24 October 2008

Prayer request

I just submitted a prayer at the Prayer Request Site. It goes like this:

Please show me a zero of the Riemann zeta function that has real part not equal to 1/2. I know that my prayer will be answered soon, as you say, in less than a month. If within a month I receive no reply to my prayer then I will know that the Riemann hypothesis is true. And therefore I can go out and claim that God told me it is so. If people ask me for a proof I will refer them to your site.

If I get a response then I will know that the Riemann hypothesis is false. If I don't then I will know it is true. Either way, I'm a winner and, I suppose, I can publish a paper titled "the Riemann hypothesis via Prayer Request Site". I will let you know what happens in a month or so.

For your information, other prayer request sites appear here. An amusing one is by the Celebrity Spotlight Ministries. In it, it is claimed that "the prayer of a righteous man is powerful and effective", followed by a picture of a mass-murderer. The most important message of the Celebrity Ministries prayer site is "PUSH". In other words, Pray Until Something Happens. (If this appears as an obvious tautology, it is because it really is.)

Update: As of to-date (11/11/08), my prayer has not been answered. So I prayed again:

I am really hoping it will be answered. The prayer request site
has direct links with the supernatural. You click it and then you communicate with the gods. Please go there and pray. It is fun! You don't need to work any more. Just pray for money. You don't need to worry any more. Just pray for happiness. In my case, I don't need to think about proofs any more. I will pray for theorems to be proved!Link

22 October 2008

Conventions and rationality

One of the basic problems is the inability of many to recognise the difference between something that is fundamental and something that is a convention.

An example of a fundamental truth is the inverse square law of gravity. It is an empirical fact, based on observation, experiment, measurements; by accepting it, we build a theory upon which we can predict other facts, understand why it takes about 365 days for the earth to go around the sun, and build satellites, among other things. This is why the law of gravity is fundamental.

An example of a convention is that a week has seven days. It surprises many people that the concept of a week is not a concept, but a rather arbitray convention. It is difficult for many to digest this because the concept has been around for, well, some four thousand years. Of course, there are many a posteriori justifications. Why, say some, the week consists of seven days because the Bible says so. This is not a justification, even if the Bible was written by a God, as many contend.

The inability to distinguish between a fundamental truth and a convention is a major problem. It suddens me to have to point this out to my students (and others), in the post-enlightenement 21st century. But if we have to point out trivialities like this, we should do so if wesave the rapid decline that will lead us to the another medieval era.

I see this inability to distinguish between fundamentals and conventions each and every day in the university. Sadly, we all blame students. We blame poor elementary education. We blame politicians. But we should blame ourselves as teachers. I do encounter many professors who themselves cannot distingush between fundamentals and conventions. They pass on the wrong message to the students. And the result is catastrophic.

Let me give an example of poor teaching that cannot distinguish between fundamentals and conventions. Suppose a lecturer in a basic Statistics course tells students that sums of large numbers behave as if they come from a normal distribution without making the effort to explain (a) the assumptions or (b) why the normal distribution is unavoidable. I am well aware of the mathematical difficulties of such a concept for a beginning Statistics course, but I am also well aware of methods that can convey the message to the students without having to go through a stern mathematical demonstration. Such methods, unavoidably, will leave students with questions. But this is the point of education: students should be left with (the right) questions so that they will seek to fill in the gaps themselves or by attending more advanced courses. Instead, what is happening in many institutions of so-called higher education (and, indeed, in mathematical sciences departments), is that lazy (or ignorant?) lecturers will fill the students' minds with wrong impressions such as: Statistics means opening an excel programme, filling in the squares with data, and pressing a few buttons to get an answer. How boring! But how convenient if your utility function is to mazimize profit by having lots of students enrol in your classes. And how well it works for those students who choose the path of least action.

21 October 2008

Religion (Greek Orthodox style) via mobile phones!

This posting requires little explanation. The sheer idiocy of (this kind, but not only...) religious practice is so blatant that only a moron would not see it.

Caption of picture on the left:
(Incidentally, the first fellow on the fifth row is Aghios Nectarios and I am at distance 2 from him, for he was the teacher of my grandfather in a boarding school. In other words, I have direct link to saints; I need no mobile phone.)

Caption of picture on the right:

Actually, the right picture poses a certain challenge. Does it actually say that one cannot communicate with God via mobile telephony? Should we blame mobile phones? Should we blame the bad use of them? Does it, perhaps, say that if one uses mobile phones in a certain manner in a church then he or she can actually communicate with God? I wouldn't be surprised if the latter was actually meant. It is well-known that, for decades now, Americans have been communicating with God via TV. So why not via a cell phone? They are both communication media based on electromagnetic waves. I would guess that God can be reached around the frequency of 10^8 Hz, although I could be wrong. It would be an interesting Science project to find out God's frequency band. If you happen to know it, please let me know. I'll pass on the information to the Greek Orthodox church.
Thank you.

19 October 2008

Physics vs Management

As I was flying back home, I spent a few hours at the Amsterdam airport. One of my favourite activities is to visit the business section of the bookshop and look at amusing titles such as "the path to leadership", "how to make money while sleeping", "management and zen", "the habits of the fifty five most successful people in Saudi Arabia", and so on. Last year I came across a book on finance (still in the business section) which contained a statement of the Black Scholes formula, followed by a programme in Excel. It is then when I realised that it is good I don't do financial mathematics.

