In his science-fictional “Foundation Trilogy”, Isaac Asimov famously hypothesized a future science called “psychohistory”, a mathematically grounded theory of generalized and predictive human action, based on an amalgamation of psychology, history, and sociology. The future galactic empire was managed by this theory and practice (look out – almost seventy year old spoilers!) except for an exceptional character that was not anticipated and essentially unpredicable.

Asimov had in mind well validated continuous and statistical theories of physics, for example for idealized gases and their laws. I was stuck by an image for an explanation of turbulence that highlighted key elements of velocity, density, pressure, and viscosity, and how it was (in my mind) analogical to antagonistic individuals, dominating leaders, submissive society, and affiliated coteries. Of course, an article below states that turbulence is still too complicated to provably model correctly at this point in time.

I had no idea that psychohistory was claimed to be an actual field of study these days, albeit being somewhat controversial in its authenticity. And it doesn’t seem to have any mathematical basis yet, as far as I know. Mathematician Dan Crisan gave an inaugural talk a few years ago that was hypothesizing using heat equations instead of fluid dynamics as a basis. Even so, we can’t seem to properly model any sort of social action so how could psychohistory be within our grasp?

In these turbulent times perhaps we should make an effort to understand ourselves a bit better, as we hope to navigate between the Charybdisian whirlpool of civil discord and environmental collapse and the Scyllaian rocks of fascism, authoritarianism, and / or totalitarianism. But hey, isn’t Apple doing an Apple TV+ series based on Asimov’s books? Let’s all tune in!

I’ve been interested in Category Theory for a substantial number of years, but have never dedicated the time necessary to learn it properly. But it seems to me that these are great days (salad days?) for learning at least the rudiments of the subject. There are now wealths of copious materials available on-line for free for one’s self-study and enrichment, as well as new and classic treatises on Categories and their theory.

Category Theory has been called “abstract nonsense”, but while it is very abstract, it is hardly nonsense. Like set theory, it can be used to study the foundations of mathematics. Like algebra, it can be used to study generalized structure and relationships in math. Like the assortment of tools that is computer science, it can be used to study the essence of logic and computation. And like calculus, it can be used for all sorts of applied and scientific purposes.

Like all good math, Category Theory (CT) generalizes concepts that lie at the heart of many branches of mathematics. These concepts allow the mathematician to see similarities between these different branches, and carry them over into others as well. You might think “maths” is all one thing, and it is, roughly, but like science, it has evolved into a myriad of subjects and specialities. Ontologically (or perhaps even categorically), CT is listed as a topic under algebra, and it in turn has its own distinct branches.

And like all good math, CT benefits from a judicious choice of definitions and properties, that balance generality with precision to great expressive advantage. This balance between abstraction and concreteness gives it the power and utility that it has enjoyed for the better part of seventy-five years. This makes CT rather a new-comer to mathematics, but please don’t mistake youth for lack of expertise.

If I wanted to represent CT emblematically, I might suggest the diagram above. At root, a category merely consists of a collection of objects and the pairwise morphisms (or arrows) between them. But additionally, the arrows must also obey a small set of conditions, so the objects usually have a similar nature. This nature can be quite abstract though, and one of the most familiar “concrete” categories is that of sets and the functions between them.

If arrows are the mappings between objects, then “functors” are mappings between categories, that once again have to obey some rules to maintain structure. So you can think analogically that arrows are to objects as functors are to categories and you wouldn’t be too wrong. Next, you can imagine generalizing to a higher level that there are mappings between functors, and “natural transformations” are indeed defined to be so.

There are many types of entities and characters that inhabit the theory that serve to propel the plot threads along. Some turn out more important than others, but most are essential to the overarching tale. Mathematics as storytelling? What an interesting and novel concept!

In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt^{} to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new way of making objects depend on other objects, namely

terms allowed to depend on types, corresponding to polymorphism.

Guided only by their feelings for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefullness.

— E. T. Bell

A smile fell on the grass.
Irretrievable!

And how will your night dances
Lose themselves. In mathematics?

Here is another identity but this time corresponding to an eight-fold: the Eight-square identity of Ferdinand Degen found about 1818. You know the drill: it states that a product of two numbers that are each the sum of eight squares is itself the sum of eight squares!

…The sum of the expressions in the eight triangles written in the diagram above. (Please consult the Wikipedia entry below for the textual formulas, as it’s too hard to write in HTML.)

Note that the expressions above have an interesting symmetry, aside from the one on the upper left. Indeed, Euler’s Four-square Identity has a similar simpler symmetry. There is also a connection with Octonions if you are interested in digging for it. If you are anticipating that there is such a formula for sums of sixteen squares, there is, but not a bilinear one, and it is much more complicated!

Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.

This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:

And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed: