algebra as cognitive science

There exists a zoo of algebraic structures in mathematics. All have emerged naturally from different areas of geometry and analysis. Yet they have profound similarities. In fact, most fall into a few general categories, some of which are so ubiquitous that one can not help but speculate that they emanate from certain human cognitive biases.

Of course, simpler structures are more likely to emerge in numerous places. But this is not the phenomenon I am referring to here... There is more to ubiquity than the prevalence of the simple. There are infinitude of possible algebraic structures out there, but human mathematicians seem to be consistently picking some in favour of others.

The discovery of the ubiquity of adjunctions is not solely a mathematical achievement. An account of it should have been published in cognitive science journals as well.

Some further observations:

1) If there are intelligent alien civilizations, then their catalogue of ubiquitous algebraic structures may be different from ours due to our cognitive differences.

2) Those, who claim that computers can generate mathematics that is of interest to human mathematicians, will first have to understand the biases and particularities of the human mind.

It must be admitted that the use of geometric intuition has no logical necessity in mathematics, and is often left out of the formal presentation of results. If one had to construct a mathematical brain, one would probably use resources more efficiently than creating a visual system. But the system is there already, it is used to great advantage by human mathematicians, and it gives a special flavour to human mathematics.

Ruelle as quoted in Leinster, Higher Operads, Higher Categories (Page 1)

Recent computer analysis of chess, working backwards, has revealed surprising results. There are positions which result in a win for one of the players in over 90 moves, none of which involves the capture of a piece or the movement of a pawn. The winning "line", moreover, is almost impossible to memorize, because there seems to be no rhyme or reason for the specific winning moves. Seemingly random moves results in a win, without anyone being able to characterize the moves as a "strategy" of one kind or the other. This is the opposite of what we find in mathematics.

Steiner, The Applicability of Mathematics as a Philosophical Problem (Page 64)

Proving human theorems via inhuman means... This is what the future computers should be instructed to do. Using non-geometric and pedantic proof techniques that humans are not good at, they should seek theorems that humans will like.

3) It is true that after the discovery of category theory, we realized that most of our mathematical structures in use are natural (i.e. adjunctional) in the precise categorical sense. But these are the structures that have managed to survive centuries of mathematical discourse. Those that failed to survive probably died off due to their categorical unnaturality. Evolution selects for optimality. What this means is that, now that we are equipped with the guidance of category theory, development of mathematics will accelerate. We can eliminate unnatural structures outright rather than waiting for time and public discourse to filter them out.

structural information inside DNA

I had always thought that structural symmetry was strictly a product of evolution due to its phenotypical advantages. Most animals for example have bilateral symmetry. Plants on the other hand exhibit other types of symmetries. In nature one rarely encounters structures that are devoid of such geometrical patterns.

While reading an article on algorithmic complexity, it immediately dawned upon me that there may be another important reason why symmetry is so prevalent.

First, here is a short description of algorithmic complexity:

Given an entity (this could be a data set or an image, but the idea can be extended to material objects and also to life forms) the algorithmic complexity is defined as the length (in bits of information) of the shortest program (computer model) which can describe the entity. According to this definition a simple periodic object (a sine function for example) is not complex, since we can store a sample of the period and write a program which repeatedly outputs it, thereby reconstructing the original data set with a very small program.

Geometrical patterns allow economization. Presence of symmetries can drastically reduce the amount of information that needs to be encoded in the DNA for the orchestration of biochemical processes responsible for the structural development of the organism. Same may be true for more complicated morphological shapes that are still mathematically simple to describe. An example:

Researchers discovered a simple set of three equations that graphed a fern. This started a new idea - perhaps DNA encodes not exactly where the leaves grow, but a formula that controls their distribution. DNA, even though it holds an amazing amount of data, could not hold all of the data necessary to determine where every cell of the human body goes. However, by using fractal formulas to control how the blood vessels branch out and the nerve fibers get created, DNA has more than enough information.