Tag Archives: Science

Grigori Perelman according to Masha Gessen

I had mentioned Grigori Perelman in a rather old post: 7 x 7 = (7-1) x (7+1) + 1. I discovered recently a new book about this exceptional mathematician, not so much about his achievements but more about his personality.

About Asperger’s Syndrome

I will not tell much Perelman here, Masha Gessen does it with talent. Let me just translate here form the French version I am reading: “It seems to me that many of the whistleblowers,” wrote Atwwod, “have Asperger’s Syndrome, I’ve met several who have applied the code of ethics of their company or government to their work and have reported wrongdoing and corruption in the workplace. All of them were surprised to see that their management and their colleagues did not understand their attitude. ”
So it is perhaps not a coincidence that the founders of dissident movements in the Soviet Union were among mathematicians and physicists. The Soviet Union was not the place for people who took things literally and expected the world to work in a predictable, logical and fair way.
[Pages 215-6, French edition]
[…]
One can also interpret the difficulties he experienced when he presented his solutions. If Perelman was suffering from Asperger’s syndrome, this inability to see “the big picture” is perhaps one of the most surprising traits. British psychologists Uta Frith and Francesca Happe talked about what they call the “low central coherence” characteristic of autism spectrum disorders. Autistics focus on details, to the detriment of the overall picture. When they manage to reconstitute it, it is because they have arranged the various elements, a little like the elements of the periodic table, in a systemic scheme that satisfies them to the extreme. “… the most interesting facts, wrote Poincaré, one of the greatest systematizing minds of all time, more than a century ago, are those who can be used many times, those who have a chance to happen many times. We have had the good fortune to be born in a world where there is are many; suppose that instead of sixty chemical elements we have sixty billion, that they are not common ones and rare ones, but they would be evenly distributed, so every time we pick up a new pebble, there would be a high probability that it would be made up of some unknown substances… […] In such a world, there would be no no science, perhaps no thought and even life would be impossible because evolution could not have developed the conservative instincts; thanks to God it is not so.”
People with Asperger’s syndrome apprehend the small pebble world by small pebble. Speaking of the existence of this syndrome in society, Attwood resorted to the metaphor of a five thousand-piece puzzle, “where normal people would have the full image on the lid” which would allow them to have global intuitions. Aspergers, they would not see this big picture and should try to nest the pieces one by one. So maybe rules like “never take off your hat” and 2lis all the books that are on the list “formed for Gricha Perelman a way to see the missing image on the lid, to encompass all the elements of the periodic table of the world It was only by clinging to these rules that he could live his life.
[Pages 217-8, French edition]

About power

Another interesting topic addressed by Misha Gessen is on page 236 of the French edition again:
– When he received the letter from the commission that invited him he replied that he did not speak with committees, said Gromov, and that is exactly what he did. They represent everything that one should never accept. And if this attitude seems extreme, it is only in relation to the conformism that characterizes the world of mathematics.
– But why refuse to talk to committees?
– We do not talk to committees, we talk to people! exclaimed Gromov, exasperated. How can we talk to a committee? Who knows who is on the committee? Who tells you that Yasser Arafat is not one of them?
– But he was sent the list of members, and he continued to refuse.
– The way it started, he was right not to answer, Gromov persisted. As soon as a community begins to behave like a machine, all that remains to do is to cut ties, and that’s all. The strangest thing is that there is no longer a mathematician who does the same. That’s what’s weird. Most people agree to deal with committees. They agree to go to Beijing and receive a prize from President Mao. Or the king of Spain, anyway, it’s the same!
– And why, I asked, could not the King of Spain have the honor of hanging a medal around Perelman’s neck?
– What is a king? Gromov asked, totally furious now. Kings are the same morons as the Communists. Why would a king award a medal to a mathematician? What allows it? It is nothing from a mathematical point of view. Same for the president. But there is one who has taken control of power like a thief and the other who inherited it from his father. It does not make any difference.
Unlike them, Gromov explains to me, Perelman had made a real contribution to the world.

It reminds me of a colleague’s quote: “There are not many statues for committees in public parks.”

