The Riemann-Davis hypothesis

Few musicians can claim a legacy the size and impact of Miles Dewey Davis III’s (26 May 1926 – 28 September 1991): from The Birth of the Cool (1957) through Kind of Blue (1959) all the way to Bitches Brew (1970) and beyond Miles Davis was not only influencial, he made at every stage of his career (cue Academic Tone) “seminal contributions to the field”; whatever stage of Miles’ career was marked by two things: organisational genius and contagious creativity. Take Kind of Blue for instance. It is known that the session musicians were only given rhythmic lines and general indications on key and tonality for each track they recorded; they had to improvise along those lines and they recorded the album in a two-day session. The result: one of the most critically acclaimed and comercially succesful albums of all time.

Miles Davis made great music and in passing influenced almost every musical genre; he’s credited as one of the creators of “jazz fusion”; he could bring together percussionists of afro-latin rhythms, classically trained pianists and jazz musicians to create music that otherwise-I venture to say-would’ve never existed:

Whatever you may think of the above piece, where do you put it? It’s not “jazz-jazz”, is it? It’s not exactly “rock”; it’s not quite “funk”… yet jazz fans can make sense of it, rock fans can relate to it and funky people might just ask for a little more up-beat to be content. Incidentally, the track above belongs to the recording sessions of what became Jack Johnson (1971) which Miles described as “the best rock band you’ve ever heard”… and in my opinion, he was right; Ali didn’t make it to the official release of the album, though.

Now take this one, called Ascenseur pour l’echefaud (Lift to the Gallows):

It’s the soundtrack of a film of the same title (1957). Background story: Miles was introduced to the director of the film and he invited him to record the soundtrack for it; Miles agreed and less than a month later he came to the studio with his band, gave them a few general directives and harmonic sketches, heard the synopsis of the film and while an edited version of it was projecting, they started recording. The music is jazz… but how do you account for that?

Allow me to repeat myself: how do you account for a bandleader that can inspire his bandmates to improvise a whole (and beautiful, in my opinion) album while the film for which it’ll be the soundtrack is playing right in front of them? It takes a special kind of genius to do that.

Trading God for numbers

Miles’ recording career as a bandleader started in 1951. A hundred years earlier, many miles away from the U.S., in Göttingen, Hannover (in what later became Germany) Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) submitted his Dissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (more or less: “foundations for a general theory of functions of a complex variable”). Granted, quite far removed from funky-jazz-rock music. Nonetheless, Riemann and Miles had two things in common: genius and virtually ever-lasting legacy.

B. Riemann, source unknown.
B. Riemann, image source unknown.

Riemann’s life falls short for a few months to Miles’ recording career (well, not exactly true: Miles took a 4-year hiatus in the late 70s… but that’s rather unimportant here). Tuberculosis killed him at the age of 39. He was the son of a Lutheran pastor in Breselenz, Hannover. He moved to Göttingen with the intention of studying theology at the university there; however in that university lied the most influential mathematician of his time (and one of the most influential of all times): Carl Friedrich Gauß (anglicised “Gauss”). Riemann, having attended some mathematics lectures (in all probability, some of them given by Gauß), asked for permission of his father to switch from theology to mathematics. Perhaps theology departments around the world don’t realise the genius they lost, but we mathematicians are grateful for the change. Then in the spring of 1847 Riemann moved to Berlin; that was a Mecca for mathematical research in those days (later Göttingen became it, thanks in no small part to the influence of Riemann’s work) and he made the most of it. Being naturally disposed to mathematics he was taken under the wing of Gustav Dirichlet, one of the leading mathematicians of the time. He had a very fertile visual imagination and immediately put it to use to study functions of a complex variable. If you ever took a calculus course imagine it using complex numbers. Complex calculus is in a sense easier that the usual calculus because for a function to be “well-behaved” in the usual (termed “real”) calculus one has to ask for a lot of conditions on the function, whereas minimal conditions are required for a complex function to be “nice”. All this is rather unimportant here, but you can find out more if you’re interested; that’s why I mention it.

In his short academic career Riemann made “seminal contributions” to almost every field of the mathematics of his day. The ideas in his dissertation later became the theory of Riemann surfaces which are still studied today; he studied problems in electrodynamics and thermodynamics that led him to new mathematics, especially in the theory of conformal maps; he single-handedly created Riemannian geometry, which is the study of curvature in spaces of arbitrary dimension, ideas that became later fundamental for grounding the theory of general relativity on sound mathematics; and he lend his name to one of the most famous open problems in contemporary mathematics: the Riemann hypothesis. I want to give you a glimpse of the latter contribution.

