Lévy's forgery theorem

I asked my students in the exam of my graduate probability class to prove Lévy's forgery theorem which, in folk terms, states that if you run a Brownian motion long enough then it will write your signature (provided your signature is performed continuously). The one-dimensional version of this can be stated in mathematical terms as follows.

Let B be a Brownian motion and let f be a continuous function (your "signature"). Assume B(0) = f(0) = 0. Then, no matter how small ε > 0 is we will for sure be able to find a time interval of length 1 such that B will differ from f on that time interval by at most ε.

To prove this, all we have to prove is that there is a positive probability that the above will happen on the time interval 0 ≤ t ≤ 1, and then appeal to ergodicity (a theorem in probability): if you toss a coin sufficiently many times, you will for sure bring a head, provided that the probability of a single head is positive (assuming that coin tosses are independent).

The idea for proving the above is as follows. Split the interval 0 ≤ t ≤ 1 into n subintervals I₁, ... , I_n of length 1/n each and pick n so large that the maximum change of f within any interval of length 1/n is at most ε. (This can be done because f is uniformly continuous.)

Then observe that on each interval  I_k, the maximum deviation of B from its value at the beginning of the interval is smaller than ε with positive probability. This requires understanding that the maximum of a Brownian motion has density which leaves no gaps.

Also, observe that the difference of B and f at the beginning of the interval I_k can also be made smaller than ε with positive probability because when we observe a Brownian motion on equally spaced times, such as 0, 1/n, 2/n,..., then we are observing a random walk with normal increments.

Putting these things together we obtain a proof.

Intuitively, the proof was based on the following observation: The maximum difference of B(t) from f(t) will be small if 
(i) the maximum deviation of B(t) from its value at the beginning of the interval I_k containing t is small,
(ii) the difference between B and f at the beginning of this interval is small, and
(iii) f does not change much on this interval.

You might say, well, yes, this result is correct, but I might have to wait really long time to see my signature. And you'd be right. The analogy is this: if the probability of heads is 10^-10 then you will have to wait on the average  10¹⁰ billion seconds (more than 300 years) to see a single head, if you toss one coin per second.

We spoke of  "an interval of length 1'' above. Why? Is there any reason? No, absolutely none. By scaling, we can prove the same thing for any length. But this means that you don't have to wait too long, provided you don't mind if your signature is written in tiny letters. That is, the following holds:

When you wake up in the morning, pick a particle and make it move like a Brownian motion. Arrange things so that the particle's motion is monitored by a computer which keeps track of all places it visits. Go make a cup of coffee and come back and look at the computer files. The trajectory of the particle is stored in there. Zoom in, and zoom, and zoom, and zoom, and move around, and you will see your name. In fact, you will see the whole Bible, both in Greek and in Hebrew.

Is that amazing or what? You may say, "I don't believe this". This is a false statement. It's not a matter of belief. It's a matter of proof. Assuming that the hypotheses hold, the conclusion is true. The catch is, of course, that the hypotheses are mathematical ones and whether they are met in reality is a different matter. Of course, there is no such thing as "complete independence'' in real life, and there is no such thing as ``particle with infinitesimal size'' in real life. (Also, there is no such thing as a straight line...) However, a physicist can assure us that, with high precision, the assumptions are not unrealistic, meaning that, yes, even in real life the above claims are true. When it comes to seeing your signature some time far in the future, yes, you will see it, but it will take long time. When it comes to seeing your signature before you have finished your cup of coffee, all you'll see is black space because when you zoom and zoom and zoom, after a while you'll hit the limit imposed by the particle's size.

There is no problem with mathematics. There is no problem with physics. However, there is a problem with the word "belief". Simply, the word does not exist.

Bertrand Russell's message to future generations

— What would you think it’s worth telling future generations about the life you’ve lived and the lessons you’ve learned from it?

