Tag Archives: Mathematics

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.

648x415_uvre-banksy-pres-jungle-calais

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.

When Science Looks Like Religion: The theory That Would Not Die.

It is the third book I read about statistics in a short while and it is probably the strangest. After my dear Taleb and his Black Swan, after the more classical Naked Statistics, here is the history of the Bayesian statistics.

mcgrayne_comp2.indd

If you do not know about Bayes, let me just add that I like the beautiful and symmetric formula: [According to wikipedia]
For proposition A and evidence B,
P(A|B) P(B) = P(B|A) P(A)
P(A), the prior, is the initial degree of belief in A.
P(A|B), the posterior, is the degree of belief having accounted for B.
the quotient P(B|A)/P(B) represents the support B provides for A.
Another way of explaining it mathematically is Bayes’ theorem gives the relationship between the probabilities of A and B, P(A) and P(B), and the conditional probabilities of A given B and B given A, P(A|B) and P(B|A).

I was never really comfortable with its applications. I was probably wrong again, given all what I learnt after reading Sharon Bertsch McGrayne’s rich book. But I also understood why I was never comfortable: for three centuries, there’s been a quasi-religious war between Bayesians and Frequentists on how to use probabilities. Are these linked to big, frequent numbers only or can they be applied for rare events? What is the probability of a rare event which may never occur or maybe just once?

[Let me give you a personal example: I am interested in serial entrepreneurship, and did and still do tons of statistics on Stanford-related companies. I have more than 5’000 entrepreneurs, and more than 1’000 are serial. I have results showing that serial entrepeneurs are not on average better than one-time, using frequency and classical methods. But now I should think about using:
P(Success|Serial) = P(Serial|Sucess) P(Success) / P(Serial)
I am not sure what will come out, but I should try!].

If you want a good summary of the book, read the review by Andrew I. Daleby (pdf). McGrayne illustrates the “recent” history of statistics and probabilities through famous (Laplace) and less famous (Bayes) scientists, through famous (the Enigma machine and Alan Turing) and less famous (lost nuclear bombs) stories and it is a fascinating book. I am not convinced it is great at explaining the science, but the story telling is great. Indeed, it may not be about science at all. But about belief as is mentioned in the book: Swinburne inserted personal opinions into both the prior hunch and the supposedly objective data of Bayes’ theorem to conclude that God was more than 50% likely to exist; later Swinburne would figure the probability of Jesus’ resurrection at “something like 97 percent” [Page 177]. It obviously reminded me of Einstein’s famous quote: “God does not play dice with the universe.” This is not directly related but for the second time in my life, I was reading about links between science, probability and religion.

Statistics: Garbage In, Garbage Out?

I have already talked about statistics here, and not in good terms. It was mostly related to Nicholas Nassim Taleb‘s works, The Black Swan and Antifragile. But this does not mean statistics are bad. They may just be dangerous when used stupidly. It is what Charles Wheelan explains among otehr things in Naked Statistics.

nakedstatistics

Naked Statistics belongs to the group of Popular Science. Americans often have a talent to explain science for a general audience. Wheelan has it too. So if you do not know about or hate the concepts of mean/average, standard deviation, probability, regression analysis, and even central limit theorem, you may change your mind after reading his book.

Also you will be explained the Monty Hall problem or equivalent Three Prisoners problem or why it is sometimes better (even if counterintuitive) to change your mind.

Finally Wheelan illustrates why statistics are useless and even dangerous when the data used are badly built or irrelevant (even if the mathematical tools are correctly used!). Just one example in scientific research (which is another topic of concern to me) “This phenomenon can plague even legitimate research. The accepted convention is to reject a hypothesis when we observe something that would happen by chance only 1 in 20 times or less if the hypothesis were true. Of course, if we conduct 20 studies, or if we include 20 junk variables in a single regression equation, then on average, we will get 1 bogus statistically significant finding. The New York Times magazine captured this tension wonderfully in a quotation from Richard Peto, a medical statistician and epidemiologist: “Epidemiology is so beautiful and provides such an important perspective on human life and death, but an incredible amount of rubbish is published”.
Even the results of clinical trials, which are usually randomized experiments and therefore the gold standard of medical research, should be viewed with some skepticism. In 2011, the Wall Street Journal ran a front-page story on what it described as one of the “dirty little secrets” of medical research: “Most results, including those that appear in top-flight peer-reviewed journals, can’t be reproduced. […] If researchers and medical journals pay attention to positive findings and ignore negative findings, then they may well publish the one study that finds a drug effective and ignore the nineteen in which it has no effect. […] On top of that, researchers may have some conscious or unconscious bias, either because of a strongly held prior belief or because a positive finding would be better for their career. (No one ever gets rich or famous by proving what doesn’t cure cancer. […] Dr. Ionnadis [a Greek doctor and epidemiologist] estimates that roughly half of the scientific papers published will eventually turn out to be wrong.”
[Pages 222-223]

When age does not hinder creativity: a rare example in mathematics

I seldom (but sometimes) talk about Science or Mathematics. Mostly when it helps me illustrate what innovation or creativity is about, and sometimes when I see analog crises in all these fields (see for example the posts on Dyson, Thiel or Smolin). And there is another related point: it is often claimed that major scientific discoveries or entrepreneurial ventures are done at a young age.