Building the building blocks

Positive integers, or whole numbers, are familiar from everyday experience; we count with them (Spanish pun very much intended). Negative integers arise when substracting a whole number from one which is stricty lesser; we use them to represent debts. For instance, if I earned $10 and owe you $15, if I give you my $10 I still owe you $5, so I end up with -5 “coins”. Among integers there are many special families. One is the sequence of even numbers: 2, 4, 6, 8… It’s easy to generate that sequence: you take any integer and multiply it by two. Likewise, odd numbers are generated by taking any whole number, multiplying it by two and substracting 1 from the result: 1->1, 2->3, 3->5, etcetera.

Perhaps the most interesting family of integers is that of prime numbers: those that cannot be divided by any other number, save 1. For instance: 17 is prime because dividing it by any other integer doesn’t return an integer as a result, whereas 14 is not prime because 14/2=7; neither is 45 because 45/5=9. A number that is not prime is called composite; when dividing a composite number by a prime number there are two possibilities: either the result is a prime number or another composite. If the latter is the case, we can divide it further by the same or another prime number, and the same two possibilities arise… and so on. For example, 210 is composite and dividing it by 2 gives 105; dividing the latter by 3 gives 35; dividing by 5 gives 7 and 7 is prime. Put another way (and this is the crucial idea) 210=2·3·5·7 (the dot denotes multiplication). It turns out that this can be done for any number; this is the so-called fundamental theorem of arithmetic: every (positive) integer (excepting 1) is a product of primes (repetitions are allowed; for example, factor 72 into primes). In this sense, prime numbers can be considered as the building blocks for all other numbers.

To the point (finally!): whereas even and odd numbers are easy to generate, there is no straightforward way to generate prime numbers. What this means is that there is so far no “recipe” to generate prime numbers in the same way there is one to generate even numbers (take a number and then multiply it by 2).

In 1859 Riemann submitted a work to the Berlin Academy of Sciences, titled Über die anzahl der Primzahlen unter einer gegegen Grösse, which translates (roughly) into “On the quantity of prime numbers less than a given quantity”. Using ideas from complex analysis (in which he was an expert) he proposed a way for measuring how many prime numbers are less than a given number x; this quantity is usually denoted by \pi(x). The Riemann hypothesis states that this quantity is controlled by a complex function, called the Riemann-zeta function, in such a way that the zeroes of this function (that is, the points at which the function returns the value zero) predict very accurately the value of \pi(x). Rephrasing this: when x is really big the difference between the number \pi(x) and the value predicted by the Riemann hypothesis becomes very small. This is not as straightforward as multiplying a given integer by 2, but it would help us to know just how many primes are there below a given number. This is an unsolved (and VERY difficult) problem in mathematics. For more information you can go here.

Abstract legacies

Of course there is no direct relation between the life and works of Miles Davis and those of Bernhard Riemann. One can draw an analogy though: both men were unrelenting in their pursuit of the discipline that gave their lives meaning. Miles said that he didn’t like dwelling in the past, regarding his past work as “overheated turkey”; he strived for experimentation, for new ways of making music he could enjoy making. Riemann traded his passion for God–though, allegedely, remained very religious throughout his life–for a new passion: mathematics. In doing so he became one of the most important mathematicians of history; he never gave up his willing of understanding the ideas behind the problems of the mathematics of his time. In his pursuit of understanding he contributed problems and theories that keep mathematicians busy to this day. Certainly, the legacies of both men are ever-lasting, though very abstract (music is abstract in the sense that, when it is not being played, it is not clear what it is). That is, in my opinion, the sine qua non characteristic of genius: exploring a discipline so thoroughly that the exploration gives rise to innovation. My hypothesis, the Riemann-Davis Hypothesis, is that genius, though scarce, arises when someone is disciplined and curious enough to transform a field anew, disregarding current opinions or practical difficulties, proposing new ideas and developing them to their utmost consequences.

Genius is a word that is gratuitously used to express admiration or preference for someone’s work. Of course it is a matter of opinion, but whether or not you like or understand the work of Miles or of Riemann I find it very hard to deny that they are works of genius. I find them both inspiring and beautiful. I hope I’ve managed to convince you they are.

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The solitude in the middle

My quarrels with ideologies.

I’m finishing reading Hanna Arendt’s book “The origins of totalitarianism”; today I reached the final chapter. If I could I would quote the entire book, but I can’t. I’ll be quoting from the Kindle edition of the 1973 edition published by Harvest Books. Here’s the first one:

Ideologies are harmless, uncritical, and arbitrary opinions only as long as they are not believed in seriously. Once their claim to total validity is taken literally they become the nuclei of logical systems in which, as in the systems of paranoiacs, everything follows comprehensibly and even compulsorily once the first premise is accepted (p. 457).

Ideologies, according to Arendt, are then a form of Weltanschauung in which the premises are changeable but the modus cogitandi is so uncompromising that it leads to “the total denial of reality” (she of course referst to totalitarian movements, concretely the Nazis and the Bolsheviks).

A very famous (or infamous, depending on your chosen side in the “science wars”) example of the influence of ideology was given by Alan Sokal’s parody article which was accepted for publication in the journal Social Text; the nutshell of this “Sokal Affair” is that cultural-studies department would (and are, and will) be willing to accept anything for publication as long as you use the correct buzz-words, the accepted and recieved ideology of Cultural Relativism, Deconstructionism and Oppression-Structures. What these are I shall not discuss, since you can just Google those words and you’ll be directed to the chasms of language-fixated ideology.

Some may challenge my interpretation of the above systems of ideas as ideologies; they are, according to their practitioners, philosophical systems. The difference, I think, is very subtle.

A philosophical system is akin to a contextual framework from which their practitioners approach reality. For instance, Idealism is the Metaphysical position that denies the existence of an objective reality, that “everything is in our heads” (a very short, incomplete and, under a harsh light, false depiction of Idealism of course). Idealist philosophers try to make sense of the world from that perspective, which led them to conlusions such as the Cartesian Dualism: the “mind” and the “body” are two separate entities and whatever happens to the mind is independent of and irrelevant to what happens to the body. This is a very enthralling issue in the Philosophy of mind and there are many theories for it. Philosophers of mind speculate on what is the nature of Consciousness, whether or not the Human Mind is completely determined by the nervous system, and that sort of things. The Philosophy of Mind is a very attractive area of research nowadays because of the advances in neurology, neuropsychology and neurophysiology. And this is the crucial point.

Philosophers of mind (or whatever any other kind of philosophers) strive to keep track (as much as possible) of all human endeavours that are relevant to their field of study; being practitioners of the philosophical tradition inherited from the Enlightenment they try to be rational and open-minded about their system’s basic tenets; that’s the origin and the role of Philosophical Critique: reexamining assumptions, giving them new light, thinking everything anew and from the top… wir mussen denken, wir werden denken (pacem Hilbert).

Ideologues, on the other hand, regard the basic assumptions of their “system” as all-encompassing and all-explaining, as the only possible and logical way of knowing reality, for they “[claim] to possess either the key to history, or the solution to all the ‘riddles of the universe’, or the intimate knowledge of the hidden universal laws which are supposed to rule nature and man” (p. 159). So evidence and facts nonwithstanding, ideologues will always know they are right because for them there is no other possibility. Summarising: the very logical framework of an ideology doesn’t even contemplate the possibility of it being wrong, or even incomplete.

Granted, uncompromising people make revolutions possible; voting rights for women could not have been possible in England without the staunch “stubborness” of the Suffragettes at the turn of the 20th Century; Irish independence was largely a consequence of de Valera’s unwillingness to compromise with London; Richard Stallman’s unwillingness to use any kind of proprietary software is the inspiration to the many working on Open Source projects. But revolutions are consequences of unbearable and untenable situations; being staunchly uncomprossing on anything leaves little room for discussion and thus to consensus; without consensus, without the possibility of agreeing to some middle ground between the extreme positions of any issue, little can be done to preserve a civilised society.

It’s not easy maintaining a rational possition in all aspects of life; there’s always going to be people or situations that will annoy you, or make you angry, or sad… but that doesn’t mean we must take everything to heart, as if there could be no understanding, or friendship, or love between people that hold different opinions on any conceivable topic.

