# 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.

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.

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.

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.

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.

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.

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.

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:

# 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:

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/

# The Black Swan and the danger of statistics

“Thought is only a flash in the middle of a long night. But this flash means everything.”
Henri Poincaré*

When I talked to friends and colleagues about The Black Swan (“BS”), they were surprised about my interest in the movie with Natalie Portman. I cannot say, I have not watched it. I was talking about Nassem Nicholas Taleb’s book and theory. Some other friends classified at it as American b… s…, these superficial books that give advice on anything and that seem to always become bestsellers; my colleagues would classify it as airport literature, not to be read in academic circles.

I read it and enjoyed it, but I have to admit Taleb is sometimes painful. Is it because he was so much frustrated by I do not know whom or what or is it because he is so proud of his certainties? I am not sure. But his ideas are certainly worth thinking about more than a minute. (Whereas you forget about airport American b… s… after 30 seconds). So back to the BS.

You’ll find great accounts of his book or of his theory, e.g.
– Nassim Taleb’s “The Black Swan” by Andrew Gelman,
– or even another essay by Taleb, the Fourth Quadrant,
so I will not try to do the same.

However defining the Black Swan might be useful! In the Fourth Quadrant, Taleb writes the following:

There are two classes of probability domains—very distinct qualitatively and quantitatively. The first, thin-tailed: Mediocristan”, the second, thick tailed Extremistan. Before I get into the details, take the literary distinction as follows: In Mediocristan, exceptions occur but don’t carry large consequences. Add the heaviest person on the planet to a sample of 1000. The total weight would barely change. In Extremistan, exceptions can be everything (they will eventually, in time, represent everything). Add Bill Gates to your sample: the wealth will jump by a factor of >100,000. So, in Mediocristan, large deviations occur but they are not consequential—unlike Extremistan. Mediocristan corresponds to “random walk” style randomness that you tend to find in regular textbooks (and in popular books on randomness). Extremistan corresponds to a “random jump” one. The first kind I can call “Gaussian-Poisson”, the second “fractal” or Mandelbrotian (after the works of the great Benoit Mandelbrot linking it to the geometry of nature). But note here an epistemological question: there is a category of “I don’t know” that I also bundle in Extremistan for the sake of decision making—simply because I don’t know much about the probabilistic structure or the role of large events. Black Swans are the unknown deviations in Extremistan.

Here are more notes taken while reading.

[Page xxii] The black swan is characterized by “rarity, extreme impact and retrospective (though not prospective) predictability” (with additional footnote: the occurrence of a highly improbably event is the equivalent of the nonoccurrence of a highly probably one.

[Page 8] The human mind suffers from 3 aliments:
-The illusions of understanding, or how everyone thinks he knows what is going on in a world that is more complicated (or random) than they realize;
-the retrospective distortion, or how we can assess matters only after the fact, as if they were in a rearview mirror; and
-the overvaluation of factual information and the handicap of authoritative and learned people – when they platonify.

[Page 15] While in the past a distinction had been between drawn Mediterranean and non- Mediterranean (i.e., between the olive oil and the butter), in the 1970s, the distinction suddenly became between Europe and non-Europe.

[Page 54] There is a major difference and often-made mistake between no evidence of something and the evidence of its non-occurence (mental bias.)

[Page 77] The answer is that there are two varieties of rare events: a) the narrated Black Swans, those that are present in the current discourse and that you are likely to hear about on television, and b) those nobody talks about, since they escape models – those that you would feel ashamed discussing in public because they do not seem plausible. I can safely say that it is entirely compatible with human nature that the incidences of Black Swans would be overestimated in the first case, but severely underestimated in the second one.

[Page 80] One death is a tragedy; a million is a statistic. […] We have two systems of thinking. System 1 is experiential, effortless, automatic, fast, and opaque. System 2 is thinking, reasoned, local, slow, serial, progressive. Most mistakes come from using system 1 when we think we use system 2.

[Page 140] We overestimate what we know and underestimate uncertainty. Another bias, ”think about how many people divorce. Almost all of them are acquainted with the statistic that between one-third and one-half of all marriages fail, something the parties involved did not forecast while tying the know. Of course, “not us” because “we get along so well” (as if others tying the know got along poorly.)”