“I should like to say two things, one intellectual and one moral. The intellectual thing I should want to say is this: When you are studying any matter, or considering any philosophy, ask yourself only what are the facts and what is the truth that the facts bear out. Never let yourself be diverted either by what you wish to believe, or by what you think would have beneficent social effects if it were believed. But look only, and solely, at what are the facts. That is the intellectual thing that I should wish to say.

The moral thing I should wish to say… I should say love is wise, hatred is foolish. In this world which is getting more closely and closely interconnected we have to learn to tolerate each other, we have to learn to put up with the fact that some people say things that we don’t like. We can only live together in that way and if we are to live together and not die together we must learn a kind of charity and a kind of tolerance which is absolutely vital to the continuation of human life on this planet.”

Bertrand Russell, 1959

The trouble with Michael Ruse

Michael Ruse, a philosopher from Florida State University, is known for his efforts in trying to compromise christianity and evolutionary theory. He describes himself both as an atheist and agnostic. But he likes to keep the other side happy. Many people claim that his knowledge is insufficient and his arguments confused. Here is an example (Ruse defends adaptationism), and here is another (Ruse on the nature of morality).

But one does not need to go to great philosophical depths to realize that Ruse's views are flawed. For very simple reasons. Let me explain one. Look at this 6.5 minute video where Ruse describes what he thinks is the trouble with Richard Dawkins. He says that Dawkins is too simplistic and that he spends no time in trying to understand how christians think. "Come on", says Ruse, "no Christian really believes in the old testament literally. The god of the old testament may very well be an ethnic cleanser, as Dawkins points out, but learned christians (St Thomas) would never consider this seriously. If you find a christian who wants to sacrifice his child because god told him so then you would take him to a psychiatrist".

Now, all that is true. I think (and hope) that only a very small minority of christians (and jews and muslims) would take all aspects of the Bible literally. But, in this case, I'd like to ask Michael Ruse, why has the Bible (or the offending parts of it) not been banned yet by the religious folk, the church, the Rabbis, etc? At which point of time has any religious leader stood up to say that parts of these texts contain gruesome, unethical, disturbing stories and, implicitly or explicitly, suggestions? If nobody believes in them or if they are not taken seriously, then they should be removed.

But last time I went into a church (a Presbyterian church to be precise), I looked in the Bible, turned to my favourite Deuteronomy and saw that the gruesome parts were still there.

Yes, nowdays, christians do not behave in the same way they did 400 hundred years ago. I can say and teach Newton's mechanics and their consequences for the motion of celestial rocks (planets, ...), without fearing that I will be burned. There have been changes, thank god [sic]. But these changes were not changes that were initiated by the church. Rather, they were natural consequences of the way people live nowadays. Church had to accept the changes, but it was not its choice.

So, this is the trouble with Michael Ruse. He doesn't see that the reconciliation he is defending is one-sided.When he describes the trouble of Dawkins, he doesn't see that there is a big trouble in what he is saying.

The moral landscape

New book by Sam Harris, dealing with the question whether science can determine morality, human values. The last bastion of theists, namely that without belief in the supernatural there is no way for people to behave ethically, is challenged by Harris who defends his position apparently well. I just bought the book and am reading it, so I am not yet in a position to offer my personal views. A review by Michael Schermer which just appeared in the Scientific American is positive. Another review by Massimo Pigliucci which just appeared in the latest edition of the Skeptic is more negative.

What is important for me is mostly that the question can be asked and the dogmatic view be challenged. Almost certainly, morality has nothing to do with these people, or these ones, or these ones, or these ones, or these ones. And, certainly, the hypocrisy of Blair is not an answer but means for providing further divide between those who believe and those who don't, whatever the verb may mean.

Oh, by the way, for those who asked me recently, I don't only read books or articles I only agree with.

Aristotle, the church, and vegetables

I never quite understood why Aristotle, out of all ancient philosophers, was Christianity's favorite child. It is said that Aristotle was widely read and taught by Christian theologians and that his works greatly influenced Orthodoxy and Catholicism alike.

