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One of the modern trend amongst mathematical philosophers has been the application of categorical foundations to logic. Lawvere is probably the best known for his use of closed cartesian categories in the various paradoxes. (source)

Have there been any notable criticisms of Lawvere's theories from a philosophical perspective?

  • +1, Interesting question - but I'm not sure if it's answerable. What would you consider as a "philosophical perspective" criticizing Lawvere? We know that you can represent boolean logic in category theoretic terms, so it's unclear what this kind of criticism could event look like. – smartcaveman May 3 '13 at 20:43
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I don't know a lot about the debate, but I know for instance Solomon Feferman has written a couple of papers on this topic, e.g. "Categorical Foundations and the Foundations of Category Theory". John Bell has also examined this issue critically in "Category Theory and the Foundations of Mathematics."

Better yet, I'm sure you could find more in the bibliography section of SEP's entry on category theory.

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Theres been a lot of ferocious critique from set theoreticians perhaps generating more heat than light as they see categorical foundations as intruding on their own turf.

Most philosophers interested in foundations are schooled in foundations in ZFC style. Category Theory hasn't made much inroads here - but this will probably change in the future as the importance of Category Theory as a structuring device in mathematics has only become more pronounced, and as more philosophers are schooled in Category Theory.

Whilst ZFC is generally taught in the first year of an undergraduate course, one only gains appreciation of the categorical view after a substantial amount of sophisticated mathematics has been absorbed, say by the end of a 1st year graduate course.

Lawvere certainly has done a great deal of good work in categorical foundations but the key concept is that of a Topos which is a generalised Set Theory, which have an implicit logic tied to it (via its internal language which is usually some form of type theory) and geometric interpretation (via sheaves).

An important observation is that the logic is not Boolean but intuitionistic, that the Axiom of Choice is not always upheld and that they do not necessarily have an infinite object. When these are asserted then of course we have a much more set-like topos. Of course there is a huge zoo of toposes rather than the unique ZFC that one supposes one has.

The n-lab a repository of category-theoretical work by higher dimensional category theorists have a reference on Foundations.

One would suppose that the philosophical school known as Structuralism would have a great deal to say about Category Theory as Structure is a notion that pervades both fields. But Structuralism is associated more with architecture, , linguistics (its originary home), literary studies and anthropology than it is with the mathematics, so the essential overlap between the two fields is actually zero.

Whereas ZFC is the endgame of reductionalism in mathematics - reducing all mathematics to sets, and then sets to logic+axioms; Category Theory takes each mathematical activity to have its on natural home, its own place - instead of a sharply hierarchical view it looks towards a more expansive relational, intrinsic and natural view - it is not Descartian.

The paper you refer to is mathematics proper rather than philosophy - the structural similarity of certain arguments in the paradoxes that Lawvere re-examines had been already noted and just generally referred to as a diagonal argument in honour of Cantors first use of this kind of argument to show that the continuum had a larger cardinality than the integers. After the invention of Category theory by Maclane it was possible to place these arguments in a host of different contexts in a systematic format.

The use of closed cartesian categories is probably a red herring too - essentially these are categories in which the notion of an exponential naturally occurs and satisfies the usual laws. Given how prevalent the exponential in ordinary mathematics it would seem 'obvious' that the same notion would be just as useful in more general contexts where mathematics are done such as the category of groups or of rings etc. This has proved to be the case. From there its just one step to investigate the what kind of beast is a cartesian closed category. Its not really, I think, a foundational issue.

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