Why I am Not religious

My Religion

This is a special way of being afraid

No trick dispels. Religion used to try,That vast moth-eaten musical brocade
Created to pretend we never die,
And specious stuff that says No rational being
Can fear a thing it will not feel, not seeing

That this is what we fear—no sight, no sound,

No touch or taste or smell, nothing to think with,

Nothing to love or link with,

The anaesthetic from which none come round. - Philip Larkin

I have read the life stories of many remarkable men, but none has struck me as forcefully as the life of Blaise Pascal, the 17th century French polymath. His was a peculiar case of supreme rational thinking coexisting with a fervent belief in mysticism to such a high order that this union was never observed in such intensity in anyone else including in the life of remarkable Dutch jurist and theologian Emanuel Swedenborg. Pascal was afflicted with myriads of physical ailments which ranged from mild to extreme, and as is the case for any genius tormented by physical ailments, his highly developed mental faculties may have had effects on his body and vice versa. Why am I digressing to the case of Blaise Pascal is that his case is one of the most singular exceptions in my knowledge, of how a supreme rational thinker embraced irrational ideals with a blind conviction. Isaac Newton was also a believer, but as far as we know from his life, he also had reasoned the non existence of a holy Trinity and in most modern ways would have been termed more a rational naturalist than a religious man. 

The likes of Pascal, Swedenborg, Newton, who can all be termed as believers, are rarities in the 21st century scientific world. I certainly think that any thinker of distinction in the current times, who may be a practicing religious believer, cannot escape from the logical inconsistencies and factual errors faced when confronting uncomfortable meta religious questions. As Nobel laureate Franck Wilczek in his insightful way has clarified, his earlier love for Catholicism naturally could not compete or substitute the sort of universal ideas such as Infinite Space, Infinite Energies, Time that Physics throw at its devotees in abundance, and therefore in no time with his questioning mind, he lost his religious faith. 

Then naturally a question arises, is it possible for a reasonable person to believe in the god as much as in rational science? Or to rephrase it, is it really possible for any one who consider themselves rational thinkers to believe in god and at the same time disbelieve in occult studies or astrology when at the base both ideas are erected on the edifice of unreasonable belief? My reasoned answer to both questions is a qualified and emphatic NO.

As children growing up in India in a Hindu culture household we were definitely religious in the conventional way where I truly believed in my prayers and thought without an iota of doubt that my praying to unseen spirits or gods so would result in something remarkable beyond normality. This was unreasonable belief in something, be it an amorphous entity such as God as much as the notion that the world is populated by good people. I certainly look back with dread at the innocent way we children used to approach elders with immense respect and duty bound listened to them without asking questions, and to think of the way some of them in the faintest sense would have wanted to take advantage of our innocence. 

There sure was a sense of excitement in praying for impossible with fervent faith. I was afflicted with brief asthmatic episode in my childhood days. We being a religious tolerant family had images of a Christian saint in the prayer corner, who was supposed to work miracles to cure disease if one prays hard enough with faith. I did just that for days on end, praying everyday for this terrible constriction in my breathing to go away so that I can go out and play and enjoy. There was s sense of pure delight when one realizes that disease has gone away and is bound to thank the divine intervention the saint has worked with God on my behalf. Also very mundane things like getting a ticket as last guy for a movie that well wanted to see etc were exciting ventures of my praying days. That was a phase of life everyone goes through with some good reason. Primitive men feared rain, thunder bolts and sun before reason gave them the tools to understand the control these entities. 

This was also before reason was absent in our dominant mode of thinking and we used to see world through given accepted facts guided by our unreasoned beliefs, often coming from our elders or books. I was religious in the conventional way throughout my early college life. When I look back, I think that the barrier to get rid off belief in an unnecessary being as the supreme doer of everything good in the world and praying before the entity to let us live good life, was a faint fear of being called non conformist and also self doubt in ones' rationality. 

I have read about a class of reluctant atheists, whose whole thinking stems from religious ideas, they sort of are in atheist camp but cannot come to terms with atheism at its fullest and still cling onto religious ideals as a moral support. The recent book by Alain De Bouton is a case in example for this sort of atheism. I am the sort of atheist that Steven Weinberg is, the dyed in wool rationalist materialistic atheist, who cannot reason or believe in any "Ghost in the machine" behind the working of universe, all of which could in principle be studied or analyzed in the language of mathematics and ideas of symmetry. All of this stems from that most basics of your character, making decision for oneself and stick to it like a man and work to bear it to its full fruition. I still regret some shortcomings in my intellectual development , one is that I did not put much efforts to build rigor in my mathematical thinking. Second I may have been an enthusiastic Physics student if I had been steered myself in that direction in my young age. 

The university where I am currently doing masters is rated as one of the best in the world and is justifiably so considering the quality of infrastructure and excellent faculty that abound here. I was an enthusiastic participant of many of the club meetings of students. I happened to see an advertisement for a debate on "Evolution and Catholicism", organized by Catholic society. I thought that Catholicism has finally come to terms with the indubitable truth of a scientific assertion, and has thoughtfully convened this little debate as crossing a final frontier to reconcile science with religion. And the two debate speakers were supposed to be from Science side and religious side, a scientist, molecular biologist at that and an ordained priest. So I emailed the group of students organizing the debate and prepared some questions to ask in the debate and was duly present at the main library theatre. The first debater, Scientist started his part with a passionate admission of the fact that evolution has passed all scientific tests to be considered as an uncontested truth and so on and he rambled on for some 10 minutes. 

The next U turn, came as a denouement and delivered with a poker face was that he didnt necessarily believe in evolution as a purely mechanical thing, but was infact a process that has something to do with God's will, no other god but Jesus Christ in that (I do not know the version of Jesus Christ they endorses). I was stunned. I peppered him with questions and questioned his assertions, it seems surprising to me altogether that he didnt think he made any drastic course of change in his thought and was adamant (justifiably so) that human consciousness need a divine explanation. So for him a human foetus isnt human unless it is conscious of itself or maybe baptized in a church? The second debateer was an even interesting gentleman priest from Spain. Yeah, he was the kind of gentle fathers we see in churches with a balding head, kind demeanor and a friendly disposition. His first slide failed to suppress laughter from my side, that he believed in Bibles account of age of universe as only some 4004 years.

I just shut off my mind and ears after first slide and just glided through his lecture. The priest generously peppered his debate portion with subtle sexual innuendos to make matters more interesting. I liked that, may that is the reason he keeps his flocks together in church and keep them interested in his preaching. As I walked out, and I balked at asking further penetrating questions to this priest as I didnt want to offend this fine man and the fine catholic team who has taken pains to organize this semblance of a debate. I realized the gulf of differences that lies between me and a believer. For me it is almost inconceivable to make me believe in anything that cannot be reasoned out in a satisfactory way. May be, it is the way I have developed myself by reading and interaction with world and its influence.

What is in a name?

Have you observed this funny thing? If you repeat a name, a very familiar one say of your city, country etc, a hundred times without pause, the name dissociates itself from the signified and will just be a bunch of sound words than anything. Often it sounds downright awkward to repeat those words devoid of meaning. So names, labels on itself signify nothing. But pradoxically, it is everything as the profoundest guy who has ever lived, Herr Wittgenstein says "The limits of language is the limits of my world"! Yeah, you can call anything anyname and insist on it till the cows come home, but does it make any difference in the least? But why did I name my blog "Diagonal Slash" and named the domain "Amlafruit"? Yeah, what I call meaning and language in this naming concept encapsulates some concepts which I think sort of describes the aims with which I started writing this blog. That is to write about some aspects of the world that interests me in person, that happens to be related a lot with mathematics and self identification. So Diagonal Slash, the most famous of mathematical techniques is about an ideal of a concept in thought that has had far reaching implications than eye meets. Also Amlafruit happens to be a fruit that purifies your body, as it is rich in vitamin C. I started this blog right after my visit to a beautiful temple called "Thirunelli Temple" in Wayanad Kerala. The temple name trasnates to "Holy Amlafruit Tree" and I went through the legends of establishments of temple, the aptness of the name struck me.

