Dase's algorithm

I just saw a TED talk about Mathemagics by Arthur Benjamin, and googled a few related things. With the notable exception of Alexander Aitken, it seems that very few calculator prodigies are able to explain how they actually calculate so quickly. The fuzziness of the explanations suggests some less-rational procedure that seems closer to intuition than to any algorithm.

One such calculator prodigy was Zacharias Dase (1824-2861). He is shortly mentioned in Hofstadter’s book, Gödel-Escher-Bach. His calculation skills were such that Gauss himself was impressed and he had him hired by the Hamburg Academy of Sciences to compile some mathematical tables. Dase had other amazing abilities, like counting sheep in a flock in a single glance.

In the MacTutor History of Mathematics archive, you can read the following sentence about Dase: "He multiplied two 20 digit numbers in 6 minutes; two 40 digit numbers in 40 minutes; two 100 digit numbers in 8 hours 45 minutes."

When you put these figures on a logarithmic plot it seems that Dase’s unknown algorithm was slightly slower than quadratic (solid line). It is likely that he was using the same algorithm as anyone of us: multiplying each digit of the first number by the second number, and taking the sum. He was very good at it, and he obviously had some tricks, but that was probably all of it.

Had Zacharias Dase lived lived today, he might have tried to use the FFT-like algorithm by Schönhage and Strassen (dotted line), which is faster for multiplying large numbers.

http://www.ted.com/talks/lang/eng/arthur_benjamin_does_mathemagic.html.

Use, misuse, and abuse of science

Cold Spring Harbor Laboratory (of which James DNA Watson was president) hosted the Eugenics Record Office during the first half of the twentieth century. A very interesting website about eugenics can be found at

http://www.eugenicsarchive.org/eugenics/.

Anything than can go wrong with science is found in the history of eugenics.

Devinette

What is this?

Try to guess.

Marcello Truzzi

"In science, the burden of proof falls upon the claimant; and the more extraordinary a claim, the heavier is the burden of proof demanded. The true skeptic takes an agnostic position, one that says the claim is not proved rather than disproved. He asserts that the claimant has not borne the burden of proof and that science must continue to build its cognitive map of reality without incorporating the extraordinary claim as a new "fact." Since the true skeptic does not assert a claim, he has no burden to prove anything. He just goes on using the established theories of "conventional science" as usual. But if a critic asserts that there is evidence for disproof, that he has a negative hypothesis—saying, for instance, that a seeming psi result was actually due to an artifact—he is making a claim and therefore also has to bear a burden of proof. "

(Marcello Truzzi, Zetetic Scholar, 12-13, 1987. )http://www.anomalist.com/commentaries/pseudo.html

Marcello Truzzi was an interesting character. He was born in 1935 in Copenhagen, in a Russian family of circus performers. The family moved to the US in the 1940, and Marcello became professor of sociology at Eastern Michigan University. He is one of the founders of the Society for Scientific Exploration (http://www.scientificexploration.org/), which aims at "the rigorous study of unusual phenomena that may be ignored or inadequately studied within mainstream science". He died in 2003.

Why would it be natural to oppose science and religion?

At first sight, religion and science have little in common, of course. One is about faith and the other is about doubts. In their nature, however, they are similar. Both are a quest of truth.

A more realistic statement would be that the purpose of science is to produce useful models of reality [1]. One can of course argue on the exact meaning of words, but this definition is apt to create a consensus. As for religion, being religious can be considered equivalent to believing in the existence of supernatural entities (referred to hereafter as gods) that rule or influence the natural world.

One can only speculate on why people have first come to believe in the existence of gods [2, 3]. It seems to me that gods contribute to make the world more understandable, if not predictable. A catastrophe occurs, say, heavy rains followed by a flood. Attributing this to the anger of some god, rather than to mere chance, is a way to preserve the comfortable idea that everything happens for a reason. Moreover, this explanation is also enticing because it provides people with a means of controlling Nature (do not irritate the gods). Therefore, humans might well have created gods because their world ought to be understandable and controllable; in other words, the inventors of gods believed in causality and in determinism.

From that perspective, gods are simply a mental representation of the principles that control Nature. So to speak, gods are a model of reality. And that early model was definitely useful in helping people accept their faith and keep living, which is sometimes described as an evolutionary asset [2]. According to the definition given above, the creation of gods can be though of as a scientific process. Another obvious usefulness of religions is that they helped people believe in causality and determinism, which beliefs eventually happened to be so fruitful in the development of our technological world during the last three centuries or so.

