The n-Category Café

In case anyone is wondering, the n-Cafe is back in operation here in its usual place.

In Lawvere’s commentary to the linked article (Probabilistic Mappings ) he notes the actual intention of an averaging monad $P(A)=$ the part of $Hom_R(R^A,R)$ consisting of those functionals … provides for an easy route to understand the algebras of the Giry monad restricted to the full subcategory of standard Borel spaces. The trick is to take $R=[0,\infty]$, which is the one point compactification of $[0,\infty)$, which is a second-countable compact Hausdorff space (and hence $V=[0,\infty]$ is a Polish space). One now simply applies the tools that Lawvere and Rosebrugh lay out in their text Sets for Mathematics concerning weakly averaging $V$-generalized points. This leads to a fully faithful functor $\mathbf{Std}_{SCvx}^{op} \xrightarrow{Y} [\mathbf{V},\mathbf{Set}]$ which spits out the algebras (barycenter maps) $G(A) \rightarrow A$ using the fact that $\mathbf{V}$, the category with the single object $V$, is codense in the category $\mathbf{Std}_{SCvx}$. (The whole point of $V$-generalized points is to obtain such a fully faithful functor.) The details are written up on the nLab page for the Giry monad.

Of course, I advised against it, after having verified mathematically that the proposed system was unfeasible.

I wonder how one would mathematically prove that a plan to defuse guerilla tactics is not viable?

Ell oh ell.

Only a guy as mathematically sophisticated and practically naive as Lawvere could

i) take a few absorbing Markov chains and make them so inscrutable that DoD would decide “let’s dangle this in front of other militaries so it will cook their brains”

and

ii) miss the obvious fact that he was being cut loose, and most likely also being used by both the Americans and French in collaboration with each other versus independently.

Bill explained to me in a private conversation at the Matematisk Institut in Aarhus that it was his horror at the ‘war games’ he encountered in his youth at the Rand corporation that drove him toward Marx.

So many words enter the language as acronyms and thereafter different rules of capitalization apply. I am not sure whether RAND Corporation has reached that yet.

Bill’s conversation took place after Peter Johnstone had given a talk about Conway’s Numbers and Games. Bill suddenly exploded at the word ‘games’ to the audience’s surprise. Saunders came to the rescue and serenity was restored. But this incident persuaded me that Bill had suffered genuine trauma.

The category Lawvere introduced in 1962 does indeed look the same as the one described by Chentsov (and presented by Kolmogorov) in a short article in 1965 (the term “Markov category” does not appear, but “Markov morphism” does) and later used throughout his 1972 monograph (which calls the category of measurable spaces and “Markov morphisms” the “decision rule category”). This is peculiar, especially given that the latter monograph is often cited by modern books on information geometry as one of the foundational works.

I didn’t know about this episode in Lawvere’s life, but I find it interesting that it was connected to his work on categories of stochastic relations.

It was envisaged that a protocol would involve three tiers of verification: Space, Stratosphere, and On-site. The passage from one tier to the next would follow probabilistically from continuing observations. What would be the mathematical framework under which this whole fantasy would function? Someone described it as a “network of probabilistic mappings”.

I’m fond of the idea that probability and statistics can be seen as a branch of logic, as the word “inference” suggests. (The University of Barcelona used to have a department of Probability, Logic and Statistics, in that order.) Going further, I’m suspicious of any attempt to apply logic outside mathematics that doesn’t involve some probabilistic aspect. So it’s interesting that, in Lawvere’s telling, probability was built into their mathematical model from the start. His word “fantasy” suggests that he still took the whole thing with a big pinch of salt, even so.

Mathematicians are used to operating in a world of precise definitions and watertight chains of reasoning. I think we can be particularly prone to naively exporting that mode of argument to the real world, where nothing is precise or watertight — and then making it worse by attaching ourselves too confidently to our conclusions. This can lead us to overlook factors such as ambiguity, irrationality, stupidity, accidents, unforeseen events, hidden agendas, power struggles, dishonesty, deception, misinformation, factionalism, whims, corruption, laziness, and plain incompetence.