It’s also worth mentioning here an article from the New Yorker that Gessen mentions too: Manifold Destiny. A legendary problem and the battle over who solved it by Sylvia Nasar and David Gruber. On a related topic, the authors quote Perelman whom they met: He mentioned a dispute that he had had years earlier with a collaborator over how to credit the author of a particular proof, and said that he was dismayed by the discipline’s lax ethics. “It is not people who break ethical standards who are regarded as aliens,” he said. “It is people like me who are isolated.” We asked him whether he had read Cao and Zhu’s paper. “It is not clear to me what new contribution did they make,” he said. “Apparently, Zhu did not quite understand the argument and reworked it.” As for Yau, Perelman said, “I can’t say I’m outraged. Other people do worse . Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.”

The Gene by Siddhartha Mukherjee – a Symphony of Research

Progress in science depends on new techniques, new discoveries and new ideas, probably in that order – Sydney Brenner
All science is either physics or stamp collecting – Ernest Rutherford

I did not know much about genetics, except my data here and there about biotechnology startups. But thanks to his book The Gene, Siddhartha Mukherjee makes me feel I know much more. His brilliant storytelling is like a symphony, describing the early days of genetics, with Darwin and Mendel and the latest developments of this fascinating science. There are brilliant quotes like the two above, respectively page 202 and 221. And there are marvelous portraits.

I will only give a few of them, the pictures I mean, as a quiz…

But before letting you discover the names below, here is a short extract from The Gene, a famous anecdote about how Genentech was born…

And a great analysis of what science, technology and biology are:
“Do not be lulled by that description. Do not, gentle reader, be tempted to think – “My goodness, what a complicated recipe!” – and then rest assured that someone will not learn to understand or hack or manipulate that recipe in some deliberate manner.

“When scientists underestimate complexity, they fall prey to the perils of unintended consequences. The parables of such scientific overreach are well-known: foreign animals, introduced to control pests, become pests in their own right; the raising of smokestacks, meant to alleviate urban pollution, releases particulate effluents higher in the air and exacerbates pollution; stimulating blood formation, meant to prevent heart attacks, thickens the blood and results in an increased risk of blood clots to the heart.

“But when non-scientists overestimate complexity – “No one can possibly crack this code” – they fall into the trap of unanticipated consequences. In the early 1950s, a common trope among some biologists was that the genetic code would be so context dependent – so utterly determined by a particular cell in a particular organism and so horribly convoluted – that deciphering it would prove impossible. The truth turned out to be quite the opposite: just one molecule carries the code, and just one code pervades the biological world. If we know the code, we can intentionally alter it in organisms, and ultimately in humans. Similarly, in the 1960s, many doubted that gene-cloning technologies could so easily shuttle genes between species. By 1980, making a mammalian protein in a bacterial cell, or a bacterial protein in a mammalian cell, was not just feasible; it was, in berg’s words, rather “ridiculously simple”. Species were specious. “Being natural” was “often just a pose”. [Page 408] […]

“Technology, I said before, is most powerful when it enables transitions – between linear and circular motion (the wheel), or between real and virtual space (the Internet). Science, in contrast, is most powerful when it elucidates rules of organization – laws – that act as lenses through which to view and organize the world. Technologies seek to liberate us from the constraints of our current realities through those transitions. Science defines those constraints, drawing the outer limits of the boundaries of possibility. Our greatest technological innovations thus carry names that claim our prowess over the world: the engine (from ingenium, or “ingenuity”) or the computer (from computare, or “reckoning together”). Our deepest scientific laws, in contrast, are often named after the limits of human knowledge: uncertainty, relativity, incompleteness, impossibility.

“Of all the sciences, biology is the most lawless; there are few rules to begin with, and even fewer rules that are universal. Living beings must, of course, obey the fundamental rules of physics and chemistry, but life often exists on the margins and interstices of these laws, bending them to their near-breaking limits. The universe seeks equilibriums; it prefers to disperse energy, disrupt organization, and maximize chaos. Life is designed to combat these forces. We slow down reactions, concentrate matter, and organize chemicals into compartments, “It sometimes seems as if curbing entropy is our quixotic purpose in the universe,” James Gleick wrote.” [Page 409]

And now the answer the the picture quiz:

Charles Darwin Gregor Mendel William Bateson Thomas Morgan Alfred Sturtevant
Calvin Bridges Hermann Muller Hugo de Vries Ronald Fisher Theodosius Dobjansky
Frederick Griffith Oswald Avery Max Perutz George Beadle Edward Tatum
Linus Pauling James Watson Francis Crick Rosalind Franklin Maurice Wilkins
Arthur Pardee François Jacob Jacques Monod Edward Lewis Christiane Nüsslein-Volhard
Eric Wieschaus Sydney Brenner Robert Horvitz John Sulston Paul Berg
Arthur Kornberg Janet Merz Herbert Boyer Stanley Cohen Frederick Sanger
Walter Gilbert Craig Venter Allan Wilson Luigi Luca Cavalli-Sforza Shinya Yamanaka
Jennifer Doudna Emmanuelle Charpentier

What we cannot know

It’s the 3rd book I read by Marcus du Sautoy. After the Music of Primes and Finding Moonshine: A Mathematician’s Journey Through Symmetry, here is What we cannot know.

Seven frontiers of knowledge according to du Sautoy: Randomness and Chaos, Particle Physics and the Infinitely Small, Space and Quantum Physics, The Universe and the Infinitely Big, Time and Gravity, Consciousness, Mathematics.

To illustrate some of this, here are tww short extracts:

Du Sautoy asks, what is the B. in Benoit B. Mandelbrot and the answer is Benoit B. Mandelbrot. Nice!

And quite nice too about the “purity of fields” by xkcd.com

If you love science(s) or mathematic(s), a clear must-read!

Claude Shannon, an honorable mathematician?

A Mind at Play is a very interesting book for many reasons. The subtitle “How Claude Shannon Invented the Information Age” is one reason. It is a great biography of a mathematician whose life and production are not that well-known. And what is Information? I invite you to read these 281 pages or if you are too lazy or busy, at least the Shannon page on Wikipedia.

What I prefer to focus on here is the ever going tension between mathematics and engineering, between (what people sometimes like to oppose) pure and applied mathematics. Pure mathematics would be honorable, applied mathematics would not be, if we admit there is such a thing as pure or applied maths. So let me extract some enlighting short passages.

The typical mathematician is not the sort of man to carry on an industrial project. He is a dreamer, not much interested in things or the dollars they can be sold for. He is a perfectionist, unwilling to compromise; idealizes to the point of impracticality; is so concerned with the broad horizon that he cannot keep his eye on the ball. [Page 69]

In Chapter 18, entitled, Mathematical Intentions, Honorable and Otherwise, the authors dig deeper: Above all [the mathematician] professes loyalty to the “austere and often abtruse” world of pure mathematics. If applied mathematics concerns itself with concrete questions, pure mathematics exists for its own sake. Its cardinal questions are not “How do we encrypt a telephone conversation?” but rather “Are there infinitely many twin primes?” or “Does every true mathematical statement have a proof?” The divorce between the two schools has ancient origins. Historian Carl Boyer traces it to Plato, who regarded mere computation as suitable for a merchant or a general, who “must learn the art of numbers or he will not know how to array his troops.” But the philosopher must study higher mathematics, “because he has to arise out of the sea of change and lay hold of true being.” Euclid, the father of geometry, was a touch snobbier “There is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to gibe the student threepence, ‘since he must make gain of what he learns’.”
Closer to our times, the twentieth-century mathematician G. H. Hardy would write what became the ur-text of pure math. A Mathematicians’ Apology is a “manifesto for mathematics itself,” which pointedly borrowed its title from Socrates’ argument in the face of capital charges. For Hardy, mathematical elegance was an end in itself. “beauty is the first test,” he insisted. “There is no permanent place in the world for ugly mathematics.” A mathematician, then, is not a mere solver of practical problems. He, “like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” By contrast, run-of-the-mill applied mathenatics was “dull,” “ugly”. “trivial” and “elementary”
And one (famous) reader of Shannon’s paper dismissed it with a sentence that would irritate Shannon’s supporters for years: “The discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author’s mathematical intentions are honorable.” [Pages 171-2]

This reminds me of another great book I read last year Mathematics without apologies with one chapter entitled “Not Merely Good, True and Beautiful”. Shannon was a tinkerer, a term I discovered when I read Noyce‘s biography, another brilliant tinkerer. He was a brilliant tinkerer and he was a brilliant mathematician. He had himself strong vues about the quality of scientific research (pure or applied – who cares really?): we must keep our own house in first class order. The subject of information theory has certainly been sold, if not oversold. We should now turn our attention to the business of research and development at the highest scientific plane we can maintain. Research rather than exposition is the keynote, and our critical thresholds should be raised. Authors should submit only their best efforts, and these only after careful criticism by themselves and their colleagues. A few first rate research papers are preferable to a large number that are poorly conceived or half-finished. The latter are no credit to their writers and a waste of time to their reader. [Page 191]

A brilliant tinkerer as the video below shows…

and it seems he designed and built the (or one of the) first computer that played chess. He was a juggler and a unicycler.