I think that the theologians who studied Aristotle never bothered to study his works too carefully; or that they skipped the parts they didn't like.

I am referring, in particular, to several paragraphs in Aristotle's Metaphysics (Book 4) where an argument is made about those who cannot understand that we cannot claim that something and the negation of it are simultaneously true.

Aristotle writes:

εἰσὶ δέ τινες οἵ, καθάπερ εἴπομεν, αὐτοί τε ἐνδέχεσθαί φασι τὸ αὐτὸ εἶναι καὶ μὴ εἶναι,  καὶ ὑπολαμβάνειν οὕτως. (There are some who, as we said, assert that it is possible for the same thing to be and not to be, and they accept this.)

And concludes:

ὅμοιος γὰρ φυτῷ ὁ τοιοῦτος ᾗ τοιοῦτος ἤδη. (Any such person is therefore no better than a vegetable.)

When you tell religious people that there are contradictions in their arguments, in their statements, in the way they behave, in the things they believe, in their sacred texts..., they reach a point when deus ex machina comes to save them: this is due to “faith”, to “mystery”, to something that I cannot understand because I don't believe what they believe. (How could I? Even if I was willing to believe blindly, whose belief should I espouse? Well, it is a mystery...)

Daniel Dennet uses this Aristotelian quote to make a point:

All parties to a reasonable conversation have to agree at the outset to set aside any trump cards their religion commends. So what if the Bible, or the Quran, says something? Since not everybody accepts that these texts are infallible, citing them as if they were is just rude.

Those who believe that their holy texts are infallible have a tough task ahead of them: convincing the rest of us, point by point, that they are right, starting from common ground.

Indeed, they have to. Otherwise, I can, using their argument, claim that a scribbling done by Kanzi (the famous bonobo ape) is, according to my belief, sacred, and proves whatever I want to prove.

Dennet concludes:

People whose religion does not permit them to engage in such open-minded discourse are in an important sense disabled: They may be the nicest people in the world, but they are incompetent participants in an open forum, and must be excused. Perhaps somebody else can be found to take on the task of representing their point of view while abiding by the basic rules of inquiry.

I agree. They are nice guys and gals, I've met many of them and share many common interests, values and passions. But they better get someone else to argue for them. (And good luck in finding this person...)

Combinatorial species and a masterpiece in the philosophy and practice of mathematics research

I recently discovered the book Combinatorial Species and Tree-like Structures by Bergeron, Labelle and Leroux, a systematic treatment of combinatorial species, a rigorous formalization of the concept of a discrete structure.

What I would like to discuss here is the extremely interesting, for many reasons, foreword written by Gian-Carlo Rota. He talks about the dynamics of progress in mathematics, pointing out of the ways that the disciplines moves forward.

The first way, Rota writes, occurs when a long-standing problem is solved; e.g., Bieberbach's conjecture or Fermat's last theorem. These are holy grails in mathematics, problems which puzzle generations of mathematicians, leading to surprising developments in the field, until, one day, someone finally gets credit for the solution. While the person who finally obtains the solution rightly deserves the credit, Rota points out that the `genius' does not, simply, belong to one individual but it is a collective, cumulative intelligence belonging to generations of hard-working people.

The second way is also very interesting. There are ideas circulating in mathematics for years and years, collective intuition, one might say, things that people work on, use, but no one dares to put down rigorously for fear (perhaps) of formalizing a triviality, or, simply because nobody knows how to formalize the intuition, or because nobody wants to do that. The second way that mathematics advances is when a commonplace idea finally finds a proper, solid, rigorous home. Rota tells us that mathematicians are reluctant to publicize this second way that the field advances. But when it happens, properly so, that is, it opens a new window into a new way of thinking.

