Posts Tagged ‘Mathematics’


December 30, 2020

There are two languages: common language, which is messy, and mathematics, which is much more precise, and contains basic physics. Indeed, all common languages are more or less isomorphic (same shape and preoccupations). 

Mathematics is the part of common language which, given precise axioms, is the simplest and irreducibly deduced from those simplest notions (in physics, thus nature). “Physics” is a compendium of how nature looks, for sure, or how it works, de facto. How nature looks, as deduced from experiments, has varied in the last 100 million years… and that description is getting increasingly precise, as demonstrated by our ever greater power in making nature do as we wish. 

But how nature works inside brains has become ever more powerful and precise ever since there are brains, and they have grown. Neurology is an emergent part of nature. Thus it is factual, being natural, and we also call its basic architecture mathematics, when we describe it. For example, basic category theory looks like the simplest abstraction of basic neurology restricted to the simplest axons…

Thus elucidated, counting becomes a matter of neural networks. 1 + 1 = 2 can be directly envisioned as a semantic description of a (very useful) neural network which has appeared in advanced species. That makes “2” a description of some neuronal architecture. There is no free will there. “2” is just the label for a particular type of neural network found in nature.

Proof that our brains contain Quantum Physics; because they contain a description of light, and the best there is, no more, no less… OK, to get Maxwell equations, something like dF = 0 & d*F = C, we would have to do more math, but you get the idea… It’s not just that mathematics describe better, mathematics leads to physics (the right math, that is…).

As a result of being the product of emerging neuronal networks, there is no more free will in “2” than in the Iron nucleus (Fe 56). And so on it goes: “pi” is the length of the circumference of a circle of radius 1. No free will there, either. 

Nor is there for multiplication of real numbers. Even better: one gets in complex numbers by trying to build a multiplication in the plane which generalizes the multiplication of real numbers. There is a way to do this (multiplying distances to the origin, adding angles from the real axis): it enables us to get square roots of negative numbers… some numbers which multiplied by themselves, have a negative square. Not much freedom there. But then something spectacular happens: this gives the best description of light (including momentum, energy and polarization)… And as such becomes the basic language of Quantum Physics.

How could that all be?

Does that mean that our brain and how we build networks there, is not free from Quantum Physics? Indeed. Let’s inverse the question: how could the brain be free of Quantum Physics, considering, well, that Physics, Nature in Greek, is Quantum? Would that not be considering that brains are not natural? 

If somehow there is no free will in the nature of the neural networks (and thus mathematics) we build, where could free will be? Well, in which kind of networks we decide to build, then? The networks themselves, at their simplest, are mathematics, and thus mathematics is digital… So is language. Being digital, and finite (in its mode of construction) make languages and mathematics, limited and pre-ordained. But Quantum Physics itself is based on a continuum, and that brings the freedom… of the butterfly effect. Free will is a subtle thing.  

The famous mathematician Richard Dedekind said numbers were the work of God, and the rest of mathematics the work of man. It is probably wiser to acknowledge that we, or at least our mathematics, are the work of physics… self-describing…

Patrice Ayme


January 25, 2017

Mathematically Built Brain: The Example of Grid Cells, Incarnating Algebraic Geometry.

Understanding how the cognitive functions of the brain arise from its basic physiological components has been the final frontier in logic and rational science for thousands of years. (As I tried to explain yesterday, the superstitious religious fanatics tried their best to bury all of science, and the scientific mindset, the essence of humanity; they nearly succeeded!)

The 2014 Nobel was given to John O’Keefe (a “half”!), the rest jointly to May-Britt Moser and Edvard I. Moser “for their discoveries of cells that constitute a positioning system in the brain.” I will develop here the philosophical viewpoint, which is broader (O’Keefe’s career was steered by the influence of Hebb, the famous psychologist, who got the idea of the outside patterns imprinting the neurocircuitry of the brain).

Here is Hebb: “Let us assume that the persistence or repetition of a reverberatory activity (or “trace”) tends to induce lasting cellular changes that add to its stability.[…] When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.”

Well it turns out that evolution has had even more imagination than that. I will even propose Patrice’s Neural Theory, a vast generalization.

Galileo famously said the language of nature was written in mathematics. It turns out that it is much more than that. Our brain is mathematically organized. What Descartes consciously discovered, a coordinate frame in which to set-up calculus, is automatically generated in the brain. This is the meaning of grid cells.

Grid cells are neurons that fire when an animal moving of its own free will traverses a set of small regions (firing fields) which are roughly equal in size and arranged in a periodic triangular array that covers all of the available environment. They were discovered in 2005 by a couple (literally) of Norwegian researchers, the Mosers, and rewarded by the Nobel Prize in 2014 (shared with O’Keefe, from London, who invented the basic experimental technique, and discovered “place cells)

Once set, navigation can be done in the dark, blinded. Scientists’ discovery that rodents, bats and nonhuman primates have a system in the brain for so-called “dead reckoning navigation”… “Dead reckoning” refers to the ability to navigate without external cues. The term comes from ship navigation. A crew will “take a sighting” via cues such as the stars or landmarks to determine where the ship is on a map. Then, when the ship moves, ‘dead reckons’ to update location on the map paying attention to speed and direction. The Greco-Romans already had such systems, with little paddled wheels counting the distance covered over the sea. It turns out that ‘dead reckoning’ is enabled by the grid cell system, inside the brain. 

Recording Of Grid Cells Activity Inside Rat Brain (Jeffery Lab and others.)

Recording Of Grid Cells Activity Inside Rat Brain (Jeffery Lab and others.)

Kate Jeffery, a professor of behavioural neuroscience at University College London puts it this way:

“The importance of grid cells lies in the apparently minor detail that the patches of firing (called ‘firing fields’) produced by the cells are evenly spaced. That this makes a pretty pattern is nice, but not so important in itself – what is startling is that the cell somehow ‘knows’ how far (say) 30 cm is – it must do, or it wouldn’t be able to fire in correctly spaced places. This even spacing of firing fields is something that couldn’t possibly have arisen from building up a web of stimulus associations over the life of the animal, because 30 cm (or whatever) isn’t an intrinsic property of most environments, and therefore can’t come through the senses – it must come from inside the rat, through some distance-measuring capability such as counting footsteps, or measuring the speed with which the world flows past the senses. In other words, metric information is inherent in the brain, wired into the grid cells as it were, regardless of its prior experience. This was a surprising and dramatic discovery. Studies of other animals, including humans, have revealed place, head direction and grid cells in these species too, so this seems to be a general (and thus important) phenomenon and not just a strange quirk of the lab rat.”