In the chapter Constructive Dissatisfaction, the topic is intelligence. It requires talent and training, but also curiosity and even dissatisfaction: not the depressive kind of dissatisfaction (of which , he did not say, he had experienced his fair share), but rather a “constructive dissatisfaction”, or “a slight irritation when things don’t look quite right.” It was a least, a refreshing unsentimental picture of genius: a genius is simply someone who is usefully irritated. He had also proposed six strategies to solving problems: simplifying, encircling, restating, analyzing, inverting and stretching. You will need to read that section pages 217-20.

He was also a good investor. In fact he was close to a few founders of startups and had a privileged access to people like Bill Harrison (Harrison Laboratories) and Henry Singleton (Teledyne) and although he used his knowledge to analyze stock markets. Here is what he has to say about investing: A lot of people look at the stock price, when they should be looking at the basics company and its earnings. There are many problems concerned with the prediction of stochastic processes, for example the earnings of companies… My general feeling is that it is easier to choose companies which are going to succeed, than to predict short term variations, things which last only weeks or months, which they worry about on Wall Street Week. There is a lot more randomness there and things happen which you cannot predict, which cause people to sell or buy a lot of stock. To the point of answering to the question of the best information theory for investment with “inside information.” [Page 241-2]

A genius, a wise man, an honorable mathematician.

March 8 – International Women’s Day

About to give my optimization class this morning, I just remembered only one woman got the Fields Medal. This was in 2014. Unfortunately she died of cancer last year

Maryam Mirzakhani (3 May 1977 – 14 July 2017) became the first Iranian and first and only woman to win the Fields Medal.

Let me add, that in the field of optimization, apparently only one woman got the Dantzig Prize, Eva Tardos.

I have to admit, I did not take the time to think of a similar name for startups and innovation. Comments welcome…

Facts, Truth, Courage…

The word Start-up is controversial. Money, business are often the synonymous and at the time of the Paradise Papers, they do not have good press, with reason! I sometimes had to “defend” and I would still defend a world that is important … And more widely, I believe, we must fight to defend the importance of facts and a “dirty word”, truth, with courage. In a world where communication is confused with information (already at the beginning of the 20th century, Schumpeter explained that capitalism could not exist without marketing and advertising) it is important that the voices defending the facts, truth with courage be heard. For this reason, I have sometimes come out of the strict framework of technology, innovation and high-tech entrepreneurship to mention intellectuals who seem important to me. Harari, Piketty, Fleury, Stieger, Picq.

A long introduction to mention a beautiful interview of Pascal Picq on France Inter this morning: “We must deconstruct male domination in Hollywood as elsewhere.” The link to the site in French is here. A brief excerpt: “Note: the bonobos, who are close to us, have not evolved like us. Homo sapiens is one of the most violent species towards its females, so women. The degrees of this violence vary according to time and culture. The dominant status is exercised with regard to Nature on the one hand, and women on the other hand. The behaviors of male dominance that are rooted in our human societies are the result of a behavioral evolution, and not of a natural and genetic determinism. In the bonobos, the female is dominant, the violence of the relations is mastered by the hedonic relations, and the bonobos live according to a model of gynocracy. The bonobos and the men did not have the same behavioral evolution. And I say that we can not blame our nature, as some do, in the principles of male dominance that have been imposed on humans. It is a behavioral evolution that has become embedded in agricultural societies and industrial societies. Historically, where foods or things are produced, men tend to take control of production and reproduction.”

I should have added that this interview is related to the Oscar granted last night to Agnès Varda for her career and coincidence or not, my previous article in French only – If you do not love me, I can tell you that I do not love you either – dealt with Sandrine Bonnaire, the actress of Agnes Varda in the magnificent Vagabond.

In all disciplines, scientific or not, complexity makes it difficult sometimes, not to say often, to assert truths. They seem sometime to be moving. But there are facts that one must always analyze as honestly as possible with courage and then we can come out with truths. Thanks to these intellectuals who work without dogma and help us understand the world.

