The axiomatic method: its origin and purpose

This article is written by Prof. S. D. Agashe. I greatly enjoyed it when I was a graduate student in department. I hope, some of you will enjoy reading this too.

Euclidean geometry and the axiomatic method

Euclid’s Elements constitutes the earliest extant substantial presentation of a body of material in the axiomatico-deductive form . Through it the subject of geometry got permanently associated with axiomatico-deductive formulation which was then viewed as a method, so much so that the expression ‘more geometrico’ (the geometric way) became synonymous with axiomatico-deductive formulation. Thus arose the general belief, especially in methodological quarters, that Euclid’s Elements and, in particular, Euclid’s geometry were merely instances of the application of a previously thought out/discovered/known method, and, thus, that the axiomatico-deductive method existed prior to the axiomatico-deductive formulation of geometry. Using Euclid’s Elements as my principal evidence, I want to suggest that the true state of affairs is the other way round. The axiomatico-deductive formulation of geometry emerged out of a successful attempt- most probably by some of Euclid’s predecessors – to solve some geometrical problems. Once this was done, it was seen by these geometers and also, of course, by Euclid as an instrument of open-ended discovery. Only, then, could the germs of a method be seen in it. My view of the genesis of the axiomatic method emboldens me to suggest further that in general a method, which is something consciously conceived, arises as the result of reflection on an activity that is already being pursued ‘intuitively’. Again, once the method is consciously conceived, it can engender new activity being pursued consciously in accordance with the method, i.e. methodically.

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Primitive Network Theory of Human relationships

We intend to differentiate males and females. In fact, we are going to make them complementary to each other. Surely, this is an oversimplification for both of them exhibit certain behaviors common to each other. This is more or less an Utopian view of human race. This is not to say that ‘if men are from planet X then women are from planet Y.’ We only say that both are from the same planet but given an option would chose different directions.

Humans are not complete without emotions. In fact, these emotions make them impossible to define in plain and simple terms. Here, we categorize emotions into two sets. When an action of a single human effects an individual emotional state and creates an emotion, we say it is an ‘individual induced emotion’ (IIE) e.g. jealousy or envy after seeing someone’s car, hate and love for an individual, anger towards an individual because he/she has done or said something etc. When an emotion is a byproduct of an action of a large group, we call it ‘group induced emotion’ (GIE) e.g. Hate for Muslims after 9/11, caste based discrimination in India, hate for alien cultures, envy for some nation’s wealth, respect for a brave race etc.

Similarly, we also classify actions. Any individual action which causes IIE (or a GIE is a very small group) is classified as ‘individual oriented actions’ (IOA) e.g. showing off wealth in neighborhood, shopping useless but fancy products, public display of affection etc. If an action by an individual is done to influence or create GIE is called ‘group oriented actions’ e.g. actions of politicians, saints, intellectuals, scientists etc.

We are confused where to put humor. We consider it as ‘zero’ element of our emotions. It is present in both GIE and IIE. The purpose of ‘humor’ is also not well defined. We can not say why individual uses humor while making an argument and why people who are humorous can say many things which they would not dare say in plain terms.

Lemma 1 : A human male is like a capacitor. 

Capacitors are usually not coupled. Charging and discharging of one capacitor does not affect the charge on other capacitor (unless they are coupled). This is to say that a male in a group is not at all interested in others IIE. He is mostly interested in GIE for his or his group benefits. A group of men can live together without causing IIE. This is an oversimplification.

Lemma 2 : A female is like a inductor.

Inductors have this property of ‘mutual and self inductance’. They are mostly coupled to each other. Any instant of charging or discharging of an inductor changes the state of other inductors. This is to say, in a group of females, any action of one female emotionally influences females in group. They causes IIE. This is to say, they mostly talk about personal stuff in for the sole purpose of enhancing or reducing IIE’s of the member of groups. This is also an simplification.

NOTE:  Group induced emotions such as motivation, inspiration, patriotism (love for fellow citizens), nationalism (hate for foreigners), enlightenment, humor are caused by GOA. GOA changes ‘intellectual emotions’ (if I can say it that way) while IOA targets IIE (‘personal emotions’). We are not saying that female only communicate in a way which only causes individual effects. We meant to emphasis on the fact that one can clearly see the females have inclination for this kind of behavior.
 
