## February 5, 2007

### 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:

1. The study of mathematics as a rational tradition of enquiry, with special attention to value judgements.
2. That changes in the ‘foundations’ of mathematics brought about by $n$-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, $U$, to the collection of truth values. Think of the concept ‘cat’. As we pass through $U$, 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 $U$, as it allowed his ‘set of all sets which don’t contain themselves’ paradox. But leaving this to one side, our $n$-category theoretic ears should have pricked up. A concept is a function with codomain the set of truth values, or if you like the $0$-category of $(-1)$-categories (or should that be $0$-groupoid of $(-1)$-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 $A_1$ and $A_2$, produces an appropriate “conjunction” of those two sentences, perhaps having the appearance “($A_1$ & $A_2$)”.

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 $X$ to $Y$ if $X$ is a descendant of, or identical to, $Y$. 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?

Posted at February 5, 2007 11:58 AM UTC

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### 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.

Posted by: Jonathan Vos Post on February 5, 2007 5:29 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

It also seems slightly odd to me to make a codomain so completely dependent on a timeframe. In 1750, an airplane would definitely not make it into the realm of possibilia, whereas nowadays, it’s a perfectly natural component.

Similarily, dragons seem to have left possibilia during the last couple of centuries; and time travel still is not there.

Posted by: Mikael Johansson on February 5, 2007 5:52 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

One must be careful not to mix epistemology and ontology, i.e., knowledge and what is. We may not recognise something as possibly existing, but if it possibly exists, then it is a possibile.

If you want to take a look at what metaphysicians do these days, take a look at an encyclopedia entry such as this one on possible objects, or you could try a top-flight paper such as this one.

Since analytic philosophy owes so much to logical positivism and logical empiricism, both inclined to dismiss metaphysics as meaningless, it should strike us as odd that metaphysics has bloomed in the past 40 years or so. Not all analytic philosophers believe this flourishing of analytic metaphysics is a good thing. Bas van Fraassen (Princeton) as an empiricist, has written against it in ‘The Empirical Stance’.

But even if we agree, we can still believe that category theory can help us with everyday and scientific language and cognition. Van Fraassen, who has stressed the importance of symmetry arguments in science, asked me once whether I thought category theory was important. And I believe some philosophers of physics are becoming interested.

Posted by: David Corfield on February 5, 2007 6:51 PM | Permalink | Reply to this

### Fiction and Imaginary Logic; Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

Thank you for those links, David. I agree with you that it is good that some philosphers are interested in Category Theory. I agree with the deficiencies of the Empirical stance. I agree that most scientists have a default metaphysics which is dumbed down Logical Positiivism. I agree that some mathematcians are forced to adopt a more sophisticated metaphysical stance.

The Stanford Encyclopedia of Philosophy has it right again. “One popular approach regards the notion of a possible object as intertwined with the notion of a possible world…. Another category of object similar to that of a possible object is the category of a fictional object.”

This is why, on another thread, I brought up N.A.Vasiliev’s imaginary logic. See, for instance,

www.logic.ru/LogStud/02/No2-08.html

staff.ulsu.ru/bazhanov/english/vas_gent.pdf

I was serious in mentioning Science Fiction and Fantasy, as both deal with fictional worlds.

Recent advances in extending Vasiliev’s notions into good axiomatic metalogic (his earlier exponents missed his central point about multiple universes) allow a logician in universe 1 using logic 1 to correctly and cosistently reason about beings in universe 2 using logic 2 or even, more subtly, beings in universe 2 alleged by other beings in universe 2 to be using logic 3.

Posted by: Jonathan Vos Post on February 5, 2007 7:48 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

I might have been responsible for the confusion, Mikael, when I only mentioned the epistemic interpretation of the modalities in my lecture. This is the one which runs:

$p$ is necessary $\to$ I know $p$
$p$ is possible $\to$ I do not know $\not p$

But this is not a metaphysical interpretation. See the Wikipedia entry for a range of versions of necessity and possibility: logical, physical, metaphysical.

