Douglas Hofstadter on Recursive Functions and The Abstraction Ceiling

... and physics, ... given in April 2015.

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This was given in September 2006:

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The curious thing is that at the beginning of that talk he describes analogy in the same way Euclid described it (following Eudoxus) as "A is to B as C is to D" and if you write that as a ratio you get A:B = C:D. More generally you have A:B::C:D = A:C::B:D which holds for irrationals and ἀναλογία as well as irrationals (See Proposition 16 of Book V of Euclid's Elements). This is analogous to Hofstadter's "Interchange" relation (22:35) he describes in the first talk above (14:25).

Yesterday I was wondering about duality (see Dual Spaces, Lie Groups and Conservation Laws) and whether there could be any reason behind why it seems so ubiquitous in formal reasoning. I think analogy must be something fundamental in our ability to learn and interpret language. So our ability to spot analogies and use them to interpret sentences we have never heard before is what gives rise to the syntactic (i.e. grammatical) structures we see in all languages. Dualities such as those which hold in vector spaces, finite sets and propositional logic have a purely syntactic structure, but they are only considered significant because that syntactic transform happens to correspond to something 'real'. So for example the duality Aristotle refers to as "proportionals alternate" is the statement "If A is to B as C is to D then A is to C as B is to D" is only significant because it holds for any finite magnitudes whatsoever: numerical, rational, rational in square only, irrational or ἀνάλογον. So I think that the reason dualities are ubiquitous is that they are stable under reinterpretation. Ultimately it is because we only ever have access to linguistic descriptions in order to compare experience, and this process naturally produces adjunctions and their associated dualities.

See What is Formal Logic?

I think he should talk to Sophie Maclean. This problem has a lot of the kinds of patterns in the first talk above:

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It's funny. This morning I was just thinking that there ought to be an award for people over the age of 40 who have done good maths. The Fields medal is for people younger than 40 I think. Then I thought that this medal, rather than being given to just a few people each year, ought to be given to about 40 per year, and maybe it should be possible to get one after you're dead, but then I think it'd be many years before it was awarded to anyone still alive, and in the mean time all those candidates would probably have died!