[Page 174-179] Poincaré is a central personality of Taleb’s theory, in particular through the 3-body problem. According to Taleb, “Poincaré angrily disparages the use of the bell curve.” Now the next figure simply illustrates the concept of sensitivity to initial conditions.

Predicting

Operation 1: imagine an ice cube and consider how it may melt.
Operation 2: consider a puddle of water. Try to reconstruct the shape of the ice-cube.
The forward process is generally used in physics and engineering, the backward process in nonrepeatable, nonexperimental historical approaches. And the backward is much more complex to analyze.

[Page 198] While in theory it is an intrinsic property. In practice, randomness is incomplete information. Nonpractitioners do not understand the subtlety. A true random process does not have predictable properties. A chaotic system has entirely predictable properties, but they are hard to know.
a) There are no functional differences in practice between the two since we will never get to make the distinction.
b) The mere fact that a person is talking about the difference implies he has never made a meaningful decision under uncertainty – which is why he does not realize that they are indistinguishable in practice.
Randomness in practice, in the end, is just unknowledge. The world is opaque and appearances fool us.

[Page 204] Trial and error means trying a lot. In the Blind Watchmaker, Richard Dawkins brilliantly illustrates this notion of the world without grand design, moving by small incremental random changes. Note a slight disagreement on my part that does not change the story by much: the world, rather moves by large incremental random changes. Indeed, we have psychological and intellectual difficulties with trial and error and with accepting that series of small failures are necessary in life. “You need to love to lose”. In fact the reason I felt immediately at home in America is precisely because American culture encourages the process of failure, unlike the cultures of Europe and Asia where failure is met with stigma and embarrassment.
[It’s really Taleb writing and not the blog’s author, but I fully agree !]

[Page 207] When you have a very limited loss, you need to be as aggressive as speculative and sometimes as unreasonable as you can be. Middlebrow thinkers sometimes make the analogy with lottery tickets. It is plain wrong. First lottery tickets do not have a scalable payoff. Second, lottery tickets have known rules.

The economics of superstars

[Page 24] Who is this book written for? You need to understand who your audience is and amateurs write for themselves, professionals write for others. [This irony of the author’s is stimulating. I experienced it, I’m an amateur. But are the masterpieces not then written by amateurs? The Black Swans (The Lord of the Rings, Harry Potter) look often like a work of amateurs. The Yevgenia Krasnova example provided by Taleb is also stimulating]

[Page 214] Someone who is marginally better can easily win the entire pot. The problem is the notion of “better.” People take from the poor to give to the rich. An initial advantage follows someone through life and keep getting cumulative advantages. Failure is also cumulative. The advent of modern media has accelerated these cumulative advantages. The sociologist Pierre Bourdieu noted a link between the increased concentration of success and the globalization of culture and economic life.

[Page 221] Taleb claims new comers mitigate the cumulative advantages. “of the five hundred largest US companies in 1957, only seventy-four were still part of that select group, the S&P 500, forty year later. Only a few hundred had disappeared in mergers; the rest either shrank or went bust.

Actors who win an Oscar tend to live on average five years longer than their peers who don’t. People live longer in societies that have flatter social gradients.

[Page 277] What is poorly understood is the absence of a role for the average in intellectual production. The disproportionate share of the very few in intellectual influence is even more unsettling than the unequal distribution of wealth- unsettling because, unlike the income gap, no social policy can eliminate it. Communism could conceal or compress income discrepancies, but it could not eliminate the superstar system in intellectual life. [I am not sure]

Skepticism

Taleb defines himself as a skeptic and his mentor are Hayek and Popper. He links it with humility in the following: [Page 190] Someone with a low degree of epistemic arrogance is not too visible, like a shy person at a cocktail party. We are not predisposed to respect humble people, those who try to suspend judgment. Now contemplate epistemic humility. Think of someone heavily introspective, tortured by the awareness of his own ignorance. He lacks the courage of the idiot, yet has the rare gust to say “I don’t know”. He does not mind looking like a fool or, worse, an ignoramus. He hesitates, he will not commit, and he agonizes over the consequences of being wrong. He introspects, introspects, and introspects until he reaches physical and nervous exhaustion.