This is one of the most unique temples I have ever visited, with an unmistakeable northern Kerala charm with the unmatched beauty of it being in the midst of valley surrounded by forests, the temple and its centuries long history, the strong reputation and everything attracts me to it. So I decided to name my domain Amlafruit to that effect.  

My best dozen

I normally boast to my friends that I am a voracious reader of good many number of books. Fact is that, I was a voracious reader of books only in my late teens. Both my parents worked as professors and are book lovers to boot. My father has built a huge collection of engineering books, bound volumes of collected works of famous writers in Malayalam, science books and dictionaries in our home (I do not know the motive for having 20 different English- Malayalam dictionaries in any case). My mother, who has a PhD in subaltern studies, also reads a lot in her free time. In fact her collection of books now attracts me the most as it consists of  works by Jaquese Derrida, Edward Said, Michele Foucault, Gayatri Chakravorty Spivak, Ranajit Guha et al who convey their ideas with intricate use of English language that seems so much incomprehensible and alien to my line of thinking that it is interesting. (Order of Things - by Foucault for example)  My father is also a member of the district public library housed in British era building of grand proportions with tall arch way as entrance and tall roof to accommodate old dusty wooden racks creaking with weight of books.

                                                         
                                          Thrissur Town Hall, where the public library is located

I used to regularly visit the Thrissur public library to read magazines and borrow books. So even in the midst of quite hectic exam preparation for entry into top engineering colleges in India (Luckily I managed to get admission into one of the best possible), I spent an equal time reading books from this public library so much so that the small account book in which borrowed book entries are noted got filled within 2 years of my use (That must be reading an average of 3 books per week for 2 years, or 300+ books).

Now I have sort of lost the mojo for long concentrated reading, a bye product of age I say (I am not that old!)? I was reflecting on the best books I have read as this question came to me in the midst of a chat with a dear friend of mine. She is a voracious reader and has also completed a double degree in Computer Science and Philosophy. As I was never myself asked this question before about the life changing books in my life, I had to think a little hard to come up with a list.

The following are the best 12 books I have read. Take word from me that, all of the books are immensely worth reading for the new vistas it opens in your thinking. I do not profess to have developed any special skills from reading these books, but it sure have enhanced my horizon of knowledge and have since then colored my thinking and philosophy to a great extent. I am a rationalist libertarian humanist. I was convinced of the futility of believing in god by ruminating on a stimulating book about Complexity Theory. From the face of it, if I am able to understand any of these books to its entirety, I would be content to consider myself a learned man.

(Disclaimer: This list in no way an endorsement that I only read very serious books. I Just wanted to blog about these list of books keeping in my mind the goal that someone, younger than me, of my taste of interests can take note. Infact, I have read most of the thrashy stuff that everyone reads, but is no longer interested in reading pulp fiction type for enjoyment sake. I think I have grown in a way in my reading selection, but I do not know for sure that has anything to do with growing maturity and intellectual development as an adult. Now I am more inclined to take/buy Science books, adventure or expeditions, light philosophy and classic/modern fiction of good pedigree if I choose to read.)

1) Emperors New Mind, Concerning Computers Minds and the Laws of Physics - Roger Penrose

In my opinion, this is quite the best book on popular exposition of Physics, Mathematics & Computer Science written by a great master of the subject. It is quite deep and wide ranging in its subject content and this book is considered one of the bonafide classics in Science writing. When I first encountered the book in my college library, I brushed it aside as I had heard of Roger Penrose as a famous theoretical Cosmologist and disingenousness of him writing a popular science book put me off from reading it (Martin Gardner, in preface also states this). But curiosity got better off me, I was hooked when I read the first chapter - where such fundamental questions as What is mind? Do computers think? What is intelligence? are all discussed with such ease that we fail to grasp that we are being led to his magnum opus with the best possible introduction ever. Once Penrose has taken his stand in first chapter that he is a skeptic of the boastfulness of proponents of strong AI, he slowly and surely develops the background for his argument in the next few chapters, in possibly the best ever introduction to computer science and QM. 

His thesis is that Godels Theorem, Turing solution to Halting problem, non computability of classical systems and the mysteriousness of collapse of QM wavefunction (classical and QM world collide) are all indications that current theories are insufficient to describe the operations of mind. He also touches on the incompatibility between general theory and QM as a source of revolutionary new workings in science of mind. And also that human mind is no algorithmic computer. Human thought process transcends strict logical algorithms through what he calls as "insight" - which he terms as the non algorithmic part of mathematics. Penrose also proposes a vignettes of his novel explanation of the working of mind and intelligence in the last two chapters of the book. 
This is a carefully argued book with concise and precise introductions of most of modern physics and computer science. Second and fourth chapters are little demanding for non specialists. The book has everything in range to cater to an informed enthusiast and also to the most proficient practitioner of the subjects. (Try derive the Turing number of a UTM!!) Mr. Penrose will come across as an individual with strong belief in the methods of science but, as the boy in the preface is, unafraid to ask some uncomfortable questions at the current state of thinking related to some of the most fundamental human interest conundrums. I do agree that strong AI is now not as prominent as it was in late 60s till 80s, but the relevance of the book according to me is the timeless nature of his explanations of physics and computers in his peerless classic pedagogical language, which makes it a classic of excellent science writing. I have observed that this book being sold in even some very ordinary book stalls, which itself attests to the continued popularity of the classic.
After reading this book twice and having it in my book shelf for last 6 years, I couldnt but marvel at the breadth and the authority with which the material is covered in this book, so much so that if I ever get stranded in an island and can have only one scientific book with me, it would be "Emperors New Mind".

2) Dreams of a Final Theory - Steven Weinberg

The deepest but the most lucid among the list of books here, I marveled at the depth to which Weinberg has gone to formulate his arguments for rejection of philosophical doctrines and also the lucid Physics the book explains. Steven Weinberg is my intellectual idol.

3) Godel Escher Bach, an Eternal Golden Braid - Douglas Hofstadter

One of the most unique and profound books written by a very singular man. The book is a mine of ideas and can stand alone even as a fine literary exercise.

4) Randamoozham (Second Turn) - M T Vasudevan Nair

In the pantheon of the most talented prose writers ever to grace Literary scene in Kerala, M T Vasudevan Nair towers over his contemporaries as one of the most versatile writers in Malayalam. His masterpiece is the epic novel Randmoozham. I got hold of this book in my late teens. The novel starts with the dramatic scene of the destruction of the city of Dwaraka as Pandavas look on when the insatiable waves slowly devour the city. Unable to even protect the honour of the ladies and heartbroken after the death of their beloved Krishna, Pandavas decided to go for Vanavasa on their march to heaven and salvation. Draupadi was the first one to fall on the way to Himalayas. Tired and grief stricken, she falls by the way side as other Pandavas march on. In the fertile imagination of M T, as a retelling of Mahabharatha in the eyes of Bhima, who is the most powerful yet fallible and humane of all Panadavas, all episodes of the grandest of the Hindu epics receive an incisive treatment and retelling. M T is not inventing any stories, but is using all liberties bestowed on a talented writer who reads and analyzes the original prose of Mahabharatha and reconstructs what the great first author has implied by meaningful silences. Bhima is the only one of the Pandavas who runs back to help Draupadi get up. He is not concerned about the distant call of heavens and eternal life there. Bhima lives in the real world with its agonies and ecstasies and doesnt hesitate none a bit to rush and help her as she lies there dying. In an eternal twist, Bhima remembers his life as one of the Pandavas and there starts the story of Mahabharatha in the eyes of Bhima. M T hits a pinnacle in the use of the most intricately contructed prose and use of words and phrases that only the genius masters can ever put in words (this novel is yet to be translated into any other language, depsite being acclaimed one the best ever novels in Malayalm attests to this intricacy of language) to retell a story that we all know, in a manner that makes it stand along side the original is, in my opinion, one of the grandest and boldest literrary experiments ever done in Malayalam.