Rota gives a few examples belonging to the second way. The first is the formalization of group theory. The second is category theory. And, of course, Rota points out that the topic of the Combinatorial Species book is, precisely, an advancement of the second kind: someone (André Joyal) found a correct way of associating a combinatorial structure to a generating function and formalized the notion of combinatorial species. Rota points out that species relate to generating functions in much the same way that random variables relate to distribution functions.

My favorite example of the second kind of advancement is Stokes' theorem. Stokes' theorem is a generalization of the fundamental theorem of calculus to higher dimensions and, indeed, in a geometric setup. It states that the integral of a differential form over the boundary of a smooth oriented manifold equals to the integral of the derivative of the form over the manifold. The proof of the theorem is a `triviality'. It is a triviality that takes lots and lots of pages of setting up the scene properly: multilinear algebra, differential forms, manifolds. Once the scene is established, and once dozens of `trivial' lemmas are proven, Stokes' theorem comes out easily.

When progress of the second kind occurs in mathematics, Rota points out, it is met with distrust until many papers are written, using the theory and showing to the old fogies that things are done in a much nicer way using the newly established setup. After this happens, the old fogies will take notice. (Some will pretend they "knew that all along''.)

At first, the old fogies will pretend the book [Bergeron et al.] does not exist. This pretense will last until sufficiently many younger combinatorialists publish papers in which interesting problems are solved using the theory of species. Eventually, a major problem will be solved in the language of species, and from that time on everyone will have to take notice.

He makes an analogy:

Those probabilists of the thirties who held on to distributions, while rejecting random variables as “superfluous,” were eventually wiped out, and their results are not even acknowledged today.

People, including mathematicians, are very protective of their way of doing things. I have met mathematicians who are completely reluctant in accepting a new way of seeing things, a different point of view. Once, when I was a fresh MSc student at Berkeley, Eugene Wong told me that there are two ways of thinking: one is geometric, the other is analytical; but the best progress is made by people who can use both.

Now, I daresay add something more to Rota's prediction. You see,  once the distrust phase is gone and everybody is happily using the ``new math'', it is the old, pedestrian, way of doing things that is forgotten: everybody (even the enemies of the new field) has converted. But there still remain problems which are best attacked by the good-old intuitionistic way, the one used before the formalization occurred. At this point, the ones who are going to have an advantage are those who can effortlessly combine both points of view.

Here is then the exact article by Rota. It is taken from Bergeron's webpage:





Forward [to Combinatorial Species and Tree-like Structures by Bergeron, Labelle and Leroux]
by Gian-Carlo Rota

Advances in mathematics occur in one of two ways.

The first occurs by the solution of some outstanding problem, such as the Bieberbach conjecture or Fermat’s conjecture. Such solutions are justly acclaimed by the mathematical community. The solution of every famous mathematical problem is the result of joint effort of a great many mathematicians. It always comes as an unexpected application of theories that were previously developed without a specific purpose, theories whose effectiveness was at first thought to be highly questionable.

Mathematicians realized long ago that it is hopeless to get the lay public to understand the miracle of unexpected effectiveness of theory. The public, misled by two hundred years of Romantic fantasies, clamors for some “genius” whose brain power cracks open the secrets of nature. It is therefore a common public relations gimmick to give the entire credit for the solution of famous problems to the one mathematician who is responsible for the last step.

It would probably be counterproductive to let it be known that behind every “genius” there lurks a beehive of research mathematicians who gradually built up to the “final” step in seemingly pointless research papers. And it would be fatal to let it be known that the showcase problems of mathematics are of little or no interest for the progress of mathematics. We all know that they are dead ends, curiosities, good only as confirmation of the effectiveness of theory. What mathematicians privately celebrate when one of their showcase problems is solved is Polya's adage “no problem is ever solved directly.”

