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After an interesting conversation with somebody recently I was looking around and aggregating simple mathematical facts that have somewhat crazy proofs. Like for all such questions, I found a great MathOverflow thread and decided to share this gem from there:

Fact: \sqrt[n] {2} is irrational for any integer n \geq 3.

Proof: Suppose it is not. Then \displaystyle \sqrt[n] {2} = \frac{p}{q}, then 2 q^n = p^n , or p^n = q^n + q^n contradicting Fermat’s Last Theorem.

Although a commenter there mentions that the argument is essentially circular (which I find fascinating), but other than that it made me laugh, what I find interesting about this answer and an accompanying comment by Greg Kuperberg is that it made me realize that Fermat’s Last Theorem is not strong enough to imply the irrationality of \sqrt{2}.

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Not only is it not right, it’s not even wrong! – Wolfgang Pauli

I just found a delightful joke concerning Pauli while rummaging through my email, thought it was worthy of sharing!

The phrase “Not Even Wrong” ofcourse was famously coined by Wolfgang Pauli, who was known to be particularly acerbic to sloppy thinking. The wiki entry for the phrase has the following story on how it originated. Rudolf Peierls writes that “a friend showed Pauli the paper of a young physicist which he suspected was not of great value but on which he wanted Pauli’s views. Pauli remarked sadly,”Not only is it not right, it’s not even wrong!”

Coming to the email which centers around being “Not even wrong”:

Wolfgang Pauli

Exactly, Pauli could be pretty scathing in his reviews. Visiting physicists delivering a presentation would dread seeing him in the audience. Pauli would sit and listen and scowl, arms crossed, and shake his head. The faster he shook his head, the more he disagreed with you.

The joke goes that when Pauli died he asked God why the fine structure constant has the value 1/(137.0) … God went to a blackboard and began scribbling equations. Pauli soon started shaking his head violently…

Note: I didn’t write this but apparently I read it somewhere a few years ago and mailed it to somebody. I googled for parts of it, but couldn’t locate the source. If you happen to know, then please link me up!

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I presume that a lot of people who drop by this blog are familiar with Doron Zeilberger‘s opinions already. Even though a lot of people who know me personally get linked frequently to some or the other opinions of Zeilberger, I thought it would be a good idea to blog about them in any case, for I believe more people should know about them, even if the number is not high enough.

For a one line introduction, Doron Zeilberger is a noted Israeli mathematician who is presently a professor at Rutgers. He maintains a “blog” which has the title “Dr. Z’s Opinions” in which there are an assortment of views on topics broadly related to Mathematics. Zeilberger certainly has a flair for writing and oftentimes makes hard-hitting points which might outrage many (his latest writing on Turing for example is sure to make many people shake their heads in disagreement – me included) which usually could be seen as chipping away at commonly held opinions. All the interestingness about his opinions aside, his sense of humour makes them entertaining in any case. Even if one disagrees with them I would highly recommend them as long as one exercises some discretion in sifting through these Indiscrete Thoughts.

I found his opinions many years ago while searching for something witty about weekly colloquiums which I could send to some of my colleagues who somehow took pride in not going for them. Skipping colloquiums is a habit that I have not understood well. He wrote the following about it (Opinion 20):

Socrates said that one should always marry. If your spouse would turn out to be nice, then you’ll be a happy person. If your spouse would turn out to be a bitch/bastard, then you’ll become a philosopher.

The same thing can be said about the weekly colloquium. If the speaker is good, you’ll learn something new and interesting, usually outside your field. If the speaker is bad, you’ll feel that you have accomplished something painful, like fasting, or running a marathon, so while you may suffer during the talk, you’ll feel much better after it.

What prompted me to blog about his “blog” was a recent opinion of his. Some months ago when Endre Szemeredi won the Abel Prize, I got very excited, almost like a school boy and the next morning I went to the college library to see what the national dailies had to say about the achievement. To my surprise and dismay none of the dailies seem to have noticed it at all! Three or four days after that the New York Times carried a full page advertisement by Rutgers University having a great photo of Szemeredi, however that doesn’t count as news. I was delighted to see that Doron Zeilberger noticed this too and wrote about it (see his 122nd Opinion)

Let me conclude by wishing Endre, “the computer science professor who never touched a computer”, many more beautiful and deep theorems, and console him (and us) that in a hundred years, and definitely in a thousand years, he would be remembered much more than any contemporary sports or movie star, and probably more than any living Nobel prize winner.