Having unrestricted access to information and to forums in which to vent opinions (like this one) makes things even worse: every video, every tweet, every blog is a veritable and reliable source nowadays. The unwillingness (the uncompromising refusal, if you will) to take an extreme position on anything may result on shunning or loss of friends since everyone is a scholar and an ideologue these days. But again, let us think before judging anything that lies outside of our intellectual comfort zone; and let us not take everything so seriously (not even this post).

The impossible librarian

The futile search for meaning in the digits of numbers.

Every year on the 14th day of March a lot of people celebrate π day (written Pi day). Now Pi is by far one of the most interesting numbers in all of mathematics. The reason is because it belongs to a very large class of interesting numbers and it is conjectured to belong to another very interesting class. The first class is that of transcendental numbers; the second is that of normal numbers. Whatever those two words mean, I want to state now the message of this piece: trying to find meaning in the digits of Pi makes no sense.

First things first: one of the reason there’s such a craze about Pi is because of the digits after the decimal point. Let’s recall that Pi starts like this: 3.1415926536… The ellipsis serve a very specific yet very vague purpose: they indicate that the expansion continues, but it is not possible to predict what digit will appear after the last computed digit; in other words: the sequence of digits after the decimal point is “random”. This is the key word. The sort of reasoning that makes Pi interesting for a lot of people with what I call “mystical inclinations” is the fact that randomness seem to entail the possibility of finding any possible sequence of numbers in the digits of Pi; another leap of reason leads to the following “conclusion”: if we codify Joyce’s Ulysses, or Milton’s Paradise Lost, or the Book of Psalms in King James’ Bible as a string of numbers (and there are many ways of doing this) then the digits of Pi will contain them all. Yet another leap of faith leads to the “conlusion” that any possible story, work of art, biography or musical composition will be found within the digits of Pi. This is what I propose to show that is nonsense.

Now to our first digression: transendental numbers.

Different kinds of numbers

Very early in our mathematical education we get acquainted with these old chaps: 1,2,3,4,5,… (the positive integers). The ellipsis here is very precise: we know what number comes next, just add 1 to the last one appearing in the sequence. Among the many things one can do with these numbers is dividing them:

\displaystyle \frac{1}{16}=0.0625,\quad\frac{17}{3}=5.666666\ldots,\quad,\frac{22}{7}=3.142857142857\ldots

These three examples will suffice to make my point; I want to discuss what happens after the decimal point in each of these numbers. In the first one things are as simple as can be: some digits (in this case, four) and that’s it, no more. The second case is just an “infinite” string of sixes; this means we know exactly what digit to expect after the last one: the same one. The last one repeats after 6 digits, so we also know exactly what digit will appear next: we just have to count in the repeating string. By the way: this last number is one of the first of the approximations of Pi and it dates back to Archimedes in the third century before Christ.

For the three numbers above there are knowable, exact patterns to compute the decimal expansion (that is, the string of digits after the decimal point): it is either finite or it becomes repetitive after a certain number of digits (which can be VEEEERY large; try dividing 355 by 113). It so happens that this is always possible for numbers that result from the division of two integers; they are called rational numbers. These numbers, then, are the ones for which the decimal expansion is completely determined and knowable (though it may take a long time to compute it in practice). Another way of saying this is that the decimal expansion is not random.

A number that is not rational, meaning it is not the result of dividing two integers, is called irrational. Pi is irrational. So from what we said above it follows that the decimal expansion of Pi is not determined exactly, we have to settle for some approximation. Restating this: if we are given a long string of digits of Pi it is not possible to know which digit comes next just by looking at all the digits we are given, other methods are necessary (but I won’t talk about them here). A number is irrational if its decimal expansion is infinite and non-repeating.

Another irrational number is \sqrt{2} which is approximately 1.414213. The difference between this number and Pi is that \sqrt{2} can be computed as the solution to the equation x^{2}-2=0 (which also returns its negative); Pi on the other hand is not the solution of such an equation; of course, Pi is the solution of x-\pi=0, but that is like saying something like “the blue sky is blue”; what I meant earlier is that Pi cannot be expressed as the solution of a polynomial equation with integer coefficients, such as

\displaystyle 155836x^{14}+35x^{2}-35561235=0

Meaning: equations in which powers of an unkown quantity \displaystyle x appear multiplied by only integer numbers (the sort of thing one does in an algebra class). Such numbers form the first class Pi belongs to: transcendental numbers are those which are not solution to equations with integer coefficients (like the one above, or x^3+4x^{2}-135x+1965=0).

The second class I referred to above, that of normal numbers, is a little more tricky, so here goes a second digression: probability.

Probability of things

If I put three yellow marbles and four blue ones on a bag, shake it and without peeking into it take a marble out, will it be yellow or blue? There is no way of knowing for certain the colour of the marble but one thing is certain: it is more “probable” that a blue one comes out, because there are more blue marbles than yellow ones. More precisely: there are seven marbles in total of which three are yellow and four are blue. In this case the probability of getting a yellow marble is

\displaystyle \frac{\text{number of yellow marbles}}{\text{total number of marbles}}=\frac{3}{7}=0.42857...

or about 42.8%; similarly the probability of extracting a blue one is \frac{4}{7}=0.571428... or about 57.2% (I cheated here; both numbers’ decimal expansion repeat the sequence before the ellipsis; I rounded them up in percentage so they add 100%). The conclusion is that the probabilites are not equal.

If you toss a coin there are only two possible outcomes: heads or tails, each with the same probability (50-50); if you toss the coin a few times (three, five, seven) you may not notice anything particular about how many heads and how many tails appeared; but if you toss it a very large number of times (usually around twenty or thirty will suffice) the number of tails and the number of heads are more or less the same; that is, if we toss the coin seven times and register the events: heads, tails, tails, heads, heads, tails, tails, then evidently the number of times heads appears is not equal to the number of tails; however, doing it many times both events tend to even out. Another way of saying this is that we have two events with the same probability and if we repeat the experiment (tossing) long enough, the appearance of these two events will be approximately equal, that is they will appear with the same probability. This is crucial for what follows.

An irrational number is said to be normal if each digit appears with the same probability as any other digit, namely \frac{1}{10}=0.1 because there are ten digits: 0,1,2,3,4,5,6,7,8 and 9; I’ll explain what that means below. In the numbers above (all rational) there is no chance of normality: the digit 9 never appears in them (even in the ones that have an infinite decimal expansion); so 9 occurs with probability 0 (a mathematically pedantic way of saying it doesn’t occur… with some caveats, that is not what probability zero actually means in math). Now take the following number:

0.12345678901011121314151617181920212223242526272829303132...

This is called Champernowne’s number and its just writing the sequence of positive whole numbers in order. Now, is the sequence of digits “random”? Not in the sense that we cannot predict what number follows, although for doing so we need to check all the digits preceding the one that we are interested in, which in practice is virtually impossible. However this number is normal, since all the digits will appear “equally often”. What this means is that if we take a very big chunk of the sequence of digits, every digit will appear with almost the same probability (recall the experiment of tossing the coins a large number of times); the bigger the chunk the more equal the probability of every digit will be. Of course the sequence doesn’t repeat itself so this number is irrational. This is a normal number.

Digressions over, back to Pi.

“O time thy pyramids”

So, “the randomness of the digits of Pi means that we can find anything in them”, right? Well, it depends what you mean by that. First of all, it is not known whether Pi is normal or not. Assuming it is normal, then it would follow that the digits of Pi are a random sequence of numbers; but remember, randomness only means in this sense that any digit will appear with the same probability as any other digit. Now, does this means it is possible to find “any given string of numbers” (and “therefore” any possible text, painting, symphony or Grateful Dead song) in the decimal expansion of Pi? Well, not really. In Champernowne’s number you will find it, but only because every single integer will appear there. Consider the number
0.101100111000111100001111100000\ldots
which is not normal as we defined normality, but is normal in base 2; this means that all possible binary digits (namely: 0 and 1; the pattern is: one 1, one 0, two 1s, two 0s, three 1s, three 0s…) will appear equally often in the above number; what we actually defined above was normality in base 10. Nonetheless it is not possible to find the following string in the above number:
0100011111111110001010101010100100100101000111111
So normal numbers don’t necessarily enjoy this property of “having the meaning of life” within their digital expansion.

