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, by.us, 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’



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39 Responses to “WE ARE MATHEMATICS”

  1. Gmax Says:

    Deep. However you have a typo, you wrote the second time 3016 meaning 2014 for the Nobel of the Moser pair

  2. Paul Handover Says:

    Stop getting me all fired up, as in mentally excited, when the bedside light is going out in twenty minutes! 😎

    Actually, there are some fabulous personal anecdotes I could bore you and your readers with concerning DR; Dead Reckoning. The history of the ship’s log for example. Or how the mind, even when asleep, can wake one because the self-steering has dropped out, or there’s been a course upset. How out in the ocean, when there are no land masses to interfere gravitationally, one can easily sense local magnetic North even with one’s eyes closed!

    Some other time! Sleep calls!

    • Patrice Ayme Says:

      Hahaha. You sleep more than our Dear Leader, who gets only 3-4 hours of sleep a night, to his chief of staff’s distress… But Trump does not need to get very deep at that point, just do stuff…
      Yeah, dead reckoning has an instinctive aspect. I remember running in northern European forests in winter, under deep clouds, gloom, and little light. One could not tell at all where the sun was. Somehow I did not get lost in these flat forests. That always amazed me.
      There is no boredom from you, go ahead.

      (I could have given the example of some carnivorous mammals like genets, who learn a room and all its branches, slow and careful. Then they can run all around it, even blinded…)
      Anyway, please rest your math! 😉

  3. Gmax Says:

    So, if I understand you well, you are predicting that there will be found more sophisticated organizations in the brain reflecting higher mathematics?

  4. The conscious, mathematical brain! – Learning from Dogs Says:

    […] is of a most fascinating post published by Patrice Ayme on the 25th. January. It was called WE ARE MATHEMATICS. But there were parts of Patrice’s post that I struggled with so I am offering it to you with […]

  5. Paul Handover Says:

    https://learningfromdogs.com/2017/01/30/conscious-mathematical-brain/ is my contribution to the cause! 😎

  6. Patrice Ayme Says:

    From Paul Handover’s Learning From Dogs:
    “The Conscious, Mathematical Brain!”

    ….more of a reposting today than a pure guest post. That reposting is of a most fascinating post published by Patrice Ayme on the 25th. January. It was called WE ARE MATHEMATICS. But there were parts of Patrice’s post that I struggled with so I am offering it to you with a rather long introduction.

    I hope you enjoy it.


    The sound of an object falling to the floor of the shower, suddenly and without warning, nearly caused me to jump out of my skin.

    I was washing my hair and had my eyelids tightly closed lest the shampoo suds got into my eyes. Inadvertently, I had felt my right elbow dislodge something from the top of the small corner shelf that held the bar of soap and the bottle of hair shampoo.

    I let the flowing warm shower water rinse the suds from my face, opened my eyes and looked down. The object that had made such a sudden, loud noise was a plastic brush maybe three or four inches long. It had fallen to the wet shower floor some five feet below the corner shelf where the brush normally lived.

    As I stared down at the brush, the warm water cascading comfortably down my body, I reflected that in the space of a fraction of a second my mind had computed the distance that the unknown object had fallen and offered me a sense of the speed it must have been traveling when it hit the floor.

    Now don’t get me wrong! I didn’t come up with a precise answer to that question of how fast the brush was going but in that moment of thought I sensed both the distance the brush had fallen, five feet; plus or minus, and the effect of gravity in accelerating that brush even over such a small distance. (Later I calculated the brush hit the shower base going at around 10 fps.)

    Now it would have never occurred to me that my brain was capable of almost instantaneous calculations, as in mathematical calculations, if I hadn’t read in the previous twenty-four hours a recent essay from Patrice Ayme. An essay that convinced me completely that, in Patrice’s words:

    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.
    Patrice had opened my eyes, more accurately opened my mind, to something that was then immediately clear to me and will be to you, dear reader: Our brains have an intuitive and instinctive sense of space. Not space in some abstract sense of the term but space in the sense of spatial awareness.

    Think how easily, how quickly, you understand distance. Whether it is a measure of distance in your own home or assessing how far away that bird is flying towards and setting down on a high branch of a tall pine tree.