There is a second way by which mathematics advances, one that mathematicians are also reluctant to publicize. It happens whenever some commonsense notion that had heretofore been taken for granted is discovered to be wanting, to need clarification or definition. Such foundational advances produce substantial dividends, but not right away. The usual accusation that is leveled against mathematicians who dare propose overhauls of the obvious is that of being “too abstract”, As if one piece of mathematics could be “more abstract” than another, except in the eyes of the beholder (it is time to raise a cry of alarm against the misuse of the word “abstract,” which has become as meaningless as the word “Platonism.”)

An amusing case history of an advance of the second kind is uniform convergence, which first made headway in the latter quarter of the nineteenth century. The late Herbert Busemann told me that while he was a student, his analysis teachers admitted their inability to visualize uniform convergence, and viewed it as the outermost limit of abstraction. It took a few more generations to get uniform convergence taught in undergraduate classes.

The hostility against groups, when groups were first “abstracted” from the earlier “group of permutations” is another case in point. Hadamard admitted to being unable to visualize groups except as groups of permutations. In the thirties, when groups made their first inroad into physics via quantum mechanics, a staunch sect of reactionary physicists, repeatedly cried “Victory!” after convincing themselves of having finally rid physics of the “Gruppenpest.” Later, they tried to have this episode erased from the history of physics.

In our time, we have witnessed at least two displays of hostility against new mathematical ideas. The first was directed against lattice theory, and its virulence all but succeeded in wiping lattice theory off the mathematical map. The second. still going on, is directed against the theory of categories. Grothendieck did much to show the simplifying power of categories in mathematics. Categories have broadened our view all the way to the solution of the Weil conjectures. Today, after the advent of braided categories and quantum groups, categories are beginning to look downright concrete, and the last remaining anticategorical reactionaries are beginning to look downright pathetic.

There is a common pattern to advances in mathematics of the second kind. They inevitably begin when someone points out that items that were formerly thought to be “the same” are not really “the same,” while the opposition claims that “it does not matter,” or “these are piddling distinctions.” Take the notion of species that is the subject of this book. The distinction between “labeled graphs” and “unlabeled graphs” has long been familiar. Everyone agrees on the definition of an unlabeled graph, but until a while ago the notion of labeled graph was taken as obvious and not in need of clarification. If you objected that a graph whose vertices are labeled by cyclic permutations – nowadays called a “fat graph” – is not the same thing as a graph whose vertices are labeled by integers, you were given a strange look and you would not be invited to the next combinatorics meeting.

The correct definition of a labeled graph turned out to be more sophisticated than the definition of an unlabeled graph. A labeled graph – or any “labeled” combinatorial construct – is a functor from the groupoid of finite sets and bijections to itself. This definition of a labeled object is not “abstract”: on the contrary, it expresses in precise terms the commonsense idea of “being able to label the vertices of a graph either by integers or by colors, it does not matter,” and it is the only way of making this commonsense idea precise. The notion of groupoid, which is one of the key ideas of contemporary mathematics, makes it possible to withhold the assignement of a specific set of labels to the vertices of a graph without making the graph unlabeled.

Joyal’s definition of “labeled object” as a species discloses a vast horizon of new combinatorial constructions, which cannot be seen if one holds on to the reactionary view that “labeled objects” need no definition. The simplest, and the most remarkable, application of the definition of species is the rigorous combinatorial rendering of functional composition, which was formerly dealt with by handwaving – always a bad sign. But it is just the beginning.

Species are related to generating functions in much the same way as random variables are related to probability distributions. Those probabilists of the thirties who held on to distributions, while rejecting random variables as “superfluous,” were eventually wiped out, and their results are not even acknowledged today.

I dare make a prediction on the future acceptance of this book. At first, the old fogies will pretend the book does not exist. This pretense will last until sufficiently many younger combinatorialists publish papers in which interesting problems are solved using the theory of species. Eventually, a major problem will be solved in the language of species, and from that time on everyone will have to take notice. The rewriting, copying and imitating will start, and mathematicians who capitulate to the new theory will begin to tell us what species really are. Considering the speed at which mathematics progresses in our day, that time is more likely to come sooner than later.