And to finish on a “lighter” touch, here are two works by Obey whose artistic fight is magnificent!

Hopeful Monsters

‘What are hopeful monsters?’ I said ‘They are things born perhaps slightly before their time; when it’s not known if the environment is quite ready for them.’
Hopeful Monsters, by Nicholas Mosley [P. 71]

Hopeful Monsters could have been startups, but it is a novel, a marvelous novel written in 1990 and that I am reading again these days. I had read it in another century, when there were only books in paper and independent bookstores still existed. I had bought it in the late Black Oak Books in Berkeley, California.

Bruno held out his hands to the flames and talked to them in an unintelligible language. Minna said ‘What do you say to the fire?’
Bruno said ‘I say “Come on up! Do as I say or I’ll punish you!” ’
Minna said ‘And does it?’
Bruno said ‘If it wants to.’

The first 3 chapters begin this way:
Chapter I – if we are to survive in the environment we have made for ourselves, may we have to be monstrous enough to greet our predicament?
Chapter II – if we are talking about an environment in which the acceptance of paradoxes might breed, then this can happen in an English hot-house, I suppose, as well as in a melting-pot of Berlin streets.
Chapter III – if, for the sake of change, old ground has to be broken up, one or two seeds lie secret – what terrible opportunities there were during those years!

I had never read a novel which mixes philosophy and science with beautiful story-telling. Not an easy read. Not sure it is a masterpiece either, though…

Mathematics again: Unexpected, Inevitable and Economical

“La libertad es como un número primo.” Roberto Bolaño, Los Detectives Salvajes

Michael Harris’ mathematics without apologies, I said it elsewhere, is a must-read if you are interested in mathematics. And probably even more, if you are not. But again, it is not an easy reading.

After the claim in his Chapter 3 that mathematics was “Not Merely Good, True and Beautiful”, Harris goes on with provocative and thoughful arguments about the relations that mathematics have with Money (Chapter 4 – Megaloprepeia), with the Body (Chapter 6 – Further Investigations of the Mind-Body problem), with Foundations (Chapter 7 – The Habit of Clinging to an Ultimate Ground) and even with tricks (Chapter 8 – The Science of Tricks), Harris finally comes back to Apologies after a personal chapter about inspiration and work (Chapter 9 – A Mathematical Dream and Its Interpretation).

The author made me discover, shame on me, that “apology” does not mean only praise, but also excuse or defense. Difficulty and confusion of the vocabulary, indeed a recurrent theme of Harris’ book. Let me be quite clear again. I did not understand everything and I imagined Harris could have created a new index. As you may know if you read my blog, I mention Indices from time to time, like the Erdős Index, the Tesla Index. This new Index could be 0 for Maths Giants or Supergiants, humans who could be awarded the Fields Medal, the Abel Prize or equivalent, 1 for those who can understand (everything) that has been written in mathematics by those with 0 Index; then 2, for those who can understand (everything) that has been written in mathematics by those with 1 Index, etc… I do not know where the index would stop and perhaps it already exists… I woudl like to believe that I was at the Index 3 but not sure! But then I made my discovery about “apology”, I put myself down at Index 5…

Harris goes even stronger than Hardy with his “No Apologies” even if he quotes him: Irony has not spoken its last word on the flight from utility [of science], even when utility is understood, with Hardy, as that which “tends to accentuate the existing inequalities in the distribution of wealth”. [Page 296] I think harris has written a very useful book about mathematics. I add another example on the nature of mathematical beauty: “there is a very high degree of unexpectedness, combined with inevitability and economy” [Page 307].

When looking for more information about harris, I found his web page which begins with the quote i give above from Bolaño. When I discovered Bolaño a few years ago, it was such a shock that I read everything I could find. Again without understanding everything. But if you read Harris’ chapter 9, you will undestand that “not understanding everything” may not be that important, compared to the impact that (apparent) confusion may create…

PS: I could have added that while I was reading Harris, a controversy arose around a new solution for the P vs. NP problem. More about this in a detailed pdf and on its author’s blog. I also should have mentioned the Langlands program and Alexander Grothendieck, whom I also mentioned here. But again Harris book is so rich…

Mathematics – Not Merely Good, True and Beautiful

“It’s not the marbles that matter. It’s the game.” Dutch proverb

“In mathematics, the art of proposing a question must be held of higher value than solving it.” Cantor

Mathematics can be made simple, even obvious; and beautiful, and even useful. Just read my previous post about Ian Stewart’s 17 Equations That Changed the World. But there are other more provocative views. You just need to read Michael Harris’ mathematics without apologies.

