What is the mind made of? We have progressed enormously as far as the brain objects are concerned: neurons, axons, dendrites, glial cells, neurohormones, various organs and substructures in the brain, etc.

But is there a broad mathematical framework to envision how this is all organized? There is! Category Theory! It turns out it’s a good first order approximation of mind organization. At least, so I claim.

Category Theory is about diagrams. Category Theory has been increasingly replacing advantageously Set Theory. It’s not only because Category Theory does not have to ponder the nature of objects, elements, sets.

Category Theory was long derided as “abstract nonsense” and “diagram chasing”. But it gives very deep, powerful theorems.

I claim the powerful theorems of Category Theory should translate directly into… neurology.

Amusingly, although I accused Aristotle to have demolished democracy and fostered plutocracy through his beloved pets, the mass murdering criminal plutocratic psychopaths, Alexander and Antipater, I recognize humbly that it’s the same Aristotle who invented categories (thus making him a great thinker, and justifying an Aristotle cult among those who need to have cults to feel good about themselves)…

Aristotle’s meta-idea about categories was just to talk about the most fundamental notions:

The present essay was suggested, and is an extension of what the honorable Bill Skaggs seems to have wanted to say, in Scientia Salon, in his “Identity A Neurobiological Perspective”. (As far as I can comprehend.)

However, forget Theseus’ ship and Hollywood’s Star Trek “Transporter”. As I said in “Quantum Identity Is Strong”, Quantum Identity is not erasable, and makes those time honored examples impossibly disconnected with reality. The notion of identity has thus to be found elsewhere (as we intuitively know that there is such a notion).

According to modern Quantum Field Theory, we are made, at the most fundamental level, of fluctuating fields. They come and go, out of nowhere. So, that way, we are continually been deconstructed and rebuilt. The question naturally arises: what is preserved of me, as a set of Quantum Fields? Well, the most fundamental mathematical structure is preserved.

The same seems to hold, to a great extent, in neurobiology, as neuro circuitry, to some extent, seems to come, go, and come back.

Thus we are all like old wooden Greek ships, perpetually falling apart, and rebuilt.

To some extent, this is what happens to species, through reproduction: cells split, and reproduce themselves, thanks to DNA.

A species has identity. Yet that identity is made of DISCONTINUOUS elements: the individuals who incarnate the species, who are born, and then die. And others appear, just the same, sort of. How is that possible?

A species’ identity is its structure. Just as a neurology, or an elementary particle identity is its structure. Not just a geometric structure, not just a topological structure, but its structure, as the most fundamental notion, as a category.

So what is preserved? Shape. And how to morph said shapes… Naturally (there is a notion of natural transformation, in Category Theory).

Historically, analyzing shape was systematized by the Greeks: Euclidean geometry, cones, etc. Then, at the end of the Nineteenth Century, it was found that geometry studied shapes mostly by studying distance, and yet, even if distance was denied consideration, there was a more fundamental notion of shape, topology. That was the structure of shapes as defined by neighborhoods.

Two generations later, Category Theory arrived. Category Theory is about morphisms, and the structures which can be built with them. Please listen to the semantics: structures, morphing… This is all about shapes reduced to their most basic, simplest symbolic expression. It’s no wonder that it would come in handy to visualize neurological structures.

A morphism is a pair of “objects” (CT leaves unspecified what the “objects” are). To model that neurologically, we can just identify ‘objects’ to neurons (or other neurological structures), and morphisms to axons (although dendrites, and more, could be included, in a second stage, when the categoretical modelling become more precise).

The better model is category theory. When are two diagrams equivalent? When are they IDENTICAL? Cantor defined as of the same cardinal two sets in a bijection (a bijection is a 1 to 1, onto map).

Category Theory defines as identical the same diagram (a drawing reduced to its simplest essence). Say: A>B>C>D>A is the same as E>F>G>H>E.

Thus, when are two diagrams identical in category theory? When they are modelled by the same neuronal network. (Or, more exactly, axonal network: make each arrow “>” above, into an axon.) And reciprocally!

