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November 30, 2008

Tom Leinster in The Reasoner

Posted by David Corfield

For an editorial by me and an interview with Tom Leinster in our in-house journal here at Kent, read this month’s edition of The Reasoner. Posted at November 30, 2008 11:54 AM UTC

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Re: Tom Leinster in The Reasoner

Great interview! When it suddenly ended and the subject turned to the omnipotence of God, for a second I thought we’d be hearing Tom’s view on this issue.

I’m not quite sure what David meant by the word ‘oblique’. It’s actually an interesting issue, how we (and Tom in particular) reconcile our interest in nn-categories with our interest in various other specific subjects. I remember being disappointed at first when, after writing his book on nn-categories and higher operads, Tom ‘suddenly switched’ to working on fractals. But now I see how nicely it fits into Tom’s broader interest in categorification, and I think it was a very good move to make.

Posted by: John Baez on November 30, 2008 6:42 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

We did the interview in one ‘take’, as it were, so I didn’t have time to specify what was meant by ‘oblique’.

I meant to suggest that Tom’s path was rather unusual, off the beaten track, avoiding the better trodden paths of representation theory, homotopy theory, algebaraic geometry, etc.

I was wondering about linking this to the oddness of some forms of British graduate education. I have the impression you’re left to your own devices rather more than had you been at a US grad school, Chicago say.

My own Part III education was a quirky mix of modules, including John Conway giving his own personal take on representation theory.

Posted by: David Corfield on November 30, 2008 8:41 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

I had the benefit of both systems =
Princeton and Oxford. cf. two theses
which appeared in print as
Homotopy Associativity of H-spaces I and II
Depending on your supervisor at Princeton, you could also be left pretty much on your own or at the meercy of fellow grad students, though there were courses and Topology seminars - at the U and at the IAS.

Biggest contrast was in preparation of the final document - Princeton had standard US requirements like paper and margins, etc. Oxford required the thesis be bound but as to format therein, my last superviosr when I asked responded: well, if you are going to submit a sonnet, it should have 14 lines!

Posted by: jim stasheff on November 30, 2008 8:57 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

It was a very nice interview; thanks.

Regarding whether category theory interacts with fluid mechanics: I thought I had heard somewhere that for a number of years, Eduardo Dubuc was actively involved in the study of fluid mechanics (or something like that). But I haven’t been able to corroborate that independently.

Posted by: Todd Trimble on November 30, 2008 8:41 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Yeps, great interview, gives me lots of ideas for how to speak about higher category theory to a broader audience. I too was caught out by the abrupt ending, and was quite surprised (in a good way!) to see Tom suddenly talking about the problem of the omnipotence of God.

Posted by: Bruce Bartlett on December 1, 2008 6:14 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

in re: the omnipotence of God

since it’s a question of logical inconsistency that’s invoked,
wasn’t something like this debate
in the earliest univesities?

Posted by: jim stasheff on December 1, 2008 11:32 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

It was thoroughly hashed over in the Middle Ages, though perhaps the simplest solution — there is no omnipotent being! — was overlooked.

Posted by: John Baez on December 2, 2008 12:01 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

My usual response to the “too big to lift” question is: “Lift above what?”. The whole question appears to be ill defined.

Oh, and is the collection of all things an Omnipotent being can do a set or a proper class?

Posted by: Mark Biggar on December 2, 2008 12:40 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

I wonder if there’s more than a hint of similarity between the anti-omnipotence argument and the argument of Baer against Finsler’s notion of a maximal system. Each wants to say that there’s a contradiction in the assumption that there can be a maximal thing. Just as Finsler argues that Baer’s construction is invalid, we have Mark here questioning the lifting construction. John is being Baer by suggesting the no omnipotent being position.

An analogy wouldn’t be so surprising if we remember that Cantor held that there was an absolute infinite far beyond anything we paltry humans could attain with the powerset construction. He saw that absolute infinity as some aspect of God.

Must dash off now to begin my treatise – God from the coalgebraic point of view.

Posted by: David Corfield on December 2, 2008 9:20 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

And maybe the devil from the algebraic point of view? Does this mean that analysis is God and algebra is the devil? Some other guy thought geometry was God, but he agreed that algebra is the devil.

Posted by: James on December 3, 2008 1:42 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Some other guy being Alain Connes, perhaps? Cf. this paper.