The main point today of this comment is the following image, taken as Schiphol a few hours ago:

Imagine seeing these two books, in the same series, next to each other at the formative age of, say, fifteen. And suppose you had no hunch for science. You'd instantly think that management is as deep as physics or, at least, something that can be studied at the same level as physics. You go to university, you have a choice: Should I study this or that? You have forgotten you've seen these two covers, but it's embedded in you. You believe that there is a choice. And that it is, merely, a choice between equivalent subjects.

This is what the modern university is like. This is what our administrators promote. Science has become equivalent to making money for, hm... for what, really?

18 October 2008

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?

9 October 2008

Substitutes for religion

Many people say: "What can we do without religion? How can we have moral principles without it?"

Well, just substitute your prayer with a poem recitation. It is much better, because you don't have to subscribe to arbitrary dogmas, and, besides, poetry can be re-written and revised, if necessary, unlike religious texts that can never change.

Here is an example of a substitute for a religious practice then: recite this poem:

An excellent recitation, by Sean Connery, of an English interpretation
of this most famous poem of Constantine Cavafy

Σα βγεις στον πηγαιμό για την Ιθάκη,
να εύχεσαι νάναι μακρύς ο δρόμος,
γεμάτος περιπέτειες, γεμάτος γνώσεις.
Τους Λαιστρυγόνας και τους Κύκλωπας,
τον θυμωμένο Ποσειδώνα μη φοβάσαι,
τέτοια στον δρόμο σου ποτέ σου δεν θα βρεις,
αν μέν’ η σκέψις σου υψηλή, αν εκλεκτή
συγκίνησις το πνεύμα και το σώμα σου αγγίζει.
Τους Λαιστρυγόνας και τους Κύκλωπας,
τον άγριο Ποσειδώνα δεν θα συναντήσεις,
αν δεν τους κουβανείς μες στην ψυχή σου,
αν η ψυχή σου δεν τους στήνει εμπρός σου.

Να εύχεσαι νάναι μακρύς ο δρόμος.
Πολλά τα καλοκαιρινά πρωιά να είναι
που με τι ευχαρίστησι, με τι χαρά
θα μπαίνεις σε λιμένας πρωτοειδωμένους·
να σταματήσεις σ’ εμπορεία Φοινικικά,
και τες καλές πραγμάτειες ν’ αποκτήσεις,
σεντέφια και κοράλλια, κεχριμπάρια κ’ έβενους,
και ηδονικά μυρωδικά κάθε λογής,
όσο μπορείς πιο άφθονα ηδονικά μυρωδικά·
σε πόλεις Aιγυπτιακές πολλές να πας,
να μάθεις και να μάθεις απ’ τους σπουδασμένους.

Πάντα στον νου σου νάχεις την Ιθάκη.
Το φθάσιμον εκεί είν’ ο προορισμός σου.
Aλλά μη βιάζεις το ταξείδι διόλου.
Καλλίτερα χρόνια πολλά να διαρκέσει·
και γέρος πια ν’ αράξεις στο νησί,
πλούσιος με όσα κέρδισες στον δρόμο,
μη προσδοκώντας πλούτη να σε δώσει η Ιθάκη.

Η Ιθάκη σ’ έδωσε τ’ ωραίο ταξείδι.
Χωρίς αυτήν δεν θάβγαινες στον δρόμο.
Άλλα δεν έχει να σε δώσει πια.

Κι αν πτωχική την βρεις, η Ιθάκη δεν σε γέλασε.
Έτσι σοφός που έγινες, με τόση πείρα,
ήδη θα το κατάλαβες η Ιθάκες τι σημαίνουν.

As you set out for Ithaca,
hope that you journey is a long one,
full of adventure full of discovery.
Laistrygonians and Cyclops,
and great Poseidon do not be afraid of them,
you'll never find things like that on your way,
as long as you keep your thoughts raised high,
as long as a rare sensation touches your spirit and your body.
Laistrygonians and Cyclops,
wild Poseidon you won't encounter them,
unless you bring them along inside your soul,
unless your soul sets them up in front of you.

Hope that your journey is a long one.
May there be many summer mornings
when with what pleasure, what joy
you come into harbour scene for the first time;
may you stop at Phoenician trading stations,
to buy fine things,
mother-of-pearl, coral, amber, and ebony,
sensual perfume of ever kind,
as many sensual perfumes as you can;
and may you visit many Egyptian cities,
to learn and learn again from those who know.

Keep Ithaca always in you mind.
Arriving there is what you are destined for,
But do not hurry the journey at all.
Better if it lasts for years;
so that you are old by the time you reach the island,
wealthy with all you have gained on the way,
not expecting Ithaca to make you rich.

Ithaca gave you the marvellous journey.
Without her you would not have set out.
She has nothing left to give you now.

And if you find her poor, Ithaca won't have fooled you.
Wise as you will have become, so full of experience,
you will have understood by then what these Ithacas mean.

9/10/08... Happy Yom Kippur to my Jewish friends!


What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant