Anecdotes -II

4. More Paul Erdős

a) On one occasion, Erdös met a mathematician and asked him where he was from. "Vancouver," the mathematician replied. "Oh, then you must know my good friend Elliot Mendelson," Erdös said.

The reply was "I am your good friend Elliot Mendelson."

b) He had the habbit of phoning fellow mathematicians over the whole world, no matter what time it was. He remembered the number of every mathematician, but did not know anybody's first name. The only person he called by his Christian name was Tom Trotter, whom he called Bill.

c) This one's definitely a fabricated urban legend, but what the hell :-). There was a storm with thunder and lightening. Little Paul Erdos was in bed, frightened and fretting and his mother couldn't calm him. Then, as mothers seem to instinctively do, she found the right words. "It's all right dear", she said, stroking his shiny head, "there's always a prime between n and 2n".

After that, little Paul drifted off into a blissful sleep.

5. This is a story that I remember reading in E. T. Bell's excellent book 'Men of Mathematics'

This story is about the number 2^67-1, the 67th Mersenne number (Numbers, Mersenne had claimed to be prime, which was proven to be non-prime in 1903 by Frank N. Cole (1861-1927). In the October meeting of the American Mathematical Society (AMS), Cole announced a talk "On the Factorisation of Large Numbers". He walked up to the blackboard without saying a word, calculated by hand the value of 2^67, carefully subtracted 1. Then he multiplied two numbers(which were 193707721 and 761838257287). Both results written on the blackboard were equal. Cole silently walked back to his seat, and this is said to be the first and only talk held during an AMS meeting where the audience applauded. There were no questions.

It took Cole about 3 years, each sunday, to find this factorisation, according to what he said.

6. The mathematician G. H. Hardy was to give a keynote speech at a conference. Asked for an advance summary, he said he would present a proof of the Rieman zeta hypothesis -- but they should keep it under their hats. When he arrived, though, he spoke on a much more prosaic topic. Afterwards the conference organizers asked why he said he'd talk about the theorem and then didn't. He replied this was his standard practice, just in case he was killed on the way to the conference.

It was part of his tactics against God - in that he thought God would not allow him to die on the sea trip, because then everyone would think that Hardy had solved this great theorem. Hardy had other anti-God tactics, including always taking an umbrella, and some grading or other boring work, with him to the cricket games. For an athiest Hardy certainly spent a lot of effort against God.

Apparently Hardy's ambitions were:

1. Prove the Riemann Hypothesis
2. Score the winning play in an important game of cricket
3. Murder Mussolini
4. Prove the non-existence of God
   

Selfish to the core - Part 3

Many believe that the living world would have originated as a chaotic society of tiny molecules constituting what is referred to as the primeval soup. Among these molecules arose a particularly formidable character called the replicator (which we have referred to earlier) whose uniqueness lay in the fact that it could make copies of itself and spread through the primeval soup. The DNA of today, in character, is an example of a replicator that survived over the ages and functioned as the basis for the synthesis of higher organisms. There would have been many replicator molecules that would have fought each other over dominion of the natural world over the ages because common and limited resources for survival necessitated competition. Some replicators would have come together and recognised prudence in complicity against others in the soup thus forming even bigger replicator molecules. Indeed, the DNA that we find today in living organisms may be very much unlike these original replicators that existed and propagated in the primeval soup, but the principle on which it functions and propagates has not changed. As Dawkins puts it succinctly, the three characteristics that a replicator must possess if it has to survive in the natural world are longevity, fecundity and copying-fidelity. Is there a basic motive (purely functional in nature and nothing to do with consciousness, as defined in Part 1) at a more fundamental level of which the three aforementioned properties of a replicator are direct consequences? The DNA of any living organism in the world is a map of its psychobiological characteristics. Living beings have evolved over ages up the ladders of natural selection- able bodied organisms of today from the chaotic molecules of the primeval soup through extremely elaborate evolutionary processes over many millions of years - and it is known that it is the replicator (or simply put, the DNA) indeed that holds the masterplan over generations. Thus, can we identify a suitable 'motive' for these replicators that will unveil the process of evolution through the eyes of natural selection?

