Category: Mathematics

It stretches the mind beyond imagination!

The most incredible story of all!

I first read the story early yesterday morning in The Guardian Newspaper.

But then I saw another version of the same story on the BBC News site, from which I republish it in its entirety.

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First ever black hole image released

By Pallab Ghosh
Science correspondent, BBC News

The first ever picture of a black hole: It’s surrounded by a halo of bright gas.

Astronomers have taken the first ever image of a black hole, which is located in a distant galaxy.

It measures 40 billion km across – three million times the size of the Earth – and has been described by scientists as “a monster”.

The black hole is 500 million trillion km away and was photographed by a network of eight telescopes across the world.

Details have been published today in Astrophysical Journal Letters.

Prof Heino Falcke, of Radboud University in the Netherlands, who proposed the experiment, told BBC News that the black hole was found in a galaxy called M87.

“What we see is larger than the size of our entire Solar System,” he said.

“It has a mass 6.5 billion times that of the Sun. And it is one of the heaviest black holes that we think exists. It is an absolute monster, the heavyweight champion of black holes in the Universe.”

The image shows an intensely bright “ring of fire”, as Prof Falcke describes it, surrounding a perfectly circular dark hole. The bright halo is caused by superheated gas falling into the hole. The light is brighter than all the billions of other stars in the galaxy combined – which is why it can be seen at such distance from Earth.

The edge of the dark circle at the centre is the point at which the gas enters the black hole, which is an object that has such a large gravitational pull, not even light can escape.

Taking the temperature of black holes

Hawking: Black holes store information

Dozen black holes at galactic centre

DR JEAN LORRE/SCIENCE PHOTO LIBRARY I have suspected that the M87 galaxy has a supermassive black hole at its heart from false colour images such as this one. The dark centre is not a black hole but indicates that stars are densely packed and fast moving.

The image matches what theoretical physicists and indeed, Hollywood directors, imagined black holes would look like, according to Dr Ziri Younsi, of University College London – who is part of the collaboration.

“Although they are relatively simple objects, black holes raise some of the most complex questions about the nature of space and time, and ultimately of our existence,” he said.

“It is remarkable that the image we observe is so similar to that which we obtain from our theoretical calculations. So far, it looks like Einstein is correct once again.”

But having the first image will enable researchers to learn more about these mysterious objects. They will be keen to look out for ways in which the black hole departs from what’s expected in physics. No-one really knows how the bright ring around the hole is created. Even more intriguing is the question of what happens when an object falls into a black hole.

What is a black hole?

  • A black hole is a region of space from which nothing, not even light, can escape
  • Despite the name, they are not empty but instead consist of a huge amount of matter packed densely into a small area, giving it an immense gravitational pull
  • There is a region of space beyond the black hole called the event horizon. This is a “point of no return”, beyond which it is impossible to escape the gravitational effects of the black hole
Presentational white space

Prof Falcke had the idea for the project when he was a PhD student in 1993. At the time, no-one thought it was possible. But he was the first to realise that a certain type of radio emission would be generated close to and all around the black hole, which would be powerful enough to be detected by telescopes on Earth.

He also recalled reading a scientific paper from 1973 that suggested that because of their enormous gravity, black holes appear 2.5 times larger than they actually are.

These two previously unknown factors suddenly made the seemingly impossible, possible. After arguing his case for 20 years, Prof Falcke persuaded the European Research Council to fund the project. The National Science Foundation and agencies in East Asia then joined in to bankroll the project to the tune of more than £40m.

The eventual EHT array will have 12 widely spaced participating radio facilities

It is an investment that has been vindicated with the publication of the image. Prof Falcke told me that he felt that “it’s mission accomplished”.

He said: “It has been a long journey, but this is what I wanted to see with my own eyes. I wanted to know is this real?”

No single telescope is powerful enough to image the black hole. So, in the biggest experiment of its kind, Prof Sheperd Doeleman of the Harvard-Smithsonian Centre for Astrophysics, led a project to set up a network of eight linked telescopes. Together, they form the Event Horizon Telescope and can be thought of as a planet-sized array of dishes.

KATIE BOUMAN Information gathered is too much to be sent across the internet. Instead the data was stored on hundreds of hard drives which were flown to a central processing centre.
JASON GALLICCHIO

Each is located high up at a variety of exotic sites, including on volcanoes in Hawaii and Mexico, mountains in Arizona and the Spanish Sierra Nevada, in the Atacama Desert of Chile, and in Antarctica.

A team of 200 scientists pointed the networked telescopes towards M87 and scanned its heart over a period of 10 days.

