Rotaisms

The questioner does not want to learn any mathematics when he asks the question "What is mathematics?".

The opposite is true: the questioner wants to rid himself of the need of learning any mathematics whatsoever. He wants to add to his conversational repertoire some brilliant answer that will permanently excuse him from any further dealings with the subject.

One cannot escape the duty of giving a nutshell answer to the question "What is mathematics?", despite the dishonesty of all short answers.

Definitions are dangerous stuff in mathematics.

I did not make my brain, but I'm helping finish it.

John Updike, in his memoir Self-Consciousness,describes the act of stuttering as the process of “trying, with the machete of the face, to hack my way through a jungle of other minds’ thrusting vines and tendrils.”

According to Updike, however, his stuttering wasn’t just an aggravating mental hiccup which got him teased in school. Instead, the affliction was responsible for his lifelong interest in words. He was hurt into writing

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Though I do not really like giving entrance exams, I'm now compelled to do so. But that is tolerable, given the little torture that the United States of America is subjecting me to for entry into its Eden (of mathematics). But I cannot forget what the gifted mathematician, Vladimir Arnold had to say about GRE. After writing about the way French students are made to think in a rigid way (and turn into narrow-minded folks) to solve a problem (even an incorrectly framed one), Arnold says:

"The United States has a different danger. No Russian professor is able to solve correctly the problem they give in the Graduate Record Examination, the official entrance examination for graduate studies: find the closest pair to (angle,degree) among the pairs: (time, hour), (area,square inch), and (milk, quart).

Every American immediately solves it correctly. The official explanation for the correct response (area, square inch) is: one degree is the minimal measure of angle, one square inch is the minimal measure of area, while an hour contains minutes and a quart contains two pints.

I always wondered how it is possible for so many Americans to overcome such difficulties and become great mathematicians. One physicist in New York who
solved the problem successfully told me that he had the correct model of the degree of stupidity of the authors of such problems."

I did know that there was someone named Milan Kundera. But never knew he could get this close to describing my perennial question and nearly answering it as well...

If it is not possible to change this world which does not deserve love, then what is left to us? Not to allow to be cheated. To see and to know. To know how to see.

How can I forget Feynman...

You see, one thing is, I can live with doubt and uncertainty and not knowing. I think it's much more interesting to live not knowing than to have answers which might be wrong. I have approximate answers and possible beliefs and different degrees of certainty about different things, but I'm not absolutely sure of anything and there are many things I don't know anything about, such as whether it means anything to ask why we're here, and what the question might mean.

When I was not yet an adult, I often wondered if I have to pick up that lottery ticket at the corner of the school street. Strangely, thinking now, I wanted to have the finest juice in the world. Why? I had a peculiar instinct that juices make people strong to fight away bullies. I still do have juices and rather feel pleasant having such healthy alternatives than feel a little raise on my bi/triceps.

In any case, given the stupid I was then, it never crossed my mind why on earth lotteries are such a hit. And on the top of it, why do the poor buy up so many of those tickets. By definition, they should not. I intuitively felt because they are genuinely greedy to improve their living standards. But, I had my doubts over what they would do with the won money.

Latest research suggests the poor take a liking to lotteries as they feel poor.

“In Experiment 1, participants were more likely to purchase lottery tickets when they were primed to perceive that their own income was low relative to an implicit standard.

In Experiment 2, participants purchased more tickets when they considered situations in which rich people or poor people receive advantages, implicitly highlighting the fact that everyone has an equal chance of winning the lottery.

The study neatly illuminates the positive feedback loop of government-run lotteries. The games naturally appeal to poor people, which causes them to spend disproportionate amounts of their income on lotteries, which helps keep them poor, which keeps them buying tickets…”

So are the poor condemned to suffer this fate? Can something be done to mollify this situation?

The behavioural economist, George Lowenstein suggests:

“…states could promote and offer more games that appeal to wealthier players, such as Powerball, and not those popular with poorer players, such as instant scratch-off tickets.”

Ah! We keep on scratching those tickets and feel wealthier, while becoming lot poorer.

Lowenstein adds:

“financial institutions [can] issue investment instruments that have lottery-like qualities (for example, offered in small amounts, available at many convenient points of purchase, provide a small chance of a large upside) but offer a positive rate of return, providing the pleasure of playing the lottery without the steep cost. In many other countries “prize bonds” or other savings instruments are available that pay lottery winnings in place of, or in addition to, regular interest. Regulations in the United States have stymied the development of such offerings.”