In this confessedly small exposition (though I realise now that it is far from that), I suddenly find myself juggling at a single place with a number of concepts that were developed over three elaborate chapters in "The Selfish Gene". But I have, to the best of my abilities, tried to maintain a logical chronology of thought in the way I that I received these ideas albeit putting them through a lot of condensation. Coming back to where we were, I had mentioned in my last post that a DNA was divided into sections called the chromosomes which in turn were further divided into genes. Thus, any modern replicator is nothing but a conglomeration of genes. A more enterprising and indeed interesting way to look at it is that it were these unitary genes that were the original replicators in the primeval soup. The properties of longevity, fecundity and high copying-fidelity that were mentioned earlier are indeed the properties that genes would desire if they were to pass the test of survival in nature. A gene, if it has to survive in the gene pool, has to ensure that its copies spread faster than any of its rival genes competing for the same resources. Complicity with other genes may be a really smart option and we shall come to this when we discuss 'survival machines' in the next paragraphs. Fundamentally though, the natural propensity of a gene must be to act for its own interests. Even complicity with other genes, that may seem altruistic at the surface are really acts of forwarding self-interest, if investigated more closely. This is, simply put, the principle of gene selfishness or the 'gene centric view of evolution' or 'gene selection'. The fundamental 'motive' that we were talking about in the last paragraph is indeed selfishness in the sense that a gene must try and propagate its kind in the gene pool at all costs. Let me repeat once again that this selfishness is not a conscious motive and that the reader should not make a conclusion that genes are ruthlessly malevolent entities. Nor is this statemet a harbinger for my crescendo statement where I would issue a coup de grace on humankind as being selfish thanks to the selfish genes that inhabit them. No , I am neither a devil's chaplain nor (or even worse) an evangelist in disguise.

The terms selfishness and altruism that appear here are purely technical definitions- biological motives independent of consciousness and very similar to computer algorithms at the very basic level of the constitution of the living. Selfishness of a gene encompasses all those actions that enable it to spread its copies in the gene pool, even if this required it to inhibit the spread of other genes especially its rivals that are called alleles. Pure Altruism on the other hand refers to acts that demand the gene to facilitate the spread of copies of other genes (rivals or not) at the cost of inhibiting its own spread. There have been instances (the example of Jonathon Livingston Seagull in Part 1), where acts of seeming altruism by an organism on the surface is really selfishness at the core- gene selfishness. The basic statement of the theory of 'gene selection' is that at the fundamental level, genes are basically selfish and this selfish nature empowers them to serve as units of natural selection in all living beings along the evolutionary chain. Can we support this hypothesis with observable evidence from the natural world - observations that cannot be explained under the tenets of 'group selection' or 'individual selection' theories of evolution?( to be continued and concluded)

Selfish to the core (part 2)

As nature would have it, it seems that natural selection really takes place at the macromolecular level of the genes that constitute us. Before trying to exposit the workings of a gene as the de facto unit of natural selection, I shall define it for the benefit of the reader. The bodies of living beings are really macromolecular machines, in the sense that all living beings are made up of one or more cells that contain different types of molecules performing various functions. Now what grants each one of us our respective individuality is a particularly long molecule called the DNA. The DNA is like a blueprint to the psychobiological construct of the living organism that it inhabits. Each organism has a unique DNA which may differ relatively less from a corresponding DNA molecule of another member of its species but will nonetheless differ more when compared to the DNA of an organism belonging to a different specie class. This molecule is unique in many aspects, the foremost being that it belongs to a very remarkable class of molecules called the replicators. Such molecules have the ability to make copies of themselves pretty rapidly. This makes them ideal choices as 'blueprint archives' inside an organism, for a DNA can make numerous copies of itself that can get distributed throughout the cells in a body. Thus each part of the body can have in its possession, a consistent set of rules that correspond to various behavioral and physical characteristics of the individual.

Now a DNA itself is divided into pairs of macromolecular sections that are called the chromosomes. There are 23 such chromosomal pairs in the case of humans making it a total of 46 chromosomes that constitute what is referred to as our genome. Other plants and animals also contain chromosomes of their own but in a number different from that of humans. Now these 46 chromosomes are like 46 volumes of formatted and indexed information that constitute the 'blueprint' corresponding to an individual living machine. During sexual reproduction (in most cases) the offspring inherits half of its 46 chromosomes from its mother and the other half from its father. There is a reason we find it more convenient to refer to them as 23 chromosomal pairs rather than 46 chromosomes and the reason is as follows. The two volumes in each pair contain alternative instructions for a particular function or motif. If we suppose that pair number 16 contains information about 'the color of the eyes', 'the shape of the ear', ' literary aptitude' among many other things, then volume 16a (inherited from the mother) might contain the instructions 'brown', 'small' and 'Shakespeare' and 16b(inherited from the father) might say 'green', 'long' and 'George W. Bush Jr.'. Only one instruction in each case can be selected ( I exclude the case of multiple personality disorders for the sake of brevity :P) and the one selected is referred to as dominant. The alternative that is not selected is referred to as recessive and though that part of the chromosome does not perform any function for the individual in which it resides, it may be passed on to future generations.