The information they gathered was too much to be sent across the internet. Instead, the data was stored on hundreds of hard drives that were flown to a central processing centres in Boston, US, and Bonn, Germany, to assemble the information. Prof Doeleman described the achievement as “an extraordinary scientific feat”.

“We have achieved something presumed to be impossible just a generation ago,” he said.

“Breakthroughs in technology, connections between the world’s best radio observatories, and innovative algorithms all came together to open an entirely new window on black holes.”

The team is also imaging the supermassive black hole at the centre of our own galaxy, the Milky Way.

Odd though it may sound, that is harder than getting an image from a distant galaxy 55 million light-years away. This is because, for some unknown reason, the “ring of fire” around the black hole at the heart of the Milky Way is smaller and dimmer.

Follow Pallab on Twitter

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One of the most remarkable things about this story is that it continues to validate the theories of Albert Einstein (1879-1955). That is doubly impressive.

The film, How to see a Black Hole: The Universe’s Greatest Mystery,    is a most interesting account of the skills that were utilised by the team, and the luck of that same group in pulling it all together.

This is clearly the start of a new journey in astronomy.

I will leave you by repeating the image of the black hole.

The first ever picture of a black hole: It’s surrounded by a halo of bright gas.

Just a number, or is it!

I can do no better than republish in full the following:

(Simply because I scarcely understand it!)

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Why the number 137 is one of the greatest mysteries in physics

Famous physicists like Richard Feynman think 137 holds the answers to the Universe.

By PAUL RATNER,  31st October, 2018.

  • The fine structure constant has mystified scientists since the 1800s.
  • The number 1/137 might hold the clues to the Grand Unified Theory.
  • Relativity, electromagnetism and quantum mechanics are unified by the number.

Does the Universe around us have a fundamental structure that can be glimpsed through special numbers?

The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should “worry about”. He called it “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man”.

That magic number, called the fine structure constant, is a fundamental constant, with a value which nearly equals 1/137. Or 1/137.03599913, to be precise. It is denoted by the Greek letter alpha – α.

What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure.

Physicist Laurence Eaves, a professor at the University of Nottingham, thinks the number 137 would be the one you’d signal to the aliens to indicate that we have some measure of mastery over our planet and understand quantum mechanics. The aliens would know the number as well, especially if they developed advanced sciences.

The number preoccupied other great physicists as well, including the Nobel Prize winning Wolfgang Pauli (1900-1958) who was obsessed with it his whole life.

“When I die my first question to the Devil will be: What is the meaning of the fine structure constant?” Pauli joked.

Pauli also referred to the fine structure constant during his Nobel lecture on December 13th, 1946 in Stockholm, saying a theory was necessary that would determine the constant’s value and “thus explain the atomistic structure of electricity, which is such an essential quality of all atomic sources of electric fields actually occurring in nature.

One use of this curious number is to measure the interaction of charged particles like electrons with electromagnetic fields. Alpha determines how fast an excited atom can emit a photon. It also affects the details of the light emitted by atoms. Scientists have been able to observe a pattern of shifts of light coming from atoms called “fine structure” (giving the constant its name). This “fine structure” has been seen in sunlight and the light coming from other stars.


The constant figures in other situations, making physicists wonder why. Why does nature insist on this number? It has appeared in various calculations in physics since the 1880s, spurring numerous attempts to come up with a Grand Unified Theory that would incorporate the constant since. So far no single explanation took hold. Recent research also introduced the possibility that the constant has actually increased over the last six billion years, even though slightly.

If you’d like to know the math behind fine structure constant more specifically, the way you arrive at alpha is by putting the 3 constants h,c, and e together in the equation —

As the units c, e, and h cancel each other out, the “pure” number of 137.03599913 is left behind. For historical reasons, says Professor Davies, the inverse of the equation is used 2πe2/hc = 1/137.03599913. If you’re wondering what is the precise value of that fraction – it’s 0.007297351.

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Now, as I said in my introduction, I don’t understand this. But it doesn’t stop me from marvelling at the figure.

Saturday special!

Natural fractals!

Back in April, Mother Nature Network carried a wonderful item about the amazing fractals that can be found in nature.

Nothing to do with dogs but all to do with loving and caring for our planet!

I am not going to republish the full article with all the wonderful photographs so if the following piques your curiosity then go here to read the full piece.

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14 amazing fractals found in nature

Take a tour through the magical world of natural fractals and discover the joy of simple complexity.