We should have looked for Plato’s cave. It turned out that this cave has been built, is being built inside our heads all along! This cave is built-in two ways: automatically (grid cells) and as a response to the environment,, from the outside, from the environment, in.

(So it matters what our brain experienced before to mold afterwards what comes in anew from the outside! No experience is a neutral experience!)

That cave is both a topology (what’s near and what’s not, the logic of place), and a basic geometry (the grid and its grid cells). To have a grid built automatically is the equivalent of having a reference frame in mathematics. It makes sense if one wants to make mathematics!

And not just mathematics, but even Infinitesimal Calculus! It is indeed clear that animals such as dogs have a mastery of calculus: experiences have shown this, and anybody with a dog throwing a stick sideways in water will see the dog running along the shore a bit, and then jump in the water, so as to minimize the time to reach the stick, a typical calculus problem. Dogs can do calculus, because they can make algebraic geometry in their brains, having a reference frame made of these grid cells! (If they had no grid cells, they would not be able to do calculus.)

Thus Descartes rediscovered, consciously, something which had been found, evolved and calculated by evolution half a billion years ago (or more!). The reference frame, also known now as the neuronal grid cell system, is basic to all of mechanics, even Poincare’-Lorentz Relativity.  (An open question: Quantum Physics uses even more general reference systems, Hilbert spaces; I will therefore predict that the brain has also that sort of organization!)

The world is not as astonishingly understandable, as Einstein would have it. Neuronal grid cell studies show that we are the world. Understanding the world is understanding ourselves.

The world is not just written in mathematical language, as Galileo found out. We are made mathematically. We think mathematically, because we are made of math. We are mathematics.

We are not just looking at shadows in a cave, as Plato would have it. And the cave was not given to us by the gods, as Socrates had it. We are the cave, we, and our personal history, built it.

Any new experience, idea or emotion, taught or experienced, is another brick in that wall of perception and analysis, we better consider it carefully, before indulging in it. Call that the Principle of Mental Precaution But that Principle extends also to what we chose NOT to experience, which can be just as bad, if not worse.

You are not just what you think. You mentally are what you were submitted to, and what you decided to submit to. Fate is written in mathematical patterns, one theorem made out of neurons, their axons, dendrites and supporting glial cells, at a time.

Such theorems are written with the physics of minds, just as sturdy as the physics of stars. Just as hopeful, just as ominous.

Plato thought mathematics were “forms”, out there, outside of the physical world. This is not what science is finding. There are not “forms” out there, and physics, nature, somewhere else. Our minds are literally made of math.

So here is my theory:

Whatever exists in mathematics exists in the brain. And reciprocally.”

Patrice Ayme’


Mathematical Terror

May 7, 2016

Mathematics is dangerous. It has endowed a creature from the Sol system to acquire ever greater powers, including jumping off planet. Fortunately for the future of cockroaches, idiots are striking back. Contemplate the following: an associate professor of economics at the University of Pennsylvania was doing what I have done for longer than him: scribbling equations. Then…

This Post Was Deleted. Why? Fear Of Fighting Back? Fear That The Expression Of Fighting Back Is Too Offensive To The Idiots?

This Post Was Deleted. Why? Fear Of Fighting Back? Fear That The Expression Of Fighting Back Is Too Offensive To The Idiots?

Our world is doomed… Except if it is saved by the honor of the human spirit, raw intelligence unleashed. That will entail partial differential equations propelled deep down inside by the most revolutionary philosophy.

The power of the mind has never been greater. This can be seen: watch Obama and Clinton lie about who they helped under their watch (hint: the most useless financial  types, the hedge fund managers, the brokers, those who are behind Brexit). How do they do that? By hiding behind the complexity of Quantitative Easing (what they did was Quantitative Easing for the useless part of finance… Instead of Quantitative Easing for We The People).

This can be seen: watch Putin invading Georgia, Ukraine, chuckling about his “little green men”, denying they are Russian soldiers and then confirming that, indeed, they are.

Hitler used to laud what he called the “Big Lie” technique. Correctly, though, Big Lies should build Big Faith. Hitler observed that: “It is always more difficult to fight against faith than against knowledge.” Now we have the Big Hypnosis technique. It requires collaboration not just from the Main Stream Media, but the entire intellectual class. Take for example the fascination for novels: what can novels do that this crazy world is not already doing? Indeed, this is why (good) science fiction is precious: because it looks at possible worlds, instead of just arcane details of Conventional Wisdom.

People are fearing Islam, while having been told it was racism to do so. Islam has become a division of the minds.

Meanwhile, in the French Republic, Joan of Arc is getting ever more popular. There, once again, just as with Islam, Obama, Quantitative Easing, it’s all about not knowing what really happened, or what is going on. Ironically, it is the same problem as with Brexit. The real problems in today’s Great Britain have little to do with the European Union. Similarly what Joan of Arc helped to solve was the alliance between London and Paris: Joan of Arc was a Brexiter with a sword, who hacked her way into Paris, so as to separate Paris from London. I told you: reality beats fiction.

But how does one learn real history, and real facts, when all what matters is the fake passion of sports scores? The other day, I passed by a public transportation bus in a large city, and, where the destination should have been written, instead could be seen in huge letters: “Go Warriors!” I never heard of “Warriors” before. Obviously the local sport team. All I know is that this was free public advertising on public transportation. Governments carefully organize fake passions to divert attention from what they are doing. They, their friends, clients, bosses and patrons.

Menzio denounced a “broken system that does not collect information efficiently.” He is troubled by the ignorance of his fellow passenger, as well as “A security protocol that is too rigid–in the sense that once the whistle is blown everything stops without checks–and relies on the input of people who may be completely clueless.”

Mr. Menzio adds: “What might prevent an epidemic of paranoia? It is hard not to recognize in this incident, the ethos of [Donald] Trump’s voting base.” Education my dear Menzio, education. So how come average US citizens are so ignorant? Could it have to do, by any chance with educational inequality? And even “Cognitive Inequality“?

However, professor Menzio works in a university system where people have to pay a fortune to attend, and Mr. Menzio is happy to get a much higher salary than he would get in his native Italy, so his complaints about ignorance are (unwittingly, or should I say cluelessly) hypocritical.