Now that I have (to the best of my engineering abilities) provided a working definition of the DNA and the chromosome, let me try to do the same for a gene which is the pivot of all my purpose here. I have myself come across two different definitions of a gene which though not definitely contradictory, are not necessary equivalent either. One definition is that a gene is that unit of a chromosome which corresponds to the synthesis of a particular protein or enzyme. For example when the body is in need of glucose and there is lactose present in the environment, a particular gene in the body is instructed to synthesise an enzyme called beta-galactosidase, which is responsible for converting the lactose into usable glucose. The other definition of a gene is that it is that part of a chromosome which corresponds to a particular bodily function or characteristic of the individual say colour of the eyes or height. But in order to go ahead with this small exposition, I shall state Richard Dawkin's definition of a gene which he believes is a working definition for the gene. But that and my last bit on gene selfishness and gene selection in the next and concluding post.

Note: Though pretty late and pretty ungrateful to provide a disclaimer, I shall nonetheless go ahead with this one. All of what I have written is based upon my understanding of most of what is written in one book, little of what is written in a second book and almost none of what is written in a third book. These books are (in order) The Selfish Gene by Richard Dawkins, On Aggression by Konrad Lorenz and The Origin of Species by Charles Darwin. The first among these is highly recommended for a fantastic exposition on the 'gene selection' theory. It is succulent with multifarious examples of animal behavior and each point or claim made is substantiated with an extremely articulate line of reasoning. The second was referred to extensively in the first and I chanced upon it in my institute library. I didn't have patience to read more than a couple of chapters. The third book is an undisputed classic and an inspiration to many ethologists and biologists after its author and its eulogised importance compelled me to buy it from Crossword (many thanks to penguin paperback for bringing the Classics to the masses). But I found the arcane english prose extremely dry and soporific in parts. Plus, the very size of the book was intimidating and I chose suicide over enlightenment. But I do hope to evolve into a formidable reader myself someday so that I can confront that book. After all ,Darwin is the master of all braggarts mouthing evolutionary innuendo, isn't he? ;).

Selfish to the core

Our understanding of life and life processes in the natural world has been maximally contributed by Darwin's theory of evolution. Since the days Darwin had impressed upon humankind his articulate and extremely well researched yet elusively simple theory, much of future investigations in the life sciences have only reinforced his original idea. It is conventional wisdom today to know that we humans have evolved from the apes who in turn had evolved from other primates before them. Every other organism inhabiting the natural world is supposed to have evolved out of its primeval counterpart(s) and is known to possess physical and social characteristics that enable its kind to propagate through time. 'Survival of the fittest' is a fairly common term that appears in high school textbooks in chapters that expound Darwin's theory of evolution. Though scarcely anybody today would challenge 'evolution' as a principle characteristic of our construct, the underlying processes that evolution entails have not yet been clearly understood.

Most famous approaches to answering this problem hypothesise a basic biological motive that translates itself into the behaviour of the organism, pervading its consciousness and its actions. For instance, if 'nourishment' is a self-motive that has been programmed into my biological structure, then that would cause me to eat to my fill rather than choosing the alternative to starve. Of course, this is only a crude example to illustrate a 'biological motive' and a much learned lateral thinker would immediately point out the fact that if 'nourishment' was not a motive and 'non-nourishment' was, then life would end before it even began! But 'seeking nourishment' is a behavioral need that has evolved out of 'natural selection', considering the fact that our bodies would violate the second law of thermodynamics if they lived healthily without a balanced and nourishing diet.

The example of 'seeking nourishment' as a biological motive to explain 'natural selection' is perhaps too trivial, almost to the point of being a misplaced illustration. But I believe it will facilitate an understanding that while 'natural selection' itself is an indispensable weapon for adaptation and propagation of life, every physical or behavioral motif cannot be so easily linked with natural selection as easily and trivially as 'seeking nourishment' can. For example, straight and sinewy beaks are 'natural selection' for woodpeckers but would be useless for a parakeet, for which 'natural selection' bequeathed a short and curved beak (almost like a betel-nut cracker) that makes it easy to consume a fruit.