Posted by: Todd Trimble on December 3, 2008 3:15 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Since Connes’ vision of NCG relies heavily on finding the algebraic dualizations of geometrical/topological/measurable notions, I’m surprised if he actually said that in all seriousness…

Besides, in the (interesting) paper you link to, a quick skim suggests that said phrase only occurs as as a section title. Isn’t it a quote of, or nod to, someone else? Weyl? Weil?

Posted by: Yemon Choi on December 3, 2008 4:13 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

I thought it was Hilbert, but I’m not sure.

Posted by: Kea on December 3, 2008 4:35 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Weyl perhaps:

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.

Posted by: David Corfield on December 3, 2008 9:00 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

This may be a good excuse to repeat a lesson I was taught about by Todd, and which I am very fond of:

This vague

duality

Geometry | Algebra

which Connes mentions on p. 6 is what Lawvere identified in exact terms as the duality between Space and Quantity, namely that between presheaves and co-presheaves.

With this in hand, further nice things follow. For instance, regarding Connes’ section 3.2 titled “Cohomology”. In regard of the above Space/Quantity-duality there is the following profound statement to be made:

we can ask if Staces=presheaves “glue”. The failure of that is measured by cohomology.

we can ask if Quantities=co-presheaves “co-glue”. The failure of that is measured by homotopy.

Posted by: Urs Schreiber on December 3, 2008 7:56 AM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Fred Almgren in response to a serious question:

bubbles are homology and ballons are homotopy

or maybe it was the dual

Posted by: jim stasheff on December 3, 2008 2:06 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Thanks for the link
I had hesitated to say Medieval being unsure of my time frame. The portrait indicates Muslim philosophers also considered the paradox? How about the classical Greeks?

cf G and S: a pretty taste for paradox

Posted by: jim stasheff on December 3, 2008 2:01 PM | Permalink | Reply to this

Re: Tom Leinster in The Reasoner

Since the ancient Greeks weren’t monotheistic, it’s hard to imagine them taking the possibility of an omnipotent being seriously enough to worry about the paradoxes such a thing would lead to. A paradox is only fun if you believe the assumptions.

Posted by: John Baez on December 6, 2008 5:07 AM | Permalink | Reply to this

Coalgebra and Modal Logic

I just felt the need to point out the relationship between modal logic and coalgebras. As modal logic is seen as the logic of coalgebras. Modal logic is not only important in computer science but ever since Kripke used his semantics to form an argument against materialism, modal logic has been of great interest to philosophy.

So the question is.. what is the coalgebraic point of view on the mind-body problem?

Posted by: Paul on December 2, 2008 9:02 PM | Permalink | Reply to this

Obvious answer; Re: Coalgebra and Modal Logic

“… what is the coalgebraic point of view on the mind-body problem?”

Dual to the algebraic point of view on the comind-cobody problem? Or is the structure twisted so that it is dual to the cobody-comind problem? Or perhaps the comind-cobody coproblem?

Posted by: Jonathan Vos Post on December 3, 2008 5:05 AM | Permalink | Reply to this

Re: Coalgebra and Modal Logic

What relationship between modal logic and coalgebras do you mean? Certainly the modal operator ‘necessity’ can be described in terms of comonads, but it sounds like you’re suggesting some other connection: what does the statement “modal logic is seen as the logic of coalgebras” refer to?

Posted by: Todd Trimble on December 3, 2008 12:54 PM | Permalink | Reply to this

Re: Coalgebra and Modal Logic

No doubt we’ll need to read Modal Logics are Coalgebraic.

Posted by: David Corfield on December 3, 2008 2:08 PM | Permalink | Reply to this

Re: Coalgebra and Modal Logic

I’d also recommend Algebras and Coalgebras by Yde Venema from the Handbook of Modal Logic.

Posted by: Paul on December 3, 2008 8:13 PM | Permalink | Reply to this

Re: Coalgebra and Modal Logic

Great! Thank you, David and Paul.

Posted by: Todd Trimble on December 3, 2008 9:55 PM | Permalink | Reply to this

Re: Coalgebra and Modal Logic

Barwise and Moss discuss this subject in “Vicious Circles”. It’s a really fun book as well.

Posted by: Dan Piponi on December 6, 2008 1:02 AM | Permalink | Reply to this

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