SHEA GUNTHER   April 24, 2013

A chambered nautilus shell is an example of a fractal found in nature. (Photo: Jitze Couperus/flickr)

When you think of fractals, you might think of Grateful Dead posters and T-shirts, all pulsating with rainbow colors and swirling similarity. Fractals, first named by mathematician Benoit Mandelbrot in 1975, are special mathematical sets of numbers that display similarity through the full range of scale — i.e., they look the same no matter how big or how small they are. Another characteristic of fractals is that they exhibit great complexity driven by simplicity — some of the most complicated and beautiful fractals can be created with an equation populated with just a handful of terms. (More on that later.)

(Photo: Wikimedia Commons)

One of the things that attracted me to fractals is their ubiquity in nature. The laws that govern the creation of fractals seem to be found throughout the natural world. Pineapples grow according to fractal laws and ice crystals form in fractal shapes, the same ones that show up in river deltas and the veins of your body. It’s often been said that Mother Nature is a hell of a good designer, and fractals can be thought of as the design principles she follows when putting things together. Fractals are hyper-efficient and allow plants to maximize their exposure to sunlight and cardiovascular systems to most efficiently transport oxygen to all parts of the body. Fractals are beautiful wherever they pop up, so there’s plenty of examples to share.

Here are 14 amazing fractals found in nature:

(Photo: Rum Bucolic Ape/flickr)

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To view the other 13 fractals then go across to here.

Aren’t they beautiful! Or, to pick up on a sentence in the article: “It’s often been said that Mother Nature is a hell of a good designer,”

Not only a good designer but the provider of life as we know it!

What a great man he was!

I am, of course, referring to the recent death of Stephen Hawking.

There’s no way that I can add anything to the widespread reporting of the very sad death of the theoretical physicist, cosmologist and author Professor Stephen Hawking.

Except, possibly, this interesting quirk of fate.

For this great man died yesterday: March 14th.

The very same day that another very famous man, the German-born Albert Einstein, was born. As in March 14th. Albeit, Stephen Hawking’s death being 139 years after the birth of the 1921 winner of the Nobel Prize in Physics.

Did you also know that Professor Hawking was a great dog lover!

I was very pleased that The Conversation blog site released a wonderful tribute to Stephen Hawking. The item opens, thus:

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Acclaimed British theoretical physicist, cosmologist and author Stephen Hawking has died aged 76. Hawking is best known for his work on black holes, which revolutionised our understanding of the universe.

Hawking passed away today peacefully at his home in Cambridge, his family confirmed in a statement:

We are deeply saddened that our beloved father passed away today. He was a great scientist and an extraordinary man whose work and legacy will live on for many years.

His courage and persistence with his brilliance and humour inspired people across the world. He once said, “It would not be much of a universe if it wasn’t home to the people you love.” We will miss him forever.


Read more: A timeline of Stephen Hawking’s remarkable life


Hawking was born on January 8, 1942, in Oxford, England. In 1963 he was diagnosed with ALS, a form of Motor Neurone Disease, and later confined to a wheelchair and forced to communicate via a computerised voice. But he continued his theoretical work and was outspoken on many things over much of his life.

Tributes have been pouring in on social media for the scientist, who made complex science accessible to everyone in his 1988 bestselling book A Brief History of Time.

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Do read the rest of that article. I will take the tribute from Alice Gorman that closes The Conversation article to close today’s post.

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Alice Gorman, Senior Lecturer in archaeology and space studies, Flinders University

There are few scientists who reach as far into popular culture as Stephen Hawking did. His research tackled the biggest of big questions – the nature of time, space and the universe we live in.

Sometimes it feels like science is losing ground in the modern world, but people still look to the stars for answers about who we are and how we come to be here.

Hawking’s bestselling A Brief History of Time made cosmology accessible to people and brought black holes out of the shadows and into the public imagination.

Personally I’ll miss his appearances on The Big Bang Theory, where he could out-nerd the nerds, and also provide some often necessary common sense. It was always great to see a world-class scientist just having fun.

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What a very great man he was!

The aesthetic beauty of mathematics!

Sorry! Did you say the beauty of mathematics?

Those of you that read this blog fairly regularly know that from time to time I drift away from all things dog and potter in the garden of simply fascinating ideas.

Such is the case today.

It is an article on mathematics that was sent to me by Jim Goodbrod. He had read it in The New York Times in April.

Read it and see if you, too, find it as fascinating as I did!

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The World’s Most Beautiful Mathematical Equation

John Zande

A stirrer of brain cells!

I have just finished reading John’s recently published book On the Problem of Good. I am writing a review of the book that, fingers crossed, I will publish on Friday. But many of you that are recent converts to this blog (you poor, lost souls!) will not remember my review of John’s first book and my reaction to that book when I was only just into it.

