# 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.”

# Homo Deus : a Brief History of Tomorrow by Yuval Noah Harari (Part 2 – the Future)

I remember hesitating to buy Homo Deus. I never really appreciated people trying to analyze what the future might be. I have similar concerns with Harari’s book. I am not the only one as the New Yorker was not all positive: Then he announces his bald thesis: that “once technology enables us to re-engineer human minds, Homo sapiens will disappear, human history will come to an end, and a completely new process will begin, which people like you and me cannot comprehend.” Now, any big book on big ideas will inevitably turn out to have lots of little flaws in argument and detail along the way. No one can master every finicky footnote. As readers, we blow past the details of subjects in which we are inexpert, and don’t care if hominins get confused with hominids or the Jurassic with the Mesozoic. (The in-group readers do, and grouse all the way to the author’s next big advance.) Yet, with Harari’s move from mostly prehistoric cultural history to modern cultural history, even the most complacent reader becomes uneasy encountering historical and empirical claims so coarse, bizarre, or tendentious. […] Harari’s larger contention is that our homocentric creed, devoted to human liberty and happiness, will be destroyed by the approaching post-humanist horizon. Free will and individualism are, he says, illusions. We must reconceive ourselves as mere meat machines running algorithms, soon to be overtaken by metal machines running better ones.

If I feel the same unease, I still beleive Harari asks important questions and he might even have been misunderstood in his real motivation… I link this reading to my recent great readings of Piketty, Fleury and Stiegler.

Whatever some more extracts:

“Because science does not deal with questions of value, it cannot determine whether liberals are right in valuing liberty more than equality, or in valuing the individual more than the collective.” [Page 281]

A strange section is the following: The experiment changed Sally’s life. In the following days, she realised she has been through a ‘near-spiritual experience… what defined the experience was not feeling smarter or learning faster: the thing that made the earth drop out from under my feet was that for the first time in my life, everything in my head had finally shut up… My brain without self-doubt was a revelation. There was suddenly this incredible silence in my head… I hope you can sympathise with me when I tell you that the thing I wanted most acutely for the weeks following my experience was to go back and strap on those electrodes. I also started to have a lot of questions. Who was I apart the angry bitter gnomes that populate my mind and drive me to failure because I’m too scared to try? And where did the voices come from?’
Some of those voices repeat society’s prejudices, some echo our personal history, and some articulate our genetic legacy.
[Page 289] Again individual, society and evolution…

We see then that the self too is an imaginary story, just like nations, gods and money. Each of us has a sophisticated system that throws away most of our experiences, keeps only a few choice samples, mixes them up with bits from movies we saw, novels we read, speeches we heard, and from our own daydreams, and weaves out of all that jumble a seemingly coherent story about who I am, where I came from and where I am going. This story tells me what to love, whom to hate and what to do with myself. This story may even cause to sacrifice my life, if that’s what the plot requires. We all gave our genre. Some people live a tragedy, others inhabit a never-ending religious drama, some approach life as if it were an action film, and not a few act as in a comedy. But in the end, they are all just stories.
What, then, is the meaning of life? Liberalism maintains that we shouldn’t expect an external entity to provide us with some ready-made meaning. Rather, each individual voter, customer and viewer ought to use his or her free will in order to create meaning not just for his or her life, but for the entire universe.
The life sciences undermine liberalism, arguing that the free individual is just a fictional tale concocted by an assembly of biochemical algorithms. Every moment, the biochemical mechanisms of the brain create a flash of experience, which immediately disappears. The more flashes appear and fade, in quick succession. These momentary experiences do not add up to any enduring essence, The narrating self tries to impose order on this chaos by spinning a never-ending story, in which every such experience has its place, and hence every experience has some lasting meaning. But, as convincing and tempting as it may be, this story is a fiction.
[Pages 304-5]

At the beginning of the third millennium, liberalism is threatened not by the philosophical idea that ‘there are no free individuals’ but rather by concrete technologies. We are about to face a flood of extremely useful devices., tools and structures that make no allowance for the free will of individual humans. Can democracy, the free market and human rights survive this flood? [Page 306]