Even granting that a normal number will contain any sequence of numbers whatsoever, the question remains: how do we read those numbers? Encoding a text, a piece of music or a painting depends on the choice of code (it can be binary, hexadecimal, scrambling letters around or you can use a set of symbols of your own); so even if you can find whatever sequence you want inside the digits of Pi (as consecutive digits, of course) that leaves you with the daunting and virtually impossible task of encoding all of world literature ever written so you can “find” it in the digits of Pi (or any other normal number). The digits of Pi, by themselves, mean nothing; they’re just numbers that appear in the decimal expansion of one of the most important and useful numbers in math and science. There is no mysticism, no cosmic meaning, no interpretation of your dreams (dry or otherwise) in that string of numbers; if there is, it is you who puts it there, not Pi. I don’t believe in that sort of thing; I don’t have a problem with people that do, but I think they should not forget that those interpretations are not a property of the numbers themselves. In the film Pi the mentor of the main character (a mathematical genius obsessed with finding patterns in the digits of Pi) at one point warns him peremptorily: “once you discard scientific rigor you are no longer a mathematician, you’re a numerologist!”. Caveat credentes.

A lot of people draw the analogy of having whatever meaning encoded in Pi’s digits from Borges’ story The Library of Babel. In it, a library is described that contains any possible text; Borges mentions a volume that is “a mere labrynth of letters” but contains the sentence “O time thy pyramids” near the end; but this was Borges, he could do that sort of thing and besides, he (nor the narrator) doesn’t draw the conlusion that the library holds meaning; actually quite the contrary… but I wouldn’t deprive you of the pleasure of reading it by yourself. The idea that by randomly concatenating every individual on a set of symbols (be those decimal or binary digits, letters of the latin alphabet or the set of nonce words in a Monty Python sketch) you can obtain any possible meaning is not a new one: Jewish mysticism invented the Kabbalah (and incidentally, the theory of permutations) many centuries ago. Trying to attach meaning to things that intrinsically have none is a sort of mysticism, whether that meaning is of a mystic/religious/occult nature or not. Mysticism by itself is harmless but it breeds fanatism and reality-denial more often than not. Again, belivers beware!

Numbers are not the guardians of perennial wisdom, nor they hold the key to past, present or future; they are devices, tools we use to understand things around us. The Librarians of Babel, if there are such beings, are not numbers.

Black, Brown and Beige, and this blog

After a very good night sleep (“night” is a mental state that, today, ended at about 12:30 in the afternoon) I read my first post. I don’t have any idea what it is about. So let’s give it another try.

First, when I wrote that first piece I was actually listening to Rachmaninoff’s Vespers. Right now I’m listening to this:

Perhaps Ellington’s most worked-over piece and definitely a jewel in 20th century music. Special attention is to be paid to Mahalia Jackson’s intervetion; religious verses which she wrote herself ex profeso for BB&B.

So about this blog: as a product of insomnia it is a way of venting anything I have in mind at any particular moment. But I also want it to be somewhat coherent, so this post is an attempt at explaining what anyone coming here should expect to read.

First of all: the title. I’m a big fan of Umberto Eco’s work. I’ve read all his novels, re-read some of them from time to time (especially Baudolino and Foucault’s Pendulum) and above all I’ve loved every piece he wrote on semiotics, politics, pop-culture… he touched almost all topics imaginable. A book which is now impossible to get (because it’s out of print) is Lector in Fabula, meaning “The reader in the story”, which is a way of saying “Speaking of the reader…” when a reader suddenly appears. The book is about the role the reader plays in giving meaning to a literary work (and thus the English title of an abriged version of the work: The Role of the Reader, Indiana U. Press, 1979).

As for the name of this blog: “once upon a time I wanted to prepend ‘Dr.’ to my name”. I wanted to do math; now I do math. In the eight years that have ellapsed since I finished my undergraduate studies and now that I’m just waiting to finally become a bloody Doctor, many things have changed, especially my views of an academic carrer: it doesn’t sound that much exciting anymore. There are A HUGE LOAD of blogs, articles and books you can read about that particular topic so let’s glaze over it. So the Doctor in the story of my life (whatever that means) is finally arriving and instead of having a parade with a marching band I have a big bag of crisps, a six-pack of beer and several (some bad, some good) news to deliver him.

