Archive for the ‘Category Theory’ Category


April 22, 2015


After demolishing erroneous ideas some 25 centuries old, some brand new, I explain why Mathematics Can Be Made To Correspond To A Subset Of Neurology. And Why Probably Neurology Is A Consequence Of Not-Yet Imagined Physics.

Distribution of Prime Numbers Reworked Through Fourier Analysis: It Nearly Looks Like Brain Tissue

Distribution of Prime Numbers Reworked Through Fourier Analysis: It Nearly Looks Like Brain Tissue


Einstein famously declared that: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

Well, either it is an unfathomable miracle, or something in the premises has to give. Einstein was not at all original here, he was behaving rather like a very old parrot.

That the brain is independent of experience is a very old idea. It is Socrates’ style “knowledge”, a “knowledge” given a priori. From there, naturally enough aroses what one should call the “Platonist Delusion”, the belief that mathematics can only be independent of experience.

Einstein had no proof whatsoever that”thought is independent of experience”. All what a brain does is to experience and deduct. It starts in the womb. It happens even in an isolated brain. Even a mini brain growing in a vat, experiences (some) aspects of the world (gravity, vibrations). Even a network of three neurons experiences a sort of inner world unpredictable to an observer:

Latest Silliness: Smolin’s Triumph of the Will:

The physicist Lee Smolin has ideas about the nature of mathematics:


“the main effectiveness of mathematics in physics consists of these kinds of correspondences between records of past observations or, more precisely, patterns inherent in such records, and properties of mathematical objects that are constructed as representations of models of the evolution of such systems … Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that, once they come to exist, they don’t change”

Patrice Ayme: Smolin implies that “records and mathematical objects are human constructions which are brought into existence by exercises of HUMAN WILL; neither has any transcendental existence”. That’s trivially true: anything human has to do with human will.

However, the real question of “Platonism” is: why are mathematical theorems true?

Or am I underestimating Smolin, and Smolin is saying that right and wrong in mathematics is just a matter of WILL? (That’s reminiscent of Nietzsche, and Hitler’s subsequent obsession with the “will”.)

As I have known Smolin, let me not laugh out loud. (“Triumph of the Will” was a famous Nazi flick.)

I have a completely different perspective. “Human will” cannot possibly determine mathematical right and wrong, as many students who are poor at mathematics find out, to their dismay!

So what determines right and wrong in mathematics? How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?

I sketched an answer in “Why Mathematics Is Natural”. It does not have to do with transcendence of the will.



Neurology, the logic of neurons, contains what one ought to call axonal logic, a sub-category.

Axonal logic is made of the simplest causal units: neuron (or another subset of the brain) A acts on neuron (or brain subset) B, through an axon. This axonal category, a sub-category, corresponds through a functor, from neurology to mathematical logic. To A, and B are associated a and b, which are propositions in mathematical logic, and to the axon, corresponds a logical implication.

Thus one sees that mathematics corresponds to a part of neurology (it’s a subcategory).

Yet, neurology is vastly more complicated than mathematical logic. We know this in many ways. The very latest research proposes experimental evidence that memories are stored in neurons (rather than synapses). Thus a neuron A is not a simple proposition.

Neurons also respond to at least 50 hormones, neurohormones, dendrites, glial cells. Thus neurons need to be described, they live, into a “phase space” (Quantum concept) a universe with a vast number of dimensions, the calculus of which we cannot even guess. As some of this logic is topological (the logic of place), it is well beyond the logic used in mathematics (because the latter is relatively simplistic, being digital, a logic written in numbers).

The conclusion, an informed guess, is that axons, thus the implications of mathematical logic, are not disposed haphazardly, but according to the laws of a physics which we cannot imagine, let alone describe.

And out of that axonal calculus springs human mathematics.



If my hypothesis is true, mathematics reduces to physics, albeit a neuronal physics we cannot even imagine. Could we test the hypothesis?