The great decoupling

The practical developments might make this belief [liberalism] obsolete:
1. Humans will lose their economic and military usefulness, hence the economic and political system will stop attaching much value to them.
2. The system will still find value in humans collectively, but not in unique individuals.
3. The system will still find value in some unique individuals, but these will be a new elite of upgraded superhumans rather than the mass of the population.
[page 307]

Humans are in danger of losing their value, because intelligence is decoupling from consciousness. [Page 311]

Intelligence is mandatory, but consciousness is optional. [Page 312]

The current scientific answer to this pipe dream can be summarized in three simple principles:
1. Organisms are algorithms. Every animal – including Homo Sapiens – is an assemblage of organic algorithms shaped by natural selection over millions of years of evolution.
2. Algorithmic calculations are not affected by the materials from which you build the calculator. Whether you build an abacus from wood, iron or plastic, two beads plus two beads equal four beads.
3. Hence there is not reason to think that organic algorithms can do things that non-organic algorithms will never be able to replicate or surpass.
[Page 319]

As algorithms push humans out of the job market, wealth might become concentrated in the hands of the tiny elite that owns the all-powerful algorithms creating unprecedented social inequality. [Page 323]

Is there too much story telling with Harari’s new book. Is the cashier replaced by a robot, or by the customer? What about the Google flu tool, that is mentioned page 335. Did it really work?

The Ocean of Consciousness

The new religions are unlikely to emerge from the caves of Afghanistan or from the madrasas of the Middle East. Rather, they will emerge from research laboratories. Just as socialism took over the world by promising salvation through steam and electricity, so in the coming decades new techno-religions may conquer the world by promising salvation through algorithms and genes. Despite all the talk of radical Islam and Christian fundamentalism, the most interesting place in the world from a religious perspective is not the Islamic State or the Bible Belt, but Silicon Valley. [Page 351]

The humanist revolution caused modern Western culture to lose faith and interest in superior mental states, and to sanctify the mundane experiences of the average Joe. Modern Western culture is therefore unique in lacking a special class of people who seek to experience extraordinary mental states. It believes anyone attempting to do so is a drug addict, mental patient or charlatan. Consequently, though we have a detailed map of the mental landscape of Harvard psychology students, we know far less about the mental landscape of Native American shamans, Buddhist monks or Sufi mystics. And that is just the Sapiens mind. Fifty thousand years ago, we shared this planet with our Neanderthal cousins. They didn’t launch spaceships, build pyramids or establish empire. They obviously had very different mental abilities, and lacked many of our talents. Nevertheless, they had bigger brains than us Sapiens. What exactly did they do with all those neurons? We have absolutely no idea. But they might well have many mental states that no Sapiens had ever experienced. [Page 356]

Techno-humanism faces an impossible dilemma here. It considers the human will to be the most important thing in the universe, hence it pushes humankind to develop technologies that can control and redesign our will. After all, it’s tempting to gain control over the most important thing in the world. Yet once we have such control, techno-humanism would not know what to do with it, because the sacred human will would become just another designer product. We can never deal with such technologeis as long as we believe that the human will and the human experience are the supreme source of authority and meaning.[Page 366]

The Data Religion

[Dataism] is very attractive. It gives all scientists a common language, builds bridges over academic rifts and easily exports insights across disciplinary borders. Musicologists, political scientists and cell biologists can finally understand each other. […] Dataists are sceptical about human knowledge and wisdom, and prefer to put their trust in Big Data and computer algorithms. [Page 368]

Capitalism won the Cold War because distributed data processing works better than centralised data processing, at least in periods of accelerating technological changes. [Page 372] The Industrial Revolution unfolded slowly enough for politicians and voters to remain one step ahead. […] Technological revolutions now outpace political processes, causing MPs and voters alike to lose control. [See Stiegler again] […] The Internet is a free and lawless zone that erodes state sovereignty, ignores borders, abolishes privacy and poses perhaps the most formidable global security risk. [Here see Beaude] […] The NSA may be spying on your every word, but to judge by the repeated failures of American foreign policy, nobody in Washington knows what to do with all the data. [Page 374]

Dataism is also missionary. Its second commandment is to connect everything to the system, including heretics who don’t want to be connected. And ‘every-thing’ means more than just humans. It means every thing. My body, of course, but also the cars on the street, the refrigerators in the kitchen, the chickens in their coop and the trees in the jungle – all should be connected to the Internet-of-All-Things. […] Conversely, the greatest sin is to block the data flow. What is death, if not a situation when information doesn’t flow? […] Dataism is the first movement since 1789 that created a really novel value: freedom of information [Page 382] which has Harari correctly explains in neither freedom nor freedom of expression.