I firmly belive now that Grad-Students play one of the biggest roles in giving Academic Research meaning. Consider this: you’ve got a lot of people producing and storing (mostly useless) knowledge (pace biologists, engineers and the lot; though “useful” most of what you do needs a good implementation to actually work “out there”). At some point this knowledge has to be passed on, otherwise it made no sense producing it in the first place. Hence the scientific/academic journals and books, congress memoirs, etc. Those are written (mostly) for specialists; the General Public has little idea (and interest) in such things. A Masters or Ph.D. student, on the other hand, has an obligation of reading a substantial (but by any means large) part of that literature in order to produce some more of the same. Without the certainty that there will always be Mathematicians, Philosophers, Biologists, Sociologist, LitCrits and the lot of academics, producing this awsome amount of knowledge (alledgelly, all of it relevant to the subject matter) makes little sense. The production of knowledge presupposes that there will always be need for more of it. Teaching all this specialised and highly-technical knowledge to the General Public is a daunting task; most of them will understand mostly nothing of it. NOT BECAUSE THEY’RE NOT SMART ENOUGH, mind you, but because understanding the relevant concepts, ideas and methods of all that amount of knowledge TAKES A LOT OF BLOODY TIME (hence the fossils on Grad School). Including cutting-edge research into Primary, Secondary or Higher Education is thus very unlikely. The solution: training people that will become the new specialists in the field, thus securing the continuity of it. Q.E.D.: Grad Students perpetuate Academia. Of course, a lot of considerations are absent and this is not the whole story, but I am sure the argument stands. And hence the title of this blog.

So I’m a “Mexican Mathematician and Aspiring Philosopher”. Naturally I’m interested in the philosophy of mathematics; I’m also interested in hermeneutics and its possible interaction with semiotics; so you’ll probably find stuff about those topics in here. Also stuff about mathematics, whether it is a nice theorem or an interesting concept. My “area of expertise” is differential geometry, which studies geometric “figures” (in a very broad sense) using differential calculus; I’m also very interested in logic but not as a foundational issue… so I’ll very likely write about that also.

I consider myself a rational person (save when I’m hungry or haven’t had my morning cup of coffee) and I dislike ideologies, in the sense of a set of ideas that pretend to be all-encompassing and all-explaining (racism, postmodern feminism, marxism, etc); I consider myself to be “in a middle lane” of sorts; a lot of people (especially subscribers to any of the above ideologies, or any other for that matter) consider this to be a clear lack of conviction; I think not. It requires, in my opinion, the conviction that rational arguments trump conspiratory elucidations (“the inherent different of races”, “patriarchy”, “the burgoise oligarchy”, etc) against which there is no rational defense precisely because they aspire to be all-explaining… anyway, I’ll probably write a lot about that, having in mind that “one must not work out a definitive, concluded system, like a piece of architecture, but a sort of mobile system […]” (Eco, “In praise of Saint Thomas”, in “Travels in Hyperreality”, Harcourt, 1986).

As any other person there are authors, musicians, artists and lots of other people I admire a lot; also I have pet-peeves, favourite books, films, stories, etc. They’ll parade through here as well. Especially jazz, choral music and the poetics of the popular Mexican song, a topic very close to my heart. I also like films.

And that’s about it, save the Latin motto “Quid non intellegit aut tasceat aut discat”: that which you don’t understand either be silent of or find out about. So as Mahalia Jackson finishes singing her praises to the Lord Above I must go back to my math. And hopefully I will get a proper sleep tonight. Cheers!

Vespers (or too late to be writing)

Insomnia breeds blogs.

I should be sleeping; I should’ve been writing but instead I spent some time re-reading Logicomix; as the name suggests, it’s a comic book about logic. It is actually a comic book about Bertrand Russell speaking about his quarrels with logic, the foundations of mathematics, certain knowledge and life in general. But this is not a praise of Logicomix. It’s rather a praise of Rachmaninoff’s Vespers.

The Vespers are the evening prayers in most Christian lithurgies. It is the most “popular” canonical hour because of course it is the end of the labour day. Nevertheless Rachmaninoff’s Vespers (opus 37) is an a-cappella setting of an All-Night Vigil, which is exactly what it sounds like: staying up all night, praying-in this case-to the Virgin Mary.

I’m not a Christian. What I am right now is insomniac, waiting for the approval of my thesis so I can defend it, and a little worried about life in general and my life in particular. My worries are mundane: what to do next? Is an academic carrer worth the shambles it brings about? The usual stuff a finishing Ph.D. student worries about. But I digress. I’m not a religious person but nevertheless I find compositions like Rachmaninoff’s Vespers truly beautiful. You must listen to it in order to understand:

Staying up all night doing something as praying sound to many-a-one like a complete waste of time. Praying-they’d say-is futile and superstitious. I agree, more or less. I don’t pray. But I do other stuff, like staying up late reading and/or writing Mathematics; reading-and trying to write-Philosophy; cleaning my room (yes, I’m most productive at night; I think most grad-students can relate). All this sounds also as a complete waste of time. Those activities-among others-are my prayers.