Maybe, that is why when kids win on TV, they are given certificates to be redeemed when they grow up. But, are we the really grown-ups any better “rationally”?

“Philosophy is to be studied, not for the sake of any definite answers to its questions since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind also is rendered great, and becomes capable of that union with the universe which constitutes its highest good.”

Thus, philosophy does not give definitive answers. It was a profound discovery of the 1930’s, that even a highly precise subject like mathematics, can also pose seemingly accurate questions for which there are no mathematical answers. One of the greatest achievements of 20th century mathematics is a fundamental theorem of Godel. This theorem is set against the background of what we would call logical thinking and deductive reasoning. When we speak about the truth of a mathematical proposition, we imply that it can be deduced by applying a set of “rules”, if you will, or to be more accurate, axioms, to a proposition which you have deemed to be known, to derive the new proposition.

Godel’s theorem tells us that reasoning has limitations in the following sense. In any axiom system, one may write down propositions which can neither be proved or disproved. In other words, they cannot be derived by applying the axioms, nor can their negations be so derived. In the light of G¨odel’s theorem, is the question of arriving at all knowledge placed before us by the Upanishads a valid one? Can we even rely on reasoning to achieve this purpose? This question can be addressed in various ways.

Reading through the Upanishads, it becomes clear that detailed knowledge is not what is meant when one asks for all knowledge. Rather, one implies the essence of knowledge, or the underlying principle of knowledge. To this end, inquiry is a powerful and viable tool, not to be abandoned, even though its limitations are acknowledged in the Upanishads. In addition, language and even the mind are acknowledged as limited for this experience. The claim is made that mind must be transcended through mind by a refinement of inquiry.

Way back in 2005, I first heard of a wobbly-sounding word, phenomenology. Sounded like too much of a thing to swallow; big names bombarded my brain: Edmund Husserl, Franz Brentano, Paul Ricouer, etc. Plainly, it is the study of phenomena which color our conscious experience. But the finest practitioner of phenomenology seems to be Gian-Carlo Rota. Because Rota maybe the only thinker (and definitely, the only mathematician) who “strove to keep his eyes wide open and then tell it the way he saw it – without pretense and often without prejudice. With wit and flair”. That very well sumps up what phenomenology is about and strangely, through the man who best practiced it.

Early this month, I visited a city north to where I live for the first time. Purpose: to do something for the tribal population who are right now living on the cusp of change, so to say. Not yet “modern”, but sufficiently smart to live life their way. They are amused by entertainment: every other week they have some festival to celebrate. Either it is the change in weather, onset of a season or plain, birth of a new animal. Given this appetite for entertainment, it is an easy guess what they would pick up from the modern amusement arsenal. Yes, it’s the bloody idiot box. But, on their terrain, there’s no power. Tribals have even found a way out: they run TVs on batteries. Honestly, I never would have been so innovative.

By the way, they are also fine singers and hummers, with ease. Their sonata resonates through the ears very well and hits the right chord, often. Often, their taxonomy of animals they live with is classified according to the chirps, crackles, hums, buzz, rattles, squawks and many more.

There were many stunning scoops in the air. Tribals do not milk the cows as they think all the milk belongs to the calf. Quite different from my misguided sense of all that is in the world is meant to be used by me. Of course, polygamy is prevalent and wives are bought and sold. I reserve my comments.

I was reminded of a semantic controversy that was on the air recently. What should we call these people: isolated (to be neutral) or uncontacted (a neologism); or ‘previously unknown’ or ‘recently discovered’? Uncontacted it seems was the term preferred by Survivor International, a group that documents and helps such people survive. But “the term plays into the hands of powerful actors who wish to deny that these groups exist at all. The public can become confused when they learn that the tribes are not really ‘uncontacted,’ in the strict sense, and think then that the groups don’t even exist or don’t need protection.

Unfortunately for the isolated groups…not only live in rainforests with prized trees, some also sit atop oil reserves…”

“In truth, our reactions to and perceptions of these people reveal far more about us than about them. We easily believe that a band of hostile [tribes] confronting an airplane from a clearing do so out of ignorance and fear. But the likely truth is harder to face: The tribe might have threatened the observers precisely because they had encountered some of the worst aspects of our culture before, and suffered grievously. These images of a people courageously standing against us are not symbols of their ignorance, but of ours.”