One of my all time favourite opinions of his is Opinion 62, which compares the opposing styles of genius of two men I have had the highest respect for – Israel Gelfand and Alexander Grothendieck. I often send it to people who I think are highly scientifically talented but somehow waste time in expending energy in useless causes than trying to do science (especially if one doesn’t have an intellect comparable to some fraction of Grothendieck’s)! I take the liberty of reproducing the entire opinion here –

I just finished reading Allyn Jackson’s fascinating two-part article about the great mathematical genius Alexandre Grothendieck (that appeared in the Notices of the Amer. Math. Soc.) , and Pierre Cartier’s extremely moving and deep essay `Une pays dont on ne conaitrait que le nom: Le “motifs” de Grothendieck’. (that appeared in the very interesting collection “Le Reel en mathematiques”, edited by P. Cartier and Nathalie Charraud, and that represents the proceedings of a conference about psychoanalysis and math).

In Pierre Cartier’s article, in addition to an attempt at a penetrating “psychoanalysis” he also gives a very lucid non-technical summary of Grothendieck’s mathematical contributions. From this it is clear that one of the greatest giants on whose shoulders Grothendieck stood was Israel Gelfand, whom I am very fortunate to know personally (I am one of the few (too few!) regulars that attend his weekly seminar at Rutgers). I couldn’t help notice the great contrast between these two Giants, and their opposing styles of Genius.

Myself, I am not even an amateaur psychoanalyst, but motives and psi aside, I can easily explain why Grothendieck stopped doing math a long time ago (hence, died, according to Erdos’s nomenclature), while Gelfand, at age 91, is as active and creative as ever.

First and foremost, Grothendieck is a dogmatic purist (like many of the Bourbakists). He dislikes any influences from outside mathematics, or even from other subareas of math. In particular, he always abhored mathematical physics. Ironically, as Cartier explains so well, many major applications of his ground-breaking work were achieved by interfacing it with mathematical physics, in the hands of the “Russian” school, all of whom were disciples of Gelfand. As for Combinatorics, forget it! And don’t even mention the computer, it is du diable. As for Gelfand, he was always sympathetic to all science, even biology! In fact he is also considered a prominent theoretical biologist. Gelfand also realizes the importance of combinatorics and computers.

Also people. Grothendieck was a loner, and hardly collaborated. On the other hand, Gelfand always (at least in the last sixty years) works with other people. Gelfand is also very interested in pedagogy, and in establishing math as an adequate language.

Grothendieck spent a lot of energy in rebellious political causes, probably since in his youth he was an obedient bon eleve. On the other hand, Gelfand was already kicked out of high-school (for political reasons), so could focus all his rebellious energy on innovative math.

So even if you are not quite as smart or original as Gelfand and Grothendieck (and who is?), you will still be able to do math well into your nineties, if you follow Gelfand’s, rather than Grothendieck’s, example.

Zeilberger also seems to have a lot of respect for G. J. Chaitin, something that I certainly find very interesting. I mention this because I have been reading and re-reading Chaitin these days, especially after discovering some of his very recent work on meta-biology.

PS: Zeilgerber was featured in a BBC Documentary on infinity (not a great one, though) in which he talked about his ultrafinitist viewpoint.

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An Overcoming

A Random Post on silence.

The Silence that lives in Houses, Matisse (1947)

One should speak only when one may not remain silent; and then speak only of that which one has overcome—everything else is chatter, “literature,” lack of breeding. My writings speak only of my overcomings: “I” am in them, together with everything that was hostile to me, ego ipsissimus, indeed, even if a yet prouder expression be permitted, ego ipsissimum.

Nietzsche – Maxims & Opinions (1886)

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The solution of the problem of life is seen in the vanishing of the problem. (Is not this the reason why those who have found after a long period of doubt that the meaning of life became clear to them have been unable to say what constituted that sense?) (6.521)

Whereof one can not speak, one must pass over in silence (7)

Wittgenstein – Tractatus Logico Philosophicus (1922)

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You need not leave your room. Remain sitting at your table and listen. You need not even listen, simply wait, just learn to become quiet, and still, and solitary. The world will freely offer itself to you to be unmasked. It has no choice; it will roll in ecstasy at your feet.

Kafka – Aphorisms (1918)

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A mildly personal post.

The title does not imply that the lines quoted below correspond to the exact origin of Kolmogorov Complexity, though they are related and give away the essence.

Andrey Kolmogorov

Information theory must precede probability theory and not be based on it. By the very essence of this discipline, the foundations of information theory have a finite combinatorial character.

Andrey Kolmogorov

With my background in Electrical Engineering I had the opportunity to take courses in Information Theory and Coding which made the idea of Shannon’s Information Theory quite familiar. But there was a time when I had enough background to started noticing conversations that were perhaps relegated to the background before. Simply because I didn’t know enough to make any sense of them and hence these conversations were more or less noise to me. But these happened to be on Kolmogorov Complexity. I hadn’t sat down and studied it. But had been reading articles here and there that mentioned it even with ideas such as The Godel Incompleteness theorems and the Halting Problem. It created the impression that this area must be fundamental but not clearly why.