It is natural to search for guidance in the way the discovery, and invention, of Celestial Mechanics proceeded.

The Ancient Greeks had made a gigantic scientific mistake, by preferring Plato’s geocentric hypothesis, to the more natural hypothesis of heliocentrism proposed later by Aristarchus of Samos.

The discovery of impetus and the heliocentric system by Buridan and his followers provides guidance. Buridan admitted that, experimentally heliocentrism and “scripture” could not be distinguished.

However, Buridan pointed out that the heliocentric theory was simpler, and more natural (the “tiny” Earth rotated around the huge Sun).

So the reason to choose heliocentrism was theoretical: heliocentrism’s axiomatic was leaner, meaner, natural.

In the end, the enormous mathematical arsenal to embody the impetus theory provided Kepler with enough mathematics to compute the orbit of Mars, which three century later, definitively proved heliocentrism (and buried epicycles).

Here we have a similar situation: it is simpler to consider that mathematics arises from physics we cannot yet guess, rather than the Platonic alternative of supposing that mathematics belong to its own universe out there.

My axiomatic system is simpler: there is just physics out there. Much of it we call by another name, mathematics, because we are so ignorant about the ways our mind thinks.

Another proof? One can make a little experiment. It requires a willing dog, a beach, and a stick. First tell the dog to sit. Then grab the stick, and throw it in the water, at 40 degree angle relative to the beach. Then tell the dog to go fetch the stick. Dogs who have practiced this activity a bit will not throw themselves in the water immediately. Instead they will run on the beach a bit, and then go into the water at an angle that is less than 90 degrees.

A computer analysis reveals that dogs follow exactly the curve of least time given by calculus. Dogs know calculus, but they did not study it culturally! Dogs arrived at correct calculus solutions by something their neurology did. They did not consult with Plato, they did not create calculus with their will as Smolin does.

It’s neurology which invents, constructs the mathematics. It is not in a world out there life forms consult with.

Patrice Ayme’

Causality Explained

March 29, 2015


What Is Causality? What is an Explanation?

Pondering the nature of the concept of explanation is the first step in thinking. So you may say that there is nothing more important, nothing more human.

I have a solution. It is simplicity itself. I go for the obvious model:

Mathematics, logic, physics, and the rest of science give a strict definition of what causality, and an explanation is.


Through systems of axioms and theorems.

Some of the sub-systems therein have to do with logic (“Predicate Calculus”). They are found all over science and common sense (although they will not be necessarily present in systems of thought such as, say, poetry, or rhetoric).


A and B are propositions. They do not have to be very precise.

Precision Is Not Necessarily The Smartest. Semantic Web Necessary.

Precision Is Not Necessarily The Smartest. Semantic Web Necessary.

As it turns out, except in Classical Computer Science as it exists today (Classical CS by opposition to Quantum CS, a subject developing in the last 20 years), propositions are never precise (so a degree of poetry is everywhere, even in mathematics!) Propositions, in practice, depend upon a semantic web.

A could be: “Plate Tectonic” and B could be “Continental Drift”. That A causes B is one of axioms of present day geophysics.

Thus I define causality as logical implication.

To use David Hume’s example: flame F brings heat H, always, and so is supposed to cause it: F implies H. Hume deduced causality from observation of the link (if…then).

More detailed modern physics shows that the heat of flame F is agitation that can be transmitted (both a theorem about, and a definition of, heat). Now we have a full, detailed logos about F and what H means, and how F implies H, down to electronic orbitals.

Mathematicians are used to make elaborate demonstrations, and then, to their horror, discover somewhere something that cannot be causally justified. Then they have to reconsider from scratch.

Mathematics is all about causality.

“Causes” in mathematics are also called axioms. In practice, well known theorems are used as axioms to implement further mathematical causality. A mathematician using a theorem from a distant field may not be aware of all the subtleties that allow to prove it: he would use distant theorems he does no know the proof of, as axioms. Some mathematician’s, or logician’s axiom is another’s theorem.