YitangZhang
Yitang Zhang

You probably never heard of Yitang Zhang who has stunned the world of mathematics last month by proving a centuries-old problem. He is a totally unknown mathematician and more surprising, he is (over) 50-year old. For those interested in the problem, you can read Nature’s First proof that infinitely many prime numbers come in pairs. Basically, Zhang proved that there are infinitely many pairs of primes that are less than N apart. Mathematicians still dream to prove that N is equal to 2 – the twin prime conjecture -, but Zhang was first to prove that N exists … even if N is 70 million!

Imagination/Intuition versus Logic/Reason

As Guillermo Martinez said rightly in one of his essays, “it’s well-known that there is only one more effective way to kill conversation in a waiting room than to open a book, and that is to open a book of mathematics”. Still you may read more than this first sentence!

Even in high tech. innovation and entrepreneurship, the topic of imagination vs. reason, which could be translated by technology push vs. market push, is recurrent. So when I read books about creativity, whether it is scientific or artistic, I am always looking for links with innovation. I had the opportunity to check it again with Guillermo Martinez’s Borges and Mathematics. Borges is probably one of the “poets” who put the most mathematics in his literary work. Guillermo Martinez who is both a novel author and a mathematician has recently published in English this nice little book about Mathematics in Borges’ short stories. I already talked about mathematics in a recent post so let me add here a few things about what I liked.

borges-and-mathematics

Martinez quotes Borges who quotes Poe: “I – naively perhaps – believe Poe’s explanations. I think that the mental process he adduces corresponds to the actual creative process. I’m sure this is how intelligence works: through changes of mind, obstacles, elimination. The complexity of the operation he describes doesn’t bother me; I suspect that the real approach must have been even more complex and much more chaotic and hesitant. All this does not mean to suggest that the arcana of poetic creation were revealed by Poe. In the links, that the writer explores, the conclusion he draws from each premise is logical of course but not the only one necessary.” Borges in The genesis of Poe’s “The Raven”.

And then he adds more about the process of creativity: In the discussion of “divine, winged” intuition versus the prosaic, tortoise pace of logic, I would like to contradict a myth about mathematics: the process Borges describes is exactly the same as what happens in mathematical creation. Let’s consider the mathematician who has to prove a theorem for the first time. Our mathematician sets out to prove a result without even knowing if such a proof really exists. He gropes his way through an unknown world, proving and making mistakes, refining his hypothesis, starting all over again and trying another approach. He too has infinite possibilities within his grasp and with every step he takes. And so each attempt will be logical, but by no means the only one possible. It is like the moves of a chess player. Each of the chess player’s moves conforms to the logic of the game in order to entrap his rival, but none is predetermined. This is the critical step in artistic and mathematical elaboration, and in any imaginative task. I don’t believe there is anything unique to literary creation as far as the duality of imagination/intuition versus logic/reason is concerned.

I strongly believe that innovation is very similar to the process of artistic or scientific creation. But in another essay, Martinez says more about creation: “It’s the same feeling of euphoria you get when, after many years of struggling with your own ignorance, you suddenly understand how to look at something. Everything becomes more beautiful, and you have the feeling you can see farther than before. It’s a glorious moment, but you pay a great price for it, which is your obsession with the problem, like a constant wound or a pebble in your shoe. I wouldn’t recommend that sort of life to anyone. Einstein had a close friend, Michele Besso, with whom he discussed many details of the theory of relativity. But Besso himself never accomplished anything important in science. His wife once asked Einstein why, if in fact her husband was so gifted. “Because he’s a good person!” Einstein replied. And I think it’s true. You have to be a fanatic, an that ruins your life and the lives who are close to you.” Again you might meditate about the high rate of divorce in Silicon Valley and the fanatism creativity requires.

For those really interested in mathematics, I cannot avoid mentioning some other topics Martinez addresses: Gödel’s incompleteness theorem is one of the greatest achievements in mathematics ever, though it is complicated to understand. In a very simplistic ways, even in mathematics, there are things which are true but cannot be proven. Russell’s paradox is nearly as mesmerizing but simple to grab: (From Wikipedia): There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the Barber paradox supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge. According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell’s paradox. Symbolically:

russel-paradox-formula

7 x 7 = (7-1) x (7+1) + 1

Well yesterday I noticed this strange formula. Would it be that 7 is a magic number and I would go from rational to irrational – though start-ups are often irrational aventures too? No: 7 is not alone, the formula applies to 5 [25=24+1], 3 [9=8+1], and so on: 11, 17. So prime numbers? Not even, true for any integer… I felt a little stupid when I found it is just a particular application of a^2 – b^2 = (a-b) x (a +b)!!

I love maths, but maths is not just magical numbers, it’s much broader. And I love to read books on the topic. There is poetry and beauty in math, for sure. To conclude this unusual post, here is a list of books I enjoyed reading in the past. In no particular order, but thematic.

There are still “many” unsolved problems in mathematics. The most famous one is probably proving the Riemann hypothesis. Here are 2 books developing the story:

(Please click on image for a link to the book)

Indeed there is a million-dollar prize offered to 7 such problems by the Clay Institute. And the first solved one is the Poincare Conjecture by Grigori Perelman. Perelman declined the prize but this is another story!

Before the Millenium problems, there were the Hilbert Problems. At the time, the Fermat theorem was probably the most famous challenge!

And as 2 last examples, but I could mention so many more, here are two biographies of extremely strange geniuses, Srinivasa Ramanujan and Paul Erdös

Maybe one day, I’ll be back with more on the topic of math and more broadly about popular science books! Don’t hesitate to give me examples and advice 🙂

NB: if you want to check the French versions, go to the article: http://www.startup-book.com/fr/2012/11/19/7-x-7-7-1-x-71-1/