Humanism thought that experiences occur inside us. Dataists believe that experiences are valueless if they are not shared. Twenty years ago Japanese tourists were a universal laughing stock because they always carried cameras and took pictures of everything in sight. Now everyone is doing it. […] Writing a private diary sounds to many utterly pointless. The new moot says: ‘If you experience something – record it. If you record something – upload it. if you upload something – share it.’ [Page 386] Should I blog?

Dataism is neither liberal nor humanist. It isn’t anti-humanist. [Page 387] By equating the human experience with data patterns, Dataism undermines our main source of authority and meaning, and heralds a tremendous religious revolution. […] ‘Yes God is a product of the human imagination, but human imagination in turn is the product of biochemical algorithms.’ In the eighteenth century, humanism sidelined God by shifting from a deo-centric to a homo-centric world view. In the twenty-first century, Dataism may sideline humans by shifting from a homo-centric to a data-centric view. The Dataist revolution will probably take a few decades, if not a century or two. But then the humanist revolution too did not happen overnight. [Page 389]

A critical examination of the Dataist dogma is likely to be not only the greatest scientific challenge of the twenty-first century, but also the most urgent political and economic project. Scholars in the life sciences and social sciences should ask themselves whether we miss anything when we understand life as data processing and decision-making. Is there perhaps something in the universe that cannot be reduced to data? Suppose non-conscious algorithms could eventually outperform conscious intelligence in all known data-processing tasks – what, if anything, would be lost by replacing conscious intelligence with superior non-conscious algorithms? Of course, even if Dataism is wrong and organisms aren’t just algorithms, it won’t necessarily prevent Dataism from taking over the world. Many previous religions gained enormous popularity and power despite their factual mistakes. If Christianity and communism could do it, why not Dataism? [Page 394]

Humans relinquish authority to the free market, to crowd wisdom and to external algorithms partly because they cannot deal with the deluge of data. [Page 396]

As a conclusion, Harari ends his book with the 3 following questions:
1. Are organisms really just algorithms, and is life really just data processing?
2. What is more valuable – intelligence or consciousness?
3. What will happen to society, politics and daily life when non-conscious but highly intelligent algorithms know us better than we know ourselves?

If reading is not for you, you can still listen to Harari in a recent Ted Talk: Nationalism vs. globalism: the new political divide.

# Homo Deus : a Brief History of Tomorrow by Yuval Noah Harari (Part 1 – the Past)

I wrote here how much I enjoyed reading Sapiens. Harari’s new book, Homo Deus: A Brief History of Tomorrow, is just as good.

In Death Is Optional, the exchange between Daniel Kahneman and the author, which summarizes many of Harari’s most original ideas, here is one of the most interesting ones – in relationship to start-ups: “in terms of history, the events in Middle East, of ISIS and all of that, is just a speed bump on history’s highway. The Middle East is not very important. Silicon Valley is much more important. It’s the world of the 21st century … I’m not speaking only about technology.” One may not like it, but it is interesting.

As usual, a few extracts:
“Most studies cite tool production and intelligence as particulararly important for the ascent of humankind. […] Humans nowadays completely dominate the planet not because the individual human is far smarter and more nimble-fingered than the individual chimp or wolf, but because Homo Sapiens is the only species on earth capable of co-operating flexibly in large numbers.” [Pages 130-1]

“Animals such as wolves and chimpanzees live in a dual reality. On the one hand, they are familiar with objective entities outside them, such as trees, rocks and rivers. On the other hand, they are aware of subjective experiences within them, such as fear, joy and desire. Sapiens, in contrast, live in triple-layered reality. In addition to trees, rivers, fears and desires, the Sapiens world also contains stories about money, gods, nations and corporations. As history unfolded, the impact of gods, nations and corporations grew at the expense of rivers, fears and desires. There are still many rivers in the world, and people are still motivated by their fears and wishes, but Jesus Christ, the French Republic and Apple Inc. have dammed and harnessed the rivers, and have learned to shape our deepest anxieties and yearnings.” [Page 156]