Mr. Menzio complains about ignorance, but he seems himself blissfully ignorant of the fact that he is himself part of the system which generates ignorance, the plutocratic university system, where, to attend, one needs more, in tuition, than the median family income. He can write all the PDEs he wants, but, without the correct philosophy, they cannot bring real understanding of the socioeconomy.

In Isaac Asimov’s first novel, which he wrote in his teens, a planet in a six suns system does not ever know night. A rather primitive (human) civilization eeks a living… until, as astronomers predicted, at a particular time, all suns are on one side, and night comes. Then the savages make a mob, and go kill the astronomers. This could very well be our future if we don’t react fiercely to the savages who confuse beautiful monuments of the past, as in Palmira, or differential equations, as the work, even the world, of the devil.

But reacting fiercely means terminally offending the savages. Tolerance cannot extend to the intolerant ones. This is where it becomes delicate and subtle. It is not just the Devil who is in the details, it is also philosophy itself.

Patrice Ayme’




December 29, 2015

Thesis: Quantum Waves themselves are what information is (partly) made of. Consciousness being Quantum, shows up as information. Reciprocally, information gets Quantum translated, and then builds the brain, then the mind, thus consciousness. So the brain is a machine debating with the Quantum. Let me explain a bit, while expounding on the way the theory of General Relativity of Ontological Effectiveness, “GROE”:


What is the relationship between the brain and consciousness? Some will point out we have to define our terms: what is the brain, what is consciousness? We can roll out an effective definition of the brain (it’s where most neurons are). But consciousness eludes definition.

Still, that does not mean we cannot say more. And, from saying more, we will define more.

Relationships between definitions, axioms, logic and knowledge are a matter of theory:

Take Euclid: he starts with points. What is a point? Euclid does not say, he does not know, he has to start somewhere. However where that where exactly is may be itself full of untoward consequences (in the 1960s, mathematicians working in Algebraic Geometry found points caused problems; they have caused problems in Set Theory too; vast efforts were directed at, and around points). Effectiveness defines. Consider this:

Effective Ontology: I Compute, Therefore That's What I Am

Effective Ontology: I Compute, Therefore That’s What I Am

Schematic of a nanoparticle network (about 200 nanometres in diameter). By applying electrical signals at the electrodes (yellow), and using artificial evolution, this disordered network can be configured into useful electronic circuits.

Read more at:

All right, more on my General Relativity of Ontological Effectiveness:

Modern physics talks of the electron. What is it? Well, we don’t know, strictly speaking. But fuzzy thinking, we do have a theory of the electron, and it’s so precise, it can be put in equations. So it’s the theory of the electron which defines the electron. As the former could, and did vary, so did the latter (at some point physicist Wheeler and his student Feynman suggested the entire universe what peopled by just one electron going back and forth in time.

Hence the important notion: concepts are defined by EFFECTIVE THEORIES OF THEIR INTERACTION with other concepts (General Relativity of Ontological Effectiveness: GROE).


NATURALLY Occurring Patterns Of Matter Can Recognize Patterns, Make Logic:

Random assemblies of gold nanoparticles can perform sophisticated calculations. Thus Nature can start computing, all by itself. There is no need for the carefully arranged patterns of silicon.

Classical computers rely on ordered circuits where electric charges follow preprogrammed rules, but this strategy limits how efficient they can be. Plans have to be made, in advance, but the possibilities become vast in numbers at such a pace that the human brain is unable to envision all the possibilities. The alternative is to do as evolution itself creates intelligence: by a selection of the fittest. In this case, a selection of the fittest electronic circuits.

(Selection of the fittest was well-known to the Ancient Greeks, 25 centuries ago, 10 centuries before the Christian superstition. The Ancient Greeks, used artificial and natural selection explicitly to create new breeds of domestic animals. However, Anglo-Saxons prefer to name things after themselves, so they can feel they exist; thus selection of the fittest is known by Anglo-Saxons as “Darwinian”. Hence soon we will hear about “Darwinian electronics”, for sure!)

“The best microprocessors you can buy in a store now can do 10 to the power 11 (10^11; one hundred billions) operations per second and use a few hundred watts,” says Wilfred van der Wiel of the University of Twente in the Netherlands, a leader of the gold circuitry effort. “The human brain can do orders of magnitude more and uses only 10 to 20 watts.  That’s a huge gap in efficiency.”

To close the gap, one goes back to basics. The first electronic computers, in the 1940s, tried to mimic what were thought at the time to be brain operations. So the European Union and the USA are trying more of the same, to develop “brain-like” computers that do computations naturally without their innards having been specifically laid out for the purpose. For a few years, the candidate  material that can reliably perform real calculations has been found to be gold.

Van der Wiel and colleagues have observed that clumps of gold grains handle bits of information (=electric charge) in the same way that existing microprocessors do.

Clump of grains computing operate as a unit, in parallel, much as it seems neurons do in the brain. This should improve pattern recognition. A pattern, after all, is characterized by dimension higher than one, and so is a clump operating together. A mask to recognize a mask.

Patterns are everywhere, logics itself are patterns.



So what am I saying, philosophically? I am proposing a (new) foundation for ontology which makes explicit what scientists and prehistoric men have been doing all along. 

The theory of the nature of being is ontology, the “Logic of Being”. Many philosophers, or pseudo-philosophers have wrapped themselves up in knots about what “Being”. (For example, Heidegger, trained as a Catholic seminarian, who later blossomed as a fanatical professional Nazi, wrote a famous book called “Zein und Zeit”, Being and Time. Heidegger tries at some point to obscurely mumble feelings not far removed from some explicit notions in the present essay.)

Things are defined by what they do. And they do what they do in relation with other things.

Where does it stop? Well, it does not. What we have done is define being by effectiveness. This is what mathematicians have been doing all along. Defining things by how they work produce things, and theories, which work. The obvious example is mathematics: it maybe a castle in the sky, but this castle is bristling with guns, and its canon balls are exquisitely precise, thanks to the science of ballistics, a mathematical creation.

Things are what they do. Fundamental things do few things, sophisticated things do many things, and thus have many ways of being.

Some will say: ‘all right, you have presented an offering to the gods of wisdom, so now can we get back to the practical, such as the problems Europe faces?’

Be reassured, creatures of little faith: Effective Ontology is very practical. First of all, that’s what all of physics and mathematics, and actually all of science rest (and it defines them beyond Karl Popper’s feeble attempt).