However complex they may seem, nowadays models - like general relativity, or DNA - are far simpler than the psychology of an irascible god. Simple models are useful only because their domains of applicability are very limited. The laws of physics explain only a very small fraction of the physical world because they are all conditional [4]. No prediction can be made unless the initial state of a system is known, and most human questionings about Nature concern the initial conditions that physics says nothing about. Even more fundamentally, the way physics tries to grasp the notion of time is via evolution laws, which are somehow opposed to the truly creative evolution that is central to anybody’s life [5].

Science accumulated spectacular achievements by restricting its models to more and more limited domains of applicability. The evolution of religions is quite opposite: they evolved towards more and more general principles, such as Good and Evil, which generalization culminates in monotheism. The achievements of religions, and notably of monotheistic religions, are unquestionable given the civilizations they contributed to develop and the way they still shape the human mentalities all over the world. The appeal of their generalizing and unifying ideals to the human nature makes no doubt.

Not so surprisingly is now science moving towards more general models and paradigms than in the past. The hypothetical grand unification theory between nuclear interactions and electromagnetism is just an example. The appeal of multi- or trans-disciplinary researches is also very similar. Trans-disciplinary researches, by attempting to enlarge the initial domains of applicability of models, have already resulted in shaking some of the founding ideas of modern science. Reductionism – which is a basis of the experimental method – is now known to be of limited applicability as some systems are subject to emergence [6].

In summary, the self-confidence of the positivists of the 19th century is not defensible at the beginning of the 21st century. Moderate religion and lucid science are just two different aspects of the same questioning of mankind, torn between the need of simple models for making accurate predictions and the appeal of general principles that would give life a meaning. The endless lines about creationism –just to mention one- that are superficially presented as an opposition between religion and science are merely a clash between fanaticism and intellectual honesty.

[1] This nice definition was in a previous version of the entry about Science in Wikipedia. It seems that the article in question was recently victim of much vandalism.
[2] R. M. Henig, Darwin’s God, New York Times Magazine, March 4 2007.
[3] U. Eco, God isn't big enough for some people,

http://www.umbertoeco.com/id-49/Umberto_Eco_About_God.html.
[4] E. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics 13 (1960);
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
[5] I. Prigogine, From Being To Becoming, Freeman: 1980.
[6] P.W. Anderson, More Is Different, Science 177, (1972) pp. 393-396.

Bohmian mechanics and the organized skepticism

A few days ago, I read the paper about Bohmian mechanics on the website of the Stanford Encyclopaedia of Philosophy (SEP) [1]. The last time I came across the name of David Bohm [2] was during my stay in Princeton in 2005. I was subletting a house of Princeton University and people in the neighborhood had the habit of leaving on a shelf at the laundry any book they didn’t want to carry along when they were moving out. In one of these, there was a chapter about scientists in Princeton who had been the victims of McCarthy. David Bohm was one of them.

David Bohm had worked on the Manhattan project, he was assistant professor at Princeton University in 1949. After being suspected of being a communist, he was fired from Princeton and he could not find any position in any other American university. He worked in several countries -notably in Brazil- and he eventually settled in London, where he died in 1992.

The exile of David Bohm was actually twofold: not only was he persona non grata in his own country, he was also left out of the international clique of physicists who mattered. Not surprisingly is Bohmian mechanics unorthodox; it is the work of an outcast.

What I realized with the SEP article is that the Copenhagen interpretation of quantum mechanics (QM) is not all there is about QM. David Bohm proposed a causal (deterministic) interpretation of quantum mechanics, the predictions of which are identical with the orthodox QM [3]. Philosophically, however, the two theories are at odds. In Bohmian mechanics the particles have a well defined position and momentum, and the wave function acts as a potential (a so-called pilot wave) that guides the particles. Such an approach was claimed to be impossible, and proved to be so by a famous theorem due to John Steward Bell [4]. Interestingly enough, here is what Bell himself wrote in 1987 about Bohmian mechanics [1]:

But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? ... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

This is the story of David Bohm and of his unorthodox theory. Yet another sad example of organized skepticism turned de facto into dogmatic denial.

[1] http://www.science.uva.nl/~seop/entries/qm-bohm/
[2] http://en.wikipedia.org/wiki/David_Bohm
[3] D. Bohm, Phys. Rev. 85 (1952) 166-180.
[4] http://en.wikipedia.org/wiki/Bell%27s_Theorem

Perspective

Reminder: these rays are physically all parallel.

Reductionism, holism, causality, and entropy

Reductionism is the idea according to which a complex system can be analyzed by decomposing it into smaller and simpler entities. For instance, the complex properties of macroscopic systems can often be understood from the simpler laws that govern the atoms and molecules they are made of. The opposite idea is holism, according to which the properties of an entity are best understood in the context of the larger system it is part of. Somehow, the theory of evolution, in which the morphology of individuals is shaped by their environment, is a holistic one.