Prayer is a fundamental part of monastical work-life; for us Ph.D.s-to-be our work is the only part of our monastical life… well, not quite monastical in the strict sense (we’re far from saints); I mean a life dedicated to wasting time. Speaking of which, insomnia is also a waste of time and the reason I started this blog. Hurrah for insomnia (?).

Another thing that keeps me awake lately is G.H. Hardy’s Ramanujan. It’s not a biography of Srinivasa Ramanujan, but rather “twelve lectures on subjects suggested by his life and work” (official subtitle of the book). If you don’t know who Hardy or Ramanujan were, you certainly are not a mathematician (or math-aficionado). Suffice it to say that they were two of the most accomplished mathematicians of the beginning of the 20th century. More precisely, what keeps me awake in that book lately is the following series:

\displaystyle\frac{2}{\pi}\left(\frac{2}{B_2}\frac{\log x}{2\pi}+\frac{4}{3B_{4}}\left(\frac{\log x}{2\pi}\right)^{3}+\ldots\right)

which is a formula for approximating… but it doesn’t matter. I’m a mathematician, I study things like the above… well, not quite; it’s not a book of my area so perhaps that’s one of the reasons it keeps me awake. I find Hardy’s book fascinating precisely because I understand very little of it. And thus I arrive to my point (if there is one): we spend a lot of time doing very useless things because we find them challenging, fun, interesting and, above all, meaningful; perhaps “fulfilling” would be a better term. Mathematics is all of the above, and much more… at least to us mathematicians.

My point is the following: we mathematicians (and most people doing research in academia) find what we do full of meaning; some feel that this meaning resides in the possibility of applying mathematical knowledge to understand reality; some others think it is a “knowledge-for-knowledge-sake” kind of thing; yet others insist that some mathematics is not only useful but that ALL mathematics is eventually useful… there’s an opinion to suit anybody’s need. Disregarding the whys and the hows it is a fact that we all find what we do meaningful, with “meaning” to be specified by each mathematician. My own take is that math is a very effective kind of knowledge, not in its form but in its “method”; for me it came a moment in which the world started to make sense; I don’t claim I can understand anything you put in front of me, but doing a Ph.D. in math has helped me to strive for clarity, to go to sources, to ask whatever I don’t understand, and (perhaps more importantly) to find out whom to ask, where to find, what kind of sources to consult. I wanted to be a philosopher; that didn’t work out, I couldn’t even start (perhaps some day I’ll write about that). Now I am a mathematician, by trade if not by title; as such I can now say what kind of problems in Philosophy interest me (perhaps I’ll also write about that) and I can more securely find my way through the vast web of knowledge that is Philosophy (and yes, also Mathematics). Also now I know I can do other stuff, not only math. But the path to clarity is filled with turmoil and at some point you just want to end it all (see, for instance this excellent article about the turmoil of doing a Ph.D.) So whatever comes next… let’s see what happens.

Bertrand Russell spent decades trying to use logic to set mathematics on a rock-solid footing thereby dissipating all possible doubts about its validity… which were only noticed by people working on the foundations of mathematics. A very good friend of mine (Mark Spivakovsky) offered this analogy:

Logicians (and people who work on foundations of mathematics) are akin to a society of spiders living in the basement of a very old yet very stout castle; these spiders are no ordinary: their cobwebs are quite fantastic in shape and complexity; these spiders are convinced that it is their intrincate cobwebs that keep the castle in place and standing, so everytime a strong wind blows their webs they hurriedly and painstakingly start weaving them again so the castle doesn’t reduce to rubble.

Indeed, at the turn of the 20th century mathematics was perhaps at the peak of its performance so far; very few working mathematicianas bothered to even study the most basic questions of foundations. They thought: “if it ain’t broken, why fix it?”. So why did Russell et al. spent so much time weaving mathematics from the bottom up? Because they found it compelling, meaningful, challenging, fun (maybe)… just like I like reading a book about number theory even when I have near to zero knowledge of it; or just as people are willing to stay awake until late praying to the Virgin Mary. Well, if that vigil includes Rachmaninoff’s Vespers count me in.