If we invest money in research, then scientific breakthroughs will accelerate technological progress. New technologies will fuel economic growth, and a growing economy could dedicate even more money to research. With each passing decade we will enjoy more food, faster vehicles and better medicines. One day our knowledge will be so vast and our technology so advanced that we could distill the elixir of eternal youth, the elixir of true happiness, and any other drug we might possibly desire – and no god will stop us. […] Modern life consists of a constant pursuit of power within a universe devoid of meaning. [Page 201]

Interesting comparison between the Scientific revolution, where Knowledge = Empirical Data X Mathematics, and the Humanist revolution led by Knowledge = Experiences X Sensitivity. In medieval Europe, Knowledge = Scriptures X Logic. [Pages 235-7]

Humanist revolution according to Harari [Pages 232-3]

Harari is sometimes too long in the development of his ideas, but it is worth following him. On pages 247-76, he explains how humanism is not a coherent view of the world. Three schisms have occurred: liberalism (where liberty is the most important value), socialism (where equality is first) and evolutionary humanism (where conflict is the raw material pushing evolution forward).

“By 1970 the world contained 130 independent countries, but only thirty of these were liberal. […] And then everything changed. The supermarket proved to be far stronger than the gulag. […] As of 2016, there is no serious alternative to the liberal package. […] China is the most promising ground for the new techno-religions emerging from Silicon Valley. […] God is dead. […] Religions that lose touch with the technological realities of the day lose their ability even to understand the questions being asked.” [Pages 264-8]

“Numbers alone don’t count for much in history. History is often shaped by small groups of forward-looking innovators. […] In 1881, Muhammad Ahmad bin Abdallah, […] meanwhile in 1875, Dayananda Saraswati in India, […] Pope Pius IX, in Europe […] or thirty years before, Hong Xiuquan […] Hundreds of millions clung to their religious dogmas. […] Hong led the deadliest war of the nineteenth century, the Taiping Rebellion. From 1850 to 1864, at least 20 million people lost their lives.” [Pages 270-1]

“Most societies failed to understand what was happening, and they therefore missed the train of progress”. [Page 273] Ask yourself what was the most influential discovery, invention or creation of the twentieth century? That’s a difficult question […] antibiotics, […] computers, […] feminism. […] What did religions bring? This is a difficult question too because there is so little to choose from. [Page 275]

And as a conclusion of chapter 7: “Since humanism has long sanctified the life, the emotions and the desires of human beings, it’s hardly surprising that a humanist civilisation will want to maximize human lifespans, human happiness and human power.” [Page 277]

# Sapiens – A Brief History of Humankind by Yuval Noah Harari

Sapiens: A Brief History of Humankind is an extraordinary book. Very similar to Guns, Germs, and Steel: The Fates of Human Societies by Jared Diamond. It may not be directly related to innovation and start-ups, but below are some extracts I found striking. This is a must-read book…

By It is believed that the cover art can or could be obtained from the publisher.
Fair use, https://en.wikipedia.org/w/index.php?curid=48907980

“Consider the following quandary: two biologists from the same department, possessing the same professional skills, have both applied for a million-dollar grant to finance their current research projects. Professor Slughorn wants to study a disease that infects the udders of cows, causing a 10 percent decrease in their milk production. Professor Sprout wants to study whether cows suffer mentally when they are separated from their calves. Assuming that the amount of money is limited, and that it is impossible to finance both research projects, which should be funded?

There is no scientific answer to this question. There are only political, economic and religious answers. In today’s world, it is obvious that Slughorn has a better chance of getting the money. Not because udder diseases are scientifically more interesting than bovine mentality, but because the dairy industry which stands to benefit from the research, has more political and economic clout than the animal-rights lobby.

Perhaps in a strict Hindu society, where cows are sacred, or in a society committed to animal rights, Professor Sprout would have a better shot. But as long as she lives in a society that values the commercial potential of milk and the health of its human citizens over the feelings of cows, she’d best write up her research proposal so as to appeal to those assumptions. For example, she might write that ‘Depression leads to a decrease in milk production. IF we understand the mental world of dairy cows, we could develop psychiatric medication that will improve their mood, thus raising milk production by up to 10 percent. I estimate that there is a global market of \$250 million for bovine psychiatric medication.’ […] In short, scientific research can flourish only in alliance with some religion or ideology.” [Pages 304-305]

So how science developed in apparently useless fields?

