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July 16, 2007

Mathematical Imperatives

Posted by David Corfield

I like the way Yuri Manin generally throws a little ‘philosophy’ into his papers. In his Generalized operads and their inner cohomomorphisms with D. Borisov, they write:

One can and must approach operadic constructions from various directions and with various stocks of analogies. (p. 4)

That ‘must’ is interesting to think about. You might look to deontic logic for help, and be relieved that Kant’s Law (‘must implies can’) is satisified. But perhaps the more interesting question is ‘Must, or what will happen?’

Perhaps, something like: you’ll fail to understand operadic constructions fully, which would be failing in your duty as a mathematician.

They continue,

In this paper, we look at operads, especially those with values in abelian categories, as analogs of associative rings; collections are analogs of their generating spaces. We imagine various noncommutative geometries based upon operads, and are interested in naturally emerging symmetry and moduli objects in these noncommutative geometries.

But of course there are many more different intuitive ideas related to operads.

a) Operads provide tools for studying general algebraic structures determined by a basic set, a family of composition laws, and a family of constraints imposed upon these laws.

b) Operads embody a categorification of graph theory which can be used to study knot invariants, Feynman perturbation series etc.

c) Operads and their algebras are a formalization of computational processes and devices, in particular, tensor networks and quantum curcuits [sic]… With this in mind, we describe general endomorphism operads in 2.5 below.

It is interesting to notice that the classical theory of recursive functions must refer to a very special and in a sense universal algebra over a non-linear “computational operad”, but nobody so far was able to formalize the latter. Main obstacle is this: a standard description of any partially recursive function produces a circuit that may contain cycles of an a priori unknown multiplicity and eventually infinite subprocesses producing no output at all.

For another Maninian gem, see here.

Posted at July 16, 2007 11:48 AM UTC

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Re: Mathematical Imperatives

Then why one can’t describe “operads with cycles”? Or describe computations as “equations over operads” (in order to provide recursion)?

Sorry again for my dilettantic questions.

Posted by: osman on July 16, 2007 12:58 PM | Permalink | Reply to this

Re: Mathematical Imperatives

I imagine the problem is that some functions are only partially defined, i.e., for some inputs a program need not halt. So how to represent with operads, say, the Turing machine which just loops around forever whatever the input?

Presumably something would be left out were one to resort to the trick of telescoping the function so that its output is ‘undefined’, having expanded the domains to include an undefined element.

Posted by: David Corfield on July 17, 2007 9:42 AM | Permalink | Reply to this

Re: Mathematical Imperatives

Reminds me of the subtitle of the famous book by my friend and coworker Ted Nelson, inventor of Hypertext and Hypermedia:

Computer Lib: You can and must understand computers now/Dream Machines: New freedoms through computer screens–a minority report (1974), Microsoft Press, rev. edition 1987: ISBN 0-914845-49-7

Posted by: Jonathan Vos Post on July 16, 2007 10:55 PM | Permalink | Reply to this

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