Tag: Mathematics

The conscious, mathematical brain!

A new week!

As I have frequently mentioned, I so enjoy having guest posts being sent to me.

They give you, dear reader, a break from yours truly and so very often they offer a new and interesting perspective on dogs, on us, and on the world.

This week there are three guest posts, in various guises, lined up and, who knows, there may be a couple more heading in.

But to the first of those guest posts.

Well, technically, 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.

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Smack!

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.

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WE ARE MATHEMATICS

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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I do so hope that this essay from Patrice fires you up as it did me. If it leaves you with questions, then offer them to me as a comment to this post and I will take it upon myself to have Patrice answer them.

Finally, did you pick up on the fact that it isn’t just our human brains that are mathematical organs, it applies to the brains of dogs as well!

Mathematics in action. (Photo courtesy of Pinterest)
Mathematics in action. (Photo courtesy of Pinterest)

The core relationship; with ourself.

A reposting of an item from this place some three years ago.

On Sunday, in recognition of Valentine’s Day, I posted a selection of articles under the post title of Loving Relationships.

Then yesterday my day that should have been quiet and uneventful turned out to be anything other than that! So it was well after 3pm that I sat down in front of my PC wondering what to publish today. I thought that as there had been a steady flow of new readers signing up to follow this place (and a huge thank you – it really does mean a lot to me) I might see what I published three years ago.

To my surprise it was a post about the most important relationship of all; the one with ourself.

So please do enjoy what was published on February 15th, 2012.

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Do you or I really know who we are?

The strangeness of this species Homo sapiens.

My writings of the previous three days have explored the nature of man. The many ways that we struggle to understand so many issues in our lives. In particular the biggest issue of them all since we abandoned the life of the hunter-gatherer. Our very survival.

It would be so easy to beat oneself up. To stare in the mirror and despair at all the unfinished ideas that one has about being ‘sustainable’ shortly before jumping on one’s shiny new tractor, yet another symbol of our industrial civilisation. The hypocrisy, the double standards!

New tractor being delivered last December.
New tractor delivered last December.

But the mistake in any attempt at self-awareness is the assumption that you know who you are! Therein lays the problem.!

Marcus Peter Francis du Sautoy is a very smart person. This is how WikiPedia describes him.

sautoy
Prof. Marcus Sautoy

Marcus Peter Francis du Sautoy, OBE (born in London, 26 August 1965)[3] is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. Formerly a Fellow of All Souls College, and Wadham College, he is now a Fellow of New College. He is President of the Mathematical Association.

Prof. Sautoy came to the realisation that the thoughts that make us feel as though we know ourselves are easy to experience. But where do those thoughts come from? Marcus Sautoy acknowledged that they are notoriously difficult to explain.

So, in order to find out where they come from Marcus subjects himself to a series of probing experiments. With the help of a hammer-wielding scientist, Jennifer Aniston and a general anaesthetic, Professor Marcus du Sautoy goes in search of answers to one of science’s greatest mysteries: how do we know who we are?

He learns at what age our self-awareness emerges and whether other species share this trait.

Next, he has his mind scrambled by a cutting-edge experiment in anaesthesia. Having survived that ordeal, Marcus is given an out-of-body experience in a bid to locate his true self. And in Hollywood, he learns how celebrities are helping scientists understand the microscopic activities of our brain.

Finally, he takes part in a mind-reading experiment that both helps explain and radically alters his understanding of who he is.

All of this is covered in a fabulously interesting episode from Horizon, the excellent and long-running BBC TV science and philosophy series. Thankfully, it made its way onto YouTube.

(NB: In the intervening period, that BBC Horizon programme has been removed from YouTube for copyright reasons. That’s a great shame. However, the following documentary from the good Professor will, I am sure, be equally fascinating.)

The Secret Rules of Modern Living Algorithms

Published on Oct 30, 2015

Without us noticing, modern life has been taken over. Algorithms run everything from search engines on the internet to satnavs and credit card data security – they even help us travel the world, find love and save lives.
Mathematician Professor Marcus du Sautoy demystifies the hidden world of algorithms. By showing us some of the algorithms most essential to our lives, he reveals where these 2,000-year-old problem solvers came from, how they work, what they have achieved and how they are now so advanced they can even programme themselves.

As Confucius reportedly wrote: Real knowledge is to know the extent of one’s ignorance.

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Magic!

The old and the new.

Like thousands of others, Jean and I are regular viewers of the TED Talks.

So first the old. Here’s a reminder of the inspiring nature of mathematics; in this case Fibonacci numbers.

Published on Nov 8, 2013

Math is logical, functional and just … awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

Now to the new. Innovation at its very best.

Published on Jul 11, 2013

The development of new medicine is problematic because laboratories cannot replicate the human body’s environment, making it difficult to determine how patients will respond to treatment. At TEDxBoston, Geraldine Hamilton demonstrates how scientists can implant living human cells into microchips that mimic the body’s conditions. These “organs-on-a-chip” can be used to study drug toxicity, identify potential new therapies, and could lead to safer clinical trials.

Do you or I really know who we are?

The strangeness of this species Homo sapiens.

My writings of the previous three days have explored the nature of man.  The many ways that we struggle to understand so many issues in our lives. In particular the biggest issue of them all since we abandoned the life of the hunter-gatherer.  Our very survival.

It would be so easy to beat oneself up.  To stare in the mirror and despair at all the unfinished ideas that one has about being ‘sustainable’ shortly before jumping on one’s shiny new tractor, yet another symbol of our industrial civilisation.  The hypocrisy, the double standards!

New tractor being delivered last December.
New tractor delivered last December.

But the mistake in any attempt at self-awareness is the assumption that you know who you are!  Therein lays the problem.!

Marcus Peter Francis du Sautoy is a very smart person.  This is how WikiPedia describes him.

sautoy
Prof. Marcus Sautoy

Marcus Peter Francis du SautoyOBE (born in London, 26 August 1965)[3] is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. Formerly a Fellow of All Souls College, and Wadham College, he is now a Fellow of New College. He is President of the Mathematical Association.

Prof. Sautoy came to the realisation that the thoughts that make us feel as though we know ourselves are easy to experience.  But where do those thoughts come from? Marcus Sautoy acknowledged that they are notoriously difficult to explain.

So, in order to find out where they come from Marcus subjects himself to a series of probing experiments.  With the help of a hammer-wielding scientist, Jennifer Aniston and a general anaesthetic, Professor Marcus du Sautoy goes in search of answers to one of science’s greatest mysteries: how do we know who we are? ,

He learns at what age our self-awareness emerges and whether other species share this trait.

Next, he has his mind scrambled by a cutting-edge experiment in anaesthesia. Having survived that ordeal, Marcus is given an out-of-body experience in a bid to locate his true self. And in Hollywood, he learns how celebrities are helping scientists understand the microscopic activities of our brain.

Finally, he takes part in a mind-reading experiment that both helps explain and radically alters his understanding of who he is.

All of this is covered in a fabulously interesting episode from Horizon, the excellent and long-running  BBC TV science and philosophy series.  Thankfully, it made its way onto YouTube.

It is just under an hour long but I promise you it will capture you from the very first moment.

Enjoy.  Even if you end up realising, as I did, what a strange person you are!

As Confucius reportedly wrote: Real knowledge is to know the extent of one’s ignorance.