The fact that the two approaches (reductionism and holism) can both be used to define a causality relation is puzzling. When passing from holism to reductionism, one almost switches the cause and the effect. When two facts are related by some logical link, which one is to be considered as the cause? And which one is the effect? It seems to me that we always chose the simplest fact as the cause. In order words, it is easy for us to understand that simple causes can have complex effects, but we generally do not accept the other possibility. I would be most happy to discuss this with anyone!

Causality is obviously related to the notion of time. The cause always comes before the effect, and this is basically how the direction of time is defined. Now, if we really tend to chose the simplest facts to be the cause of complex effects, we tend to bias the direction of time towards an increasing complexity. Isn’t that oddly related to the impossibility of decreasing the entropy of an isolated system, as time increases?

La tomographie électronique expliquée à ma fille

Découpez le patron ci-dessus et pliez-le le long des lignes pointillées. Vous obtiendrez un très joli modèle en papier du matériau sur lequel je travaille à Utrecht. La prochaine étape consiste à générer une série de patrons, pour faire un jeu de cubes, à la manière d'un puzzle 3D qui représenterait un vrai tomogramme. Après tout, on fait bien ça avec Winnie l'Ourson, pourquoi pas avec NiO/SBA-15?

Pour ceux que ça intéresse, je tiens à votre disposition le code Matlab qui vous permettra de faire ça à partir de n'importe quelle matrice 3D.

Bricolages inspirés de Escher

J'ai fabriqué ce puzzle en argile avec mes filles, c'est inspiré d'un des nombreux pavages du plan de Escher. Je suis surpris que personne ne propose ce genre de chose pour carreler une salle de bain. Avec un emporte-pièce ça ne doit pas être trop compliqué à faire.

Les faits

"Ce sont des hypothèses honnêtes qui rendent compte des faits: mais je sens bien qu'elles viennent de moi, qu'elles sont tout simplement une manière d'unifier mes connaissances (...) Lents, paresseux, maussades, les faits s'accommodent à la rigueur de l'ordre que je veux leur donner; mais il leur reste extérieur." (Jean-Paul Sartre, La Nausée)

The language of Nature

Every practitioner of physics knows that the laws of physics, in their mathematical form, do not enable to make predictions about the real world. Quite often, solving the equations of mathematical physics provides two solutions, of which one is rejected on the basis of non-mathematical arguments.

A first example is that of the retarded and advanced potentials of classical electromagnetism. Only the former are considered, the latter being in contradiction with causality. A second example is the common situation in which one finds two possible functions: one that diverges at infinity and another one that converges to zero. Only the latter is kept, the former being termed “unphysical”.

What is interesting in the two examples is that mathematics predicts one thing as well as the opposite: causality and anti-causality, divergence and convergence. We choose the solution on the basis of non-mathematical arguments.

Mathematics may be the language of Nature, but we take some liberty in its interpretation.

Soda-bottle water rockets

To build a water rocket [e.g. 1, 2], you need:
(i) a plastic soda-like bottle, (ii) a rubber tap with a plastic flexible pipe through it, and (iii) a bike pump.

Fill half the bottle with water, close it with the tap, turn it upside down, and pump air through the pipe until the tap plops off . The water is pushed out of the bottle by the air pressure, and there you go!

Below are some pictures of a few launches with a 2 l mineral water bottle. The pressure in the bottle is about 4 atmospheres when the tap plops (my bike pump has a manometer). The camera we had did not enable us to take more than 5 pictures per second. In each case, the water expulsion (i.e. the burnout) is shorter one fifth of a second!

On the first picture I am hiding behind the cabin and pumping. Less than 0.2 seconds later, the bottle has expelled all its water and it is 5 meters high. With some rough kinematics (velocity = acceleration*time, distance = 1/2 acceleration*time squared), one finds that the average acceleration of the bottle is more than 25 g’s, and its velocity after 0.2 s is more than 50 m/s!

The inset shows the nicely even bouncing of the water jet on the ground at the rocket’s takeoff. The cloud in the second picture (indicated by an arrow) is formed when the bottle empties entirely from all its water. From the second and third pictures, it is interesting to note that, although the water is being expelled downwards, the water of the cloud is moving upwards. The reason for this is that the water is expelled downwards relative to the bottle. As the bottle itself is climbing faster than the water is descending compared to it, the absolute velocity of the water is still upwards.