Discussing the mind will involve discussing the most fundamental structures constituting it. What better place to start, than the most basic of maths? Especially if it looks readily convertible in neural networks.

Category Theory is the most fundamental theoretical structure we know of. It is the essence of identity, and identification. In conclusion, two objects are identical, neurologically, and in fundamental physics, if they are so, in category theory.

Time to learn something categorically new!

Patrice Ayme’


Note: No True Isolated Rocks: In other news, and to address a point of Bill Skaggs, whether a rock can be truly isolated is an open problem, experimentally speaking.

According to the theory of gravitation of Einstein and company, a rock cannot be isolated. Why? Because the rock is immersed in spacetime. Spacetime is animated by gravitational waves: this is what the Einstein Field Equation implies. Now, according to an unproven, but hoped-for principle of fundamental physics, to each force field is associated a particle. In the case of gravity, that hoped-for particle is called the graviton. “Particle” means a “particular” effect. Thus, an isolated rock, according to established theory, and hoped-for theory, ought to be adorned occasionally with a new particle, a new graviton, thus ought not to be isolated.

In my own theory, Objective Quantum Physics, on top of the preceding standard effect, resolving Quantum Entanglements, ought to create even more particles in “isolated” rocks.


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11 Responses to “CATEGORIZING the MIND”

  1. gmax Says:

    Gee! Another ground breaking work? We have to learn mathematics now?

  2. Bill Skaggs Says:

    @Patrice Ayme: Category theory?! See

  3. Steven Johnson Says:


    October 27, 2014 • 6:56 pm

    Patrice Ayme, An interest in ancient history leads me to nitpick. It was Cassander, Antipater’s son, who was educated by Aristotle, along with other King’s Friends (probably a rank rather than a relationship.)

    In talking about Aristotle as a person it might be of interest that his connection to the Macedonian court was inherited from his father, physician to Amyntas (the third, i think.) His father-in-law Hermias of Atarneus was reportedly tortured to death for information about Philip’s invasion plans. And of course, his pursuit of favor from Philip and Alexander resulted in the restoration of his native city of Stagira, which had been destroyed and its population dispersed…by Philip.

  4. Mario Roy Says:

    Mario Roy

    October 27, 2014 • 6:51 pm

    “Amusingly, although I accused Aristotle to have demolished democracy and fostered plutocracy”…

    Well, you are flat out wrong. Aristotle had nothing to do with that failure as some commenters, me included, have explained to you.

    “I recognize humbly that it’s the same Aristotle who invented categories (thus making him a great thinker, and justifying an Aristotle cult)”…

    We don’t know if Aristotle invented the categories, so far we know that he studied them but it isn’t clear if Aristotle invented them or he got it from somebody else. This is an open question.

  5. Patrice Ayme Says:

    Yes, Bill Skaggs. Category Theory used to be called, derogatively, “Abstract Nonsense”. The derisive expression dates from when normal mathematicians could not figure out what Category Theory was about, and what it could do.

    Now Category Theory is at the forefront of Quantum Computing, QFT, C* Algebra theory, Knot Theory, Non Commutative geometry, and many other fields. It has become an indispensable instrument, for the reasons I gave. Some physicists swim in it, as if it were a hospitable sea. There is zero chance that it will not become dominant in neurology.

    @ Mario: ““Amusingly, although I accused Aristotle to have demolished democracy and fostered plutocracy”…Well, you are flat out wrong. Aristotle had nothing to do with that failure as some commenters, me included, have explained to you.”

    Well, your “explanation” consists into well known excuses for the inexcusable. You cover new ground, and read my essay on the subject, and will find therein the teachings of Aristotle AGAINST democracy (which he repeatedly called “deviant”). I don’t know what kind of flat that makes you.