So, for both today and tomorrow I am republishing two blog posts. Today, one that was originally published on the 16th September, 2015, and tomorrow the post that was first published on 1st October, 2015.

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Of paradoxes, and headaches!

The interconnectedness of everything – even beyond our wildest imagination.

A while ago John Zande signed up to follow Learning from Dogs. Naturally, I went across to John’s blog to thank him. There I discovered that John is an animal lover and an author. For he states, referring to his book, that, “BUY IT. ALL PROCEEDS GO TO ANIMAL RESCUE AND SHELTER IN BRAZIL”. Fabulous!

John Zande cover_zpsz7wuq9cc

(I did buy the book, am about 20% through it and finding it very stimulating, – if you would like to buy it then click the image of the book on John’s home page.)

Anyway, a few days later we watched the BBC Horizon programme on multiple universes. Here’s how the BBC introduced the programme:

Which Universe Are We In?

Horizon, 2014-2015 Episode 17 of 19

Imagine a world where dinosaurs still walk the earth. A world where the Germans won World War II and you are president of the United States. Imagine a world where the laws of physics no longer apply and where infinite copies of you are playing out every storyline of your life.

It sounds like a plot stolen straight from Hollywood, but far from it. This is the multiverse.

Until very recently the whole idea of the multiverse was dismissed as a fantasy, but now this strangest of ideas is at the cutting edge of science.

And for a growing number of scientists, the multiverse is the only way we will ever truly make sense of the world we are in.

Horizon asks the question: Do multiple universes exist? And if so, which one are we actually in?

Horizon is always great to watch but this episode was incredibly stimulating and interesting. Later, in a exchange of comments to one of John’s posts, where I referred to that programme, John wrote:

The mulitverse is actually the more reasonable explanation for why there is something, and although I don’t understand the maths, the people who do say its simplistically beautiful. Matt Rave is an associate professor of physics and comments here regularly. He has a great book on it all, Why is There Anything?

raveThat lead me to purchasing Matthew Rave’s book that, likewise, is a most fascinating and unusual approach to this topic. His Amazon author’s page reveals that, “Dr. Matthew Rave is an assistant professor of physics at Western Carolina University, in the mountains of North Carolina. His research interests include interpretations of quantum mechanics, the geometric phase, solid state physics, and physics education.” Matthew Rave’s blogsite is here.

Matthew Rave’s book further illustrates the paradox, to my mind, that comes from thinking about why are we here, are we here and, if so, how do we know we are here?

So if that isn’t enough for you and me, then very recently The Conversation blogsite published the following from Geraint Lewis who is Professor of Astrophysics at the University of Sydney. It is republished here within the terms of The Conversation. Did I mention paradoxes and headaches!

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We are lucky to live in a universe made for us

Geraint Lewis, University of Sydney

To a human, the universe might seem like a very inhospitable place. In the vacuum of space, you would rapidly suffocate, while on the surface of a star you would be burnt to a crisp. As far as we know, all life is confined to a sliver of an atmosphere surrounding the rocky planet we inhabit.

But while the origin of life on Earth remains mysterious, there are bigger questions to answer. Namely: why do the laws of physics permit any life at all?

Hang on, the laws of physics? Surely they are a universal given and life just gets on with it?

But remember that the universe is built of fundamental pieces, particles and forces, which are the building blocks of everything we see around us. And we simply don’t know why these pieces have the properties they do.

There are many observational facts about our universe, such as electrons weighing almost nothing, while some of their quark cousins are thousands of times more massive. And gravity being incredibly weak compared to the immense forces that hold atomic nuclei together.

Why is our universe built this way? We just don’t know.

But what if…?

This means we can ask “what if” questions. What if the electron was massive and quarks were fleeting? What if electromagnetism was stronger than the nuclear strong force? If so, what would that universe be like?

Let’s consider carbon, an element forged in the hearts of massive stars, and an element essential to life as we know it.

Initial calculations of such stellar furnaces showed that they were apparently inefficient in making carbon. Then the British astronomer Fred Hoyle realised the carbon nucleus possesses a special property, a resonance, that enhanced the efficiency.

But if the strength of the strong nuclear force was only fractionally different, it would wipe out this property and leave the universe relatively devoid of carbon – and, thus, life.

The story doesn’t end there. Once carbon is made, it is ripe to be transmuted into heavier elements, particularly oxygen. It turns out that oxygen, due to the strength of the strong nuclear force, lacks the particular resonance properties that enhanced the efficiency of carbon creation.

