An "entropological" approach of mathematical "categories" #1

As I'm learning about categories, by reading the book "Conceptual mathematics, a first introduction to categories" by F. William Lawvere & Stephan H. Schanuel, this is a good opportunity for me to write my comments directly in english.

This will avoid any mistake by translating the concepts in french (because I just discover them in english) and, perhaps the most important, it is time to translate my personal approach in order to broadcast it more efficiently than in french.

In this exercise, I presume that my hypothetical english reader had never heard anything about my previous developments. So, I have to start from the beginning, in order to let me a chance to be understood ...

Abstract:

I use the term "entropology" suggested by Lévi-Strauss to name a new science the aim of which would be to analyze the dynamics of human activity, in terms of energy, as defined in physics.

The related domain could be very large: from the individual (how he uses his energy in order to constitute his proper feeling to be an individual) to the society (how the individuals are gathering their individual actions).

An human is a speaking being. We start our analyze by this ability, and we have first to define which are the initial characteristics of this phenomena. After famous philosophers, Lacan develops his own approach of the psychoanalysis from the same initial statement, and we continue on the same track.

• We develop our representation of the world by a series of dichotomies : good / bad; male / female and so on.
• We structure our "reasonable" discourse (i.e.: when we try to explain: this will be later specified) by respecting the "non contradiction principle". It means that an object cannot be in a particular state and, in the same time, according to the same criteria, in the opposite state.

I for Imagination

The no contradiction principle leads to various paradoxes (as for example the liar paradox). In order to overcome those contradictions rising in our discours, we progressively structure it in separate levels. The idea came from Freud who saw our imagination like a stack of sheets. And in that stack, a defined Imaginative level (let's say Ik), is a metalanguage regarding a lower one (i.e.: Ik-1). As an example; a set of axioms (at level Ik) will define all the theorems of the mathematical domain they defined (at a Ik-1 level of the language).

• When our language is situated on a particular, defined level, we say our discussion is "synchronic";
• When we discuss about the influence of one level regarding others levels, our discussion is "diachronic".

This difference, and the terms to express it, are Ferdinand de Saussure's ones.

R for Real:

If the Imagination is structured and can be the subject of a discussion or a theory, we can't say nothing about the Real except that we are in contact with it, just when it hurts us. Moreover: the Real is what doesn't match with the image we have of it. The Real is out of our imagination.

S for Symbolic

At the other extremum of our Imagination, the Symbolic is what determines our thought, but exceed our Imagination. Something which gives a coherence to our Imagination, of which we are unable to give a defined or a clear expression.

It is of major interest to define precisely the respective position of the speaker, related to his speech. To do so, we define the "diachronic axis". If we see our imagination as a stack of "synchronic levels", then, the diachronic axis is orthogonal to that stack. And we will refer to the position of a particular imagination level by its diachronic coordinate. By convention, a language at "k" level is in a meta-position with respect to another language at "k-1" level.

The expression Ik-1 < Ik minds that the language at the Ik level determines in some way the language at a Ik-1 level.

Now, as the individual imagines himself at a certain level (Im the "self level"), then this level Im can be spotted with regard to the various levels of his langage as Ik and/or Ik-1 (and vis versa, a particular language can be spotted with regard to the level Im where the speaker represents himself, his "consciousness").

Speaker's position:

There are 3 basic positions of the speaker relative to his speak:

• Ex-post : the speaker is in a meta-position regarding his speech: he handles the rules which determine his language. This is the position of a master explaining something. In mathematics, as an example, the master (Im) expresses the axioms (Ik) of a mathematical domain, and he demonstrates some theorem (Ik-1) : Ik-1 < Ik < Im
• Ex-ante : the speaker is determined by his speech. That is the position of the priest, for instance, when he has some expectation regarding a principle which can't be imagined, but determines whom he is : Im < S
• Synchronic : the subject is on the moment, in the acting. This is the "game position".

By definition we have always R < I < S.

As we are now discussing about mathematics, our position is essentially the ex-post one :

Imathematics < Im

Another important remark is the following :

• When we move in the diachronic axis downstream, the Imagination becomes poorer, looses its sens, the concepts are withering, while we go in the details.
• On the reverse, when we move in the diachronic axis upstream, the speech takes more sens, while we lose the details of our descriptions.

The most important consequence of the above is that (when we are in an ex-post position !!!), the elementary representation of time is a diachronic concept related to 2 different levels (let's say between Ik and Ik+1) while the space (or any avatar) is at the lower synchronic level (i.e.: Ik), and the speed or "motion", at the upper level (i.e.: Ik+1).

In that sens, when you move downstream the diachronic axis, there is always a time where the "motion" you are describing degenerates in a certain set of two concepts one synchronic and the other diachronic. At that point appears a quantum uncertainty. Regarding this point, please refer to my previous article: "le principe d'incertitude d'Heisenberg sans les maths".

The other major point is that this "Imaginative structure" involves a maximum speed limit, please refer to my article : "relativité du temps et vitesse limite".

A simple way to represent the time is just by counting the beatings of a pendulum. The pendulum is moving at the Ik level, and the sheet of paper where you draw a line to represent each beat is as an Ik+1 imaginary level. The measure of the time is a count of the number of lines at this level. The difference between Ik and Ik+1 is that in Ik, you have not yet the concepts necessary to do the count, even if you can see the motion. As a matter of fact, a dog can see the motion (in Ik), like you, but it cannot measures the time (in Ik+1); because its imagination doesn't reach this level.

We don't need anything else to start our reading of this introduction to mathematics.

I have already done such development in this article: "regard entropologique sur les maths". But I was led by my initial mathematical education, focused on the concept of "group structure". It appears that the concept of "categories" is more basic, at a lower Imaginative level. So it seems very interesting to reorganize our approach from this lower level, in order to reinforce our approach.

We'll start on the next article.

Hari

PS: je ne suis vraiment pas à l'aise en anglais, aussi, merci par avance pour toutes les remarques dont vous voudrez bien me faire part afin de rendre ce texte plus lisible.