5) Life a Users Manual - Georges Perec

One of the best novels published. I was fortunate to find it in my NITC library and read it at a very impressionable age.

6) One Hundred Years of Solitude - Gabriel Garcia Marquez

Most haunting and beautiful of all novels I have ever read, nearly moved me to tears after reading the part of Aureliano Buendia's downfall and the sad fate of his sons. As rich with ideas as to be a universe on its own.

7) Life and Times of Michael K - J M Coetzee


I nearly cried after reading the book. One of the most poignant novels of 20th century.

8) Complete Works of Vaikkom Muhammed Basheer - Vaikkom Muhammed Basheer


Genius of Malayalam literature. Incomparable talent in world play, philosophy and simple story telling. A true master.

9) The Mind of God - Paul Davies



Once of the great exercises undertaken by a very learned scientist. I liked the book for its wealth of ideas discussed, which could be used as  starting points to any explorations in philosophy of physics.

10) The Blind Watch Maker - Richard Dawkins

Logical, lyrical, and exhortative classic of fine science writing for a cause, which I side with quite closely.

11) The Perfect Symmetry, the Search For the Beginning of Time - Heinz R Pagels

I read this book quite early in my life and started my dalliance with brilliant science books. I still remember the prose part where Pagels explains the dramatic Einstein Bohr debates and specially the part when Bohr skillfully demolishes Einsteins beautifully constructed thought experiments one by one.

12) 1 2 3 Infinity - George Gamow

Others which i would mention are 1) Midnight's Children 2) Tin Drum 3) Elegant Universe 4) Argumentative Indian 5) Foucualt's Pendulum 6) Fabric of Reality 7) India After Gandhi

Speech by an Economist

This blog post is a transcript of a speech by Professor Raghuram Rajan (Professor of Finance, Chicago Booth School of Business, and formerly governor Reserve Bank of India) about the root causes of the global financial crisis and the emerging new world order at the event, "New Insights in Business Finance" (organized by the newspapers De Tijd and L'Echo, with the support of Ernst & Young and Petercam).

I do not consider this speech to reflect the very best of Prof. Raghuram Rajan, on the contrary I find the speech as full of half developed ideas and confused rhetoric not backed by fact based arguments. I chose to transcribe this speech mainly for my personal understanding of the global financial crisis (Raghuram Rajan was one of the earliest known voices of the growing resentment in the unregulated explosion in derivatives trading and piling debt in US economy. He rightly predicted the consequences of tail risks should anything untoward happens.). I am quite confident that, reading this lecture in its entirety will give reader some useful ideas as starting points in further intellectual exploration.

This is not a verbatim transcription.....All errors are mine

We do not know for sure what is going to be the new world order in the future?  What we know is about the forces that are emerging to define the new world order. We can also surmise that a study of evolution (as the forces collide and compete) of these systems of forces will lead to better understanding of the new system of world order and help define some of its broad characteristics. 

We have some broad understanding of the forces that led to the current financial crisis. But superficial facts such as a bunch of greedy bankers, conflict of interests and liberal regulators fed by dose of Ayn Rand philosophy as responsible, fail to answer (and the charge that financial sector were essentially unregulated) the following  important questions. Why did the crisis happen now, and why did it occur in one of the most sophisticated and competitive market systems in the world? More sophisticated market systems are supposed to solve such conflicts of interests by self evolved system of checks and balances, regulations and governance, then why did this system fail? Also what was the subprime  why did the crisis start here? What were the reasons behind universal affinity for subprime loans? If it was just a matter of a bunch of greedy bankers and lax regulations, then some exemplary punishments and tightening of regulations would have solved the issue. So this crisis is a much deeper problem with a set of socio economic developments acting as precedents. It can be argued that, at least three major forces came together to cause the present financial crisis. Two of them are specific to United States economy and third one is a more global emerging force. Force Number 1 is the rise in income inequality in United States and how owned housing was used as palliative to ease the pain caused by widening income gap, without addressing the root cause of the issue. In the last 30 years, income inequality in United States has widened. If examined in detail we can find that top 10 percentile of earners as a group has seen their income rising much more rapidly than the middle and lower income group. So it is not just the super rich who grew richer. There is a social reason behind this. In the race with technology and education, education is falling behind in the United States. Disruptive growth in technology in 20th century was one of the biggest catalysts in the industrialized growth of rich countries. Telecommunications, revolution in transportation etc affected lives and caused enormous changes and chartered nature and course of economies of the industrial world. Consequent to the growth in technology was the growing need for knowledge based workers. This caused an explosion in education and United States was one of the first countries to take advantage of advanced education to expand in the tide brought out by new 20th century technology. US became the most educated country in the world in 1920s when most proportion of its men and women went through high school. 

Jobs can be broadly divided into a 2*2 matrix classified based on skills and scale  of routine work. The classifications are unskilled routine jobs, unskilled non routine jobs, skilled routine jobs and skilled non routine jobs. Unskilled routine jobs were increasingly substituted by new technology, which displaced huge number of jobs in the 70s-90s. Also routine skilled jobs were transformed by competition, technology and globalization. Jobs, that once were populated by clerks accountants etc were seen increasingly shifted abroad. For example, a skilled work such as analysis of balance sheet can be done much more cost effectively in Shanghai or in Mumbai, India. So what left out were the non skilled non routine jobs, which cannot be transferred anywhere, jobs such as drivers, cooks, gardeners, farming etc – where wages grew much slower on an average and the non routine high skilled jobs, which demanded much more specialized university level education. And the need for education also shifted from high school to workers with university degrees who can do skilled jobs with non routine tasks, and be creative and think in their roles as consultants, professors, lawyers etc. This has caused the hollowing out of jobs in the middle level who were employed doing routine skilled jobs and an increasing number of such people are finding themselves unemployed or doing less skilled jobs with lower pay. To give some statistics, in the age group of 26--54 males with no high school degree, 35% are unemployed in US. For all people aged between 24-54 age groups  24% are unemployed. These are staggering number of unemployed working age people in an industrialized economy. The broad level of education hasn't been changed in last 20 years. Number of people with university degree remained stagnant in the last 40 years with only one silver lining that more women are getting higher degrees than men. These social changes lead to stagnant incomes of middle and lower middle class in US and put tremendous pressure on politicians to find a workable solution to the rising income inequality. The gulf between the have and have-nots, those who grew along with technology and better educations and those who were left behind, is still a worrying concern in the States. Who will not want to fix the problems of common man when his next month paycheck remained stagnant for years on end? This was also a key moment in American politics with transition from Reagan years to Clinton years. The prevailing policy to tinker with taxation to redistribute income was resisted from both camps. In fact taxes were cut. So how did the politicians try to solve the issue of lower growth of income and subsequent decline in spending power of an average American family? The enlightened answer was to expand credit. Predictably, no one objected to this easiest of solutions and also felt no guilt in giving liberal housing credit as an ingenious solution to increase in assets and equity. This enabled the home owners, though in debt, to borrow against their house simultaneously feel that they are borrowing against an asset, not just piling debt on debt. So the end result of this debt enabled social changes were that consumption inequality did not increase in line with income inequality. As people started getting loans, US household saving rate went to zero and even touched negative. Clinton government pushed for affordable housing with the mandate aim was to enable more teachers, firemen, policemen to buy houses, by extending credit to people with lower repayment capacity. Both sides of political spectrum supported this housing push and Bush government went even further towards an ownership society. It also helped that the property owners were deemed to be conservative in outlook and thus were favored to vote for republicans. Thus they pushed for more housing credit by using the instruments of government such as Fannie Mae and Freddie Mac to drive housing credit often with government money.  Thus fault line number one was the much looser credit as the beginning roots of the problems.