This prevents all of the carbon being quickly consumed. The specific strength of the strong force has thus resulted in a universe with an almost equal mix of carbon and oxygen, a bonus for life on Earth.

Death of a universe

This is but a single example. We can play “what if” games with the properties of all of the fundamental bits of the universe. With each change we can ask, “What would the universe be like?”

The answers are quite stark. Straying just a little from the convivial conditions that we experience in our universe typically leads to a sterile cosmos.

This might be a bland universe, without the complexity required to store and process the information central to life. Or a universe that expands too quickly for matter to condense into stars, galaxies and planets. Or one that completely re-collapses again in a matter of moments after being born. Any complex life would be impossible!

The questions do not end there. In our universe, we live with the comfort of a certain mix of space and time, and a seemingly understandable mathematical framework that underpins science as we know it. Why is the universe so predictable and understandable? Would we be able to ask such a question if it wasn’t?

Our universe appears to balance on a knife-edge of stability. But why?

We appear to be very lucky to live in a universe that accommodates life. Zdenko Zivkovic/Flickr, CC BY

One of a multiverse

To some, science will simply fix it all. Perhaps, if we discover the “Theory of Everything”, uniting quantum mechanics with Einstein’s relativity, all of the relative masses and strengths of the fundamental pieces will be absolutely defined, with no mysteries remaining. To others, this is little more than wishful thinking.

Some seek solace in a creator, an omnipotent being that finely-tuned the properties of the universe to allow us to be here. But the move from the scientific into the supernatural leaves many uncomfortable.

There is, however, another possible solution, one guided by the murky and confused musings at the edge of science. Super-strings or M-theory (or whatever these will evolve into) suggest that the fundamental properties of the universe are not unique, but are somehow chosen by some cosmic roll of the dice when it was born.

This gives us a possible explanation of the seemingly special properties of the universe in which we live.

We are not the only universe, but just one in a semi-infinite sea of universes, each with their own peculiar set of physical properties, laws and particles, lifetimes and ultimately mathematical frameworks. As we have seen, the vast majority of these other universes in the overall multiverse are dead and sterile.

They only way we can exist to ask the question “why are we here?” is that we happen to find ourselves in a universe conducive to our very existence. In any other universe, we simply wouldn’t be around to wonder why we didn’t exist.

If the multiverse picture is correct, we have to accept that the fundamental properties of the universe were ultimately dished out in a game of cosmic roulette, a spin of the wheel that we appear to have won.

Thus we truly live in a fortunate universe.

The ConversationGeraint Lewis, Professor of Astrophysics, University of Sydney

This article was originally published on The Conversation. Read the original article.

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How to follow that, eh?

Let me give way to Hariod Brawn and part of an extensive comment she left back then:

John Zande is most certainly one of the most thoughtful, perceptive, well-informed and sharp-witted bloggers I have ever come across, and I wish him well with his book, which by the way, appears so far to have been met only with a deluge of 5-star reviews on Amazon. I daresay that you and I will both lengthen that list.

Here! Here!

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)

Tensions abound in many societies.

Reader alert: This has nothing to do with dogs!

Then as soon as I started to write today’s post (as of yesterday afternoon) I realised the error in my sub-title. For one might argue that this does have a connection with how our dogs behave as a cohesive group. But I’m going to be ‘an arse’ (‘ass’ in American speak) and ask you to hold out until the end of today’s post to read the ‘doggie’ connection. (Note that today’s post is Part One. Part Two continues tomorrow.)

american-gridlock_0I am in the middle of reading American Gridlock written by H. Woody Brock and published in 2012. Here’s an extract of what the book is about, courtesy of Amazon:

A sensible solution to getting our economy back on track

Pessimism is ubiquitous throughout the Western World as the pressing issues of massive debt, high unemployment, and anemic economic growth divide the populace into warring political camps. Right-and Left-wing ideologues talk past each other, with neither side admitting the other has any good ideas. In American Gridlock, leading economist and political theorist H. Woody Brock bridges the Left/Right divide, illuminating a clear path out of our economic quagmire.

Arguing from first principles and with rigorous logic, Brock demonstrates that the choice before us is not between free market capitalism and a government-driven economy. Rather, the solution to our problems will require enactment of constructive policies that allow “true” capitalism to flourish even as they incorporate social policies that help those who truly need it.