    Think how even with our eyes closed we can navigate around a familiar part of our lives. Think how the sailors of ancient times (and trust me not so ancient times) used ‘Dead Reckoning’ (DR) to navigate safely and securely across vast oceans.

    Our brains could only do this if they were computing these spatial assessments mathematically.

    OK, that’s enough from me. Here’s that essay from Patrice.

    • Patrice Ayme Says:

      Thanks Paul for re-publishing my essay! Your introduction is absolutely excellent I will reproduce as a comment to my essay, as it illustrates it so well. It is true that such subjects call on long meditation, as you point out. I had plenty of time to ponder the essence of mathematics, as a researcher in the discipline… And I think you studied electrical engineering, which calls on mathematics such as infinitesimal calculus, so it’s easier for people like us…

  7. mincs1 Says:

    January 30, 2017 at 04:39
    Interesting to think of ourselves (and dogs) as mathematical beings – a little mind bending thought to contemplate 🙂

    • Patrice Ayme Says:

      Dear Mincs1: It’s not just a thought, but a flower which has been picked up. An unexpected blossom! It’s for us to look at, and to go further, then, where no thought, or even feeling, has been before

  8. petspeopleandlife Says:

    January 30, 2017 at 09:41
    I’m so sorry but math does not remotely interest me, I hate math, in fact. Guess that is why I’m so dumb.

  9. Susan Leighton Says:

    One of the basic ways to communicate is through mathematics. This is the go to language for future space exploration and possible interaction with intelligent life forms. Sounds sci-fi but it is based in reality. Terrific article, Patrice!

    • Patrice Ayme Says:

      Hello Susan, and thanks! The idea is that mathematics is not just a language (that idea was written, black on white, by Galileo. He spoke of circles, triangles, lines: the ellipses of Kepler had no sunk in, and Galileo did not know about brand new Descartes’ algebraic geometry and even newer Fermat’s calculus). The idea is that our very substance is mathematics itself.
      Our minds are unknown, because our maths are unknown.
      As far as SETI is concerned, I am rather skeptical (I have personal addition to the Drake equation…)

  10. Sue Dreamwalker Says:

    Sue Dreamwalker
    February 1, 2017 at 07:59
    I had half written a comment here Paul and I enjoyed Patrice’s take on the Conscious Mathematics.. Then got called away..
    We are all of us upon this planet part of the Jigsaw within the mathematics of Energy.. Our Energy is constantly emitting and radiating outwards at the speed of thought.. 🙂 As we move.. Energy around us moves and all particles within our space then move also.. For there is no Empty Space.. But its full of invisible Energy

    I am sure Brian Cox the scientist could explain it much better than I.
    I feel each of us contribute to the Conscious Maths of the Universe.. As we give out either our Negative or Positive rays of thought..
    Each thought we think is energy, we too are Energy.. and are generating our future with what we think right now..
    One only has to look at the paint upon our walls, the fabric of our clothes.. All once someones thought put into motion via Action projected into our reality..

    Which is why we should be careful what we wish for.. For we really do attract like for like as energy is energy it neither knows what is Good or bad.. it will gather momentum regardless and follow the pattern of mathematics like a magnet.. Positive or Negative..

    And this is why I try to keep my vibration in the energy of Love, Unity, Harmony and Peace…

    A very interesting Read..

    • Patrice Ayme Says:

      Thanks Sue!
      The claim is that our unconscious is mathematically organized. That’s where Plato’s “forms” are found, in the brain…

      As far as Brian Cox and Al., exposing modern physics, the truth is that we know some things, which are astounding. But we are far from knowing the full story. We know a spectacular bit: plenty of energy in empty space. BUT we don’t know how MUCH there is (some attempted theories are off by a factor of 10^60, a one followed by 60 zeroes…). And we don’t know what causes “it”, whatever “it” is: the theory of virtual particles is controversial (I debated this subject with the creator of the theory, Richard Feynman!!!!!). By the way, I have my own theory.

      Thus, really grand extrapolations by grand physicists are just like claims in the primary school yard.