The second problem was the inadequate safety net in US society. So if a person loses job and he/she also loses access to such essential services as health insurance. This thin safety net had historically worked when US had short recessions often followed by sharp recoveries, which was an observed pattern of economic growth in post World War II economic recovery. This forced workers to be very active in seeking jobs, as on average within 8 months of end of recession, most jobs were back in economy. But something structurally changed in 1991 when it took 23 months for the jobs to come up. George Bush Sr ignored economy at the peril of losing his reelection. This lesson was a bitter pill to be swallowed for both parties and since then, politicians could not ignore the economy, even the slightest hint of slowing down in job growth numbers. The active intervention of government to loosen up fiscal policy by fiscal expansion, monetary expansion, in response to the 2000 recession is a case in example. The famous Greenspan put, which was a free option to the stock market, to the effect that government is there to support asset prices in case they fall, was not an exercise to arrest sliding asset prices. It was more an exercise to revive the job growth as an assurance was given to industries that, no matter what happens to general level of demand and prices, government will support companies, so that companies can concentrate on spending and creation of new jobs. Job creation was the sure way to emerge out of these recessions.  But the 2008 story turned out to be different that even most aggressive interest rate cuts and fiscal policy easing, jobs were no longer getting created in sufficient speed to recover from the recessionary spiral. So the 2000 Greenspan put changed to Bernanke put. The Greenspan put was responsible for speculation in housing market in 2001-04 which created a housing bubble. The present monetary easing has started creating bubbles in bond markets, emerging market and in farmland. Though the policy is made with very good intention, its consequences are often problematic. 

All the blame for slow economic recovery cannot be laid on US. Other export oriented economies too played their part. For example, Japan followed export led growth policy for decades. As a country torn after the devastating war, Japan rebuilt their economy entirely relying on foreign demand and created dozens of strong companies and by exporting products to stimulate domestic growth. Positive repercussions of this export led growth were the rise of extremely strong manufacturing companies who were very successful in global markets. But if we think about any prominent service oriented Japanese firm, such as a Japanese consulting firm, or hotel chain, etc we will surprisingly find that none rule the global market and none even elicit any sort of global recognition when compared to the famed manufacturing counterparts.  Japanese service sector found it hard to grow due to cartelisation. Cartelisation helped manufacturing to be disciplined for better tackling foreign competition by producing extremely good products. For domestic service sector, this combination of government supported cartelisation proved a detrimental to their growth. An interesting side story is to examine the high cost of Japanese haircuts. They are costly due to existence of a barber cartel. Yes, new entrants tried to challenge the cartel by offering cheap haircuts in Tokyo. The barber cartel decided to fight back in an ingenious way, in a fit of genius, they declared that having haircuts without having shampoo is unhygienic and made it mandatory for all barber shops to have shampooing facilities. This immediately meant that the new entrants have to invest in shampooing facilities and the resulting time for investment will put them out of business for an extended period. This is a symbol of a more general problem in service sector that has kept the domestic demand from expanding. This similar problem could emerge in an export led economy such as China, and would be a common problem for any export led nation. Lower domestic demand in export led industrialized countries forced them to rely on United States as the last resort consumer. So once the US started going into recession, their woes compounded and led them and global economy deeply into trouble.

So the pertinent question is, what is this all got to do with financial crisis? How were all the low quality mortgage securities created? Relatively price insensitive government money was used as fund this credit expansion. Money from outside US came pouring in to invest in US securities and no questions were asked. Some of the banks only cared to ask for AAA rated US securities to invest in. With all the flush of money looking for US assets, the brokers who make these loans realized that credit quality was not important.  Customers were willing to buy any loan with no questions asked and brokers were getting the fee regardless of the quality of assets, and thus it was no one’s responsibility to check the quality of underlying assets. Therefore discipline broke down in the financial sector. Beset with greed, banks ended up creating products that the buyers wanted. Another critical question to be asked is, banks who were most probably be aware of the fact of the bad credit quality of the underlying assets, continued to hold the assets even with full knowledge that these assets might default with higher probability? This is a very serious question in need of a long answer. 

Now the third factor was the slow building of a crisis in public finance which started brewing from Europe, where they had the issue of taking care of an increasingly ageing population. Public debt has increased in all G7 countries since 1970s. This is also a realization of a larger crisis. We have used the government balance sheet to moderate the volatility or economic shocks. Government balance sheet is full of debt and a significant rethink is needed to effect and to move away from this over reliance on public debt, and towards a more self sustainable economy. We thought that we understood the European strategy as “lets try and muddle through for some time. when there is some daylight (or when growth returns) we will fix the problem, otherwise we can always restructure and eat the losses”, sort of thinking. Germany’s stance changed with the change in domestic political scene and with Merkel losing elections. It became apparent that private markets were not willing to fund Greece Portugal or Ireland, so in effect sovereign bonds were increasingly being bought by official institutions. What was a private sector problem became a public sector problem. Assuming that bonds need to be written off, these losses were to be imposed on public money expenditure rather than on private sector. This was one of the reason for continued calls for immediate restructuring rather than wait for all private sector institutions to bail out without having to pay for the financial mess. People have toyed with this idea that the European financial crisis is likely to be contained, which is a sort of muddling through the whole issue, with the thinking that indebted countries can grow their economy and service the debt own their own. The probability of this happening is very low. Other issues is that the accommodative economic policy in US had unanticipated consequences throughout the world. With the very low interest rates in States, the emerging market rates have also stayed low, on account of which their economies are growing fast with consequent global increase in commodities prices. This is the reason for high oil prices and reduction in disposable income of average consumers. Industrial countries have serious problems, they were excessively accommodative in their fiscal and monetary policies and any tightening of which will slow down their growth.

To compensate for the slow domestic growth, industrialized countries have to see that emerging markets are rising fast. One big issue is how to take advantage of the growth in emerging markets?  US has recently become cost competitive in manufacturing, as costs are low because of depressed economy. This is quite an opportunity to produce and market products suitable for emerging markets with their consumer needs in mind. The old strategy of thinking stratification of society and looking at classes with western lifestyles and targeting them, wont work. For example, emerging market middle class is a huge market, with 600 million strong in China and 300-400m in India. The question to be answered is “how am I going to service that market, when the market is middle class by their own considerations rather than by western standards". An interesting example of selling a product suited for emerging market is the story of an innovative refrigerator for rural markets, produced and marketed by Godrej Industries in India. In most of Indian villages, electricity comes for only 9 hours a day that too in random times. The question is, how do one run a refrigerator with such an inconsistent supply of electricity? Godrej engineers found out that consumers primarily used refrigerator to keep food cool for at most a day and they had no need for ice and freezer. Once the need for ice is eliminated, the fridge can be run at 6-7C and therefore can be cooled by fans running on batteries. This type of innovative frugal technology is being developed by MNCs such as GE, who has their own product design center in India. It is an enormous challenge for the industrialized countries to move design of a product 8000 miles away which is about decentralizing controls.