Brock demonstrates how deductive logic (as opposed to ideologically driven data analysis) can transform the way we think about these problems and lead us to new and different solutions that cross the ideological divide. Drawing on new theories such as game theory and the economics of uncertainty that are based upon deductive logic, Brock reveals fresh ideas for tackling issues central to the 2012 U.S, Presidential election and to the nation’s long-run future:

It greatly influenced me and I sat down and wrote an essay. Mainly to clarify my own thinking ahead of a meeting last Saturday of our local Freethinkers Group where the topic was “Ideas for Improving our Democratic Processes”. The “our” being the US democratic process but just as valid for many other countries.

I first set out to see if there was a clear, unambiguous definition of what a democratic society is. Surprise, surprise there isn’t one. Very quickly I came up with three:

The first:

A democracy by definition is government through elected representatives. It is a form of society which favours equal rights, freedom of speech and a fair trial and tolerates the views of minorities. Civics and Citizenship website

The second:

A DEMOCRACY IS a society in which all adults have easily accessible, meaningful, and effective ways:

(a) to participate in the decision-making processes of every organization that makes decisions or takes actions that affect them, and;

(b) to hold other individuals, and those in these organizations who are responsible for making decisions and taking actions, fully accountable if their decisions or actions violate fundamental human rights, or are dishonest, unethical, unfair, secretive, inefficient, unrepresentative, unresponsive or irresponsible;

(c) so that all organizations in the society are citizen-owned, citizen-controlled, and citizen-driven, and all individuals and organizations are held accountable for wrongdoing. Democracy Watch website

And the third:

Better democracy, everywhere.

The Democratic Society (Demsoc) works for more and better democracy, where people and institutions have the desire, opportunity and confidence to participate together.

We work to create opportunities for people to become involved in the decisions that affect their lives and for them to have the skills to do this effectively. We support governments, parliaments and any organisation that wants to involve citizens in decision making to be transparent, open and welcoming of participation. We actively support spaces, places and processes to make this happen. Democratic Society website.

I went on to say in my essay:

Yes, there is some harmony between all three definitions but there are also significant differences in tone and language.

I am sure many of you are familiar with the book by H. Woody Brock American Gridlock. I started reading it a few days ago and cannot now put it down.

For the core message of the book is that we, as in society, must distinguish between inductive and deductive reasoning. Let me use the definitions as found on the Live Science website.

Deductive reasoning

Deductive reasoning is a basic form of valid reasoning. Deductive reasoning, or deduction, starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion, according to the University of California. The scientific method uses deduction to test hypotheses and theories. “In deductive inference, we hold a theory and based on it we make a prediction of its consequences. That is, we predict what the observations should be if the theory were correct. We go from the general — the theory — to the specific — the observations,” said Dr. Sylvia Wassertheil-Smoller, a researcher and professor emerita at Albert Einstein College of Medicine.

In deductive reasoning, if something is true of a class of things in general, it is also true for all members of that class. For example, “All men are mortal. Harold is a man. Therefore, Harold is mortal.” For deductive reasoning to be sound, the hypothesis must be correct. It is assumed that the premises, “All men are mortal” and “Harold is a man” are true. Therefore, the conclusion is logical and true.

That is, we predict what the observations should be if the theory were correct.

Let that really work it’s way through your consciousness. It’s an idea that is rooted in the great scientists and philosophers of many thousands of years ago. Think of Euclid, the Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BCE, and was present at his death. (I cheated and looked it up.)

It was Euclid who through Euclidian geometry came to understand the principles of angles and straight lines; as in the shortest distance between two points.

Moving on:

Inductive reasoning

Inductive reasoning is the opposite of deductive reasoning. Inductive reasoning makes broad generalizations from specific observations. “In inductive inference, we go from the specific to the general. We make many observations, discern a pattern, make a generalization, and infer an explanation or a theory,” Wassertheil-Smoller told Live Science. “In science there is a constant interplay between inductive inference (based on observations) and deductive inference (based on theory), until we get closer and closer to the ‘truth,’ which we can only approach but not ascertain with complete certainty.”

Even if all of the premises are true in a statement, inductive reasoning allows for the conclusion to be false. Here’s an example: “Harold is a grandfather. Harold is bald. Therefore, all grandfathers are bald.” The conclusion does not follow logically from the statements.

Inductive reasoning has its place in the scientific method. Scientists use it to form hypotheses and theories. Deductive reasoning allows them to apply the theories to specific situations.

