In the Footsteps of Rudolf Carnap II
Posted by David Corfield
The next day I set off East to Jena, following the path taken by Carnap, and by my host, David Green, a British mathematician who works on the cohomology of finite groups. While in Wuppertal, David had become interested in philosophy and had read my book, hence the invitation. In the afternoon I spoke to the mathematical colloquium about my work, stressing its two strands:
- The study of mathematics as a rational tradition of enquiry, with special attention to value judgements.
- That changes in the ‘foundations’ of mathematics brought about by -category theory should be explored both for themselves, and for their possible importance to philosophy.
I was very pleased to see in the audience a philosopher friend, Mike Beaney. Through his questions we established a certain ambiguity in my term ‘philosophy of real mathematics’. It could suggest that I am advocating that philosophers look to contemporary mainstream mathematics to ask and resolve completely new philosophical questions. I certainly don’t see it that way. The value judgements of a field like mathematics, and the way they are elaborated through time, in a community which supports a certain kind of dialogue are surely not radically different topics for attention. On the other hand, less common is my according this an historical dimension, in that, like scientists, mathematicians have learned how to learn, and presumably will continue to do so. An important difference boils down to how likely you think it is that mathematicians will have done important work to resolve fundamental tensions in the conceptual organisation of their field. I take it that that is precisely the first place we should look. To my mind, category theory and its higher-dimensional variety have been devised to deal with deep problems at the core of mathematical activity.
We also spoke about the second strand, and what I would need to do to attract philosophical interest. Mike is spending the year on sabbatical in Jena carrying out research for a book which will treat the many instances of mathematics’ influence on philosophy. Naturally, he does not want to rule out future occurrences. At dinner he asked me what category theory could do to take a step beyond Frege’s theory of a concept as a function. Let me elaborate. Frege, a mathematics professor in Jena, who gave logic lectures attended by Carnap, devised a logical calculus and an interpretation of this calculus which allows us to understand a concept as a function from some collection, , to the collection of truth values. Think of the concept ‘cat’. As we pass through , those entities which are cats get assigned the truth value ‘true’, while other entities, like Julius Caesar and the Eiffel Tower are assigned the value ‘false’. The number ‘2’ is then a second level concept, under which fall the concept ‘an ear of George W. Bush’ and the concept ‘astronomical bodies humans have stood upon’.
As we know from Bertrand Russell, there was a problem with this class I called , as it allowed his ‘set of all sets which don’t contain themselves’ paradox. But leaving this to one side, our -category theoretic ears should have pricked up. A concept is a function with codomain the set of truth values, or if you like the -category of -categories (or should that be -groupoid of -groupoids?). Climbing the ladder, as we do in these parts, our next step should then be to look for a functor from some category to the category of sets.
Time for a short aside. Having borrowed the term category from Kant, Mac Lane then borrowed functor from Carnap. I have never looked closely at Carnap’s use of the term. But in an article by Nuel Belnap we read:
At the abstract level that is relevant to our concerns, we think of a grammar as involving the following. Categorematic expressions, such as sentences or terms, with the idea that a semantics will then give a “value” of some kind to each categorematic expression. Syncategorematic expressions, such as “~” or “&” or “(”, which play a role in some grammatical operation. Grammatical operations, or modes of combination or functors, each of which is a (grammatical) function taking categorematic expressions as input, and producing a categorematic expression as output. Example: the operation which, given two sentential inputs and , produces an appropriate “conjunction” of those two sentences, perhaps having the appearance “( & )”.
I guess that gives us an idea of why Mac Lane chose it, although I don’t see that the notion of a functor taking arrows as arguments is present.
Returning to my Jena excursion, in a Czech bar after the meal, I talked the problem over with Mikael Johansson, a PhD student of David Green, a regular visitor to our Café, and himself a blogger. (By the way, if other regulars would like to tell us of their blogs, feel free to do so.) This is what we came up with. As domain take the category of people with a single arrow from to if is a descendant of, or identical to, . Then a richer understanding of the concept ‘ancestor’ than that provided by the function with truth values as codomain, is the functor to Sets which assigns the set of descendants to a person, and the relevant inclusion function to an arrow.
One important consideration is what to take as the codomain. Logicians and mathematicians tend to look at the world in a more typed way than most philosophers. One of the latter, Timothy Williamson, has argued for the existence of a set of all possibilia (things which can possibly exist), to provide a good domain. I suppose in our ancestor case we could pad our domain category out with a whole lot of objects, such as cars and coffee machines and Father Christmas, which will be mapped to the empty set, but this goes against the grain.
Returning to Frege’s ‘2’, we can see it as a functor from the groupoid of concepts, the equivalence relation of equinumerosity, to Sets, which is the connected components functor. Now, will that impress?
Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II
“Timothy Williamson, has argued for the existence of a set of all possibilia (things which can possibly exist)” begs the question, with an undefined use of ‘can possibly exist.’
This opens the door to the Ontological Argument, and other problematic issues. It also sounds like a piece of a definition which attempts to distinguish Science Fiction from Fantasy. For example, a human expedition to the moons of Saturn is possible; a human expedition to the Asgard of Norse myth is impossible.
Not to disrespect Carnap, a brilliant and important thinker, but his followers seem not to be as careful.