      It would be better to teach only what we really know for sure, and that’s amazing enough…

  11. Sue Dreamwalker Says:

    February 1, 2017 at 08:05
    I should also have said Patrice is so right here with his thoughts
    ” But that Principle extends also to what we chose NOT to experience, which can be just as bad, if not worse.”
    I so agree.. putting our INTENT is pouring energy into a thing.. a thought.. When we keep pouring that energy into not wanting something to happen we then can well be attracting that very same thing towards us.. because that is the prominent energy we are focusing upon

    • Patrice Ayme Says:

      Very true, Sue! Some of the Trump phenomenon just arose that way. For decades “democrats” have been busy denying what their leaders were really doing. More and more energy was put into denial and lying. The Main Stream Media, all plutocratically owned, got used to lying ever more bluntly and subtly.

      Finally all too many voters (who last voted for Obama in 2012, studies have shown) had enough… They voted for those using the same methods, but with the avowed aim to serve them. Evil has grown.

      • Sue Dreamwalker Says:

        Yes which if we look back within Europe’s History before WWII is how these leaders came into power.. and How dangerous they can be.

        • Patrice Ayme Says:

          As regards to your latest point, Sue, I am writing a little essay on this today. I have a slightly different take, even more ominous… Much more present. Not a question of particular “leaders”. Obama viewed himself as a great “leader”. In truth, he was led by the nose, by the real leaders, thus leading from behind as the average circus bear used to be… The same happened with… Hitler. What is more important is to find out who, what, the true leaders are.

  12. Gmax Says:

    Are you saying that GRID is a FORM, Plato style? And there would be others, not yet revealed?

  13. Paul Handover Says:

    February 1, 2017 at 12:35
    Sue, Patrice, looks like you two have really got going. Will one of you turn out the lights when you’ve finished! 🐶

    Seriously you two, I am delighted with the exchanges. Just read them out to Jeannie and we have both been humbled by the knowledge and intelligence displayed in your ‘conversation’. Thank you!

  14. Patrice Ayme Says:

    [Sent to Aeon, Feb 8, 2014.]
    It’s telling that no one, in these essay and comments, but for myself, mentioned the 2014 Nobel in biology… However, nobody seems to have noticed what I said, so let me reiterate:

    Mathematics is a neuronal performance. See:


  15. Patrice Ayme Says:

    Mathematics is understandable, because our brains contain such mathematical structures, themselves in-formed from the universe.

    So the unreasonable effectiveness of math (Wigner) and the amazing comprehensibility of the world (Einstein) are both explained by the math in our head (including Quantum structures!), replacing Plato’s cave.https://patriceayme.wordpress.com/2017/01/25/we-are-mathematics/

    • Peter Kinnon Says:

      Peter Kinnon
      On the contrary “the amazing comprehensibility of the world” is ascribable not to some mystical “math in our head” but rather our the position in the broad evolutionary process that can be traced back at least as far as the formation of chemical elements and extends through the evolution of technology in the shared imagination of we snoutless apes. See my latest book “The Intricacy Generator: Pushing Chemistry and Geometry Uphill”.

      • Patrice Ayme Says:

        There is nothing “mystical” about the “math in our head”. At least that’s what the Nobel Committee decided in 2014 when it attributed the Nobel to three “math in our head” biologists. Of course it has to do with an evolution started a billion years ago (most probably, if not older). Attributing the math in our head to neuronal networks is much more precise, scientific and testable, than the more general ascribing “in a position in the broad evolutioniary process”.

        • Peter Kinnon Says:

          All that you are really pointing out with your odd “brain in the head” wording is that the CNS is essentially a computing device, which is these days generally accepted.

          Moreover, that it is capable of the simple tasks of differentiation and integration should in no way surprise us.

          Even a bunch of silicon chips can solve differential equations. Such far more difficult tasks as complex pattern recognition (what flower is that?) are still beyond the reach our present day information processing devices. But not for long!

          Peter KinnonFOLLOW
          That approach has some merit, Patrice, but still does not bring formal mathematics into necessary correspondence with reality.

          (I use “formal” here to refer tor the ponderous conscious activity)

          • Patrice Ayme Says:

            Thanks for the merit, Peter! Glad to see we agree. The brain is most probably at least a Quantum computer. But it is so by incarnating, embodying. Thus mathematics is essentially the working out of the laws of physics of the brain. And physics is checking out which of these brain physics laws are operating out there too.

            I am basically claiming the brain has all the physics, and more…

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