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Recently, I have been contemplating restarting to blog. I started this blog in the naive enthusiasm of an undergrad 10 odd years ago, and slowly both grew out of it, and also felt quite embarrassed of the older posts (although I resisted the temptation to delete it altogether). My coping mechanism was to pretend it did not exist. Now I feel I might want to blog a little bit more. So till I figure out how to go about it, whether to keep it focused on a narrow stream of topics, or expanding it out a little bit more, or to abandon this old piece altogether and start anew, I’d share this intriguing reflection on tea and coffee by Ernst Jünger from Albert Hoffman’s bio that I had been reading recently (while of course sipping green tea).

“What interested me above all was the relationship of these substances to productivity. It has been my experience, however, that creative achievement requires an alert consciousness, and that it diminishes under the spell of drugs. On the other hand, conceptualization is important, and one gains insights under the influence of drugs that indeed are not possible otherwise. I consider the beautiful essay that Maupassant has written about ether to be such an insight. Moreover, I had the impression that in fever one also discovers new landscapes, new archipelagos, and a new music, that becomes completely distinct when the “customs station” [“An der Zollstation” [At the custom station], the title heading of a section in Das Abenteuerliche Herz (2d ed.) that concerns the transition from life to death.] appears. For geographic description, on the other hand, one must be fully conscious. What productivity means to the artist, healing means to the physician. Accordingly, it also may suffice for him that he sometimes enters the regions through the tapestries that our senses have woven. Moreover, I seem to perceive in our time less of a taste for the phantastica than for the energetica—amphetamine, which has even been furnished to fliers and other soldiers by the armies, belongs to this group. Tea is in my opinion a phantasticum, coffee an energeticum—tea therefore possesses a disproportionately higher artistic rank. I notice that coffee disrupts the delicate lattice of light and shadows, the fruitful doubts that emerge during the writing of a sentence. One exceeds his inhibitions. With tea, on the other hand, the thoughts climb genuinely upward.

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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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I am in the process of winding up taking a basic course on Bio-Informatics, it is offered as an elective subject for final year under-graduate Information Technology students.  I preferred taking this course as a visiting faculty on weekends as managing time in the week is hard (though i did take some classes on weekdays).

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nbt1205-1499-F1[Gene Clustering : (a) shows clusters (b) uses hierarchical clustering (c) uses k-means (d)  SOM finds clusters which are arranged in grids. Source : Nature Biotechnology 23, 1499 – 1501 (2005) by Patrick D’haeseleer]

Why Bio-Informatics?

The course (out of the ones offered in Fall) I would have preferred taking the most would have been a course on AI. There is no course on Machine Learning or Pattern Recognition at the UG level here, and the course on AI comes closest as it has sufficient weight given to Neural Nets and Bayesian Learning.

The only subject that comes nearest to my choice as AI was not available, was Bio-Informatics as about 60 percent of the syllabus was Machine Learning, Data Mining and Pattern Recognition. And it being a basic course gave me the liberty to take these parts in much more detail as compared to the other parts. And that’s exactly why taking up Bio-Informatics even though it’s not directly my area was not a bad bargain!

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The Joys of Teaching:

This is the first time that I have formally taken a complete course, I have taken work-shops and given talks quite a few times before. But never taken a complete course.

I have always enjoyed teaching. When I say I enjoy teaching, I don’t necessarily mean something academic. I like discussing ideas in general

If I try to put down why I enjoy teaching, there might be some reasons:

  • There is an obvious inherent joy in teaching that few activities have for me. When i say teaching here, like I said before I don’t just mean to talk about formal teaching, but rather the more general meaning of the term.
  • It’s said that there is no better way to learn than to teach. Actually that was the single largest motivation that prompted me to take that offer.
  • Teaching gives me a high! The time I get to discuss what I like (and teach), I forget things that might be pressing me at other times of the day. I tend to become a space-cadet when into teaching. It’s such a wonderful experience!
  • One more reason that i think i like teaching is this : I have a wide range of reading (or atleast am interested in) and I have noticed that the best way it gets connected and in most unexpected ways is in discussions. You don’t get people who would be interested in involved discussions very often, also being an introvert means the problem is further compounded. Teaching gives me a platform to engage in such discussions. Some of the best ideas that I have got, borrowing from a number of almost unrelated areas is while discussing/teaching. And this course gave me a number of ideas that I would do something about if I get the chance and the resources.
  • Teaching also gives you the limits of your own reading and can inspire you to plug the deficiencies in your knowledge.
  • Other than that, I take teaching or explaining things as a challenge. I enjoy it when I find out that I can explain pion exchanges to friends who have not seen a science book after grade 10. Teaching is a challenge well worth taking for a number of reasons!

From this specific course the most rewarding moment was when a couple of groups approached me after the conclusion of classes to help them a little with their projects. Since their projects are of moderate difficulty and from pattern recognition, I did take that up as a compliment for sure! Though I can not say I can “help” them,  I don’t like using that word, it sounds pretentious, I would definitely like to work with them on their projects and hopefully would learn something new about the area.

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Course:

I wouldn’t be putting up my notes for the course, but the topics I covered included:

1. Introduction to Bio-Informatics, Historical Overview, Applications, Major Databases, Data Management, Analysis and Molecular Biology.