The present book is the first thorough treatment in English of the theory of species. It is lucidly and clearly written, and it should go a long way to making this fundamental chapter of combinatorial mathematics available to the entire spectrum of mathematicians, computer scientists and cultivated scientists generally.

What is Enlightenment?

The motto of the Enlightenment is not a thing of the past. It is modern and will always be so:

Aufklärung ist der Ausgang des Menschen aus seiner selbst verschuldeten Unmündigkeit. Unmündigkeit ist das Unvermögen, sich seines Verstandes ohne Leitung eines anderen zu bedienen. Selbstverschuldet ist diese Unmündigkeit, wenn die Ursache derselben nicht am Mangel des Verstandes, sondern der Entschließung und des Muthes liegt, sich seiner ohne Leitung eines anderen zu bedienen. Sapere Aude! Habe Muth dich deines eigenen Verstandes zu bedienen! ist also der Wahlspruch der Aufklärung.

Enlightenment is man's release from his self-incurred tutelage. Tutelage is man's inability to make use of his understanding without direction from another. Self-incurred is this tutelage when its cause lies not in lack of reason but in lack of resolution and courage to use it without direction from another. Sapere aude! "Have courage to use your own reason!" - that is the motto of enlightenment.

So the message of the Enlightenment is not, simply, the lack of reason but the lack of ability to use one's reason. Reasoning independently and with courage is what the motto of the enlightenment is. "Sapere aude" means "dare to know" and, by extension, dare to use your own reason, independently, without the guidance of somebody else, free of external influences even (or, perhaps, especially) when these influences are part of an established "tradition".

The message of the Enlightenment should be read and understood by all, even people working in the Academia (teachers and administrators alike).

Here is the original first page from Imanuel Kant's essay "Beantwortung der Frage: Was ist Aufklärung? -- Berlinische Monatsschrift, Bd. 4, 481-494, 1784".

Don't Cite Works You Haven't Read, II

As previously mentioned, the philosopher Bernard-Henri Lévy made a ridiculous blunder giving a citation to a non-existent philosophical work: he had not read his references. This is not a serious scientific approach.

But is Bernard-Henri Lévy serious? No, according to a Greek philosopher who says (personal comments):

You suspected well: Bernard-Henri Lévy is not serious. He belongs to a generation of philosophers promoted by the media in the 80s as "new philosophers". In trying to critique the soviet totalitarianism, they developed a right-wing approach; behind their pompous rhetoric there was no philosophical substance.

Further information on Lévy from wikipedia:

• Some of his professors there included prominent French intellectuals and philosophers Jacques Derrida [remember the Sokal hoax?] and Louis Althusser.
• Returning to Paris, Lévy became famous as the young founder of the New Philosophers (Nouveaux Philosophes) school. This was a group of young intellectuals who were disenchanted with communist and socialist responses to the near-revolutionary upheavals in France of May 1968, and who articulated a fierce and uncompromising moral critique of Marxist and socialist dogmas.
• Lévy was one of the first French intellectuals to call for intervention in the Bosnian War in the 1990s.
• When his father died in 1995, Lévy became the manager of the Becob company, until it was sold in 1997 for 750 million francs to the French entrepreneur François Pinault.
• He drew controversy due to his support of the Iraq War, in 2003.
• Le Testament de Dieu or L'Idéologie française faced strong rebuttals, from noted intellectuals such as historian Pierre Vidal-Naquet, philosophers Cornelius Castoriadis, Raymond Aron and Gilles Deleuze, who called Lévy's methods "vile".
• Lévy's writing and speaking style is regularly lambasted as grandiloquent and smug by fringe essayists and popular satirical TV puppet show Les Guignols de l'info, in which Lévy has his own puppet.
• Lévy is, with his third wife, a regular fixture in Paris Match magazine, wearing his trademark unbuttoned white shirts and designer suits. Some have attributed to Lévy a reputation for narcissism. One article about him coined the dictum, "God is dead but my hair is perfect.
• He once said that the discovery of a new shade of grey left him "ecstatic."
• He is a regular victim of the "pie thrower" Noël Godin, who describes Lévy as "a vain, pontificating dandy".
• Lévy is proudly Jewish, and he has said that Jews ought to provide a unique Jewish moral voice in world society and world politics.
• Lévy was one of six prominent European public figures of Jewish ancestry that were targeted for assassination by a Belgium-based Islamist militant group in 2008.

