Jeffrey Morton - 2006-01

Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of N as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of N[x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic oscillator in which the group of phases, U(1), plays a special role. We describe a general notion of “M-Stuff Types” for any monoid M, and see that the case M = U(1) provides an interpretation of time evolution in the combinatorial setting, as well as other quantum mechanical features of the harmonic oscillator.

Robert K. Niven - 2007-04

This study critically analyses the information-theoretic, axiomatic and combinatorial philosophical bases of the entropy and cross-entropy concepts. The combinatorial basis is shown to be the most fundamental (most primitive) of these three bases, since it gives (i) a derivation for the Kullback-Leibler cross-entropy and Shannon entropy functions, as simplified forms of the multinomial distribution subject to the Stirling approximation; (ii) an explanation for the need to maximize entropy (or minimize cross-entropy) to find the most probable realization of a system; and (iii) new, generalized definitions of entropy and cross-entropy - supersets of the Boltzmann principle - applicable to non-multinomial systems. The combinatorial basis is therefore of much broader scope, with far greater power of application, than the information-theoretic and axiomatic bases. The generalized definitions underpin a new discipline of “combinatorial information theory”, for the analysis of probabilistic systems of any type.

Jaynes’ generic formulation of statistical mechanics for multinomial systems is re-examined in light of the combinatorial approach, including the analysis of probability distributions, ensemble theory, Jaynes relations, fluctuation theory and the entropy concentration theorem. Several new concepts are outlined, including a generalized Clausius inequality, a generalized free energy (“free information”) function, and a generalized Gibbs-Duhem relation and phase rule. For nonmultinomial systems, the generalized approach provides a different framework for the reinterpretation of the many alternative entropy measures (e.g. Bose-Einstein, Fermi-Dirac, Rényi, Tsallis, Sharma-Mittal, Beck-Cohen, Kaniadakis) in terms of their combinatorial structure. A connection between the combinatorial and Bayesian approaches is also explored.

It bums me out that the school system is basically set up to tell kids who struggle with the technical aspects of math & science that they are bad at math & science.

Most of the important progress in math & science has come from conceptual, rather than technical, insight.

Most of the important products of math & science are primarily characterized by creativity rather than skill.

Can someone with a background in art comment on the degree to which the above comments also apply to art?

How and to what degree can the axiomatization of geometry (by Euclid, c.  300BC), the invention of the number zero (by various people independently at various times), and/or the invention of category theory (by Eilenberg & Mac Lane, c. 1945) be described by physics?

Can the techniques of information theory or the theory of computation be brought to bear on these questions? Can the relationships between these fields and physics then tell us something about these (ultimately physical) processes, or about what physics should be like given that they seem to have occurred?

Will an answer along these lines help us reproduce or, better yet, automate this kind of phenomenon?

Could it tell us more about the physics of minds in general?

so the existential and universal quantifiers turn out to be the left and right adjoints (respectively) to weakening, which is the logical operation whereby you add an extra hypothesis (often called a “dummy” proposition/variable) to an entailment statement

as in if you had a statement like “A entails B” you could weaken it by saying “(A and C) entails B”

who knew

(via thatmathblog)

Who wants to talk or write about mathematical physics with me? I am primarily interested in the following ideas and in particular their potential interplay:

• categorical quantum mechanics à la Abramsky & Coecke et al.
• topos-theoretical physics à la Isham et al.
• homotopy type theory as foundations à la Voevodsky et al.

The Princeton Companion to Mathematics

(chapter on algebraic geometry)

SMBC #2245 by Zach Weiner

Mathematicians often joke that the only problem physicists know how to solve is the simple harmonic oscillator. This is, of course, only true to first order.

me lately