The second big issue that has become relevant is about rethinking of governments in rich countries. In United States, there is enormous distrust in elites from both left and right sides of political spectrum, which he calls (Raghuram Rajan’s term) Sarah Palinization of politics. It is a fact that nobody has paid the price for the economic mess that US in and there is fear that the elite (Washington and Wall Street) might further collude together.Sarah Palin emphasizes the fact that she is not from an elite background and most of her family upbringing strikes a chord with an average American. This is causing a polarization and a lack of structured dialogues in divergent political views, such as the left arguing for taxing the rich and a proper redistribution of income, and right arguing for cutting back on spending. This political polarization is accompanied at the same time by the skilled and unskilled disparity and the spread of disparity between the haves (those who are benefiting from globalization and technological changes) and have nots. It is a dialogue that could spread to other countries. These are enormous challenges faced by today's government. The promise of social mobility is shattering and bottom of the pyramid in developed countries are in danger of struggling for education and access healthcare, which are essential in order to create people with capabilities. So government has to spend to keep up the demand and also to raise living standards of the bottom of the pyramid by way of support, but government spending is not unlimited. This would mean radical rethinking of entire tax and incentive structure in a way such that government can work more efficiently and focus on the most important things at hand.  Also the demand to shift and reorient industrial production with an east ward look is essential. Emerging markets also have their own problems in managing their own demand, for example, over consumption and over expansion led to Latin American and Asian financial crises. There will be relative loss of position for industrialized countries, which could be hard to accept for them and can also spur the forces in bottom half to rig protectionist barriers which might stifle growth. US has resisted the call for protectionism at least for now, but years of slow growth and protectionism in capital flows and asking of banks to invest in more Government securities will be construed as over alarmist. There is a worry that such conditions will call for a revolution unless government is seen working hard providing fair chance for upward mobility via education, health care, infrastructure services for all.

In general, there is hope for a happier and better world.

Brain Teasers - 1

Anyone with some interest in mathematics and logic will find solving brain teasers a very rewarding way to pass time. Yes, it is fraught with some dangers of over exertion on some very fiendish puzzles, as is the case with most puzzle solvers worth their salt, they just cannot shut down the process of thinking about a puzzle unless they solve it in some satisfactory way. I too am a little enthusiastic about solving puzzles, though I claim no special intelligence or special skills to demand to be any exceptional in this case. My experience of solving any puzzle has taught me some valuable lessons in structured thinking in solving general problems. For example I will just describe how the following puzzle was solved by me and the corresponding learning from the method employed to solve it.

When a brain teaser or puzzle is posed, Ones' initial aim is to find the key idea to solve this problem. A problem could be solved based on combinatorial reasoning, probabilistic reasoning, pigeon hole principle, logic etc, which is almost always clearly asked for in the problem statement itself. But in some cases, a translation of the problem to some analogical problem only will lead to clues as to the general method to solve it. Some of the guidelines, as followed in good puzzle books, to solve brainteasers are 1) Try to start from simple cases and then generalize to find patterns 2) Try to start from an analogical problem based on your experience in solving puzzles 3) Trying to work backward from the result to construct a method to solve the problem 4) Use of conditions in a clever way, unanticipated in a superficial reading of problem, to lead the way to the key of the puzzle etc etc

Puzzle

There are 60 blue ribbons in a box such that  all 120 ends are hanging out and you cannot see which ends belong to which ribbons. You randomly join all 120 ends together into pretty bows and dump out the box. Depending on chance you will form anywhere from 1 to 60 loops. How many loops would you expect?

Ok. This sounds like a difficult puzzle. Why? This problem sounds very neat and quite simple to understand though. Atleast according to me, a superficial thinking about the puzzle will leave one clueless about the mechanism of increase in number of loops in each step. There are 120!/2^30 different ways of joining the ends of 60 ribbons, a number so large that any cursory glance at solving sigma(P)E(P) will be an undoubtedly daunting task beyond your limit. So we evidently cant solve this problem, using pen and paper by brute force method. (We could very well simulate it in computer but I think this is cheating in a way to solve a puzzle). I grappled with the creation of loops in each step and the expected increase in number of loops for some time. I tried to start from an arbitrary step, where "k" number of ribbons are joined with E(k) number of loops and finding expectation of E(k+1) based on transitional probability. This didnt work without a key idea. That is - How many free ends can one expect at the end of "k" number of ribbons are joined? It is a simple idea, the solving of which was the key to the puzzle. Suffice to say, I wrestled with steps 1, 2 and 3 (as indicated above) in many combinations and performed some messy calculations till step 3 to no avail. I started from backwards, middle (general case), try to think of analogical puzzles (none came to my mind) and was stumped for some time. The key idea still eluded me and thus the solution to puzzle.

But working out the solution using different ways finally had its result.

Pattern recognition is one key component of successful solution of may problems. For example E(1) = 1/119 and E(2) = 2*118/119*117 or 1/119+1/117 which points to a way to solution. Another key thing is to understand the physicality of the whole problem and try visualize the knotting of ribbons and forming of loops in an abstract way. By this time, it became apparent to me that, whether a new ring in formed or not, at every step, the number of free ribbons reduces by 1 each. This was the key idea that I was waiting for, then the expectation calculation can be easily derived for a general case.

Now, coming back to all of earlier reasoning, it was a simple exercise to write a recursion relation for expected knots at the end of tying k ribbons together as follows

E(k) + 1*(60-k)/2*(60-k)*(119-2k)/2 = E(k+1) ------>; (1)

Simplifying

E(k) + 1/(119-2k) = E(k+1) -----------> (2)

Now E(0) = 0. E(1) = 1/119

Thus E(60) = 1/119+1/117+.....+1/3+1 = 3.028933 loops on average, as the answer.

This is a simple and elegant answer which makes immediate sense.

For example, we can see that on an average it is very difficult for the rings to form in beginning, but ring formation will be more common as we reach the end of this exercise.

Some of my thoughts about the concept of Time

"Eternity is time
Time, eternity
To see the two as opposites
Is Man's perversity"

The Book of Angelus Silesius

What is time - Musings based on Physical Philosophy?

I have thought (ok for some years atleast) that, what we measure in seconds or minutes is not time and time is among the most mysterious concepts forever within us but remain elusive in its complete understanding. For me, it is quite amazing to reflect that it took an Einstein and his genius to rewrite the history of Physics with his audacious attack on fixed Newtonian scheme of universe. Which core idea pile drived the startling dismantling at the base of foundations of Physics? It was nothing but the radical rewriting of the concept of time!

Time is an amorphous concept inherent in our selves which we perceive as flux in the observable world and its effects on us. Based on this definition, if we imagine an immobile person completely shut off in a dark room but conscious of himself, he perceives a flow of time. This is due to the person, though he cannot observe the flux of universe (similar to the state of a sense impaired person lying down motionless), through his intellect can "sense" the change of state of his body, his rhythmic breathing, hear beats, rumbling in stomach, uneasiness in some joints. If we take this argument further, it can be argued that such sort of imaginary person can sense something similar to time, but as and when new changes cease to be perceived as new and a sort of repetitiveness in the senseless dark world engulfs the person, then his sense of any reliable measurement of the elapsed times will be impaired. For him such confinement of 1 year is no different from 50 years, but the world outside may have completely changed. So left to ourselves in a completely shut off state, any sort of reliable time measure fails and we cannot self consistently talk about change of objects in the universe with any agreement on a sort of universal time within ourselves. So we all have cognitive mental images of a state of world at one reference point which is then compared at another point with the reference state of own self understood to be the time invariant base. So it is obvious that this cant be measured quantitatively. I came across a well written research paper on the individual perception of time by a mathematician, Ludger Ruschendorf. He interestingly constructs a model of time perception based on the intensity of interesting events, and in a very general setting proves that the model allows for logarithmic decrease of feeling of time. So as we grow older, the well known assertion that time moves slower is proved!

