distinction from set
> theory, and the 'thinking of being'. In doing so he sets out the basic
> onto-logical requirements of his greater and transcendental logics as
> articulated in his magnum opus, Logics of Worlds.

Book review by Andrej Bauer:

Book review by Colin McLarty:

Kind regards,
Jose M.
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