Incidentally, this is what multistage rocket systems aim at avoiding [3]. The first stage uses low exhaust velocity fuels, and high exhaust velocity fuels are used in the upper stages. Under these conditions, the exhaust velocity increases when the rocket gains speed: the fuel is always expelled backwards, which makes the propulsion very efficient.

It would be interesting to check whether the same can be obtained with a water rocket by optimizing the shape of the bottle.

Here is another launch that was pretty successful, as for the height reached by the bottle.

There is something disappointing with all these launches though: the bottle rotates. This is clear from the second picture of the first launch, and from the third picture of the second. The rotation clearly increases the aerodynamic drag on the rocket, which reduces its maximum height.

An additional issue related to the rotation of the bottle is that it is not possible to put a parachute on the rocket (which prospect made my daughters hectic). The parachute opens as soon as the bottle starts rotating, i.e. immediately after the burnout, and the launch is a fiasco.

Perhaps an appropriate shape of the bottle can make the rocket more stable, and prevent it from rotating.

How about side wings?
Surely my daughters will love it!

[1] D. Kagan, L. Buchholtz, L. Klein, “Soda-bottle water rockets”, The Physics Teacher, 33 (1995) 150-157.
[2] http://en.wikipedia.org/wiki/Water_rocket.

[3] R.H. Gowdy, “The physics of perfect rockets”, American Journal of Physics, 63 (1995) 229-232.

25/5=14

This is good fun, although it vaguely reminds me of some math lessons I attended as a kid.

25/5=14

From Ma and Pa Kettle Back on the Farm (1951).

Is it really easy to multiply a number by 10?

It seems so easy to multiply a number by 10. Just write a zero at its right, and that's it.

Well, this only means that it is easy to write down the number you obtain when you multiply any number by 10, when you use a base-10 system. Do I really know anything about the number I write down?

I can just as easily write down any mulitplication by 117 in a base-117 system (just write a zero). Does this mean that I am actually able to handle easily a multiplication by 117? I don't think so.

The ease that I feel with the multiplication by 10 is absolutely not sound. I just know the recipe I have to apply to write that number down in a base-10 system. Actually, I know no more of the number I obtain when I multiply by 10, than of the number I obtain when I multiply by 117.

It is really not clear what "calculation" means.
It looks like the so-called "calculation skills" are merely the skills needed to write down a number.

Lettre sur les ignorants à l'usage de ceux qui savent

« Ceux qui peignent les paysages se tiennent dans la plaine pour considérer la forme des montagnes et des lieux élevés ; et pour examiner les lieux bas, ils se juchent sur les sommets. De même, pour bien connaître la nature des peuples, il faut être prince ; et pour connaître les princes, être du peuple. »

(Nicolas Machiavel, Le Prince)

Il faut bien se rendre à l’évidence : l’Europe Occidentale baigne dans une mer de mysticisme. Parfois une vague obscurantiste plus grosse que les autres déferle bruyamment. La vague obscurantiste actuelle est celle du créationnisme ; elle est ponctuée notamment par la publication de l’Atlas de la Création par Harun Yahya, et le rejet par le Conseil de l’Europe du rapport Lengagne sur « Les dangers du créationnisme dans l’enseignement » en juin dernier.

De nombreuses personnalités ont réagi en rappelant combien les théories créationnistes sont scientifiquement peu justifiées, et parfois théologiquement peu pertinentes. On peut toutefois se demander dans quelle mesure ces réactions sont vraiment convaincantes. D’une part, les media confèrent souvent aux spécialistes qu’ils invitent le statut de « ceux qui savent », ce qui les auréole, parfois malgré eux, d’une autorité morale de même nature que celle dont jouissent souvent les auteurs des thèses qu’ils combattent. D’autre part, et c’est sans doute lié, ces mêmes spécialistes se départissent rarement de leur jargon, ce qui contribue à rendre leurs interventions incompréhensibles par ceux à qui elles s’adressent.

Après tout, si les ignorants sont incapables de faire la distinction entre la science et l’obscurantisme, c’est peut-être que les tenants de l’une et de l’autre se comportent de manière semblable : a quelques exceptions près, la plupart des émissions télévisées dites scientifiques, nombre de livres destinés à la jeunesse, et même quelques revues de vulgarisation, exposent les conclusions de recherches sans jamais parler de la démarche scientifique qui constitue leur seule valeur. En communiquant de la sorte, ceux qui savent s’exposent à ce que certains ignorants soient créationnistes plutôt qu’évolutionnistes, puisque le seul argument qu’ils avancent en faveur de l’évolution est leur autorité de savants, si fragile face à une quelconque autorité religieuse.