And then I came across the above rather cryptic lines by Kolmogorov. Used to the idea of entropy (defined in terms of probability) as information, they made my brain hurt. I spent a couple of days thinking about them and suddenly I realized WHY it was so fundamental. And things started making more sense. Ofcourse I didn’t know anything about it as such, but the two day thinking session convinced me enough, that in a sense it was as fundamental as calculus for me given the things I was interested in (along with Shannon‘s and Fisher’s ideas). It also convinced me enough to want to know more about it no matter what projects I was involved in and immediately bought a book that I have been trying my way through as an aside to what I have been working on (linked below).

I find such insightful one liners that happen to cause almost a phase transition or a complete change in the way you look at some thing (information theory in this case) quite remarkable, making the new view very beautiful. Ofcourse there is a “right” time for them to occur but this was certainly one of them. The lines below had an auxiliary effect too:

The applications of probability theory can be put on a uniform basis. It is always a matter of consequences of hypotheses about the impossibility of reducing in one way or another the complexity of the descriptions of the objects in question. Naturally this approach to the matter does not prevent the development of probability theory as a branch of mathematics being a special case of general measure theory.

The concepts of information theory as applied to infinite sequences give rise to very interesting investigations, which, without being indispensable as a basis of probability theory, can acquire a certain value in investigation of the algorithmic side of mathematics as a whole.

– Andrey Kolmogorov (1983)

While the above was a more personal story there are many other famous examples of cryptic one liners changing a view. Here’s a famous one:

A Famous Cryptic Comment:

Robert Fano

I remember reading a story about the great mathematician and electrical engineer Robert Fano. Around the same time the father of Cybernetics, Norbert Wiener was at MIT and was famous at the time for wandering around campus and then talking to anybody about anything that caught his fancy. There are stories on how graduate students would run away when Wiener was sighted coming to save their time. Wiener’s eccentricities are famous (recommendation [2] below) but let me not digress. In one of his routine days he appeared in the office of Fano and made a cryptic comment:

You know, information is entropy.

Fano spent a good time thinking about what this might mean and he has himself remarked that it was in part responsible for his developing, completely independently the first law of Shannon’s theory. Claude Shannon even cited Fano in his famous paper.

I can’t help thinking that such one liners are perhaps the best examples of information compression and Kolmogorov complexity.

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Recommendations:

1. An Introduction to Kolmogorov Complexity and its Applications – Ming Li and Paul Vitanyi (on the basis of the first third)

2. Dark Hero of the Information Age – Conway and Siegelman

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Earlier Related Post:

1. Ray Solomonoff is No More (has a short discussion on Solomonoff’s ideas in the same. It is noteworthy that Solomonoff published the first paper in what is today called Kolmogorov Complexity. His approach to the area was through Induction.  Kolmogorov and Chaitin approached it from randomness).

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Hello Uncle Erdős!

Not very long ago I wrote rather enthusiastically about Paul Erdős. While Erdős  has inspired me since my high-school days, I never really thought I could have an Erdős Number of 2 or 3. Hence it was a pleasant surprise when it was pointed to me that the acceptance for a recent paper for publication would get me an Erdős number of 2! This paper has now been accepted. And though it will take a while to appear on the AMS collaborative distance page given the time it takes to get published, it is something that got me pretty excited last month!  This paper on Graph Clustering was written with Dr Gábor Sárközy. He wrote this paper with Erdős . Dr. Sarkozy is also the son of András Sárközy, it is noteworthy that Prof. András Sárközy wrote 62 papers with Erdős, the maximum by anyone.

My Erdős number is unlikely to drop further, unless this happens (via XKCD):

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This is a first for this blog, and hence worth mentioning.

I came across a paper that is to appear in the proceedings of the IEEE Conference on Computer Systems and Applications 2010. Find the paper here.

This paper cites an old post on this blog, one of the first few infact. This is reference number [2] on the paper. It was good to know, and more importantly, a boost to blog to discuss small ideas that are otherwise improper for a formal presentation.

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Since it is lame to write just the above lines, I leave you with a couple of talks that I watched over the friday night and I would highly recommend.

There was a talk by Machine Learning pioneer Geoffrey Hinton some years ago at Google Tech Talks that became quite a hit. This talk was titled The Next Generation of Neural Networks that discusses Restricted Boltzmann Machines, and how this generative approach can lead to learning complex and deep dependencies in the data.

There was a follow up talk recently, that I had long bookmarked, but just got around to seeing yesterday. This like the previous is a fantastic talk that has completed my conversion to begin exploring deep learning methods. :)

Here is the talk –

Another great talk that I had been looking at last night was a talk by Prof Yann LeCun

Here is the talk –

This talk is started by the late Sam Roweis. It feels good at one level to see his work preserved on the internet. I have quite enjoyed talks by him at summer schools in the past.

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