I also mentioned in the lecture that by categorification one passes from propositional logic to predicate logic and from there to a form of modal logic. When I worked it out with John Baez we came up something close to $S 5_n$, a logic which has been used to model the epistemic states of multi-agent systems. Timothy Porter has written about this logic in Geometric Aspects of Multiagent Systems. (He also talks there of groupoid atlases, which we should probably learn about here.)

It seems likely we’d pick up something $S 4_n$-like if we thought in terms of directed homotopy.

Posted by: David Corfield on February 6, 2007 12:20 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

Quite probably: I’m still very much guessing more than thinking when it comes to the philosophical arguments: my logic education stopped just ahead of modal logic; and I still have problems keeping ‘epistemology’, ‘metaphysics’, ‘extensiality’ et.c. straight and separated from each other… I’d blame the fact that almost all my philosophy education comes from reading Russel’s History of Western Philosophy…

I still feel skeptical toward the realm of possibilia as such. It seems to either carry hidden traps, or be a cop-out: we don’t want paradoxes, so we state that we only look at this that aren’t paradoxical; which seems to me to be … questionable at best.

Posted by: Mikael Johansson on February 6, 2007 12:55 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

I’m certainly sympathetic to van Fraassen’s chapter Against Analytic Metaphysics, and would rather avoid talk of ‘possible worlds’, a key topic for metaphysicians. There might, however, be something worthwhile in thinking about modality in the context of classical mechanics, e.g., the path of least action from the set of all possible paths. Jeremy Butterfield has thought about this kind of thing.

Posted by: David Corfield on February 6, 2007 1:12 PM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

I also mentioned in the lecture that by categorification one passes from propositional logic to predicate logic and from there to a form of modal logic.

Do you have a link to this? This sounds quite interesting. And I am somewhat surprised that you did not end up with intensional logic – perhaps that arises after yet more categorification?

Posted by: JacquesC on February 10, 2007 12:57 AM | Permalink | Reply to this

### Re: Here there be Dragons; Re: In the Footsteps of Rudolf Carnap II

I think John noted down what we did in a notebook. Let me see what I remember.

Propositional logic has propositional symbols. In a model these take truth values. Given a composite sentence there is a set of truth value assignments for which the sentence is true.

Typed predicate logic has a collection of types, and a collection of typed relation symbols. A valuation assigns a set to each type, and a subset of the relevant product to each relation. A sentence has a groupoid of models which satisfy it.

Typed quantified modal logic has three layers of syntax: a collection of metatypes, and a collection of metatyped relational types, and a collection of typed relations. The metatypes each correspond to a groupoid of worlds. There are variables for worlds within metatypes, and for individuals within worlds.
A sentence has a 2-groupoid of models which satisfy it.

Hmm, I’d have to think a bit harder about how it all worked.

Posted by: David Corfield on February 11, 2007 3:15 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

Having borrowed the term category from Kant, …

By the way, does anyone except me feel that the term is not all that good, actually?

I am a complete ignorant of the relevant history, but my impression is that the use of the word “category” for categories is motivated from those categories that come to as in the form of collections of sets with extra structure and structure preserving maps between these.

Maybe I am wrong about that. You’ll correct me. I guess I am saying that “category” is motivated as meaning something like “concept”.

For instance: the category of vector spaces $\leftrightarrow$ the concept of a vector space (set with such and such a structure).

Maybe what I am saying is this: the word “category” is well motivated from the point of view of category theory as “meta-mathematics”: it allows us to categorize lots of mathematical concepts.

The term seems to be much less well motivated (to me, that is, but maybe it’s a language issue anyway) for categories that we are interested in already as entities in themselves. Like a group. Or a poset. Or the pair groupoid of some set.

For all these concepts, the word “category” seems ill-motivated to me.

Even if I could turn back the wheel of time and intervene when MacLane baptizes categories, I wouldn’t really have a good alternative to offer though.