Moreover, watch Europe. Some, including learned, yet nearly hysterical commenters who have graced this site, are desperately yelling to be spared from a “Federal Europe“, the dreaded “European Superstate“. The theory of Effective Ontology focuses on the essence of Europe. According to Effective Ontology, Europe is what it does.

And  what does Europe do? Treaties. A treaty, in Latin, is “foedus. Its genitive is foederis, and it gives foederatus, hence the French fédéral and from there, 150 years later in the USA, “federal”. Europe makes treaties (with the Swiss (Con)federation alone, the Europe Union has more than 600 treaties). Thus Europe IS a Federal State.

Effective Ontology has been the driver of Relativity, Quantum Physics, and Quantum Field Theory. And this is precisely why those theories have made so many uncomfortable.

Patrice Ayme’


April 25, 2015

Abstract: A new view is seen (“theo-ry”) for the relationship of mind and universe, and mathematics is central. The Mathematical Mind Hypothesis (MMH). The MMH contradicts, explains, and thus overrules Platonism (the ruling explanation for math, among mathematicians). The MMH is the true essence of what makes the Mathematical Universe Hypothesis alluring.


What’s the nature of mathematics? I wrote two essays already, but was told I was just showing off as a mathematician, and the subject was boring. So let me try another angle today.

The nature of mathematics is a particular case of the nature of thinking.

For a number of reasons, deep in today’s physics, as I have (partly) explained in “Einstein’s Error”, many physicists are obsessed with the “Multiverse”, an extreme version of which is the “Mathematical Universe Hypothesis” (MUH), exposed for example by Tegmark, a tenured cosmologist at MIT. Instead of telling people what happened in the first second of the universe, as if I considered myself to be god, I prefer to consider dog:

Dogs LEARN To Choose “y” According To Least Time

Dogs LEARN To Choose “y” According To Least Time

[Dogs can also learn to solve that Calculus of Variation problem in much more difficult circumstances, if the water is choppy, the ground too soft, etc. To have such a mathematical brain allowed the species to catch dinner, and survive.]

The “Multiverse” has its enemies, I am among them. Smolin, a physicist who writes general access books, has tried to say something (as described in Massimo’s Scientia Salon’ “Smolin and the Nature of Mathematics”).

“Smolin,” Massimo, a tenured philosophy professor also a biology PhD, told me “as a counter [to Platonism], offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”

My reply:

Smolin is apparently unaware of a whole theory of “truth” in mathematical logic, and of the existence of the work of famous logicians such as Tarski. When Smolin was in the physics department of Berkeley, so was the very famous Tarski, in the mathematics department. Obviously, the young and unknown Smolin never met the elder logician and mathematician, as he is apparently still in no way aware of any of his work.

Thus, what does Smolin say? Nothing recent. Smolin says mathematics is axiomatic, and develops like games. That was at the heart of the efforts of Frege’s mathematical logic, more than 115 years ago. (Bertrand Russell shot Frege’s theory down, by applying the 24 centuries old Cretan Paradox to it; interestingly, Buridan had found a rather modern solution to the problem, in the 14C!) To help sort things out, it was discovered that one could depict Axiomatic Systems with sequences of numbers. Could not Axiomatics then be made rigorously described, strictly predictive?

Gödel showed that this approach could not work in any system containing arithmetic. Other logicians had proven even more general results in the same vein earlier than that (Löwenheim, Skolem and contemporaries). Smolin is now trying to reintroduce it, as if Löwenheim, Skolem, Gödel, and the most spectacular advances in logic of the first half of the Twentieth Century, never happened.

Does Mr. Smolin know this? Not necessarily: he is a physicist rather than a mathematician (like Tarski, or yours truly).

Smolin: “Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will.”

Smolin: Math brought into existence by HUMAN WILL. Mathematics as will and representation? (To parody Schopenhauer.)

So how come the minds of animals follow mathematical laws? Dogs, in particular, behave according to very complicated applications of calculus.

How come ellipses exist? Have ellipses been brought into existence by Smolin’s “human will”? When a planet follows (more or less) an ellipse, is that a “construction which has been brought into existence by exercises of human will”?

Some will perhaps say that the planet “constructs” nothing. That I misunderstood the planet.

Massimo’s quoted me, and asserted that there was no value whatsoever to the existence of mathematical objects:

I had said: “How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”

Massimo replied: “They don’t.”

And that’s it. It reminded me the way God talked in the Qur’an. It is, what it is, says Allah, and his apparent emulator, Massimo. Massimo did not explain why he feels that the spiral of a nautilus does not exist (or maybe, he does not feel that way, because it clearly looks like a spiral). According to Smolin, the spiral is just a “construct of human will”.

If the spiral is a construct of human will, why not the mountains, and the ocean?

I am actually an old enemy of mathematical Platonism. However, I don’t throw the baby with the bath.

I agree that the “Mathematical Universe Hypothesis”, and Platonism in general are erroneous. However that does not mean they are deprived of any value whatsoever.

Ideas never stand alone. They are always part of theories. And idea such as Platonism is actually a vast theory.

MUH is: ‘Our external physical reality is a mathematical structure.’

I do not believe in the MUH. Because of my general sub-quantic theory, which predicts Dark Matter. In my theory, vast quantum interactions leave debris: Dark Matter. That process is essentially chaotic, and indescribable, except statistically (as the Quantum is). propose a completely different route: our mind are constructed by (still hidden) laws which rule the universe. Call that the MATHEMATICAL MIND HYPOTHESIS (MMD).

Here is the MMD: Our internal neurological reality constructs real physical structures we call “mathematics”.

This explains why a dog’s brain can construct the neurological structures it needs to find the solutions of complex problems in the calculus of variations.

Dogs did not learn calculus culturally, by reading books. Indeed. Still they learned, by interacting with the universe. (It’s unconscious learning, but still learning. Most learning we have arose unconsciously.)

From these interactions, dogs’ brains learn to construct structures which solve very complicated calculus of variations problems. As explained by the Mathematical Mind Hypothesis, (hidden) physics shows up in neurological constructions we call mathematics. And those structures, constructed with this yet-unrevealed, not even imagined, physics, are not just mathematical, but they are what we call mathematics, itself. That’s why dogs know mathematics: their brain contain mathematics.

Patrice Ayme’

Technical Note: Some may smirk, and object that my little theory ignores the variation in neurological structure from one creature to the next. Should not those variations mean that one beast’s math is not another beast’s math?

Not so.

Why? We need to go back to Cantor’s fundamental intuition about cardinals, and generalize (from Set Theory to General Topology). Cantor said that two sets had the same cardinal if they were in bijection. (Then he considered order, and introduced “ordinals”, by making the bijection respect order.)