    @ Steven Johnson: Nitpicking history is good. I mentioned Cassander the son (not the brother) in my long essay on the destruction of democracy by Aristotle. The other son would have poisoned Alexander.
    It is funny that some cannot read between the lines of Wikipedia. Here is Wikipedia:

    “Antipater participated in a new division of Alexander’s great kingdom. He appointed himself supreme regent of all Alexander’s empire… ANTIPATER WAS A STUDENT OF ARISTOTLE AND ARISTOTLE NAMED HIM AS EXECUTOR-IN-CHARGE OF HIS WILL, when he died in 322 BC…[7][8]”

    Now will somebody explain to me how, having Aristotle teach AGAINST democracy, and having the worst dictator as executor of one’s will, one’s own student, and closest buddy, makes one NOT responsible? Antipater was worse than Alexander. Alexander did not subjugate and destroy Athenian democracy. Antipater did, tortured, executed, and stripped of Athenian citizenship all those who had less than 1,800 Drachma property.

    @ Mario: Dr. Alfred Rosenberg had a decisive role in shaping Nazi philosophy and ideology. The main example quoted at the Nuremberg trial was his book, “Myth of the Twentieth Century”, which was published in 1930, where he incited hatred against “Liberal Imperialism”. Aristotle incited hatred against democracy. His executor established 20 centuries of plutocracy and non-republican regimes. Alfred Rosenberg was hanged on 16 October 1946.

  6. SciSal Says:

    October 27, 2014 • 10:57 pm

    Patrice, You do understand that “democracy” in ancient Athens was more akin to mob rule, and moreover was restricted to a pretty small section of the population, right? I’m not sure I would be a Democract under those conditions, so cut some slack to Aristotle (and Plato), will you?

    • Patrice Ayme Says:

      No slack.
      Claiming democracy was “more akin to mob rule” while “restricted to a pretty small section of the population” seems self contradicting to my rigid self.
      Aristotle killed ancient direct democracy, replaced it by plutocracy. No slack. Public hanging of this tragic train of thought better.

  7. Steven Johnson Says:


    October 28, 2014 • 10:53 am

    Patrice Ayme, Antipater’s role as executor would have been justified by his rule over Macedon and Greece, which gave him the power to enforce Aristotle’s testament, a matter of civic importance for Stagira I believe. I do not believe the wikipedia can possibly be correct regarding Antipater personally being a student inasmuch as Antipater was some years older than Aristotle. Certainly Aristotle did not travel to Athens as an adult to be taught by a man younger than himself. When Aristotle taught in Macedon, Antipater was active in government and army, not a student. As for being regent? The agreed regent for Alexander’s children was Perdiccas. After Perdiccas’ murder it is true that Antipater declared himself regent but it’s hard to know whether anyone other than his allies took that seriously. But it seems doubtful.

    SciSal, Patrice Ayme is quite right that Athenian democracy was terminated by Antipater. (By the way, assertions that democracy in Athens was, mysteriously, vanishing have no merit.) As for giving Plato and Aristotle some “slack,” I do think that as to judgments on personal motivations should be based on a full consideration of all the facts, which includes trying to imagine how we might feel were we ourselves present. But overall, given the centuries of adulation, I don’t think a little irreverence is out of bounds. In particular, there is no reason to avoid drawing conclusions about their common enmity to democracy, or Aristotle’s support for monarchy. Alexander’s arete was imperial conquest. Didn’t Aristotle exalt the active life in pursuit of one’s excellence? Every indication is that Alexander consciously modeled virtue, whether sincerely or not.
    I feel compelled to point out that “a pretty small section of the population” is not enough for a respectable mob, or really even something “akin” to a mob. No mob, no mob rule. The anti-democratic thinkers may be revered but their opinions should be critically examined.

  8. SciSal Says:


    October 28, 2014 • 10:58 am

    steven, I said in the past very clearly that I’m perfectly fine criticizing the ancients when the need be. I just feel like Patrice is out on a crusade that has relatively little correspondence with the culture of the time and the historical facts. As for the mob thing, a simple majority of a relatively small group is still a “mob” in the sense relevant here. And had I been present at the time, I might actually have had some sympathies for Plato’s distrust of a democracy that had led to the ruin of the Athenian polis and to the murder of his mentor.

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