“Monoidoid” is very entertaining but arguably perverse.

“2-Set” might be good.

Posted by: urs on February 5, 2007 7:26 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

The 2-Set-suggestion makes me immediately want to interpret a 2-whatever to be a category enriched in whatevers. Now, David said here in Jena that albeit the hom’s in the category of groups only then is a group when we stay abelian, this is not the case with groupoids: so that for groupoids we really do get groupoid hom’s as well. So this makes me figure that groupoids form a category enriched in itself?

However, we do understand something else entirely under “2-groupoid” – how much of a problem would this arguably closely staged overloading be?

Posted by: Mikael Johansson on February 6, 2007 12:49 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

One ought to speak of the (strict) 2-category of groupoids being cartesian closed. Groups form a sub-2-category which is not closed.

The trouble with choosing the zeroth step of the ladder, like ‘set’, to form terminology from, is that as you haven’t got started with your climbing, it’s not so obvious what the first step, here a ‘2-set’, should be.

Posted by: David Corfield on February 6, 2007 1:28 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

The trouble with choosing the zeroth step of the ladder, like ‘set’, to form terminology from, is that as you haven’t got started with your climbing, it’s not so obvious what the first step, here a ‘2-set’, should be.

I agree. I am not seriously promoting to say “2-set” for “category”.

Posted by: urs on February 6, 2007 1:44 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

David wrote:

The trouble with choosing the zeroth step of the ladder, like ‘set’, to form terminology from, is that as you haven’t got started with your climbing, it’s not so obvious what the first step, here a ‘2-set’, should be.

Right! Especially since what seems like the ‘zeroth step’ of the ladder often turns out to be the first or second step.

Here’s the classic example:

First people realized that 3,4,5,… were numbers.

Only later did they realize that 2 was a number! And indeed, I’d still feel odd saying I have ‘a number of hands’. The special role of the number 2, perhaps due to the bilateral symmetry of the human body and the existence of two sexes, pervades our thinking. The Greeks had a ‘dual’ case separate from the ‘plural’. Even in English, we distinguish between ‘both’ and ‘all’, a ‘couple’ and a ‘few’, etcetera.

And then, later still, people realized that 1 was a number. As late as the work of Plato, the concept ‘one’ was treated as the opposite of number, or ‘many’!

And then, even later, people realized that 0 was a number. Perhaps the Indians did this first.

The exact same phenomenon happened in category theory, which is why our terminology is now saddled with $-1$-categories and $-2$-categories.

You only notice you’re climbing after you’ve been climbing a while!

Posted by: John Baez on February 7, 2007 4:35 AM | Permalink | Reply to this

### Grammatical interlude

JB said:

The Greeks had a ‘dual’ case separate from the ‘plural’.

You mean a dual number, not a dual case! Case and number are different grammatical phenomena, though they both involve nouns.

Dual number is actually quite widespread among the world’s languages. Other languages with dual number include Hebrew and Anglo-Saxon (Old English).

A much smaller number of languages have not only dual but also a “trial” (three) or “paucal” (a few) number.

Posted by: Tim Silverman on February 7, 2007 8:06 PM | Permalink | Reply to this

### Re: Grammatical interlude

Other languages with dual number include […] Anglo-Saxon (Old English).

Traces of this survive in modern English, such as the ‘both’/‘all’ distinction that John mentioned.

Posted by: Toby Bartels on February 7, 2007 10:21 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

The trouble with choosing the zeroth step of the ladder, like ‘set’, to form terminology from, is that as you haven’t got started with your climbing, it’s not so obvious what the first step, here a ‘2-set’, should be.

As John mentioned, ‘set’ isn’t really the zeroth step of the ladder. Once you see have a set is a categorified truth value, it’s not too hard to see that a 2-set is … wait for it … a groupoid. In historical reality, of course, we only noticed this after categories (and even higher n-categories) appeared.