I propose to say two neurological structure are mathematically the same if they produce the same math. (Some will say that’s obvious, but it’s not anymore obvious than, say, “Skolemization“.)

[Last point: I use “neurology” to designate much more than the set of all neurons, dendrites, synapses, axons and attached oligodendrocytes. I designate thus the entire part of the brain which contributes to mind and intelligence (so includes all glial cells, etc.). That ensemble is immensely complex, in dimensions and topologies.]


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’

Why Mathematics Is Natural

April 21, 2015

There is nothing obvious about the mathematics we know. It is basically neurology we learn, that is, that we learn to construct (with a lot of difficulty). Neurology is all about connecting facts, things, ideas, emotions together. We cannot possibly imagine another universe where mathematics is not as given to us, because our neurology is an integral part of the universe we belong to.

Let’s consider the physics and mathematics which evolved around the gravitational law. How did the law arise? It was a cultural, thus neurological, process. More striking, it was a historical process. It took many centuries. On the way, century after century a colossal amount of mathematics was invented, from graph theory, to forces (vectors), trajectories, equations, “Cartesian” geometry, long before Galileo, Descartes, and their successors, were born.

Buridan, around 1330 CE, to justify the diurnal rotation of Earth, said we stayed on the ground, because of gravity. Buridan also wrote that “gravity continually accelerates a heavy body to the end” [In his “Questions on Aristotle”]. Buridan asserted a number of propositions, including some which are equivalent to Newton’s first two laws.

Because, Albert, Your Brain Was Just A Concentrate Of Experiences & Connections Thereof, Real, Or Imagined. "Human Thought Independent of Experience" Does Not Exist.

Because, Albert, Your Brain Was Just A Concentrate Of Experiences & Connections Thereof, Real, Or Imagined. “Human Thought Independent of Experience” Does Not Exist.

At some point someone suggested that gravity kept the heliocentric system together.

Newton claimed it was himself, with his thought experiment of the apple. However it is certainly not so: Kepler believed gravity varied according to 1/d. The French astronomer Bullialdius ( Ismaël Boulliau) then explained why Kepler was wrong, and gravity should vary as, the inverse of the square of the distance, not just the inverse of the distance. So gravity went by 1/dd (Bullialdius was elected to the Royal Society of London before Newton’s birth; Hooke picked up the idea then Newton; then those two had a nasty fight, and Newton recognized Bullialdius was first; Bullialdius now has a crater on the Moon named after him, a reduced version of the Copernicus crater).

In spite of considerable mental confusion, Leonardo finally demonstrated correct laws of motion on an inclined plane. Those Da Vinci laws, more important than his paintings, are now attributed to Galileo (who rolled them out a century later).

It took 350 years of the efforts of the Paris-Oxford school of mathematics, and students of Buridan, luminaries such as Albert of Saxony and Oresme, and Leonardo Da Vinci, to arrive at an enormous arsenal of mathematics and physics entangled…

This effort is generally mostly attributed to Galileo and Newton (who neither “invented” nor “discovered” any of it!). Newton demonstrated that the laws discovered by Kepler implied that gravity varied as 1/dd (Newton’s reasoning, using still a new level of mathematics, Fermat’s calculus, geometrically interpreted, was different from Bulladius).

Major discoveries in mathematics and physics take centuries to be accepted, because they are, basically, neurological processes. Processes which are culturally transmitted, but, still, fundamentally neurological.

Atiyah, one of the greatest living mathematicians, hinted this recently about Spinors. Spinors, discovered, or invented, a century ago by Elie Cartan, are not yet fully understood, said Atiyah (Dirac used them for physics 20 years after Cartan discerned them). Atiyah gave an example I have long used: Imaginary Numbers. It took more than three centuries for imaginary numbers (which were used for the Third Degree equation resolution) to be accepted. Neurologically accepted.

So there is nothing obvious about mathematical and physics: they are basically neurology we learn through a cultural (or experimental) process. What is learning? Making a neurology that makes correspond to the input we know, the output we observe. It is a construction project.

Now where does neurology sit, so to speak? In the physical world. Hence mathematics is neurology, and neurology is physics. Physics in its original sense, nature, something not yet discovered.

We cannot possibly imagine another universe where mathematics is not as given to us, because the neurology it is forms an integral part of the universe we belong to.

Patrice Ayme’

Emotional Thinking Is Superior Thinking

March 11, 2015

By claiming that emotional thinking is superior, I do not mean that “logical” thinking ought to be rejected, and replaced by passions running wild. I am just saying what I am saying, and no more. Not, just the opposite, “logical” thinking ought to be embraced. However, there are many “logical” types of thought in existence (as Pascal already pointed out). Including the emotional type. They are entangled.

Emotional and logical thinking can be physiologically distinguished in the brain (the latter is mostly about axons; the former about the rest).

Any “logical” thinking is literally, a chain made of points. (And there are no points in nature, said a Quantum Angel who passed by; let’s ignore her, for now!)

Elliptic Geometry In Action: Greeks, 240 BCE, Understood The Difference Between Latitude & Geodesic (Great Circle)

Elliptic Geometry In Action: Greeks, 240 BCE, Understood The Difference Between Latitude & Geodesic (Great Circle). (Traditionally, one quotes Eratosthenes. However, it’s Pytheas of Marseilles who first did this elliptic geometry computation… A century earlier. Pytheas also discovered the Polar Circle, sea ice, and maybe Iceland, among other things boreal…) Whether to develop, or not, this sort of mathematics and physics was, fundamentally, an emotional decision. Involving in particular the emotional worth of the axioms involved.

Some say that hard logic, and mathematics is how to implement “correct thinking”. Those who say this, do not know modern logic, as practiced in logic departments of the most prestigious universities.

In truth, overall, logicians spent their careers proposing putative, potential foundations for logic. Ergo, there is no overall agreement, from the specialists of the field themselves, about what constitute acceptable foundations for “logic”.

It is the same situation in mathematics.

Actually dozens of prestigious mathematicians (mostly French) launched themselves, in the 1950s into a project to make mathematics rigorous. They called their effort “Bourbaki”.

Meanwhile some even more prestigious mathematicians, or at least the best of them all, Grothendieck, splendidly ignored their efforts, and, instead, founded mathematics on Category Theory.

Many mathematicians were aghast, because they had no idea whatsoever what Category Theory could be about. They derided it as “Abstract Nonsense”.