On professionalism

The word "professional'' as been used and abused. It is nowadays often confused with "bureaucrat'' or with "wearing a suit and a tie''. But let's see what Schopenhauer had to say when he contrasted professionals to dilettantes, an also confused word:

"Dilettantes! Dilettantes! - this is the derogatory cry (directed at) those who apply themselves to art or science for the sake of gain raise against those who pursue it for love of it and pleasure in it... The truth, however, is that to the dilettante the thing is the end, while to the professional as such it is the means; and only he who is diretly interested in a thing, and occupies himself with it from love of it, will pursue it with entire seriousness. It is from such as these, and not from wage-earners, that the greatest things have always come." Parerga and Paralipomena, 1851 (Essays and Aphorisms, R. J. Hollingdale, trans., London Penguin Books, 1970), p 227.

On the Platonic existence of numbers

Yesterday I came across an introductory paper by Olle Häggström, Objective truth versus human understanding in mathematics and in chess (2007), The Montana Math. Enthusiast. In it, Olle supports two ideas, both of which I have always held firmly. First that numbers exist independently of humans, and second that the human way of doing mathematics will always play a role in it and, no matter how advanced our computational machines turn out to be, they will never substitute the way we think, we prove and, more importantly, we understand mathematics (and science more generally).

Olle questions whether the first philosophical idea (the platonic ecistence of numbers) can be as dangerous as the following argument for the existence of God: " To anyone who has met God, His existence can no longer be in doubt. " I don't think so. Of course, accepting the existence of numbers as independent of humans goes beyond the realm of science and mathematics, but it is not dangerous. In fact, what Olle does not mention in his article is that we did have a whole system, a religion if you wish, that was based, precisely, on the concept of number as a divine object: Why, Pythagoras himself established his school which lasted for 600 years before it was destroyed by early Christians (they stoned to death the last of the Pythagorean mathematicians, Hypatia of Alexandria). I maintain that we would have ended up with a much better society had numbers still been the object of divinity, rather than a merciless God. I would definitely go to church every Sunday (or Saturday or Friday...) if we were to discuss numbers, and discuss I mean, rather than listen (without the possibility of asking any deep questions) to a boring priest, minister, rabbi, etc, and performal silly rituals.

Anyway, I'm digressing. Back to Olle's article, I would like to add that if we accept that numbers are independent of the human experience then we quickly reach the fact that it is merely the empty set that is the only thing that exists. Indeed, all numbers can be constructed from the integers which themselves can be constructed from the empty set, which I here denote by o. Indeed, zero is defined as o. One is defined as the set that contains o. Two is defined as the set that contains o and 1, and so on: Integer n is defined as the set containing all the previous integers. This model is, arguably, the best we have: It leads naturally to the construction of transfinite ordinals (Cantor) as well as surreal numbers (Conway). The latter ones are representations of two-player strategic games, just as the game of chess that Olle discusses in his paper.

So, the empty set is all there is then. Do you hear some reverberations of Zen Buddhism or Ancient Taoism? Hm, yes, indeed. Not that I will take these systems too seriously, but they do rely on the concepts of emptiness and nothingness, respectively.