What does Science say about time? Based on Einsteins special theory of relativity arguments, it can be proved that simultaneity as a concept has to be rewritten in the language of relativity.  This based on the proven scientific assertion that no one can agree on the simultaneity of any event. Taking this argument further, a mind boggling realization can be reached, the world as encompassed in the space time as en entity, is "frozen" - so to speak, where past present and future of the universe exists simultaneously at all time anywhere. If we care to take this argument (based on the sound logic), then any self respecting physicist would wonder "what is that the variable "t" in classical dynamics and quantum mechanics equations doing" - if change of "t" itself is illusory, so to speak? I have observed that Physicist Julian Barbour, in his thought provoking book "The end of time" and Philosopher J M E McTaggart in early 20th century essentially argues that time is illusory. If the world history and its future evolution is irrevocably etched in frozen space time what does it mean to have a time variant evolution of physical systems? But wait, isnt this similar in proposition to the already existing space and we happily measure distances between two bodies without evoking any logical incredulity in mind?
Also string theoretic and Quantum Mechanic logic dictates an ultimately quantized time with discontinuous properties. We could imagine the world forever growing and stopping an infinitely large number of times in every human second. (If quantum of time is 10^-35 s or so, then universe will start and stop 10^35 times every single second)

Lets imagine ourselves to be hovering motionless in deep space with a huge ball for company, and nothing else but you and a big ball in an empty corner of the universe. We are totally isolated and the space around us is empty with no shard of matter or energy (elementary particle, neutrino, neutralino, quarks, non hadronic matter, dark energy or whatever new types yet to be discovered) in our neighborhood. But getting a little ahead, it could argued also that complete emptiness is also a physically unrealizable concept. What does it mean to have space and time but no matter? Newtonian concept calls for absolute space and time as an entity outside of matter. According to many physicists, this is not a logically correct concept due to the inherent fuzziness in quantum nature of the entities and that any empty space will be permeated with the quantum fluctuations of matter and energy fields at all place and all time. But lets just put little physical niggle aside for the time and continue along.

Any movement is space is perceived though information transmitted from the moving object through light waves; which our mind processes as a change of state. This intertwined evolution of our observation of this part of universe and consequent change of information about it (if we assume that the quantum mechanical uncertainty is always source of new information), if normalized wrt to a quantity, I would say that is time. Any metaphysical interpretation of time without a physical basis will not affect us, and in this respect would be as good as irrelevant. What we perceive as the history and future evolution of a ball in space is completely determined by observation of the ball for a finite of amount of time. With this, the entire history of ball and its future can be accurately deduced. As information is is the uncertainty of outcome, processing of information with the help of relevant physical laws will remove uncertainty about the motion of the matter. While we do not know beforehand that the ball will hit the hovering space man, by observation and deduction we can safely calculate this possibility of such a collision. This is what is called determinism of physical laws in classical sense. As the famous thought experiment of "universes A,B, C with flashing lights at regular times" show, it is not absurd, but we can even logically think about a duration when time didnt flow, as long as it has a physical basis - a changeless universe. So change in information and change of state is the physical basis of time. Here we can escape from a logical problem by postulating ultimate bulding blocks of matter and energy. A particle or matter (eg an electron) might not change at all since creation, hence for the matter on its own does not possess a time varying property, but since it has some ultimate structure with internal change of states which affect the physical properties of its interaction, time still persists even in a basic universe with a single electron.

Buddhists (rational Buddhism) often think of the word as a series of recurrences going on for infinite time, with no beginning or end or purpose for the world, and only for the soul or energy that animate living matter induce change of state of matter and evolve and dissolve in an infinite regress. I find this unappealing for two reasons 1) We cannot logically think of an infinite physical parameter in a physical universe without producing logical fallacies - as sound logic is prelude to any sensible explanation of universe, this argument fails here 2) If an explanation of the sort which somehow incorporate a cyclical time is the case, then by definition of time as a measure of rate of change of  information of a physical object - what would they perceive as cyclical time is in its essence a linear time. Topology of time as linear in nature or even curling back to itself is counter intuitive to the notion of a cyclical time, as the curling back is not by physical characteristic produce a cyclical universe of the definition.

Ramanujan - Life of a genius

All of us are born with some gifts (skill set, whether inborn or developed), but some are born more equal than others and as if by magic to mere mortals, they live to attain such mastery over a subject as to inspire feeling of incredulousness in observers. And they are the prime movers of the human thoughts, ideas and deeds for through their work, many life times worth of effort is laid on the ever forward march of our species to in their quest to epistemological immortality. For example, mathematician Gauss alone is said to have sped mathematics development by atleast 50 years by his genius (Had all his discoveries been published the impact would have been much greater!!). Same goes for the works of legendary geniuses and polymaths such Von Neumann, Leonard Euler, Herbert Simon.

It is also often said that what a "Newton" or "Einstein" achieved was a product of the time, as the ideas were ripe to be discovered. "No mortal force can stop an idea whose time has come" - said Voltaire, French philosopher and defender of human freedom nonpareil. It can be claimed that any exceptionally astute thinker of the caliber among the greatest could have done what a Newton or Einstein had achieved in their life time. A case in point is to examine the lives of Christian Huygens and Henri Poincare, contemporaries of Newton and Einstein respectively. Then even in the highly dignified list of such greatest geniuses, some still stood on a different level on their own and did their work seemingly beyond the reach of even the most gifted in many generations. It can be rightly said that Ramanujan was one among such singular geniuses the world has ever seen. Some of his theorems could never have been known if he hadnt mysteriously derived and wrote it in his famed notebooks, as if taken out of thin air.

He was born and raised in India and was a product of the best of how Indian system has molded the thinking of the some of the ablest men of sciences.

Mathematics has a platonic existence in the sense that its reality, the interlinking of relationships between mathematical concepts, their logical order,  is independent of the task of discovery. The concept of numbers were immediately apparent to the thinking man when he was surmounted with the task of calculation of seasons for grain harvesting or the number of animals to be hunted for a daily meal for his growing family. But the elaborate edifice of number of theory and its many connections with other disciplines of mathematics is no man made object as the connections became apparent as and when the subject was explored by the intrepid mathematical explorers in their mental voyages. G H Hardy has likened the process of mathematical discovery as the process where a climber arduously scale a difficult peak and then survey the vast landscape laying below him and distant peaks lying ahead of him to be scaled later, while the large landscape of Mathematics remain a mysterious reality shrouded by the clouds of our limits of reasoning. Roger Penrose has beautifully and forcefully put forward the platonic reality of Mathematics in his discussion about how the fractals and its unique properties were discovered, and its computer based explorations was not able to scratch the surface of the immense complexity of the evolving patterns and their underlying relationships. Godel's famous theorem has put an end to the dream of mechanical theorem proving, which was akin to reducing the whole of mathematics sophisticated yet mechanical symbol play.

Here, Terence Tao's analogy also comes to mind. Mathematics as a subject consists of different levels of abstractions, all of which are attempts to model or approximate the reality of the physical world. At the base level would be the concept of the primitive objects as numbers and simple geometric concepts and then move up to sets, spaces, operations relations, functions and operators.