Now for some of you this side trip into reasoning may have seen more like a complete departure. But the point is that, as American Gridlock makes so powerfully: There are two main problems to be solved if this nation is to get back on track. First, win-win policy solutions must be identified for the five real-world problems addressed in Chapters 2 through 6. Second, the Dialogue of the Deaf must come to an end, policy gridlock with it, and these solutions must be implemented. (Pages 7-8)

I closed my essay by setting out the following proposition:

Until we have a clear, universally acknowledged definition of what a democratic society is then it is impossible and utterly futile to debate the various processes including what is the best process for American society.

Is this relevant to the world outside the USA? You bet it is. For better or for worse, what the USA does today the rest of the world does soon thereafter.

And as you will see in Part Two that comes tomorrow democratising the economy is key.

For when we look at the way that dogs, and wolves, operate as a pack in the wild there are only three animals with status:

  • The alpha female who has first choice of the male dogs for mating purposes and makes the decision, as and when necessary, to move her pack to a new territory,
  • The beta dog, always a male, whose role is to keep the pack running smoothly as a cohesive group and not letting squabbles get out of hand, and,
  • The omega dog, that can be of either gender, whose role is to keep the pack happy.

All the other animals in the pack of around fifty are of equal status and work for the benefit of the pack. Now that is something we should learn from dogs!

The certainty of uncertainty

Just wanted to share this with you.

There seems to be so much going on in the world that creates uncertainty. In browsing the web looking for some inspiration for today’s blog post, I came across this short video of the late Richard Feynman and just wanted to share it with you.

The Richard Feyman website explains:

This web site is dedicated to Richard P. Feynman (1918-1988), scientist, teacher, raconteur, and musician.  He assisted in the development of the atomic bomb, expanded the understanding of quantum electrodynamics, translated Mayan hieroglyphics, and cut to the heart of the Challenger disaster.  But beyond all of that, Richard Feynman was a unique and multi-faceted individual.

Find out about Feynman, what he was and why he remains one of the most celebrated and revered scientists of modern times.

Richard Feynman
Richard Feynman

Now to the video. The Uncertainty of Knowledge

“If you expected science to give all the answers to the wonderful questions about what we are, where we are going what the meaning of the universe is and so on then I think you can easily become disillusioned and then look for some mystic answer to these problems. How a scientist can take a mystic answer I don’t know because the whole spirit is to unders…well never mind that, anyway I don’t understand that…but anyhow…if you think of it though…I..the way i think of what we are doing is, we are exploring, we are trying to find out as much as we can about the world.

People say to me, “Are you looking for the ultimate laws of physics?” No I am not. I am just looking to find out more about the world. And if it turns out there is a simple ultimate law that explains everything so be it. That would be very nice discovery. If it turns out it’s like an onion with millions of layers and we just sick and tired of looking at the layers then that’s the way it is! But whatever way it comes out it’s nature, it’s there, and she’s going to come out the way she is. And therefore when we go to investigate we shouldn’t pre-decide what it is we are trying to do except to find out more about it. If you said…but..the problem is why we do you find out more about it, if you thought that you are trying to find out more about it because you are going to get an answer to some deep philosophical question you may be wrong and may be that you can’t get an answer to that particular question by finding out more about the character of the nature.

Hope you find this inspiring!

If you want to find out more about this amazing person then do go across to that website.

 

Make a leap!

Celestial rhythms

This seemed a rather appropriate post for today, February 29th.

Republished from here within the terms of The Conversation.

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Leap day: fixing the faults in our stars

The number 2016 divided by 4 equals 504, exactly – with no remainder, which makes the year 2016, like the upcoming years 2020, 2024 and 2028 (and beyond), a leap year. We will get an “extra” day, February 29.

This pattern will repeat until 2100, when the cycle breaks. Though 2100 is exactly divisible by 4, there is an exception – for years whose number is exactly divisible by 100. (On top of that, there’s another exception – for years exactly divisible by 400. So 2400 will be a leap year. Mark your calendars now.)

Where do these quadrennial liberties with our calendar originate?

In the stars, of course.

Celestial rhythms

One of the simplest joys of life is to watch the stars, night after night, month after month, year after year. They become old friends. They spend a season, and then move on. Or rather, it is we who move on – ever advancing around the sun toward next week’s deadlines, new constellations, new fashions and new ideas.

Orion, the annual visitor. Mouser, CC BY-SA

I imagine myself late one night, eight months from now, remembering the overfull recycling bin, at midnight on trash day. As I try to quietly dump wine bottles into the yellow-topped container, there striding over the eastern skyline is Orion. Back again is my ancient friend, telling me that winter is near, and that I have ridden this miraculous rock almost another full lap around my home star. Rigel shimmers its blue-white light, the twinkle in the eye (the knee, actually) of a companion who has visited me, annually, every place on Earth I have lived since childhood. Even to the Southern Hemisphere, the steady Orion came for a summer visit – cartwheeling upside down, feet over hands.