Instead it was rather “Abstract Sense”.

But let’s take a better known example: Euclid.

There are two types of fallacies in Euclid.

The simplest one is the logical fallacy of deducing, from emotion, what the axioms did not imply. Euclid felt that two circles which looked like they should intersect, did intersect. Emotionally seductive, but not a consequence of his axioms.

Euclid’s worst fallacy was to exclude most of geometry, namely what’s not in a plane. It’s all the more striking as “Non-Euclidean” geometry had been considered just prior. So Euclid closed minds, and that’s as incorrect as incorrect can be.

To come back to logic as studied by logicians: the logicS considered therein, are much general than those used in mathematics. Yet, as no conclusion was reached, this implies that mathematics itself is illogical. That, of course, is a conclusion mathematicians detest. And the proof of their pudding is found in physics, computer science, engineering.

So what to do, to determine correct arguments? Well, direct towards any argument an abrasive, offensive malevolence, trying to poke holes, just as a mountain lion canines try to pass between vertebras to dislocate a spine.

That’s one approach. The other, more constructive, but less safe, is to hope for the best, and launched logical chains in the multiverses of unchained axiomatics.

Given the proper axioms, (most of) an argument can generally be saved. The best arguments often deserve better axiomatics (so it was with Leibnitz’s infinitesimals).

So, de facto, people have longed been using not just “inverse probability”, but “inverse logic”. In “inverse logic”, axioms are derived from what one FEELS ought to be a correct argument.

Emotions driving axiomatics is more metalogical, than axiomatics driving emotions.


To the preceding philosophy professor Massimo Pigliucci replied (in part) that:


“…Hence, to think critically, one needs enough facts. Namely all relevant facts.”

Enough facts is not the same as all the relevant facts, as incorrectly implied by the use of “namely.” 

“It is arrogant to think that other people are prone to “logical fallacies”.”

It is an observation, and facts are not arrogant. 

“A Quantum Wave evaluates the entirety of possible outcomes, then computes how probable they are.”

Are you presenting quantum waves as agents? They don’t evaluate and compute, they just behave according to the laws of physics.

“just as with the Quantum, this means to think teleologically, no holds barred”

The quantum doesn’t think, as far as I know. 

“Emotional Thinking Is Superior Thinking” 

I have no idea what you mean by that. Superior in what sense? And where’s the bright line between reason and emotion?

“Any “logical” thinking is literally, a chain made of points”

No, definitely not “literally.” 

It may not follow from the axioms, but I am having a hard time being emotionally seductive by intersecting circles. 

“Euclid’s worst fallacy was to exclude most of geometry, namely what’s not in a plane.”

That’s an historically bizarre claim to make. Like saying that Newton’s worst fallacy was to exclude considerations of general relativity. C’mon. 

“as no conclusion was reached, this implies that mathematics itself is illogical” 

Uhm, no. 

“to hope for the best, and launch logical chains in the multiverses of unchained axiomatics” 

Very poetic, I have no idea what that means, though.”


Massimo Pigliucci is professor of philosophy at CUNY in New York, and has doctorates both in biology and philosophy. However, truth does not care about having one, or two thousands doctorates. It would take too long to address all of Massimo’s errors (basically all of his retorts above). Let me just consider two points where he clings to Common Wisdom like a barnacle to a rock. The question of Non-Euclidean geometry, and of the Quantum. He published most of the answer below on his site:

Dear Massimo:

Impertinence and amusement help thought. Thank you for providing both. Unmotivated thought is not worth having.

The Greeks discovered Non-Euclidean geometry. It’s hidden in plain sight. It is a wonder that, to this day, so many intellectuals repeat Gauss’ self-serving absurdities on the subject (Gauss disingenuously claimed that he had discovered it all before Janos Bolyai, but did not publish it because he feared the “cries of the Beotians”… aka the peasants; Gauss does not tell you that a professor of jurisprudence had sketched to him how Non-Euclidean geometry worked… in 1818! We have the correspondence.).

The truth is simpler: Gauss did not think of the possibility of Non-Euclidean geometry (although he strongly suspected Euclidean geometry was not logical). Such a fame greedster could not apparently resist the allure of claiming the greatest prize…

It is pretty abysmal that most mathematicians are not thinking enough, and honest enough, to be publicly aware of Gauss’ shenanigans (Gauss is one of the few Muhammads of mathematics). But that fits the fact that they want mathematics to be an ethereal church, the immense priests of which they are. To admit Gauss got some of his ideas from a vulgar lawyers, is, assuredly, too painful.

That would be too admit the “Prince of Mathematics” was corrupt, thus, all mathematicians too (and, indeed, most of them are! Always that power thing; to recognize ideas have come out of the hierarchy in mathematics is injurious to the hierarchy… And by extension to Massimo.)

So why do I claim the Greeks invented Non-Euclidean geometry? Because they did; it’s a fact. It is like having the tallest mountain in the world in one’s garden, and not having noticed it: priests, and princes, are good at this, thus, most mathematicians.

The Greek astronomer Ptolemy wrote in his Geography (circa 150 CE):

“It has been demonstrated by mathematics that the surface of the land and water is in its entirety a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles.”

Not just this, but, nearly 400 years earlier, Eratosthenes had determined the size of Earth (missing by just 15%).

How? The Greeks used spherical geometry.

Great circles are the “straight lines” of spherical geometry. This is a consequence of the properties of a sphere, in which the shortest distances on the surface are great circle routes. Such curves are said to be “intrinsically” straight.

Better: Eusebius of Caesarea proposed 149 million kilometers for the distance of the Sun! (Exactly the modern value.)

Gauss, should he be around, would whine that the Greeks did not know what they were doing. But the Greeks were no fools. They knew what they were doing.

Socrates killed enemies in battle. Contemporary mathematicians were not afraid of the Beotians, contrarily to Gauss.

Aristotle (384-322 BC) was keen to demonstrate that logic could be many things. Aristotle was concerned upon the dependency of logic on the axioms one used. Thus Aristotle’s Non-Euclidean work is contained in his works on Ethics.

A thoroughly modern approach.

The philosopher Imre Toth observed the blatant presence of Non-Euclidean geometry in the “Corpus Aristotelicum” in 1967.

Aristotle exposed the existence of geometries different from plane geometry. The approach is found in no less than SIX different parts of Aristotle’s works. Aristotle outright says that, in a general geometry, the sum of the angles of a triangle can be equal to, or more than, or less than, two right angles.