If mathematics is a universal construct independent of human existence, then it can be said that discovery of mathematics is also a process independent of humans. We do not yet know of existence of mathematical prowess in other animals, but is conjectured that whales and some species of Dolphins exhibit some rudimentary number skills. Some humans are better endowed with an uncanny skill to master mathematics. Infact geniuses are most commonly observed in Mathematics and Music. Legend is that, Gauss's famous pupil, Gotthold Eisenstein had remarked that he always knew Calculus. Many mathematics prodigies are also fabulous mental calculators. Familiarity and exceptional skills in number manipulation seems to be the one sure sign of giftedness in Mathematics. Also it is assumed that we come to this world with some hard wired logic and innate sense of the 3D world. This is also the case that we always believe when two statements of seemingly contradictory facts or logic are heard, we at once naturally think that only one of them should be correct.

I was familiar with Ramanujan's incredible story right from my school days. His story was an inspiration, of how genius can transcend the cultural and geographical barriers to its fullest expression. The old adage "the whole world conspiring to bring fruit to the man's effort" is apt in his case. Ramanujan was born in Kumbhakonam, Tamilnadu to a pious Hindu Brahmin family. His school days were filled with incidences of his irrepressible mathematics brilliance, stories that has become legends associated with genius in general. Other remarkable fact was the unfathomable level of concentration which Ramanujan was able to summon while doing mathematics, so much so that he forgot to study other subjects and failed them even though by all accounts he would have easily passed them with little extra effort if he cared. In this connection, I remember the statement by Erik Demaine of MIT, who is a prodigy and is the youngest professor ever appointed by MIT, said in an interview that - he considers himself remarkable only in his ability to devote long span of his undivided  attention to a subject or solution of a problem. Discovery of Ramanujan's genius and his association with famous GH Hardy is well known to many, but less is known of the height of esteem with which Hardy regarded the mathematical talent of his protege. Hardy (as is said in the biography of Ramanujan) at once noticed the remarkable skill for algebraic manipulation of Ramanujan, and compared him only with universal geniuses Euler or Jacobi. Hardy had little doubt that, with proper application of Ramanujan's genius, Ramanujan could have been the best mathematician of the world, even surpassing the leadership of great David Hilbert or Henri Poincare, both last century's greatest mathematicians and acknowledged universal geniuses. In a note published in the Current Science Magazine (Vol 65. No.1, 94-95), famous Indian statistician P C Mahalanobis recounts his friendship with Ramanujan in Cambridge, when both were students in the mathematics department. We get the picture of Ramanujan as a very simple man of somewhat "shy and quiet disposition, dignified bearing and pleasant manners". On one occasion, Mahalanobis went to Ramanujan's room to have lunch with him. Mahalanobis had a copy of Strand Magazine which at the time used to publish a number of puzzles to be solved by its readers. Ramanujan was stirring something in a pan for lunch. Mahalanobis read out the puzzle about two British officers in Paris in a long street with houses on either side, the question was about the relationship between two of the house numbers, which were related in a special way. It took some trial and error effort to reach the answer, but the answer was not difficult to reason. Ramanujan promptly answered the solution as a continuous fraction, the first term of which was the solution that Mahalanobis had obtained. Each successive term were the successive solutions for same type of relation between two numbers as the number of houses in the street increases indefinitely. Mahalanobis was amazed and asked how Ramanujan got this solution, that too the most generalized one, in a flash. Ramanujan replied thus "Immediately when I heard the problem, it was clear that the solution should obviously be a continuous fraction, I then thought which continuous fraction and the answer came to my mind. It was as simple as that"

Though born and raised in a traditional Hindu Brahmin household, Ramanujan held progressive views about life and society. He was eager to work out a philosophical theory, which he termed the theory of reality based on the fundamental mathematical concepts of 'Zero' and 'Infinity', and the set of finite numbers. He spoke about Zero as the symbol of absolute, that is the part of reality to which no qualities can be attributed, which cannot be defined or described by words and is absolutely beyond reach of human mind. With zero this symbolizes the absolute negation of all attributes. According to him Infinity was the totality of possibilities  which was capable of becoming manifest in reality and which was inexhaustible. According to Ramanujan, the product of zero and infinity would supply the whole set of finite numbers, akin to creation of all properties of the world associated with the finite numbers. We  are not sure of how far Ramanujan made progress with this mathematical philosophy, but it was clear that Ramanujan held great prestige in working out a theory of reality than building conjectures or proving theorems about esoteric aspects of number theory, as former took real effort while the latter was nigh effortless and intuitive for him.

So looking back to the infinite and strange potential of his mind and the wonderful results he made in his short  collaboration with Hardy, one could only wonder at the magnitude of  loss suffered by mathematics community in losing Ramanujan at the young age of 32 to Tuberculosis. It might be that Ramanujan could have proved the famous Riemann conjecture (This was a life's quest for G H Hardy and many other famous mathematicians of the day. Hardy, himself had mistakenly thought that he managed to solve it after years of effort, only to be jolted back to reality when he uncovered a subtle flaw in one of the lemmas that invalidated the whole edifice of the proof. Together with Littlewood, he had proved many remarkable properties of the Zeros of Riemann function. Hardy would no doubt have prodded the genius in Ramanujan to solve the greatest unsolved problem in mathematics). Or his achievements could conceivably be much lesser than what we could imagine to be, as is human nature, we extol achievements and possibilities of geniuses to some unreasonable extent to be totally off reality.

What could Galois, Abel had done had they lived to a ripe age, are some similar musings of mathematics lovers. Basing on Probabilistic reasoning, it can be argued that greatest geniuses possible in any scientific discipline could have had obtained sufficient environment to fruitfully produce what they do the best and in that respect our world has seen its share of remarkable men and women, even with the untimely loss of a few, who could have been one among the greats but not too great as to have had such monumental impact as to make others contributions irrelevant. Yes, there are remarkable people who continue to toil in unremarkable positions in unfavorable environments, but such is the incandescent nature of true genius is that the whole world will conspire to allow the genius to flower in some ways.

Now coming back to speculation with respect to the independent reality of mathematics, with the case of Ramanujan it becomes plausible that he possessed an uncanny ability to "see" the equations as if dictated by his deity, which points to a notion not very much different from discovery of truths by intuition - which is possible if the truth is lying out there to be seen, only to be later connected with the mainstream mathematics by the laborious but essential process of proof. We live in such a complicated world that many of the seemingly easy questions elude answer, where the case may point to a sort of idea world with different structure than simply constituted of logic and that an otherworldly ability to see through ideas to truth would be essential to fathom the landscape to its fullest. In Ramanujan, we could have seen a genius who was able to do just that in a way, and thus showed to the world that with genius and patient application, this is possible. I also would like to add Terence Tao's assessment of unusual abilities of Ramanujan. According to Prof Terence Tao, Ramanujan's secret for coming out with strange theorems was his unusual mental felicity to do significant mental computations and his ability to drew intuition from patterns thus observed.

Hindu scriptures are full of accounts of teachings of ancient Rishis, who often with beautiful verses utter the most profound truths of the world. "Tatvamasi - I am what is the world" is a very famous utterance by a Aithreya Muni to Yaknjavalkyan and he explains the concept with seven different examples.

I would venture to say that such a profound thinking was unthinkable in a pre AD civilization (BC 500-350) with little training in logic and scientific thinking. It could only have been possible with this mystical superhighway of transcendental thinking to the heart of truth. We do not know whether this "Gods pathway" exists, but I would love to think that our capacity to learn is limitless and such a feature of our mind, connecting to the cosmic consciousness in a flash is conceivable.