Experts

[Page 146] We know the difference between know-how and know-what. The Greeks made a distinction between techne and episteme, craft and knowledge. We have experts who tend to be experts: astronomers, pilots, physicists, mathematicians, accountants and experts who tend to be… note experts: stockbrokers, psychologists, councilors… Simply things that move and therefore require knowledge do not usually have experts and are often Black-Swan-prone. The negative effect of prediction is that those who have a big reputation are worse predictors than those who had none.

[Page 166] The classical model of discovery is as follows: you search for what you know (say, a new way to reach India) and find something you didn’t know was there (America). It’s called serendipity. A term coined in a letter by the writer Hugh Walpole who derived it form a fairy tale, “The Three Princes of Serendip” who “were always making discoveries by accident or sagacity, of things they were not in quest of.“ […] Sir Francis Bacon commented that the most important advances are the least predictable ones.

[Page 169] Engineers tend to develop tools for the pleasure of developing tools. Tools lead to unexpected discoveries. So I disagree with Taleb’s definition: A nerd is simply someone who thinks exceedingly inside the box. It may not be contradictory but I prefer the engineer-like one: “I think a nerd is a person who uses the telephone to talk to other people about telephones. And a computer nerd therefore is somebody who uses a computer in order to use a computer. [http://www.startup-book.com/2012/02/03/triumph-of-the-nerds/]
And [Page 170] Pasteur claims “Luck favors the prepared”

[Page 170] On the difficulty of predicting, just look at the failure of the Segway which “it was prophesized, would change the morphology of cities.”

[Page 184] Another example of Taleb’s target: optimization… Optimization consists in finding the mathematically optimal policy that an economic agent could pursue. Optimization is a case of sterile modeling [discussed also in Chpater 17].

Politics

[Page 16] Categorization always produces a reduction in true complexity. Try to explain why those who favor allowing the elimination of a fetus in the mother’s womb also oppose capital punishment. [Which reminds me of André Frossard : “The unfortunate thing is that the left does not believe much in original sin and that the right has not much faith in redemption.”]

[Page 52] “I never meant that the Conservatives are generally stupid. I meant to say that stupid people are generally conservative” John Stuart Mill once complained. The problem is chronic: if you tell people that the key to success is not always skills, they think that you are telling them that it is never skills always luck.”

[Page 227] Which may explain “we live in a society of one person, one vote, where progressive taxes have been enacted precisely to weaken the winners”. I am not sure if Taleb does not prefer the aristocratic world. At least he seems to favor his friends from that world.

[Page 255] True, intellectually sophisticated characters were exactly what I looked for in life. My erudite and polymathic father – who, were he still alive, would have only been two weeks older than Benoît Mandelbrot [his mentor on non-linear fractals] – liked the company of extremely cultured Jesuit priests. I remember these Jesuit visitors […] I recall that one has a medical degree and a PhD in physics, yet taught Aramaic to locals in Beirut’s Institute of Eastern Languages. […] This kind of erudition impressed my father far more than scientific assembly-line work. I may have something in my genes dirving me away from bildungsphilisters.

Globalization/Scalability

[Page 28] a scalable profession is good only if you are successful; they are more competitive, produce monstrous inequalities and are far more random. Consider the example of the first music recording, of the alphabet, of the printing press. Today a few take almost everything; the rest, next to nothing [page 30].

[Page 32] In Mediocristan,” when your sample is large, no single instance will significantly change the aggregate or the total”. In Extremistan, Bill Gates in wealth or J. K. Rowling in book selling totally change the average of a crowd. “Almost all social matters are from Extremistan.” [When giving a talk on high-tech serial entrepreneurs at BCERC last month, I was slightly criticized with a “but you are only looking at 2% of the entrepreneurs! And I replied, yes but look at the impact…”]

[Page 85] Intellectual, scientific, and artistic activities belong to the province of Extremistan. I am still looking for a single counter-example, a non-dull activity that belongs to Mediocristan.

[Page 90] You not only see that venture capitalists do better than entrepreneurs, but publishers do better than authors, dealers do better than artists, and science does better than scientists.” (I can add that gold seekers made less money than the people who sold them picks and shovels.)

[Page 102] The consequence of the superstar dynamic is that what we call “literary heritage” or “literary treasures” is a minute proportion of what has been produced cumulatively. Balzac was just the beneficiary of disproportionate luck compared to his peers.

[Page 118] The problem here with the universe and the human race is that we are the surviving Casanovas (who should not have survived and had his life without luck – no destiny].

