Tag Archives: Mathematics

Grothendieck, a genius

I’ve written about Grothendieck here before, through two books about this mathematical genius published shortly after his death: Alexandre Grothendieck, 1928 – 2014. Summer is an opportunity for listening to radio broadcasts and I had the pleasure to rediscover this extraordinary character, first of all through Alexandre Grothendieck : un mathématicien qui prit la tangente initially broadcasted in La conversation scientifique in 2016 on French radio, France Culture,

and then by listening (while writing this post) to Alexandre Grothendieck ou le silence du génie first broadcasted in 2015 in Une vie, une œuvre, on the same radio.

From one thing to another, I downloaded Récoltes et semailles, a 929-page document written between 1983 and 1986 by the mathematician. You can download the pdf in French. Just as Perelman, Gödel or Erdős, for us, simple mortals, we can believe that genius rubs shoulders with madness and the journey, the life of these creators will undoubtedly remain mysteries forever.

I read a few dozen pages of this book and Chapter 2.20 fascinated me. I suggest you read it. I find this extract quite admirable and translated it with my limited means…

2.20 A shot look at the neighbors across the street

The situation seems to me very close to the one which arose at the beginning of this century, with the emergence of Einstein’s theory of relativity. There was a conceptual dead end, even more blatant, materializing in a sudden contradiction, which seemed irresolvable. Of course, the new idea that would bring order to the chaos was one of childish simplicity. The remarkable thing (and conforms to a most repetitive scenario…) is that among all these brilliant, eminent, prestigious people who were suddenly on their teeth, trying to “save what there was to be saved”, no one thought about this idea. It had to be an unknown young man, fresh from the benches of student lecture halls (maybe), who came (a little embarrassed perhaps at his own audacity…) to explain to his illustrious elders what had to be done to “save the phenomena”: one just had to separate space from time [68]! Technically, everything was gathered then for this idea to hatch and be welcomed. And it is to the honor of Einstein’s elders that they were indeed able to welcome the new idea, without resisting too much. This is a sign that these were still a great time…
From a mathematical point of view, Einstein’s new idea was trivial. From the point of view of our conception of physical space, however, it was a profound change, and a sudden “change of scenery”. The first mutation of its kind, since the mathematical model of physical space released by Euclid 2400 years ago, and taken up as is for the needs of mechanics by all physicists and astronomers since antiquity (including Newton), to describe terrestrial and stellar mechanical phenomena.
This initial idea of Einstein was subsequently much developed, embodied in a more subtle, richer and more flexible mathematical model, using the rich arsenal of already existing mathematical notions [69]. With the “generalized theory of relativity”, this idea broadened into a vast vision of the physical world, embracing in one look the subatomic world of the infinitely small, the solar system, the Milky Way and distant galaxies, and the path of electromagnetic waves in a space-time curved at each point by the matter which is there [70]. This is the second and last time in the history of cosmology and physics (following Newton’s first great synthesis three centuries ago) that a broad unifying vision has emerged, in the language of a mathematical model, of all the physical phenomena in the Universe.
This Einsteinian vision of the physical universe was in turn overwhelmed by events. The “set of physical phenomena” which it is a question of reporting has had time to expand since the beginning of the century! There have emerged a multitude of physical theories, each to account, with varying degrees of success, for a limited set of facts, in the immense mess of all “observed facts”. And we are still waiting for the daring kid, who will find by playing the new key (if there is one…), The dreamed “cake model”, who wants to “work” to save all phenomena at once… [71]