Why can’t we just plainly accept that we are far more ignorant than we assume we are?

Wowah! Dirac has this wonderful insight (no wonder)… Of course, this insight, or hypothesis as they would call it, has not found takers in mainstream physics. But, the boldness and even soundness with which it has been framed by no other than Dirac merits attention.

"The large numbers hypothesis concerns certain dimensionless numbers. An example of a dimensionless number provided by nature is the ratio of the mass of the proton to the mass of the electron. There is another dimensionless number which connects Planck's constant and the electronic charge. This number is about 137, quite independent of the units. When a dimensionless number like that turns up, a physicist thinks there must be some reason for it. Why should it be, well, 137, and not 256 or something quite different. At present one cannot set up a satisfactory reason for it, but still people believe that with future developments a reason will be found.

Now, there is another dimensionless number which is of importance. If you have an electron and a proton, the electric force between them is inversely proportional to the square of the distance; the gravitational force is also inversely proportional to the square of the distance; the ratio of those two forces does not depend on the distance. The ratio gives you a dimensionless number. That number is extremely large, about ten to the power thirty-nine. Of course it doesn't depend on what units you're using. It's a number provided by nature and we should expect that a theory will some day provide a reason for it.

How could you possibly expect to get an explanation for such a large number? Well, you might connect it with another large number - the age of the universe. The universe has an age, because one observes that the spiral nebulae, the most distant objects in the sky, are all receding from us with a velocity proportional to their distance, and that means that at a certain time in the past, they were all extremely close to one another. The universe started quite small or perhaps even as a mathematical point, and there was a big explosion, and these objects were shot out.

The ones that were shot out fastest are the ones that have gone the farthest from us. That explains the relationship (Hubble's relationship) that the velocity of recession is proportional to the distance, and from the connection between the velocity of recession and the distance we get the age when the universe started off.

It's called the big bang hypothesis. There is a definite age when the big bang occurred. The most recent observations give it to be about eighteen billion years ago.

Now, you might use some atomic unit of time instead of years, years is quite artificial, depending on our solar system. Take an atomic unit of time, express the age of the universe in this atomic unit, and you again get a number of about ten to the thirty-nine, roughly the same as the previous number.

Now, you might say, this is a remarkable coincidence. But it is rather hard to believe that. One feels that there must be some connection between these very large numbers, a connection which we cannot explain at present but which we shall be able to explain in the future when we have a better knowledge both of atomic theory and of cosmology.

Let us assume that these two numbers are connected. Now one of these numbers is not a constant. The age of the universe, of course, gets bigger and bigger as the universe gets older. So the other one must be increasing also in the same proportion. That means that the electric force compared with the gravitational force is not a constant, but is increasing proportionally to the age of the universe.

The most convenient way of describing this is to use atomic units, which make the electric force constant; then, referred to these atomic units, the gravitational force will be decreasing. The gravitational constant, usually denoted by G, when expressed in atomic units, is thus not a constant any more, but is decreasing inversely proportional to the age of the universe.

One would like to check this result by observation, but the effect is very small. However, one can hope that with observations that will be made within the next few years, it will be possible to check whether G is really varying or not. If it is varying, then we have the problem of fitting this varying G with our previous ideas of relativity. The ordinary Einstein theory demands that G shall be a constant. We thus have to modify it in some way. We don't want to abandon it altogether because it is so successful.

…in theory, there will be a maximum size [to this universe]. This maximum size, expressed in atomic units, would give a large number which does not vary with the time. Now, all large numbers [are] to be connected with the age of the universe so that they will all increase as the universe gets older. If you have a theory giving you a large number, of the order of ten to the thirty-nine, which is constant, you must rule out that theory." So, it doesn't make sense that the universe is expanding and at a certain point, will contract and get back to its original state of a cloud of gas -- a 360-degree cycle so to say or the myth of uroboros, the snake eating its tail.

The consequence is that this would not lead to SINGULARITY!

Singularity has become really notorious after the futurist inventor, Ray Kurzweil who projected that computers will match human brain power by around the year 2030, opening the way for a rapid merging of electronic and biological intelligence. When would that be? Ray hints a few years down in 2045 and by the way, he is hitting 100 this year!