2. Sequence Visualization, structure visualization, user interface, animation verses simulation, general purpose technologies, statistical concepts, microarrays, imperfect data, quantitative randomness, data analysis, tool selection, statistics of alignment, clustering and classification, regression analysis.

3. Data Mining Methods & Technology overview, infrastructure, pattern recognition & discovery, machine learning methods, text mining & tools, dot matrix analysis, substitution metrics, dynamic programming, word methods, Bayesian methods, multiple sequence alignment, tools for pattern matching.

4. Introduction, working with FASTA, working with BLAST, filtering and capped BLAST, FASTA & BLAST algorithms & comparison.

Like I said earlier, my focus was on dynamic programming, clustering, regression (linear, locally weighted), Logistic regression, support vector machines, Neural Nets, an overview of Bayesian Learning. And then introduced all the other aspects as applications subsequently and covered the necessary theory then!

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Resources:

All my notes for the course were hand-made and not on \LaTeX, so it would be impossible to put them up now (they were basically made from a number of books and the MIT-OCW).

H0wever I would update this space soon enough linking to all the resources I would recommend.

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I am looking forward to taking a course on Digital Image Processing and Labs the next semester, which begins December onwards (again as a visiting instructor)! Since Image Processing is closer to the area I am interested in deeply (Applied Machine Learning – Computer Vision), I am already very excited about the possibility!

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I picked up these images at Wired two days ago and just could not fit in the time to put them up earlier.

There is a remarkable quote by Einstein

“Two things are infinite: the universe and human stupidity; and I’m not sure about the the universe.”

It isn’t definite to me if I liked this quotation earlier. But I am NOW wholly convinced that I love it. And this new found unequivocalness for it is due to the following:

As a kid, I used to have a large collection of encyclopedias. I remember reading about the Aral Sea in the picture atlas, and that it mentioned that there was increasing salination of the sea water and that it would disappear in some decades.

Over the years that time, we were fed with doomsday scenarios all the while. Like all the coastal cities would be soon under sea due to rising ocean levels, and that the Himalayas would soon be ice free etc etc. Over a period of time you get fed up with such idle talk and since you don’t see anyone giving convincing answers, you tend to believe that nothing like that is true. Secondly, the eternal optimist that I am, I just probably wished that what that encyclopedia said about the Aral was some “minor” problem.

I last read about the problem many years ago and after that never came across anything on it. And just a couple of days back was shocked by these images. I have only one word for them : Tragic!

The images are from 1973, 1987, 1999, 2006 and 2009. The two recent images were released by the European Space Agency, the earlier ones were taken by the United States Geological Survey.

Aral Sea - 1973

Aral Sea - 1973

Aral Sea - 1987

Aral Sea - 1987

Aral Sea - 1999

Aral Sea - 1999

Aral Sea - 2006

Aral Sea - 2006

Aral Sea - 2009

Aral Sea - 2009

[Image(s) Source : Wired Science]

The South Aral Sea, the remnant of the original lake that you can see to your left on the above image is also expected to vanish by 2020, thankfully the North Aral sea (the part on the right) has been saved due to a world bank funded dam project.

The Aral sea, once the world’s fourth largest lake at roughly around 68,000 sq kms is now just about one-tenth that size. The trouble started when it was decided by the Soviets in 1918 that the two rivers that drained into the Aral – The Amu Darya and Syr Darya would be largely diverted to the deserts to develop them into cotton growing lands. The Soviet plan worked and cotton became one of the most important exports from that area. By the 1960s massive amount of water was being diverted and the sea began to shrink steadily. And how that happened is spoken out loud by the pictures.

The death of the Aral is extremely sad. It’s death has left it’s once thriving fishing industry destroyed, the diverting of the rivers has mostly reduced the two rivers to a shadow of their former selves. The Aral served as a climate moderator in the largely arid lands there, it’s death might herald major environmental catastrophe in the region.

This is a prime example of what human stupidity could lead to and leaves me short of words to describe my anguish at the same.

It has a number of things to say:

Ignoring warnings which have clear proof is just plain stupidity. There is ample proof for example of climate change and its bad impact. For example, I have been visiting the Himalayas once every few years since 1991. And the change there is apparent, as compared to the 80s the glaciers that make up the Ganges have shrunk by several kilometers. I don’t know what the solutions are, nor am I comparing the Aral problem with it. I understand that the Aral was a different kind of a problem. Different because it was known to the Soviets that the lake would dry up from the start. Climate change can not be compared to it as we do not yet fully understand a number of things about it, so how effective the correctives would be is debatable. It would be for our good if that debate is settled soon with good and incisive scientific evidence.

It also is a comment on how totalitarian regimes can be dangerous. In such regimes, since a decision taken can not be opposed, such a decision could either lead to major dividends/progress as it would be implemented very rapidly or major catastrophe as was in the above case.  Soviet officials were aware that the Aral would sooner or later evaporate. In 1964 Aleksandr Asarin noted that :

“It was part of the five-year plans, approved by the council of ministers and the Politburo. Nobody on a lower level would dare to say a word contradicting those plans, even if it was the fate of the Aral Sea.”

Ofcourse he was right, there is rarely any way to convince or reason with or oppose supercilious totalitarian regimes even if their decisions are clearly suicidal. I am tempted to make a political comment on two present day countries (one a totalitarian state and one a liberal democracy) here, but would avoid the temptation.

Anyhow, the images above disturbed me enough to lose sleep.

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