The comparison between my contribution to the mathematics of my time, and that of Einstein to physics, was imposed on me for two reasons: both work was accomplished through a mutation of our conception of “space” (in the mathematical sense in one case, in the physical sense in the other); and both take the form of a unifying vision, embracing a vast multitude of phenomena and situations which heretofore appeared to be separate from one another. I see there an obvious kinship between his work [72] and mine.
This relationship does not seem to me to be contradicted by an obvious difference in “substance”. As I hinted earlier, the Einsteinian mutation concerns the notion of physical space, while Einstein draws from the arsenal of already known mathematical notions, without ever needing to expand it, or even upset it. His contribution consisted in identifying, among the mathematical structures known of his time, those which were best suited to [73] serve as “models” for the world of physical phenomena, instead of the dying model bequeathed by his predecessors. In this sense, his work was indeed that of a physicist, and beyond that, that of a “philosophy of nature”, in the sense in which Newton and his contemporaries understood it. This “philosophical” dimension is absent from my mathematical work, where I have never been led to ask myself questions about the possible relations between the “ideal” conceptual constructions, taking place in the Universe of mathematical things, and phenomena that take place in the physical Universe (or even, lived events taking place in the psyche). My work has been that of a mathematician, deliberately turning away from the question of “applications” (to other sciences), or “motivations” and psychic roots of my work. Of a mathematician, moreover, driven by his very particular genius to constantly expand the arsenal of notions at the very basis of his art. This is how I was led, without even noticing it and as if playing, to upset the most fundamental notion of all for the surveyor: that of space (and that of “variety”), that is our conception of the very “place” where geometric beings live.
The new notion of space (like a kind of “generalized space”, but where the points which are supposed to form the “space” have more or less disappeared) does not resemble in any way, in its substance, the notion brought by Einstein in physics (not at all confusing for the mathematician). The comparison is necessary on the other hand with quantum mechanics discovered by Schrödinger [74]. In this new mechanism, the traditional “material point” disappears, to be replaced by a kind of “probabilistic cloud”, more or less dense from one region of ambient space to another, depending on the “probability” that the point is in this region. We feel, in this new perspective, a “mutation” even more profound in our ways of conceiving mechanical phenomena, than in that embodied by Einstein’s model – a mutation which does not consist in simply replacing a somewhat mathematical model, narrow at the armatures, by another similar one but cut wider or better adjusted. This time, the new model resembles so little the good old traditional models, that even the mathematician who is a great specialist in mechanics must have felt suddenly disoriented, even lost (or outraged…). Going from Newton’s mechanics to Einstein’s must be, for the mathematician, a bit like going from the good old Provencal dialect to the latest Parisian slang. On the other hand, to switch to quantum mechanics, I imagine, is to switch from French to Chinese. And these “probabilistic clouds”, replacing the reassuring material particles of yesteryear, strangely remind me of the elusive “open neighborhoods” that populate the topos, like evanescent ghosts, to surround imaginary “points”, which still continue to cling to and against all a recalcitrant imagination…