“The key factor was that the plant-seeking botanist and the colony-seeking naval officer shared a similar mindset. Both scientist and conqueror began by admitting ignorance – they both said ‘I don’t know what’s out there.’ They both felt compelled to go out and make new discoveries. And they both hoped the new knowledge thus acquired would make them master the world.

European imperialism was entirely unlike all other imperial projects in history. Previous seekers of empire tended to assume that they already understood the world. Conquest merely utilized and spread their views of the world. […] European imperialists set out distant shores in the hope of obtaining new knowledge along new territories.” [Page 317]

Ignoramus

[Page 279] “Modern science differs (mention Steve Weinberg here?) from all previous traditions of knowledge in three critical ways:
a. The willingness to admit ignorance. Modern science is based on the Latin injunction Ignoramus – ‘we don’t know’. It assumes that we don’t know everything. Even more critically, it assumes that the things we think we know could be proven wrong as we gain more knowledge. No concept, idea or theory is sacred and beyond challenge.
b. The centrality of observation and mathematics. Having admitted ignorance, modern science aims to obtain new knowledge. It does so by gathering observations and then using mathematical tools to connect these observations into comprehensive theories.
c. The acquisition of new powers. Modern science is not content with creating theories. It uses these theories in order to acquire new powers, and in particular to develop new technologies.”

[Pages 320-2] “The first modern man was Amerigo Vespucci.” [And not Columbus who contrarily to this lesser-known Italian sailor, was always convinced he had arrived in India and not on a new continent.] […] “Columbus stuck to this error for the rest of his life.” […] “There is poetic justice in the fact that a quarter of the world, and two of its seven continents, are named after a little-known Italian whose sole claim is that he had the courage to say, ‘We don’t know’. The discovery of America was the fundamental event of the Scientific Revolution.”

[This reminds me the day of my PhD oral presentation. A colleague of mine was surprised I dare answering ‘I don’t know’ to a question of a member of the jury. My colleague had also missed the point, I think…]

Chapters 14-16 describe how science, politics and economics are interconnected. They may be less surprising but are as convincing. Here is a disturbing extract: “Conversely, the history of capitalism is unintelligible without taking science into account. […] Over the last few years, banks and governments have been frenziedly printing money. Everybody is terrified that the current economic crisis may stop the growth of the economy. So they are creating trillions of dollars, euros and yen out of thin air, pumping cheap credit into the system, and hoping that the scientists, technicians and engineers will manage to come up with something really big, before the bubble bursts. Everything depends on the people in the labs. New discoveries in fields such as biotechnology and nanotechnology could create entire new industries, whose profits could back the trillions of make-believe money that the banks and governments have created since 2008. If the labs do not fulfill these expectations before the bubble bursts, we are heading towards very rough times.” [Page 352]

# 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.

# Emerging Science and Technologies, why so many promises? (Part 4)

This is my final post about what I have learned from Sciences et technologies émergentes, pourquoi tant de promesses? (For the record here are the links to part 1, part 2 and part 3).

The last chapters of this excellent book try to explore ways to solve the problem of excessive promises that have become a system. In Chapter IV.2, it is question of “désorcèlement” (the closest term I found would be “disenchanted”); I read it as a critical analysis of the vocabulary used by those who promise. The chapter speaks at length of the transhumanist movement, the promise of promises! “[…] Describe how these actors certainly produce, but especially divert away, reconfigure and amplify these promises […] in front of passive and naive consumers.” [Page 261] and later “[but] transhumanists are first activists, mostly neither engineers nor practitioners […] attempting answers to questions not asked or badly expressed, […] hence a really caricatural corpus,” to the point of talking about a “cult” (quoting Jean-Pierre Dupuy), “a muddled, often questionable thinking.” [Page 262]

In Chapter IV.3, the authors explore unconventional approaches, a possible sign of disarray to “scientifically” react to the promises. For example, they have contributed to the creation of a comic book to answer another comic which wanted to popularize and promote synthetic biology.