Harris is certainly not as easy to read as Stewart. But it is as (maybe more) enriching. His Chapter 3 for example is entitled Not Merely Good, True and Beautiful. In this world of increasing pressure to justify the usefulness of science, the author fights back. “There is now a massive literature on the pressures facing university laboratories. These books mostly ignore mathematics, where stakes are not so high and opportunities for commercial applications are scarce, especially in the pure mathematics.” [Page 55]

But even Truth seems to be at stake.“If one really thinks deeply about the possbility that the foundations of mathematics are inconsistent, this is extremely unsettling for any rational mind” [Voevodsky quoted on page 58] and a few lines before “Bombieri recalled the concerns about the consistency, reliability, and truthfulness of mathematics that surfaced during the Foundations Crisis and alluded to the ambiguous status of computer proofs and too-long proofs.”

Finally Harris mentions some confusion about Beauty quoting Villani: “The artistic aspect of our discipline is [so] evident” that we don’t see how anyone could miss it.. immediatley adding that “what generally makes a mathematician progress is the desire to produce something beautiful.” Harris then quotes an art expert advising museum-goers to “let go of [their] preconceived notions that art has to be beautiful”. [Page 63]

Harris adds that “the utility of practical applications, the guarantee of absolute certainty and the vision of mathematics as an art form – the good, the true and the beautiful, for short – have the advantage of being ready to hand with convenient associations, though we should keep in mind that what you are willing to see as good depends on your perspective, and on the other hand the true and beautiful can themselves be understood as goods.” [Pages 63-4]

The short answer to the “why” question is going to be that mathematicains engage in mathematics because it gives us pleasure. [Page 68]

Maybe more in another post…

Instead of another post, here is a short section extracted from page 76 and added on August 27:

The parallels between mathematics and art

“Here the presumed but largely unsubstantiated parallel between mathematics and the arts offers unexpected clarity. Anyone who wants to include mathematics among the arts has to accept the ambiguity that comes with that status and with the different perspectives implicit in different ways of talking about art. Six of these perspectives are particularly relevant: the changing semantic fields the word art has historically designated; the attempts by philosophers to define art, for example, by subordinating it to the (largely outdated) notion of beauty or to ground ethics in aesthetics, as in G. E. Moore’s Principia Ethica, which by way of Hardy’s Apology continues to influence mathematicians; the skeptical attitude of those, like Pierre Bourdieu, who read artistic taste as a stand-in for social distinction ; the institutions of the art world, whose representatives reflect upon themselves in Muntadas’s interviews ; the artists personal creative experience within the framework of the artistic tradition ; and the irreducible and (usually) material existence of the art works themselves.
Conveniently, each of these six approaches to art as a mathematical counterpart: the cognates of the word mathematics itself, derived form the Greek mathesis, which just means “learning”, and whose meaning has expanded and contracted repeatedly over the millennia and from one culture to another, including those that had no special affinity for the Greek root; the Mathematics of philosophers of “encyclopedist” schools; school mathematics in its role as social and vocational filter; the social institutions of mathematics with their internal complexity and heir no-less-complex interactions with other social and political institutions; the mathematicians personal creative experience within the framework of the tradition (the endless dialogue with the Giants and Supergiants of the IBM and similar rosters); and the irreducible and (usually) immaterial existence of theorems, definitions and other mathematical notions.”

Maybe more in another post…

How much do you know (and love) about mathematics?

A tribute to Maryam Mirzakhani

From time to time, I mention here books about science and mathematics that I read. It is the first one I read by Ian Stewart. Shame on me, I should have read him a long time ago. 17 Equations That Changed the World is a marvelous little book that describe the beauty of mathematics. A must read, I think

So as a little exercise, you can have a look at these 17 equations and check how much you know. What ever the result, I really advise to read his book! And if you do not, you can have a look at the answers below…

And here is more, the names of the equations and the mathematitians who who discovered them (or invented them – depending on what you think Math is about).