One cannot be any clearer about the existence on Non-Euclidean geometry.

Actually Aristotle introduced an axiom, Aristotle’s Axiom, a theorem in Euclidean and Hyperbolic geometry (it is false in Elliptic geometry, thus false on a sphere).

Related to Aristotle’s Axiom is Archimedes’ Axiom (which belongs to modern Model Theory).

One actually finds non trivial, beautiful NON-Euclidean theorems in Aristotle (one of my preferred frienemies).

Non-Euclidean geometry was most natural: look at a sphere, look at a saddle, look at a pillow. In Ethika ad Eudemum, Aristotle rolls out the spectacular example of a quadrangle with the maximum eight right angles sum for its interior angles.

Do Quantum Wave think? Good question, I have been asking it to myself for all too many decades.

Agent: from Latin “agentem”, what sets in motion. Quantum waves are the laws of physics: given a space, they evaluate, compute. This is the whole idea of the Quantum Computer. So far, they have been uncooperative. Insulting them, won’t help.

Patrice Ayme’

Universe: Not Just Mathematical

August 14, 2014

Some claim the “Universe is mathematical”. Their logic is flawed. I show why.

Max Tegmark, a MIT physics professor, wrote “Our Mathematical Universe”. I present here an abstract I concocted of an interview he just gave to La Recherche. Followed by my own incisive comments. However absurd Tegmark may sound, I changed nothing to the substance of what he said:

La Recherche (France; Special Issue on Reality, July-August 2014): Max, you said “Reality is only mathematical”. What do you mean?

Tegmark: The idea that the universe is a mathematical object is very old. It goes all the way back to Euclid and other Greek scientists. Everywhere around us, atoms, particles are all defined by numbers. Spacetime has only mathematical properties.

La Recherche: Everything is math, according to you?

Formulation Before Revelation of Mathematization

Formulation Before Revelation of Mathematization

Tegmark: Think about your best friend. Her great smile, her sense of humor. All this can be described by equations. Mathematics explain why tomatoes are red and bananas yellow. Brout, Englert, Higgs predicted a boson giving mass to all other particles. Its discovery in 2012 at CERN in Geneva led to the 2013 Nobel Prize in Physics!

Tyranosopher [unamused]: Notice, Max Tegmark, that the “Nobel” thoroughly excites you. You brandish it, as if it were a deep reality about the universe. But, in truth, the Nobel is strictly nothing for the universe. It’s just a banana offered by a few self-interested apes to other self-fascinated apes. The Nobel has more to do with the nature of apish society, rather than that of the universe. In other words, we ask you about the nature of the universe, and you answer with the Authority Principle among Hominidae. You may as well quote the Qur’an.

Tegmark [unphazed]: There are an enormous number of things that equations do not explain. Consciousness, for example. But I think we will make it. We are just limited by our imagination and our creativity.

La Recherche: According to you, there is no reason that part of the world escape mathematics?

Max Tegmark: None whatsoever. All properties are mathematical! We potentially can understand everything!

La Recherche: As a Platonic mathematician, you consider mathematical concepts are independent of all and any conscious act?

MT: I am an extreme Platonist, as I think that not only mathematical structures are real, but they are all what reality is.

Relativity and Quantum Physics confirmed that reality is always very different from what one believes. Very strange and very different from our intuition. Schrodinger’s equation, the fundamental equation of Quantum Mechanics, shows that a particle can be in several places at the same time. Thus one does not try to describe the motion of this particle, but the probability of its presence in such and such a place.

But, a century later, physicists are still in deep disagreement about what it all means. I think this interpretation keeps dividing people, because they refuse to admit what goes against their intuition.

Tyranosopher: Notice, Max Tegmark, that you presented as a fact (“a particle can be in several places at the same time”) something you admit later is only an “interpretation”. That’s dishonest: an “interpretation” is not a “fact”.

Tegmark [livid]: The strength of mathematics comes from the fact that they have no inhibition. Strangeness does not stop them.

Tyranosopher: Indeed, that’s why, as a trained mathematician, I am very insolent.

La Recherche: Max Tegmark, is it your mathematical approach that makes you defend another controversial idea, that of multiple universes?

Max Tegmark: I really believe that human beings never think big enough. We underestimate our capability to understand the world through mathematics, but also our capacity to apprehend its dimensions. To understand that we live on a planet with a diameter of a bit more than 12,000 kilometers was a first, enormous, step. That this planet is infinitesimal in this galaxy, itself one out of billions, was another enormous step. The idea of multiverses is more of the same. We discover again, and more, that what we understand is only a speck of something much larger. That much larger thing is the Multiverses, of types I, II, III, and IV.

Tyranosopher: La Recherche’s Interview then proceeds further, but let me unleash a fundamental critique here.

I am a deadly enemy of the Multiverse, as I believe that it rests on an ERROR of interpretation of Quantum Physics (the one Tegmark presented as a fact above, before admitting that it was, well, only an interpretation). The fact that it is another desperate scaffolding erected to save the Big bang theory does not make it better.

Now for the notion that the universe being full of math. This is understood to mean that the universe is full of equations. Equations were invented in the Sixteenth Century. Many, if not most, equate mathematics with the art of equating.

What’s an equation? It’s something that says that two things independently defined, one on the left side of the equal sign, the other on the right side, are equal. Great. What could be simpler: what is different is actually the same!

Notice this, though: before you can equate, you must define what you are equating. On both sides.

An equation equates concepts independently defined. Ultimately, definitions are not mathematical (see on the Nature of Mathematics, to follow soon). At best, definition is metamathematical. Our metamathematical universe? End of Mr. Tegmark’s naivety.

When we get down to it, it’s more our philosophical universe, before it’s our mathematical universe: no definitions, no equations.

How can a physicist make such a gross logical mistake? Are they not supposed to be smart? (OK, it’s smart to sell lots of books).

What allows to make that logical mistake? Education, or lack thereof. Many a mathematician will make the same mistake too. The problem is that neither conventional mathematicians, nor, a fortiori, physicists, are trained logicians. They just play some in the media.

Who needs a multiverse? It seems the universe of science is already too large for many physicists to understand.

Patrice Ayme’

Speciation Math; Why It’s Crucial

February 16, 2014

Believing in Christianity and its Dog God barking in the sky is a fundamental element of the subjugation of the masses in the USA. In Europe, it’s natural to be an atheist (except if one is a Muslim or a Pole, and even then…). In the USA, atheism is impolite. That’s why the president ends all his homilies with an appeal to Dog God. God the Dog is watching over you, its son, the NSA, also, and bless be the United States of America.