Posted by: Toby Bartels on February 7, 2007 10:37 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

The 2-Set-suggestion makes me immediately want to interpret a 2-whatever to be a category enriched in whatevers.

It varies, sometimes a 2-foo is a category enriched over the category of foos (such as when foo = category), sometimes a category internal to the category of foos (such as when foo = group), sometimes something a bit else (such as when foo = groupoid, since a 2-groupoid is a groupoid enriched over the category of groupoids). And you usually have to throw in some weakening to get the really correct concept (as in all my examples above). So have to be open to possibilities.

Categorification, like quantisation in physics, is (in general) an art, not (except in particular circumstances) a science, if you see what I mean.

Posted by: Toby Bartels on February 7, 2007 10:33 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

I enjoyed your talk, David, and you said a number of interesting things. So let me try to clarify the question I raised about what you meant by a ‘philosophy of real mathematics’.

In the first part of your talk, you took the example of Lakatos’ discussion of the development in mathematicians’ understanding of the concept of a polyhedron and mentioned some criticisms of this discussion. Here I took one of your points to be that as philosophers we should pay more attention to the actual development of ideas about mathematical concepts. So ‘philosophy of real mathematics’ might here mean ‘philosophy of historically-situated mathematics [i.e. mathematics as it is actually practised, or actually developed]’.

In the second half of your talk you argued for the importance of understanding category theory, as a more powerful general theory, which subsumes set theory. Here I took ‘philosophy of real mathematics’ (as advocated by you) to mean something like ‘philosophy of cutting-edge mathematics’. You seemed to be saying that philosophers of mathematics were more concerned with debates about logicism, intuitionism and formalism, emerging out of late 19th/early 20th century work on mathematical logic, than on recent developments.

This leads naturally to the question we discussed over dinner. What is the philosophical significance of category theory? What new philosophical insights does or could it yield? Does it support structuralism, for example, rather than logicism, etc.? Frege’s importance in logic and philosophy lay in his use of function-argument analysis. If ‘functor’ is a broader concept than ‘function’, what philosophical implications does this have? How does category theory change (if it does) our conception of what mathematics is - and constituent conceptions such as that of number? These are the key questions.

In short, you wanted to say that philosophers of mathematics should pay attention to category theory, but didn’t spell out the potential philosophical pay-off.

Mike Beaney

Posted by: Mike Beaney on February 5, 2007 7:55 PM | Permalink | Reply to this

### Re: In the Footsteps of Rudolf Carnap II

Thanks for this, Mike. As concerns the first half of the talk, where the question is to understand what it is to act rationally in a community such as the mathematical one, my point about looking at more recent mathematics is that new considerations have been discovered since the period Lakatos treated, the neglect of which would constitute irrationality in today’s mathematicians. Furthermore, Lakatos by no means exhausted the full range of such considerations operating at the end of the nineteenth century. Indeed, one could say that he only scratched the surface, as his omission of Riemann’s very important contribution to the modern treatment of the Euler formula bears witness to. For example, he misses out on something close to what Whewell called ‘consilience’.

As for the second half of the talk, category theory supports a form of structuralism, but it’s not the structuralism of Resnik or of Shapiro. Steve Awodey has expressed it very well at the level of ordinary (1-)categories. Page 12 discusses what can be said about natural numbers.

It’s not clear to me what would constitute compelling evidence of category theory’s philosophical usefulness. For me, Awodey would have done enough, and this is only the beginning of what’s possible. As many entries on our blog have shown, it offers insight into many foundational points of physics. It also provides an excellent framework for thinking about logic and computation, and for comparing what is similar between physics and computer science.

There’s a question of the burden of proof. Think of Russell bringing back that strange German and Italian logic to England. If no-one had made an effort to understand him, and those who couldn’t understand hadn’t trusted him, how different the history of philosophy might have been.

Posted by: David Corfield on February 5, 2007 10:55 PM | Permalink | Reply to this