Physical characteristics may perhaps be easily correlated thus but theories on 'natural selection' take a beating when it comes to explaining behavioral characteristics. This is because 'sinewy beaks', 'prehensile tails', 'sharp incisors' and 'seeking nourishment' may be understood as natural selection at the most basic levels for the respective organisms. But not every behavioral aspect can be linked to such a basic need so easily and so trivially. For example, it is found that if a gull sights a predator hovering near their communion, it gives out an alarm call that sends across a message to the other gulls to evacuate/take cover immediately. But this act attracts the attention of the predator onto itself, thereby posing a threat to its own life while it can be sure that none of the other gulls are going to pull out any acts of bravado as an acknowledgment to its altruistic act. This was just one example out of many. Behavioral patterns can also be observed not just at the level of the community, but also of the family. Why does an animal feel closer to its offspring rather than to its second cousin? Why do middle aged women attain menopause whereas there isn't a similar conspicuous event in the lives of men? The question we need to ask with regards to 'natural selection' is the following:

"What aspect of our psychobiological construct governs and causes the process of natural selection unlocking the mechanism of which will enable us to understand and interpret the observed characteristics of living organisms?"

Darwin and his apostles said that it was the 'species' that governed natural selection. In other words, each member of the specie would possess physical characteristics and behave in such a way so as to benefit the species as a whole. Let me digress here and enunciate that the word 'behave' doesn't come out of any diktat that is tutored to each member of a specie, a code which he/she follows out of his/her consciousness. It is similar to a computer program with certain rules that is engraved into the biological construct of a particular specie and if the rules were the opposite of what are observed in nature for a particular specie (e.g. a deer attempting to be a carnivore), then (the biological construct being the same) the organism or others of its kind would be unable to survive. Once again, let me add a disclaimer that I don't mean to suggest we are all mechanical beings devoid of consciousness. The 'rules' I am talking about at this point are really very basic ones which govern our behaviour through the eyes of natural selection which is necessarily not antagonistic to consciousness. For example, walking in a sandstorm would make a man immediately shut his eyes (an instinctual response to a stimulus which doesn't evolve out of a conscious decision making process - natural selection for his benefit) but he can very well keep his eyes wide open and blind himself, thereby asserting the power of his consciousness and his stupidity.

Coming back to where we left, the idea that sought to explain Darwin's theory of evolution was that of "Group Selection" or "Species selection". For a long time, "Group selection" was thought to be synonymous with natural selection. The Group selection theory essentially states that members of a particular 'species' behave so as to ensure that the numbers in the specie are preserved and that they propagate. The altruistic act of martyrdom by Jonathon Livingston Seagull might seemingly appear to support this theory. Many behavioral characteristics of other species like the honeybees, baboons, ants may also appear to be easily explained by the group selection theory.

Group selection may seem to account for the reasons behind observed groupism and comradeship between animals. Nonetheless, the theory falls apart when we look at other behavioral instances of animals. Macabre cannibalism has been observed in the cases of black-headed gulls (which feed on younglings of a neighbour as soon they hatch) , the black widow spider and certain mantises (where the female feeds on the male while they mate). While the group selection theory may be able to account for observed altruism for the community (or the species) as a whole, it cannot account for the aforementioned examples of brutal selfishness. Scientists have also proposed 'Individual selection' as a foil for the group selection theory, but even this theory faces problems of its own. We shall not delve into this one other than stating that today, neither 'group selection' nor 'individual selection' is favoured as a complete theory by biologists or ethologists across the globe.

Well then, what next? The problem or predicament if simply stated, is that of identifying a suitable entity that can serve as a convenient unit of natural selection, something which 'the group' or 'the individual' failed to do. Richard Dawkins and a few other thinkers like Pinker and Dennet propose(d) that selection takes place at the genetic level. In other words it is 'gene selection' as opposed to 'group selection' or 'individual selection'. Now then you would say that's obvious! But I believe you'd be confusing gene selection with gene evolution. The view of the 'gene' as the most basic unit of evolution has been long legitimated by many scientists but the debate as to whether it is indeed the unit of selection is an ongoing one. More about genes, gene selection and gene selfishness in the next post.