The final chapter explores scenarios that may follow the explosion of promises, like the idea of ​​increasing the number of Nobel Prize. New promises?!! More concretely, the author shows that the initial promises are not followed in practice: “The wait & see phenomenon in investment, or lack of innovation, is less known, though widespread: the effect of general and diffuse promises maintains the interest of players but too much uncertainty holds back investment in cycles of concrete promises-requirements.” [Page 297] “A game is at work which continues as long as the players follow the rules, […] they are prisoners of the game. […] They may also leave it if the right circumstances occur and then the game collapses.” [Page 298]

In conclusion, beyond a very rich description of many examples of scientific and technical promises, the authors have shown how a system of promises was built through interactions between the various stakeholders (the researchers themselves, the (political, social and economic) decision makers who fund them, and the general public which hopes and feels anxiety). The relationship to time, not only the future but also the present and the past, is beautifully described, in addition to a desire for eternity. And finally, we mostly discover that the promises have led to numerous debates that were perhaps, if not entirely, useless, as we could have known that the promises can not be kept, even from the moment they were created…

# Emerging Science and Technologies, why so many promises? (Part 3)

This is my third article about the book Sciences et technologies émergentes, pourquoi tant de promesses? After the general considerations on the system of promises, the book presents contributions describing specific areas:

I.3: nanotechnologies
II.1: semiconductors through Moore’s Law
II.2: big ata
II.3: digital Humanities
III.1: neurosciences and psychiatry
III.2: The Human Brain Project (HBP)
III.3: personalized medicine
III.4: biodiversity and nanomedicine
IV.1: assisted reproduction
IV.2: regenerative medicine

Each chapter is interesting for the curious reader as it shows the dynamics between promises and expectations of stakeholders (researchers, politicy makers, general public). The chapter about the HBP is particularly interesting in the description of the disconnection between content and form. “How was it possible that the HBP “won the competition” despite the lack of evidence to establish pragmatically the scientific relevance and the legitimacy of its ambitious organizational goals? We develop the hypothesis that this deficit, criticized afterwards, was both hidden and compensated by the production of promises shaped to anticipate and / or respond effectively to the political, economic, social and health-realted stakes on the agenda of the “challenges to come”. [Page 166] The credibility of the HBP sealed by this decision has been built […] following an adaptation process and reciprocal validations in the double register of the politicization of science and the scientification of politics. In other words, we show that one of the important conditions of this credibility was the successful co-production of a strategic congruence between [scientific] promises and the agenda of policy issues. [Page 171] The connection between knowledge of the brain and forms of social life took place mainly in the domain of discourse. […] In this contrasting situation, discursive inflation around the brain and neuroscience seems to be the consequence of a lack of evidence, as if it had overcome, positively or negatively, the differences between the present and the future, the proven and the possible, the absence and the desire. This regular feature of big science projects has resulted in the development and implementation of a prophetic rhetoric that seeks to anticipate the possibility of a better future by borrowing to the notions of hope and promise.” [Page 176-77]

I come back to a quote from chapter 3 that is essential to me as a conclusion to this new post: The real progress of techno-science will less come from their ability to keep promises than from their ability to do without them, to inherit critically from the era of great technological promise. This is not to break an idol, but to learn how to inherit. [Page 111]

# Emerging Science and Technologies, why so many promises? (Part 2)

(A word of caution: my English is reaching its limits in trying to analyze a demanding book, written in French. I apologize in advance for the very awkward wording…)

So just one day after my article describing Chapter 1 of Emerging Science and Technologies, why so many promises?, here comes an analysis of the second chapter, where the relation to time is analyzed, as well as presentism, futurism and the role of time in the promise system. There is the nice following passage:

Since The New Atlantis, futuristic speculations have accompanied the development of modern science. And in the twentieth century, Jean Perrin has sealed a new alliance between science and hope. But for him, it is science that preceded and provoked hope while now, it is rather hope that drives research. Technosciences reverse, in fact, the order of the questions that was following the three critiques of Kant: “What can I know?” – the issue addressed in Critique of Pure Reason – then the question “What can I do?” – treated in the Critique of Practical Reason – finally the question “What can I expect?” – discussed in the Critique of Judgment. In contrast, in the current scientific policies, one determines what to do and what we can possibly acquire as knowledge by identifying the hopes and promises. (Page 50)

Yet there is a paradox already expressed in the first chapter between futurism with the terms of promise, foresight and prophecy that project us and presentism particularly marked by the memory that freezes the past and transforms the future as a threat that no longer enlightens neither the past nor the present. To the point of talking about a future presentification…

The chapter also deals with the question of the future as a shock, a time “crisis” due to the acceleration. To the feeling of the misunderstanding and helplessness, is added the experience of frustration and stress caused by the accelerated pace of life, the disappointment of a promise related to modernity, where techniques were supposed to save time, to emancipate.