Statistics

Taleb is not against statistics, but against Gaussian law, averages, etc. [Page 37] “The near-Black Swan are somewhat tractable. These are phenomena commonly known by terms such as scalable, scale-invariant, power laws, Pareto-Zipf laws, Yule’s law, Paretian-stable processes, Levy-stable and fractal laws.”

One thousand and one days or the story of the turkey confirms to me that an individual may not owe to the society that fed them initially!

[Page 239] Standard deviations do not exist outside the Gaussian, or if they do exist, they do not matter and do not explain much. But it gets worse. The Gaussian family (which includes various friends and relatives, such as the Poisson law) are the only class of distributions that the standard deviation (and the average) is sufficient to describe. You need nothing else. The bell curve satisfies the reductionism of the deluded. There are other notions that have little or no significance outside of the Gaussian: correlation and worse, regression. Yet they are deeply ingrained in our methods: it is hard to have a business conversation without hearing the word correlation.

[Page 240] Taleb has nothing against mathematicians, but he refers to Hardy’s views: The “real” mathematics of the “real” mathematicians, the mathematics of Fermat end Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as of “pure” mathematics).

[Page 252] A critical feature of Gaussian statistics is the inclusion of two assumptions: First central assumption: the flips are independent of one another. The coin has no memory. The fact that you got heads or tails on the previous flip does not change the odds of your getting heads or tails on the next one. You do not become a “better” coin flipper over time. If you introduce memory, or skills in flipping, the entire Gaussian business becomes shaky. (Whereas there is preferential attachment and cumulative advantage in non-Gaussian events.) Second central assumption: no “wild” jump. The step size in the building block of the basic random walk is always known, namely one step. There is no uncertainty as to the size of the step.
[…] I have not for the life of me been able to find anyone around me in the business and statistical world who was intellectually consistent in that he both accepted the Black Swan and rejected the Gaussian and Gaussian tools. Many people accepted my Black Swan idea but could not take its logical conclusion, which is that you cannot use one single measure for randomness called standard deviation (and call it “risk”), you cannot expect a simple answer to characterize uncertainty.

But Taleb goes one step further. [Page 272] “But fractal randomness does not yield precise answer. […] Mandelbrot’s fractals allow us to account for a few Black Swans but not all. […] A gray swan concerns modelable extreme events, a black swan is about unknown unknowns. […] I repeat: Mandelbrot deals with gray swans; I deal with the Black Swan. So Mandelbrot domesticated many of my Black Swans, but not all of them, not completely.

Finance

Taleb shows that the stock crashes are sometimes linked to bad modeling and is particularly critical of the Black-Scholes options. He is very much critical of the stock portfolio theories and related Nobel prizes (Markowitz, Samuelson, Hicks or Debreu, “wrecking the ideas of Keynes”. The story of the LTCM hedge fund is an illustration of Taleb’s points.

[Page xxv] Almost no discovery, no technologies of note came from design and planning – they were just Black swans. […] So I disagree with the followers of Marx and those of Adam Smith: the reason free markets work is because they allow people to be lucky thanks to aggressive trial and error, not by giving rewards or “incentives” for skill.

[Page 17] The business world – inelegant, dull, pompous, greedy, unintellectual, selfish and boring.
[…] What I saw was that in some of the most prestigious business schools in the world, the executives of the most powerful corporations were coming to describe what they did for a living and it was possible that they too did not know what was going on.

[Page 135] When I ask people to name three recently implemented technologies that most impact our world today, they usually propose the computer, the Internet and the laser. All three were unplanned, unpredicted and unappreciated upon their discovery, and remained unappreciated well after their initial use. They were consequential. They were Black Swans.

Against averages

[Page 295] Half of the time I am a hyperskeptic; the other half I hold certainties. […] Half of the time I hate Black Swans, the other half I love them. […] Half of the time I am hyperconservative; the other half I am hyperaggressive”. I could delete the quotes!

I am not fully finished with the Black Swan, I am now reading the 70-page postcript essay which Taleb added to the latest paperback edition. There might be more to say (and read if you followed me until now…)

* Poincaré is quoted in Le Monde on July 7, 2012, by Cedric Villani, who by the way also mentions Black Swans in Dans les entrailles des cygnes noirs