A la réflexion, de nombreux faits souvent qualifiés de scientifiques ont été communiqués sans prendre la peine de mentionner la moindre raison d’y croire. J’imagine que peu de gens contestent que ces petits points qui brillent dans le ciel la nuit soient de gigantesques boules de gaz incandescent. Pourtant, combien d’entre nous ont une raison d’y croire autre que « tout le monde sait ça »? La raison pour laquelle la nature des étoiles est moins contestée que l’évolution des espèces n’est pas que ceux qui savent la physique sont plus convaincants que ceux qui savent la biologie, c’est uniquement que les enjeux philosophiques sont moindres. Si vous êtes de ceux qui savent, demandez-vous quels faits ont été communiqués aux ignorants qui leur diraient pourquoi croire en l’existence de l’ADN, en la structure de l’atome, et même simplement en l’héliocentrisme du système solaire !

Ce qui fait la scientificité d’une idée, ce sont les raisons d’y croire plus que l’idée elle-même. Ce qui fait sa force, c’est qu’elle s’impose par un processus démocratique par lequel tout le monde peut en principe en vérifier le bien-fondé. N’importe qui peut la contester à tout moment en faisant valoir des contres arguments, et c’est là sa seule légitimité.

Un exemple spectaculaire de l’efficacité de la démocratie en science est l’encyclopédie Wikipedia, accessible sur Internet, et dont chaque article peut être modifié par n’importe quel lecteur, sans qu’il doive prouver une quelconque autorité en la matière. Le développement de Wikipedia résulte donc de débats continus entre d’innombrables lecteurs et rédacteurs anonymes. La méthodologie des encyclopédies traditionnelles est diamétralement opposée, elles confient la rédaction des articles à des experts (ce qui n’a pas empêché Nodier se moquer de ce que le Dictionnaire de l’Académie décrive l’écrevisse comme un « petit poisson rouge »). Une étude comparative commandée par la très sérieuse revue Nature a montré que la qualité des principaux articles était comparable dans Wikipedia et dans l’encyclopédie Britannica. Ce qui est transparent dans Wikipedia et qui ne l’est pas dans la Britannica, c’est que la science est démocratique et critique vis-à-vis d’elle-même, et qu’elle n’est pas une question d’autorité.

Bref, si nos sociétés sont un terreau fertile pour l’obscurantisme, c’est sans doute que les scientifiques eux-mêmes se comportent rarement comme des gens qui doutent et qui remettent leurs théories en question. Ce faisant, ils sont crus pour leur autorité plus qu’ils ne convainquent par leurs arguments ; ils véhiculent une image dénaturée de la science qui est difficilement distinguable des pseudosciences. Cela n’est d’ailleurs pas sans rappeler la manière dont le « hoc est corpus meum » de la liturgie catholique a été transformé au cours des siècles en un « hocus pocus » par l’incompréhension des fidèles qui y voyaient une forme de magie. L’Eglise a réagi en son temps, non pas en traitant ses fidèles d’ignorants, mais en supprimant les messes en latin et en rendant son message compréhensible. A quand le Vatican II de la science ?

Definition

Science is what we have learned about how not to fool ourselves about the way the world is.
(R. Feynman)

The unreasonable effectiveness of mathematics

«The Unreasonable Effectiveness of Mathematics in the Natural Sciences, » by E. Wigner, in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960) ;
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

« The Unreasonable Effectiveness of Mathematics, » by R.P. Hamming, in The American Mathematical Monthly, vol 87, No II (February 1980) ;
http://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html

An idea that is present in both essays is that mathematics work because they are to a large extent inspired by our perception of the world we live in. I had the following thought about natural numbers.

Why on Earth do natural numbers seem so natural? It may be the simple fact that physical objects can be distinguished from one another. If they could not be distinguished, the very concept of counting would not be relevant at all. I don’t mean if the objects were identical, I mean if there was really no way to tell “this is one object and this is another one”. In these conditions, the idea of natural numbers would probably not have come to anybody’s mind. To some extent, the obviousness of natural numbers –and presumably their properties- has something to do about physics.

In quantum mechanics, identical particles are indistinguishable, even in principle. I don’t know, however, if this means they cannot be counted. For instance, we might be able to say “there are for sure 20 photons between these two mirrors, but there is no way we can label them”. If you are an able physicist, I would definitely be interested to discuss this.

It has often been said that Euclid’s geometry would not have been discovered if our everyday environment had been the curved space of general relativity. I am now wondering whether natural numbers would have been discovered by anybody but the most insane of us if the everyday physics had been quantum mechanics.