Notes :

[68] This is a bit short, of course, as a description of Einstein’s idea. At the technical level, it was necessary to highlight what structure to put on the new space-time (it was already “in the air”, with Maxwell’s theory and Lorenz’s ideas). The essential step here was not of a technical nature, but rather “philosophical”: to realize that the notion of simultaneity for distant events had no experimental reality. This is the “childish observation”, the “but the Emperor is naked!”, which made cross this famous “imperious and invisible circle which limits a Universe”…

[69] These are mainly the notion of “Riemannian manifold”, and the tensor calculus on such a manifold.

[70] One of the most striking features which distinguishes this model from the Euclidean (or Newtonian) model of space and time, and also from Einstein’s very first model (“special relativity”), is that the global topological form of space-time remains indeterminate, instead of being prescribed imperatively by the very nature of the model. The question of what this global form is strikes me (as a mathematician) as one of the most fascinating in cosmology.

[71] One called “unitary theory” such a hypothetical theory, which would manage to “unify” and to reconcile the multitude of partial theories of which it was question. I have the feeling that the fundamental thinking that awaits to be undertaken, will have to be placed on two different levels.
1_) A reflection of a “philosophical” nature, on the very notion of a “mathematical model” for a portion of reality. Since the successes of Newtonian theory, it has become an unspoken axiom of the physicist that there exists a mathematical model (or even a single model, or “the” model) to express physical reality perfectly, without “detachment” no burr. This consensus, which has been law for more than two centuries, is like a sort of fossil vestige of a living Pythagorean vision that “Everything is number”. Perhaps this is the new “invisible circle”, which replaced the old metaphysical circles to limit the Universe of the physicist (while the race of the “philosophers of nature” seems definitively extinct, supplanted handily by that of computers…). As long as one likes to dwell on it for a moment, it is quite clear, however, that the validity [of] this consensus is by no means obvious. There are even very serious philosophical reasons which lead to questioning it a priori, or at least to providing very strict limits to its validity. It would be the moment or never to submit this axiom to a tight criticism, and perhaps even, to “demonstrate”, beyond any possible doubt, that it is not founded: that there does not exist a unique rigorous mathematical model, accounting for all the so-called “physical” phenomena listed so far.
Once the very notion of “mathematical model” has been satisfactorily identified, and that of the “validity” of such a model (within the limits of such “margins of error” admitted in the measurements made), the question of a “unitary theory” or at least that of an “optimum model” (in a sense to be specified) will finally be clearly stated. At the same time, one will probably also have a clearer idea of the degree of arbitrariness which is attached (by necessity, perhaps) to the choice of such a model.
2_) It is only after such reflection, it seems to me, that the “technical” question of identifying an explicit model, more satisfactory than its predecessors, takes on its full meaning. It would then be the moment, perhaps, to break free from a second tacit axiom of the physicist, going back to antiquity, and deeply rooted in our very way of perceiving space: it is that of continuous nature of space and time (or space-time), of the “place” therefore where “physical phenomena” take place.
Fifteen or twenty years ago, leafing through the modest volume constituting Riemann’s complete work, I was struck by a remark from him “by the way”. He observes that it could well be that the ultimate structure of space is “discrete”, and that the “continuous” representations which we make of it perhaps constitute a simplification (excessive perhaps, in the long run…) of a more complex reality; that for the human mind, “the continuous” was easier to grasp than “the discontinuous”, and that it serves us, therefore, as an “approximation” for understanding the discontinuous. This is a remark surprisingly penetrating into the mouth of a mathematician, at a time when the Euclidean model of physical space had never before been questioned; in the strictly logical sense, it is rather the discontinuous which, traditionally, has served as a technical method of approach to the continuous.
Developments in mathematics in recent decades have, moreover, shown a much more intimate symbiosis between continuous and discontinuous structures than was previously imagined in the first half of this century. Still, to find a “satisfactory” model (or, if necessary, a set of such models, “connecting” as satisfactorily as possible..), that this one be “continuous”, “discrete” “or of a” mixed “nature – such work will undoubtedly involve a great conceptual imagination, and a consummate flair for apprehending and updating mathematical structures of a new type. This kind of imagination or “flair” seems rare to me, not only among physicists (where Einstein and Schrödinger seem to have been among the rare exceptions), but even among mathematicians (and here I speak with full knowledge of the facts).
To sum up, I predict that the expected renewal (if it has yet to come…) will come more from a mathematician at heart, knowledgeable about the great problems of physics, than from a physicist. But above all, it will take a man with “philosophical openness” to grasp the crux of the matter. This is by no means technical in nature, but a fundamental problem of “philosophy of nature”.

[72] I make no claim to be familiar with Einstein’s work. In fact, I haven’t read any of his work, and only know his ideas through hearsay and very roughly. Yet I feel like I can make out “the forest”, even though I’ve never had to make the effort to scrutinize any of its trees. . .

[73] For comments on the qualifier “moribund”, see a previous footnote (note page 55).

[74] I think I understand (by echoes that have come back to me from various sides) that we generally consider that in this century there have been three “revolutions” or great upheavals in physics: Einstein’s theory, the discovery of radioactivity by the Curies, and the introduction of quantum mechanics by Schrödinger.

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.”