Tag Archives: Mathematics

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.”

Work Rules! by Laszlo Bock (part II) – the GLAT

In Work Rules!, Bock mentions briefly the GLAT (Google Labs Aptitude Tests) that were also mentioned in David Vise’s Google Story. But he just quickly says they may have been overused and sometimes a waste of time and of resources. But let me refer to his page 73:

That page begins with the image above which can be also found on google blog’s page Warning: we brake for number theory. It’s never too late solve math problems… If you solved it at the time, you got access to the following one:

The second puzzle:
f(1)=7182818284 
f(2)=8182845904 
f(3)=8747135266 
f(4)=7427466391 
 f(5)= __________

Again feel free to try… you will find answers here. Bock just adds this: The result? We hired exactly zero people.

Maybe this will help you:

2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919
3200305992181741359662904357290033429
5260595630738132328627943490763233829
8807531952510190115738341879307021540
8914993488416750924476146066808226480
0168477411853742345442437107539077744
9920695517027618386062613313845830007
5204493382656029760673711320070932870
9127443747047230696977209310141692836
8190255151086574637721112523897844250
5695369677078544996996794686445490598
7931636889230098793127736178215424999
2295763514822082698951936680331825288
6939849646510582093923982948879332036
2509443117301238197068416140397019837
6793206832823764648042953118023287825
>0981945581530175671736133206981125099

as well as this:

x = 1
2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919

x = 2
2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919

x = 3
2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919

x = 4
2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919

x = 5
2.71828182845904523536028747135266249
7757247093699959574966967627724076630
3535475945713821785251664274274663919
3200305992181741359662904357290033429
5260595630738132328627943490763233829
8807531952510190115738341879307021540
8914993488416750924476146066808226480
0168477411853742345442437107539077744
9920695517027618386062613313845830007
5204493382656029760673711320070932870
9127443747047230696977209310141692836
8190255151086574637721112523897844250
5695369677078544996996794686445490598
7931636889230098793127736178215424999
2295763514822082698951936680331825288
6939849646510582093923982948879332036
2509443117301238197068416140397019837
6793206832823764648042953118023287825
0981945581530175671736133206981125099

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…

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).

The Beauty of Mathematics

Every year I try to convey what I believe to be the beauty of mathematics when I teach convex optimization at EPFL. I have already mentioned on this blog some beautiful books, popularizing the subject. Some recent readings have convinced me even more so let me try to convince you (again)…

Alain Badiou is a rather surprising choice to talk about mathematics but I love what he has recently written: “This quasi aesthetic feeling of mathematics struck me very early. […] I think of Euler’s line. It was shown that the three altitudes of a triangle are concurrent in a point H, it was already beautiful. Then that the three vertices were also concurrent, at a point O, better and better! Finally, that the three medians were equally so, at a point G! Terrific. But then, with a mysterious air, the professor told us that we could demonstrate, as the brilliant mathematician Euler had done, that these points H, O, G were in addition all three on the same line, which evidently was called Euler’s line! It was so unexpected, so elegant, this alignment of three fundamental points, as behavior of the characteristics of a triangle! […] There is this idea of a real discovery, a surprising result at the cost of a journey sometimes a little difficult to follow, but where one is rewarded. I often compared mathematics later to walking in the mountain: the approaching walk is long and painful, with a lot of turning, slopes; we think we have arrived, but there is still a turning point … We sweat, we struggle. But when we arrive at the pass, the reward is unequaled, truly: this gratification, this final beauty of mathematics, this surely conquered, absolutely singular beauty.” [Pages 11-12]

Another source of inspiration is Proofs from THE BOOK. Written in homage to Paul Erdös, the book begins with the two pages shown above. “Paul Erdös liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdös also said that you need not believe in God but, as a mathematician, you should believe in The Book. […] We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations.”

Sometimes I try to remember the most beautiful demonstrations I have “felt” since my high school years.

– The most luminous, the proof of the sum of the n first integers by Gauss

– Two demonstrations of the Pythagorean theorem,

– There would be many others like the infinity of prime numbers, the development in series of Π (), the beautiful concept of duality for convex sets (you can look at a set through its “internal” points or through the dual “external” envelope made of its tangents).

– But the most fascinating for me, remains the use of Cantor’s Diagonal:

[From Wikipedia:]

In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following theorem:

If s1, s2, … , sn, … is any enumeration of elements from T, then there is always an element s of T which corresponds to no sn in the enumeration.

To prove this, given an enumeration of elements from T, like e.g.

s1 = (0, 0, 0, 0, 0, 0, 0, …)
s2 = (1, 1, 1, 1, 1, 1, 1, …)
s3 = (0, 1, 0, 1, 0, 1, 0, …)
s4 = (1, 0, 1, 0, 1, 0, 1, …)
s5 = (1, 1, 0, 1, 0, 1, 1, …)
s6 = (0, 0, 1, 1, 0, 1, 1, …)
s7 = (1, 0, 0, 0, 1, 0, 0, …)

he constructs the sequence s by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the nth digit as complementary to the nth digit of sn. In the example, this yields:

s1 = (0, 0, 0, 0, 0, 0, 0, …)
s2 = (1, 1, 1, 1, 1, 1, 1, …)
s3 = (0, 1, 0, 1, 0, 1, 0, …)
s4 = (1, 0, 1, 0, 1, 0, 1, …)
s5 = (1, 1, 0, 1, 0, 1, 1, …)
s6 = (0, 0, 1, 1, 0, 1, 1, …)
s7 = (1, 0, 0, 0, 1, 0, 0, …)
s = (1, 0, 1, 1, 1, 0, 1, …)

By construction, s differs from each sn, since their nth digits differ (highlighted in the example). Hence, s cannot occur in the enumeration.Based on this theorem, Cantor then uses a proof by contradictionto show that:The set T is uncountable.

