Using ‘Category Theory’ with the most liberal meaning:

Category: Error-correcting codes

a. Golay

b. Genetic

This is not to say that the genetic code is a Golay code, but apparently some, but not all genetic code errors can be corrected.

1 – Manish Gupta [Queen’s U, CA] reviews in IEEE publication ‘The Quest for Error Correction in Biology’ 2006

http://www.mast.queensu.ca/~mankg/public_html/public_html/home/doc/my_papers/ieeeemb.pdf

2 – Daniels DS, Woo TT, Luu KX, Noll DM, Clarke ND, Pegg AE, Tainer JA.

Skaggs Institute for Chemical Biology, The Scripps Research Institute

‘DNA binding and nucleotide flipping by the human DNA repair protein AGT’ 2004

I am accustomed to divide mathematics into algebra on one hand and set theory–>topology–>geometry on the other hand, and to freely use physicist’s intuition for both separately. I wonder how the axioms of topos relate to this division.

In set theory algebraic operations can be seen as maps from Cartesian square of algebra as linear space to algebra. In more algebraic context one would Cartesian square with tensor square. The latter picture would correspond to the physicist’s view about algebraic operation as a fusion of particles. Arithmetic SUSY emphasizes algebraic aspect and is certainly highly derived in an axiomatics starting from set theory view. In my mathematical innocence I am just wondering whether the SUSY picture could define a morning exercise in axiomatization;-).

]]>Yes, that’s it exactly. But I’m convinced the only solid axioms we have as a guide are the elementary **topos** ones (not counting NNO and successor). And the product/coproduct structure there is highly non-trivial. To me SUSY is a highly derived concept.

I agree that ‘factorization’ is fundamental. Please, if there’s anybody out there that knows of any interesting work on ‘number theory topos axioms’, TELL us!

]]>I believe that the philosophy underlying axiomatization should reflect very closely the process of how we become conscious about, say prime decomposition of integer. The set theoretic definition of integers does not do this.

It might be that category theory creates these problems by giving up totally the Platonistic notion of number theoretical anatomy in trying to reduce everything to arrows.

In the construction of infinite primes the number theoretic anatomy becomes fundamental and implies also generalization of the number concept.

One very interesting finding is that the quantum states corresponding to infinite primes at the first level of the hierarchy provide representations for ordinary integers, rationals and algebraics and can be mapped to them. Quite generally, higher level represents the level below it (successor again).

The basic conscious experiences about decomposition into factors are assigned with quantum jumps between infinite integers intepreted as many particle states. Number theoretical particle reactions are in question. There is single conservation law: the algebraic number assignable to the state is a multiplicative conserved quantum number. Conservation law states that total numbers of bosons and fermions in modes the labelled by ordinary primes are conserved.

Decomposition into primes becomes the primary concept in this approach. In this representation successor axiom would physically correspond to the possibility of adding arbitrary number of bosons to a given bosonic mode. Successor axiom would be formulated in terms of multiplication by saying that all powers of given prime exist.

Perhaps one could build axiomatization using super-symmetric arithmetic quantum field theory approach as starting point with emphasis on product and co-product aspects of number theoretic particle reactions.

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