The n-Category Café

Sometimes I feel I’ve finally learned British English. For example, I’ve not only learned that an “anorak” is a waterproof coat; I also understand what people mean by saying “he’s a real anorak”.

But then someone says something like “philosophically pukka” — and my bubble of hubris is rudely popped.

“Pukka” sounds bad, but from context I’m guessing it’s good.

I’m feeling sorry for David. It must be disappointing to post a blog entry about connections between probability theory and logic and trigger nothing but a discussion about the word ‘pukka’.

So, let’s talk a bit about that probability monad.

Roughly speaking, this is a functor sending any space $X$ to the space of probability measures on $X$.

‘Any space’ — but what sort of space? David said we should use certain nice topological spaces called Polish spaces. I’m sure this has advantages, but it’s a bit technical. I think we should postpone such technicalities until we understand simpler issues.

So, let’s work with a simpler monad

$$M: Set \to Set$$

sending any set $X$ to the set $M X$ whose elements are certain very nice measures on $X$: finite positive linear combinations of Dirac delta measures.

The reason I like $M X$ is that it’s the free $[0,\infty)$-module on $X$!

Here $[0,\infty)$ is not a ring, but a rig with the usual addition of real numbers as $+$ and the usual multiplication as $\times$. We can talk about $R$-modules for $R$ a rig, just as we can when it’s a ring. The free $R$-module on $X$ consists of finite $R$-linear combinations of elements of $X$. When $R = [0,\infty)$, these are the same as finite positive linear combinations of Dirac delta measures on $X$.

David should like this, because it begins to develop a bridge between the probability monad and the wonderful world of matrix mechanics over arbitrary rigs.

For any rig $R$ there’s a monad

$$T_R: Set \to Set$$

sending any set $X$ to the underlying set of the free $R$-module on $X$. I just explained why

$$M \cong T_{[0,\infty)}$$

But, it’s also interesting to take $R$ to be the rig of real numbers, or complex numbers, or truth values — or the rig of costs $\mathbb{R}^{min}$, or the temperature-dependent rig $\mathbb{R}^T$. As David knows and the links show, all these are interrelated in a wonderful way. Probability theory is just part of this big picture, which also includes classical mechanics, quantum mechanics and Boolean logic.

Or is it???

You’ll notice my monad

$$M : Set \to Set$$

sends a finite set $X$ not to the set of probability measures on $X$, but to the set of all measures on $X$.

That’s fine if we’re willing to accept the concept of ‘relative probabilities’, which aren’t necessarily normalized so their sum equals 1.

But what if we annoyingly insisted on working with probabilities that sum to 1? Then we’d get the monad

$$P : Set \to Set$$

sending any set $X$ to the set of probability measures on $X$ that are finite linear combinations of Dirac delta measures. This is a watered-down version of David’s probability monad.

Now, there’s no rig $R$ such that

$$P \cong T_R$$

The annoying condition that probabilities must sum to 1 stands in the way.

But, I believe there’s a generalized ring that does the job — a generalized ring in the sense of Nikolai Durov. This is the guy who generalized algebraic geometry to handle weird generalized rings like ‘the field with one element’. David already has already written about this.

Indeed, if you read my summary of Durov’s enormous book, you’ll see that one of his generalized rings is nothing but a monad with a certain property!

And, I think my probability monad

$$P : Set \to Set$$

simply is a generalized ring according to Durov’s definition! Unless I’m confused, it’s even a ‘commutative’ generalized ring!

So, everything David likes is part of the same big picture: the probability monad, the field with one element, the rig of truth values, the rig of costs, and so on. It’s all about linear algebra over Durov’s generalized rings. In particular, using these generalized rings, we can really talk about a ‘generalized ring of probabilities’, even though probabilities must sum to 1 — a condition we could never demand for arbitrary finite sets of elements of a mere rig.

If we’re going to combine probability theory and logic in a really nice way, maybe we should take advantage of this big picture. It may not be philosophically pukka, but it’s certainly pukka.

Perhaps we should follow up David Mumford’s proposal for a stochastic predicate calculus.

It should have the syntax of standard predicate calculus except that we have two kinds of variables in it: the ordinary predicates and constants and quantiable free variables $x$ but also a set of random constants $\underline{x}$. In addition, it comes with a truth value function $p$ mapping all formulas $F$ without free variables to real numbers between $0$ and $1$. If the formula $F$ has only ordinary variables in it, then $p(F) \in \{0, 1\}$. Formal semantics for this theory would make the random constants functions on probability spaces so that a formula would define a subset of the product of these spaces, hence have a probability. (pp. 11-12)

This has surely got to be a useful observation:

As a final example of a hyperdoctrine, we mention the one in which types are finite categories and terms arbitrary functors between them, while $P(A) = S^A$, where $S$ is the category of finite sets and mappings, with substitution as the special Godement multiplication. Quantification must then consist of generalized limits and colimits…By focusing on those $A$ having one object and all morphisms invertible, one sees that this hyperdoctrine includes the theory of permutation groups; in fact, such $A$ are groups and a “property” of $A$ is nothing but a representation of A by permutations. Quantification yields “induced representations” and implication gives a kind of “intertwining representation”. Deductions are of course equivariant maps. (Adjointness in Foundations, p. 14)