It’s unlikely a honest to goodness American will be prone to revolution, as this would implicitly recognize the primacy of man over the creation of Pluto Dog God. So no wonder creations of man such as science (a form of anti-Plutocratic revolution from excessive usage of the nervous system) are attacked at every turn.

Mammoths were very clever. Science was well started when Neanderthals made mammoth hunting plans on the plains:

Homo Sapiens Neanderthalensis: Same Species, Us.

Homo Sapiens Neanderthalensis: Same Species, Us.

[The colored eyes of Neanderthals are a proven thing, so are the facial features; the hairline is probably too low… The will to make Neanderthals beastly is obdurate…]

Such were the idle thoughts brought to me by a rant against evolution on so called “Big Think”. The conclusion struck me, with the issues it brought:

…” [Evolution Theory is] also terrible at explaining the speed at which speciation occurs. (Of course, The Origin of Species is entirely silent on the subject of how life arose from abiotic conditions in the first place.) It doesn’t explain the Cambrian Explosion, for example, sudden appearance of intelligence in hominids, or the rapid recovery (and net expansion) of the biosphere in the wake of at least five super-massive extinction events in the most recent 15% of Earth’s existence.”

This is all false or misleading, but still it’s interesting to answer.

OK, I will let pass the fact that Darwin’s book, The Origin of Species was published in 1859, about a century before laboratory experiments enabled to create organic chemistry in the lab from the sort of atmosphere, water and lightning Earth enjoyed for billions of years. 1859: that was two years before Lincoln became president. Slavery was lawful in the USA.

Reducing Evolution to Darwinism is silly: Darwin himself was an enthusiastic Lamarckist. Lamarck established biological (Lamarck’s neologism) evolution by studying the changes in fossilized mollusks over millions of years (the order of mollusks had been scientifically determined by Cuvier, Lamarck’s predecessor) . Lamarck was banned with rage and consummate fury by the very Christian universities that dominated England (Oxbridge, etc.). This explains why neither Wallace, nor Darwin, nor Spencer were university professors. The former two were not just great expositors of the (French) Theory of Evolution, but pushed it further with discoveries, from the Wallace Line, to Patagonia or the Galapagos.

Let’s go back to the stupid quote on top (excellent lemonade can be made from old lemons, though).

One statement is clearly false: Contrarily to that misleading quote, supreme intelligence took millions of years to appear in hominids. Supreme intelligence, as evidenced, more exactly, by brain size, followed the evolution of bipedalism (and, thus, attendant change of in behavior, diet and environment). We know it took at least 5 million years, from bipedalism to the apparition of brainy Homo Erectus.

What this means is that bipedalism opened a new ecological niche. Bipedalism allowed a number of related species of primates, the hominids, to roam around, and literally, dominate the landscape. As they roamed around, the opportunity for even more sophisticated behaviors arose, and caused a sort of evolutionary sucking: nature abhors a vacuum, and biological evolution, like the Quantum (and it’s probably related) tends to occupy all the space it can occupy (this is why the math of the Quantum are represented by Hilbert Spaces spanned by Quantum States; the exact equivalent in evolution are ecological niches… the complication with life is that life itself creates the niches. The same happens with renormalization in QFT!).

Thus the hominids evolved an increasing number of environmentally disruptive and manipulative behaviors, which, in turn, favored mutations favoring them. That evolutionary phenomenon is thus non-linear, a sort of exponential, it can go fast.

A new function, bipedalism, allows to bear arms, using one’s… arms. That in turn, makes bellicose and predatory behavior, let alone good eyesight, more profitable, and so on.

Actually, a similar type of mathematics is at play after mass extinction, explaining that life’s diversity tends to augment with them.

After life got wiped out of billions of ecological niches in a major extinction event, those get reoccupied quickly by maladapted species. Maladapted to said niches, that is. Thus billions of speciations will tend to occur. These speciations cannot happen in steady state, because the environment of the steady state includes a fine balance of the physical environment with the existing, entangled species. If an event wipes those out, the old steady state cannot be recreated, and each niche acts like a throw of the dice.

Thus massive extinction tends to lead to massive speciation. It’s purely mathematical.

A lot of complicated mathematics and physics underlays Evolution Theory. I presented my own to help explain the apparent disappearance of Neanderthals. My theory rests on subtle mathematics, physiological, and environmental considerations. It’s actually not a theory of Neanderthal disappearance, rather than a theory of the disappearance of a Neanderthal appearance. So it would in particular predict that Europeans are much more Neanderthal than one thought.

Why? Neanderthal had evolved into a superior species (hilarious: Nazism is coming back, but instead of being Aryan, it’s Neanderthal… The Thal of Neander is on the Dussel, basically in the Ruhr…). Neanderthal has had to have evolved in a superior species, or she would not have rules the freezing North (a related species, the Denivosan, ruled the North-East, and evolved into Chinese, or, at least Australian bushmen).

If Neanderthal was that superior, he could not be wiped out. So, instead, one needed a more subtle explanation, which I provided: “Mathematics “Extinguished” Neanderthals”. Two very recent (end January 2014) DNA computational studies support my point of view. (See note.)

Oh, by the way, mathematics is not just about equations. It’s first of all, about ideas. Anybody who had Euclidean geometry or Mathematical Logic will confirm this. The phenomenon of occupying the entire space is the essence of Quantum Physics (Me, myself and I say). It’s the math of Hilbert Space. It’s also the math of evolution.

All these mathematics of evolution are no idle pursuit. Right now, the planet is at a fulminant stage of evolution, thanks to us. Understanding what the laws that will command our destiny, are, is the quintessence of humanity.

Patrice Aymé

Note: With up to 30% Neanderthal genome, it is found that some Neanderthal traits survived very well (say about skin), while other disappeared. Take the lack of waist of Neanderthal: an advantage in very cold climate, but not anymore after clothing became good enough; and probably a disadvantage for running and combat.


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an evolving guide to practical Stoicism for the 21st century

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Artificial Turf At French Bilingual School Berkeley

Artificial Turf At French Bilingual School Berkeley

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Defense Issues

Military and general security

Polyhedra, tessellations, and more.

How to Be a Stoic

an evolving guide to practical Stoicism for the 21st century

Donna Swarthout

Writer, Editor, Berliner


Defending Scientism