Solving a question

Question:  Two players starts with the number N and play in turns. In each turn the player chooses a prime power p^m>1, which divides N and updates N to be N/(p^m). The player who sets N to be 1 - wins. What are all the winning moves from the initial state of N=1,506,009,006,001,500,000,000 ? 

Answer:

N = 1,506,009,006,001,500,000,000 = (2^8)*3*(5^9)*(7^4)*(11^4)*(13^4) (Prime Factorization of N)

Assuming that A and B are the players.The problem reduces to find winning strategy for players A & B with a set of numbers (exponents of prime factors of N in the question) written as a set (8,1,9,4,4,4) = N, where any individual member of set can be reduced whole or part in an individual move (eg: N/p^m can again be factorized and written as a set, which will reduce the exponent of "p" in resultant number by "m") and the first person to reach (0,0,0,0,0,0) wins the game.

If we assume that player A is the first mover, then A can execute a strategy which guarantees his win. (or First mover can win the game if the player plays with the following strategy)

Strategy - 

Step 1: A's first move - Divide N by 13^4 (or equivalently can divide by 7^4 or 11^4 as first move) 

Step 2: Express the exponents of each of prime factors of the resultant number in binary format. For any move by B to reduce the primary factor of any of the number (by division), A could divide by a calculated factor of the number (N) so that XOR sum of the binary representation of all the exponents of factors of the resultant number is 0000 (XOR sum without carry -> 1+1 = 0, 0+0 = 0, 1+0 = 1, 0+1 = 1 eg: 1+1+1+1 =  (1+1)+(1+1) = 0+0 = 0)

Reasoning for the logic: The final desired state of winning position can be represented by 0000 (Winning position). As a first mover, A divides by a calculated number so that the XOR of sum of binary representation of the prime exponents of the resultant number is 0000. Whenever it is A's turn to play, he makes sure that whatever move B makes, player B can never win because he is disturbing a winning position 0000 (set by A in previous move). It can be proved that A can always find moves to move to a winning position (here 0000 represented as the XOR sum of binary representation of individual prime exponents) whatever move B plays. And B can only move to a "non winning" position by his play. 

Step 3: Continue Step 2 until B is forced to execute (0,0,0,0,1,1)->(0,0,0,0,0,1) (or any related non winning move) from where A can divide the resultant N by the lone prime factor to win the game

Demonstration: 

All winning moves can be listed with this logic. But I feel that it is only necessary to state the winning logic for any move by the second player. A possible demonstration of the game is represented below resulting in A winning the game is STEP 5. N0 is the initial number. Ni(A,B) is the result of the i th step of A or B respectively. The above result is quite general and can result in winning solution for any generalized such game.

Note: The solution was inspired from an article on NIM game from the book "Adventures in problem solving" by Shailesh Shirali, published by University Press, India

Steps	Player A	Player B
1	N1A=N0/13^4	N1B=N1A/7^2 (B divides by 49)
2	N2A = N1B/9^2 (Then A divide by 81)	N2B = N2A/2^7 (B divides by 128)
3	N3A=N2B/5^9 (Then A divides by 195,312,5)	N3B=N3A/7^2 (B divides by 49)
4	N4A = N3B/9^2 (Then A divides by 81)	N4B=N4A/3 (B is forced to divide either by 2 or 3 which she does by 3)
5	N5A=N4B/2 = 1 and A wins the game (Then A divides by the remaining lone factor 2 and wins the game)

Light bulbs problem

Imagine a room with 50 light bulbs all connected to separate switches. Switches have two positions ON and OFF which controls the light bulb's state in a similar way. All switches are initially OFF. Now a restless genie is playing with the switches inside the room. Genie fiddles with any of the switches at random and toggles its state. This, the genie does 50 times and then leaves the room. Now the question is - how many light bulbs are ON on an average at the end of the genie episode?

I saw this question in a Quantitative finance discussion board.

I will sketch my solution to the problem and suggest some of properties and generalizations of the question.

Let us label the task of genie randomly playing with switches as a trial. The question asks us to find the average number of bulbs with an odd number of switch toggles after sufficiently large number of trials. (First (odd) toggle of the switch changes its state from OFF to ON, Second (even) toggle changes from ON to OFF and so on..) Let us concentrate on an arbitrary switch and corresponding light bulb. There is 1/50 chance that it will be toggled and 49/50 that it is left untouched in a single instance of an arbitrary trial. This chance is independent of the number of instances and number of trials, and also on what is happening to the other light bulbs. (I had earlier though that a single light bulb cannot be analyzed in isolation as its state at the end of a trial is thought to be dependent on what states other light bulbs are. That is a light bulb cannot be ON say 35 times, if another light bulb is ON more than 15 times in a very very freakish trial. But the average case allows us to disregard this dependence in a single trial and to find the distribution of trials of a single light bulb spread over multiple trials. This independence over large number of trials allows us to focus on state of a single light bulb.

Note that in this solution we seek the probability distribution of number of toggles of a single light bulb spread over multiple trials. Other equivalent method (which was discussed in the forum thread) is non intuitive. Here (in my reasoning) we intuitively think of what happens to a light bulb in many trials (average case calculation) and find the probability of its states.

The process followed by the single light bulb is the binomial one (as discussed - independence criterion over multiple trials allows us to model with binomial distribution) with p=49/50 (prob of no toggle) and q=1/50 (prob of toggle) and n=50. Now the question is to find the probability of this light bulb to have odd number of switch toggles at the end of a trial.  This is the sum of probabilities of 1 toggle, 3 toggle, 5 toggle etc in multiple trials. Let P(n) denote probability of n toggles. Hence Podd = P(1)+P(3)+P(5)+...+P(49)

Podd = C(50,1)q^1p^49+C(50,3)q^3p^47+.C(50,5)q^5p^45+...+C(50,49)q^49p^1

This is calculated as 1/2*((p+q)^50-(p-q)^50), where p and q are as defined.

Podd = 1/2*(1-(48/50)^50) = 0.43505710323898

Now after sufficiently large number of trials, an arbitrary light bulb is ON with probability Podd. Now the distribution ON and OFF light bulbs in each trial is again a binomial distribution. Expected number of ON light bulbs is 50*Podd = 50*0.43505710323898 = 21.7528551619

We find that on average, we expect only less than half light bulbs to be ON after a trials. This may initially seem counter intuitive - How can less than 25 light bulbs be on if I randomly twiddle with switches 50 times in a 50 bulb array? But solution is found to be correct with a simple computer simulation, with the sample code pasted below.


#include
#include

int IMAX = 50; //Global definition of Random MAX to be used in this program
int IMIN = 0; //Global definition of Random MIN to be used in this program

int main()

{
 int bulb[50],i,j,k,m; //All int variables including index are defined
double l=0,n=0; //Variables for calculation of probability
{ //We run the trial 1000 times
 for(j=0;j<1000;j++) { for(i=0;i<50;i++) {bulb[i]=0;} //initialize array before every trial
 for(i=0;i<50;i++) {k = IMIN+rand()%(IMAX-IMIN); bulb[k]++;} //random switching simulation
 for(i=0;i<50;i++) { if((bulb[i]%2)!=0) {l++;}}
}
printf("%lf",l/1000);
}
}


Now to generalize the question with an infinitely large array of light bulbs (or countable infinity of anything), we find that the Podd = (e^2-1)/e^2

What of the outcome of twiddling with a fixed array an very large number of time, say a 50 array with 1000 random switchings? What is the solution in the limit case? I give this problem to the reader who may find it trivial to solve in the above reasoning.

Hari S Warrier