It is from these celestial cycles that our concepts of time originate, and, ultimately, from which we gain the leap day.

The sidereal year is the length of time it takes for the Earth to return to the same place with respect to the “fix’d” and “constant” stars, so that Orion appears exactly in the same place in the sky, at exactly midnight, 365.2563 days later. Stellar friends like that don’t stand you up; they keep their appointments to seven-digit precision (and more).

Right over the equator: A diagram showing the sun’s position relative to the Earth at the vernal equinox. Tfr000, CC BY-SA

Our Western calendar is tied to the tropical year – the time between successive vernal equinoxes. At that moment, the sun’s position in the sky is exactly where the ecliptic (the plane of the solar system and the path that the planets take as they move through the constellations) crosses the celestial equator (the projection of the Earth’s own equator onto the celestial sphere). Straddling the celestial equator, the sun splits its time exactly between the day side and the night side of the Earth. It returns to that place again in roughly 365.24219 days. Roughly.

Now you can see where those alternating “divisible by 4, 100 and 400” leap year rules originate.

Making up the differences

At the end of 365 days, there are still 0.24219 days (just shy of six hours) to go before Earth gets back to the equinox line.

After four years, however, this fractional 0.24219 of a day adds up to 0.96876, which is pretty close to one full day. If we were using only a 365-day calendar, the stars, and more importantly the months, corresponding to the seasons – crucial for agricultural societies – would slip behind. This was apparent to the Romans in the first century, as well as to the Olmecs and the Maya on the other side of the world.

Thus decreed Julius Caesar in 46 B.C.: that every four years an extra day would be added to February. It was called the Julian calendar. But adding one day every four years, in order to make up for that 0.96876 of a day in orbital spare change, is overcompensating. Caesar’s “every four” leap year prescription adds 0.03124 of a day too much. This makes the Julian calendar run fast by just over 600 seconds per year.

Exception after exception: Christopher Clavius, in a line engraving by E de Boulonois. Wellcome Trust, CC BY

Like with the spare coin jar in our house, small change like that takes a while to add up. It wasn’t until the age of Pope Gregory XIII, in 1582, that this mismatch was becoming a problem. After consultation, presumably with God, but particularly with his astronomer, Christopher Clavius, the pope adopted Clavius’ clever solution.

The Julian calendar runs fast by 0.03124 of a day every four years; multiply both sides by 100, and see an excess of about three days after 400 years. Clavius’ solution was to make centuries exceptions – but that would lose too much, four days in 400 years, not three. So Clavius added one back, once every 400 years, starting in 1600.

This Gregorian calendar, which we use today, has the following rules:

  • Every year divisible by 4: add February 29
  • Every century (1800, 1900, 2000, 2100): do not add February 29
  • Every century divisible by 400: add February 29

Still finer measurements

Even with this refinement, there is still orbital change left over. But now we are talking about temporal shavings that are quite small. At this level of precision, other wobbles in the relation of the Earth’s rotational period (the day) and its revolution period (the year) have to be taken into account.

When a leap second is added, digital clocks tick past 23:59:59 but don’t go directly to 00:00:00. Twid

Keeping track of minute effects like this is the job of the International Earth Rotation and Reference Systems Service, which controls the addition (or deletion) of leap seconds. For example, a second was added to Coordinated Universal Time by the service on June 30, 2015, due largely to the slowing of the Earth’s rotation by the gravitational pull of the moon.

There are other sources of calendar slip: the 8.9 magnitude earthquake that triggered the Japanese tsunami on March 11, 2011, for example, shifted the planet’s mass distribution enough to decrease the length of a day by 1.8 microseconds. This will add up to about a second after 1,500 years.

Using that ‘extra’ time

Personally, I think we should make February 29, leap day, a global holiday. It should be considered a gift to ourselves, like taking that accumulated spare change to the grocery store coin-counting machine, and trading it for some easier-to-spend bills. It should be a day of celebration, a reward for saving that quarter of a day over the last four years, to be spent on something frivolous. Or it could be a special day to realign our sense of hourly routines, weekly trash pickups, the race to fulfill monthly quotas, to the celestial schedule.

Without that extra day every fourth year, our ancient friends would begin to miss their annual appointments, and start to fall behind in wishing us prompt birthday greetings, like forgetful Facebook friends. Without February 29, roughly, every four years, the “constant stars” would cease to be constant.

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So there! Now you know!

You all spend this extra day peacefully and happily.