As for whether Maths is human or not, I do agree again with Olle in that it is not the building stones of Mathematics that are human, but it is the way we do it that is. We, as humans, have to decide what to study, what to accept, what to prove, how to prove it, how to interpret and understand a proof, and how to use a certain result. Suppose we had reached a stage where we could solve all ordinary differential equations of the form
P(D)y = z
where P is a polynomial with real coefficients of degree less than 40, z=z(x) is a given function of one real variable x, y=y(x) is the unknown function, while D is the derivative operator. But say that the explicit solution required a few thousand of pages to write down. So? Would we accept it as an answer? Certainly not! Who says so? Why, everyone, mathematicians, engineers, physicists, practitioners... Only a robot would seriously maintain that a 1000-page formula is an answer. The reason for our dislike of such a formula is, precisely, because of our physical (and therefore mental) limitations. One could, possibly, imagine another world, where "humans" were 4 times as tall, with eyes 3 times as big, brain 5 times as large, and so on. Then mathematics would have been different. This is a bit of a naive explanation, but does convey my point.

Take another example. In Maths, we are interested in rates of convergence. What does this mean? It means to find out how fast a certain sequence converges. But the answer, i.e. the rate of convergence, must be given in terms of an "elementary" function, i.e. a polynomial, an exponential, and so on--only a handful of them, or in terms of another function we understand. Again, this is because of our human understanding of what constitutes elementary.

Let me also mention that (many) mathematicians are often faced with a choice: should we study this or that? Why should we accept or reject the axiom of choice? (Axioms are, roughly speaking, statements that cannot be proved or disproved, based on previously accepted statements.) If we do, we get a certain kind of Maths. If we don't we get another. What is better? Again, the answer is beyond the field of Mathematics. It has to do with us, humans, with the way we want to interpret the world, with the machines we need to construct, the tools to use to cook better food, etc.

I find the quotation, in Olle's paper, of a statement of Gowers interesting:

Namely, that we can live without the idea that an ordered pair (x,y) really is a funny set of the form {{x},{x,y}}, and that undergraduates would be confused by it.

Why should we need to take one point of view? I do agree with Gowers that, esp. nowadays, most undergraduates would get confused. And that when we introduce an ordered pair we do have to say the obvious thing, instead of reducing it to set theory. At least not immediately. But I do maintain (and have often found useful) to have the ability to reduce intuitively understood concepts to its fundamentals, to the axioms and objects of set theory, say. I take no sides. I maintain that both rigour and intuition are absolutely necessary. I am not surprised that Gowers seems to be taking on one side only. His colleague, Alan Baker, writes in the preface of his wonderful book, A Concise Introduction to the Theory of Numbers , Cambridge U. Press (1985), that "[t]there is no need to enter here into philosophical questions concerning the existence of [the integers]". He is right: his book is a wonderful speedy introduction to those aspects of number theory that lead to the solvability of Diophantine equations. Diophantus (from the works of whom--destroyed by early Christians, preserved by Arabs--all modern Algebra stems) has discovered algorithms for solving (systems) of classes of polynomial equations in integers. He didn't care about the existence of integers: Integers did exist and if you dared accept that square root of 2 was irrational you might lose your head. But Baker adds the adverb "here" in his sentence. He means, I hope, that elsewhere, at some other time, the student might wish to question the existence of integers, wonder why they should exist and convince herself or himself that they do (or do not!).

This is why Science and Mathematics is much more desirable than blind faith: we discuss, we question, we argue, we come to a conclusion, we revise, we discuss again, all along based on proofs and physical evidence.

Olle says (and I find this amazing!) that Freeman Dyson maintains that the statement "there exists a power of two, 2^n, such that, when written in decimal and read backwards it is a power of 5" is not provable!

It is quite rare that we encounter statements that are unprovable. Although, from a counting point of view, most statements within a mathematical system should be independent of its axiomatic foundation, it is very hard to bump into one. Is that not then another evidence supporting the idea that the way we do Mathematics, the way we seek to discover (for discoverers we are) truth within its vast archipelago, is, indeed, very human-based?