Another confusion: the features of planning and roadmap which are typical of technology projects slipped into research projects where is used the term of production of knowledge, while in research, it is impossible to guarantee a result. But the author shows through two examples, that this development is complex.

In the case of nanotechnology, there has been roadmap with the first two relatively predictable stages of component production followed by a third stage on more speculative systems crowned by a fourth stage which speaks of emergence, and all this by also “neglecting contingency, serendipity and possible bifurcations,” not to predict, but to “linearize the knowledge production”. The roadmap predicts the unexpected by announcing an emergence, combining a reassuring scenario of control which helps in inspiring confidence and with at the same time an emerging scenario, to create dreams. (Pages 55-56)

In the case of synthetic biology “despite a clear convergence with nanotechnology,” the rapid development occurs without any roadmap. “A common intention – the design of the living – gathers these research paths.” And it is more to redo the past (“3.6 billion years of genetic code”) than to imagine the future. The future becomes abstract and it comes as proofs of concept. And the author adds that in normal science in the sense of Kuhn, these proofs of concepts would have fallen into oblivion. The paradox is that there is no question of right or wrong, but of designing without any needed functionality.

In the first case, “prediction or forecasting are convened as the indispensable basis for a strategy based on rational choice”, “the future looks to the present. “In the second case, there is the question of “towing the present” and “fleeing out of time.” We unite and mobilize without any necessary aim. The future is virtual, abstract, and devoid of culture and humanity; the life of the augmented human looks more like eternal rest …

Ultimately, the economy of promises remains riveted on the present either by making the future a reference point to guide action in the present, or it is seeking to perpetuate the present.

Chapters three and four are less theoretical, describing on one hand new examples in the field of nanotechnology and on the other, how Moore’s Law became a law when it was initially a prospective vision of progress in semiconductor. Perhaps soon a follow-up about the next chapters…

# Emerging Science and Technologies, why so many promises? (Part 1)

Sciences et technologies émergentes, pourquoi tant de promesses? (Emerging Science and Technologies, why so many promises?) is the title of a book (in French only) from a group of authors under the direction of Marc Audétat, a political scientist and researcher at the Sciences -Society Interface of the University of Lausanne. This is not an easy reading book, it is quite demanding, but it raises important questions.

I have already reported on this blog about books that speak of a certain crisis of science, for example in The Crisis and the American model (in French), about the books “La Science à bout de Souffle” or “The University bubble. Should we pursue the American dream?” or in The Trouble With…, a book by Lee Smolin, not to mention the most violent criticism of the promises of technology by Peter Thiel in Technology = Salvation.

This new book explores the promises related to science and technology to the point of talking about an economy of promises. This is a collective work which does not make it an easy reading, but the diversity of points of views is certainly an asset. I have not finished reading it and I will certainly come back to it. In the first chapter, P.-B. Joly describes the system of techno-scientific promises. He begins by introducing the concepts of Imaginary and Vision [Page 33] This “couple of concepts takes into account how various sources of inspiration are involved in the technical creation. […] The Imaginary gives an almost tangible appearance to concepts and ideals that are a priori devoid of it … [and becomes] a common sense that founds the action into society. […] The concept of Vision is close to that of Imaginary, but on a smaller scale. It is akin to that of “rational myth” used to analyze the dynamics of collective action in changing contexts. […] Coalitions of actors form around these visions of a prospective order and contribute to their dynamics. […] If we accept this conceptual distinction between Imaginaries (at large scales – the Nation – and in the long term) and Visions (at a level of coalitions of actors and active over periods of medium length) then comes the question of the interaction between the two.”