(Hence some hostility between mathematicians and logicians, as much of what the former use the latter proved, but the former have no idea how!)

Causality, by the way, reflects the axonal geometry of the brain.

The full logic of the brain is much more complicated than mathematics, let alone Classical Computer Science, have it. Indeed, brain logic involves much more than axons, such as dendrites, neurotransmitters, glial cells, etc. And of these, only axonal geometry is simple enough to be approximated by classical logic… In first order.

Mathematics is causation. And the ultimate explanation. Mathematics makes causation as limpid we can have it.

This theory met with the approval of Philip Thrift (March 27, 2015): “I agree exactly with the words Patrice Ayme wrote — but with “mathematics”→”programming”, “mathematical”→”programmatical”, etc.”

I pointed out later to Philip that Classical Programming was insufficient to embrace full human (and quantum!) logic. He agreed.

However the preceding somehow made Massimo P , a professional philosopher, uneasy. He quoted me:

“Patrice: “To claim that mathematics is not causal is beyond belief. Mathematics is all about causality.”

Massimo: It most obviously isn’t. What’s causal about Fermat’s Last Theorem? Causality implies physicality, and most of pure math has absolutely nothing whatsoever to do with physicality.

Patrice: “Causes” in mathematics are also called axioms.”

Massimo: “You either don’t understand what causality means or what axioms are. Or both.”

Well, once he had released his emotional steam, Massimo, a self-declared specialist of “physicality” [sic] did not offer one iota of logic in support of his wished-for demolition of my… logic. I must admit my simple thesis is not (yet) in textbooks…

Insults are fundamentally poetic, illogical, or pre-logical. Massimo is saying that been totally confused about causality and explanations is a sacred cow of a whole class of philosophers (to whom he had decided he belongs). Being confused about causality started way back.

“All philosophers, “said Bertrand Russell,” imagine that causation is one of the fundamental axioms of science, yet oddly enough, in advanced sciences, the word ’cause’ never occurs … The law of causality, I believe, is a relic of bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm …”

Russell was as wrong as wrong could be (not about the monarchy, but about “causation”). He wrote the preceding in 1913, when Relativity was well implanted, and he, like many others, was no doubt unnerved by it.

Poincare’ noticed, while founding officially “Relativity” in 1904, that apparent succession of events was not absolute (but depended upon relative motions).


But, temporal succession is only an indication of possible causality. In truth causality exists, if, and only if, a logical system establishes it (moreover, said logic has to be “true”; that, assigning a truth value, is, by itself is a separate question that great logicians have studied without clear conclusions).

When an explanation can be fully mathematized, it is finished. Far from being “abstract”, it has become trivial, or so suppose those with minds for whom mathematics is obvious.

Mathematics is just like 2 + 2 = 4, written very large.

Fermat’s Last Theorem is not different in nature, from 2 + 2 = 4… (But for something very subtle: semantic drift, and a forest of theorems used as axioms to go from side of Fermat’s theorem to the other.)

To brandish mathematics as unfathomable “abstract” sorcery, as was done in Scientia Salon, is a strange, but not new, streak.

There in “Abstract Explanations In Science” Massimo and another employed philosopher pondered “whether, and in what sense, mathematical explanations are different from causal / empirical ones.”

My answer is that mathematical, and, more generally logical, explanations are the model of all explanations. We speak (logos) and thus we communicate our thoughts. Even to ourselves.

The difference between mathematics and logic? Mathematics is more poetical. For example, Category Theory is not anchored in logic, nor anywhere else. It is hanging out there, beautiful and useful, a castle in the sky, just like all and any poem.

Such ought to be the set-up on the nature of what causality could be, to figure out what causality is in the physical world. Considering that Quantum Entanglement is all over nature, this is not going to be easy (and it may contain a hidden clock).

Patrice Ayme’


October 27, 2014

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.