But let me add another extract from Badiou (page 82): “I call truths (always in the plural, there is no “truth”) singular creations of universal value: works of art, scientific theories, policies of emancipation, love passions. Let us say to cut short: scientific theories are truths concerning the being itself (mathematics) or the “natural” laws of the worlds of which we can have an experimental knowledge (physics and biology). Political truths concern the organization of societies, the laws of collective life and its reorganization, all in the light of universal principles, such as freedom, and today, principally, equality. The artistic truths relate to the formal consistency of finite works that sublimate what our senses can receive: music for hearing, painting and sculpture for vision, poetry for speech … Finally, the love truths concern the dialectical power contained in the experience of the world not from the One, from the individual singularity, but from the Two, and thus from a radical acceptance of the other. These truths are not, of course, of philosophical origin or nature. But my goal is to save the (philosophical) category of truth that distinguishes and names them, legitimizing that a truth can be:
– absolute, while being a localized construction,
– eternal, while resulting from a process which begins in a certain world and therefore belongs to the time of this world.”

Alexander Grothendieck, 1928 – 2014

What link is there between Andrew Grove (the previous article) and Alexandre Grothendieck? Beyond their common initials, a similar youth – both were born in the communist Eastern Europe they left for a career in the West) and the fact they have become icons of their world, they just represent my two professional passions: startups and mathematics. The comparison stops there, no doubt, but I’ll get back to it.

Two books (both in French) were published in January 2016 about the life of this genius: Alexander Grothendieck – in the footsteps of the last mathematical genius by Philippe Douroux and Algebra – elements of the life of Alexander Grothendieck by Yan Pradeau. If you like mathematics (I should say the mathematical science) or even if you do not like it, read these biographies.

livres_alexandre_grothendieck

I knew as many others about the atypical route of this stateless citizen who became a great figure of mathematics – he received the Fields Medal in 1966 – and then decided to live in seclusion from the world for over 25 years in a small village close to the Pyrenees until his death in 2014. I also have to confess that I knew nothing of his work. Reading these two books shows me that I was not the only one, as Grothendieck had explored lands that few mathematicians could follow. I also found the following stories:
– At age 11, he calculated the circumference of the circle and deduced that π is equal to 3.
– Later, he reconstructed the theory of Lebesgue measure. He was not 20 years old.
– A prime number has his name, 57, who nevertheless is 3 x 19.
Yes, it is worth discovering the life of this illustrious mathematician.

tableau_alexandre_grothendieck

The reason for the connection I made between Grove and Grothendieck is actually quite tenuous. It comes from this quote: “There are only two true visionaries in the history of Silicon Valley. Jobs and Noyce. Their vision was to build great companies … Steve was twenty, un-degreed, some people said unwashed, and he looked like Ho Chi Minh. But he was a bright person then, and is a brighter man now … Phenomenal achievement done by somebody in his very early twenties … Bob was one of those people who could maintain perspective because he was inordinately bright. Steve could not. He was very, very passionate, highly competitive.” Grove was close Noyce in more ways than one, and extremely rational and according to Grove, Noyce was too lax! Grothendieck would be closer to Jobs. A hippie, a passionate individual and also somehow self-taught. Success can come from so diverse personalities.

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Last point in common or perhaps a difference. The migration. Grove became a pure American. Grothendieck was an eternal stateless, despite his French passport. But both show its importance. Silicon Valley is full of migrants. I often talk about this here. We know less that what is called “the French school of mathematics” also has its migrants. If you go to the French wikipedia page of the Fields Medal, you can read:

Ten “Fields medalists’ are former students of the Ecole Normale Superieure: Laurent Schwartz (1950), Jean-Pierre Serre (1954), René Thom (1958), Alain Connes (1982), Pierre-Louis Lions (1994) Jean-Christophe Yoccoz (1994), Laurent Lafforgue (2002), Wendelin Werner (2006), Cédric Villani (2010) and Ngo Bao Chau (2010). This would make “Ulm” the second institution after the ‘Princeton’ winners, if the ranking was the university of origin of the medal and not the place of production. Regarding the country of origin, we arrive at a total of fifteen Fields medalists from French laboratories, which could put France ahead as the formative nations of these eminent mathematicians.

But in addition to Grothendieck, the stateless, Pierre Deligne, Belgian, had his thesis with him, Wendelin Werner was naturalized at the age of 9 years, Ngo Bao Châu the year he received the Fields Medal, after doing all his graduate studies in France, and Artur Avila is Brazilian and French … One could speak of the International of Mathematics, which might not have displeased Alexander Grothendieck.