“Unlike Visions and Imaginaries, for which the content of technical arrangements is essential, what is essential for the techno-scientific promises is the creation of a relation, as well as a time horizon of expectations. […] Promises are essential in technology creation, because they enable innovators to legitimize their projects, to mobilize resources and to stabilize their environment. […] Any techno-scientific promise must convince a large audience that it determines a better future than the alternatives, even if the realization of the promise requires major, sometimes painful changes.” (The author mentions the history of electricity or the green revolution as the solution to world hunger)

“Our concept of techno-scientific promises has been systematized and became in the last forty years the governance of the new techno-sciences (biotechnology and genomics, nanotechnology, neuroscience, synthetic biology, geo -engineering, etc.) The construction of a techno-scientific promise meets two conflicting constraints: the constraint Radical Novelty and that of Credibility. […] (And I interpret that) for this request to be credible, one must disqualify alternatives. [Furthermore] For a scientific theory to be credible, its validity is neither necessary nor sufficient. […] The techno-scientific promises must have the support of a circle of specialists. Otherwise, they cannot resist the opposition manifested either in scientific arenas, or in public arenas. An extreme version is observed when the specialists refer to natural laws to justify the inevitability of technological change. (Examples are Moore’s Law and Gabor’s Law.) Thus, in principle, generic promises are not subject to validity tests.”

Finally, this intensification is reinforced by three complementary elements [page 39]:
– the future is more a threat than a source of hope;
– research and innovation are often presented as the only way to solve problems;
– the research stakeholders should demonstrate their societal impacts.

This leads to pathologies [pages 40-43]:
– the myth of a public victim of irrational fears and to be educated becomes an intangible scheme;
– the promises turn into bubbles;
– the radical novelty and uncertainty create conflicting discourses, sources of mistrust because the effects of such radicalism is not predictable so that through experimentation, the technologists become sorcerer’s apprentices and society, a laboratory;
– finally promises lead to endless discussions on fictions, on issues that may have nothing to do with the reality of research.

In conclusion Joly thinks this promises system is one of the enemies of the future because of the clear separation it creates between those who make the promise and those who are supposed to accept it. The recognition of this regime and therefore these problems is a prerequisite imperative.

After reading the first chapter, I remembered the societal concerns of Cynthia Fleury, about whom I have already said a few things in a digression in the article On France Culture, Transhumanism is Science Fiction. Our democratic societies are in crisis, and the distrust of politics as well as of experts has never been stronger. The issue of research and innovation is a component of this crisis. I am eager to discover the rest of this very interesting (and important) work …

# A brilliant conversation about science between Gérard Berry and Etienne Klein

Indeed a brilliant and “crazy” conversation between Etienne Klein, the physicist, and Gerard Berry, Gold Medal of the CNRS for his work in computer science on France Culture’s Conversation Scientifique. They speak about so many beautiful “provocative” things.

On the serious side Berry talks about the difficulty of predicting and the danger of impossible promises, also about the courage in science. Physics talks about energy, computer science about information and so do the new generations, Berry claims. Berry also speaks of the machine and the human. The digital machine, the computer goes very fast, but not with much more intelligence than a steam engine. Except the computer is everywhere. But we, humans, are intuitive, there is no insight in a computer. We are very complementary. Again, I do not agree with transhumanists who believe that the machine will overtake us – in the short term at least. Berry is very annoyed by the notion of intelligence in computers. The performance is not intelligence, but there are many interesting things in AI such as learning, find people in a photo database fascinates Berry.

On the politics of science, Berry expresses great caution and wisdom. “Claiming that a research topic will happen right away is the best way to kill it.” He was answering a question about the quantum computer. “This was the case of artificial intelligence”. There are people willing to promise the moon and more people willing to believe them. They promise sensational things in interesting topics. One must look into these areas, but one should make no promise. Among the possible benefits, but unpredictable, there will be some interesting things.

He also talks about neuroscience. He is fascinated by the way children learn, which is difficult to understand; why the brain freezes after a while. His fascination is that the brain processes information, but we do not understand creativity, the brain is a huge machine which we do not understand. But again, it will be difficult, no doubt very difficult to understand how the brain works. Berry believes “no more than that” in our ability to build artificial neurons to simulate the brain mechanisms. We also discover that pleasure and boredom, motivation are essential to learning. Finally Berry started a short analysis on the current state of research. “Do not do anything new when you want to be successful. Or rather you have to fight.” (just as in any art).

I let you discover the last quarter of an hour which talks about the college of Pataphysique… but you need to understand French…