Lemma 3 : A relationship is ‘oscillation’ in a circuit formed by L and C (we are not considering R right now). Strength of a relationship is the frequency of oscillation. 


Theorem 1 : A capacitor (inductor) can be converted into a inductor (capacitor) by using a gyrator.

Proof can be located on internet easily. This is to say, with some extra circuitry called gyrator we can make a capacitor behave like an inductor and vice versa. Equivalently, a male can behave like an female and vice versa, with the help of some ‘means’ (gyrator). To keep this blog short, we leave this undefined.

Definition : A lasting ‘relationship b/w male and female’ is like sustainable oscillation in a capacitor-inductor circuit without consuming any power.


Well, to  understand this one has to take basic course in ‘Network theory’. We assume that you have taken it. Now, we restate the common wisdom in our language.

For every capacitor (male) , there exists a inductor (female), and together they form an LC circuit (a relationship) which oscillate at its natural frequency (happily ever-after) without consuming any power (with no compromise on anyone’s part). Otherwise, the oscillations (relationship) dies out after certain time. To sustain oscillation in imperfectly matching LC circuit, one has to introduce a power source (work-out on a relationships).

Its almost impossible to define a matching pair. Once defined, finding is not a tough task. Existence of such pair is mathematically guaranteed in our theory.  According to this blogger, if one chooses to form a relationship, one can look out whether they have a common and mutual understanding. If yes, then the oscillation can be sustained with little or no effort (with a weaker power source).

We still have not touched the ‘resistor’ (externalities or friction in a relationship). We now stop building this theory. We may come back to it some other day. May be someone else might like to build it up. I am convinced that one can not go very far with it. It is doomed to fail because ‘no human behave like a circuit element’. Anyway, you get the ideas.

PS : My friend pointed out that nothing changes if I consider males as L and females as C. I was thinking of considering children as small coils. Since mothers have much more influence on their children, this is only possible if ‘females’ are inductors else coupling is not possible between them. On this, she called me crazy :-p :-(


Dilawar  

Properties of Love and myth of ‘Love Triangle’

In this blog, we explore primitive mathematical properties of of love. Here we are only concerned with love between male and females. Among other things, we shall show that concept of ‘love triangle‘ is a myth unless one allows homosexuality.MATHEMATICAL PRELIMINARIESLet \mathbb{H} be the set of all humans. Let \mathbb{M} \subset \mathbb{H} is the set of all lovable Males (definition?) and let  \mathbb{F} \subset \mathbb{H} be the set of all lovable females . It follows that  \mathbb{M} \cup \mathbb{F} \subset \mathbb{H}. Sexual love, is a binary relation. Nothing more is known for sure. Sexual love is mostly depicted in movies and romantic fiction etc. Lets denote it by \heartsuit. A \heartsuit B says that ‘sexual love’ exists between A and B. A love triangle is a construction which has three vertices and ‘sexual love’ exists between any of its two node. If we draw vertices as a dot and ‘sexual love’ between two vertices as a line, it will closely resemble the triable which we all are familiar with since school days. A \heartsuit B implies that A and B belongs to different gender i.e. if A \in \mathbb{M} then B \in \mathbb{F} etc. In plain English, homosexuality is not allowed. If love is non-sexual e.g. love between mother and her child, between math and mathematician, we’ll denote it by $\clubsuit$ but we are NOT going to talk about it in this blog since it is written on Valentine day.

PROPERTIES OF \heartsuit

But by any distorted intuition or otherwise, LOVE (whether sexual or not) can be considered a ‘weak form’ of an relationshi. Lets settle on this and try to explore its mathematical properties. And hope that there is any hope to construct any algebraic structure under \heartsuit.

Property 1: A \heartsuit A i.e. one loves oneself. It is known as Reflexivity.

This is generally not true. We will assume that this is ‘not’ true.

Property 2: If A\heartsuit B then B \heartsuit A i.e. if A loves B then B loves A. It is known as symmetry.

Going by most accounts and life experiences, it is also not always true except for movies and
fiction.

Property 3: If A \heartsuit B and B \heartsuit C then A \heartsuit C. It is known as transitivity.

This is ridiculous. We’ll give a proof of it.

Proof :

If A \heartsuit B means A is male and B is female. B \heartsuit C makes C a male. Now A \heartsuit C is not possible unless one allows homosexuality since both A and C are males.

Now we can state a theorem.

Theorem :   Sexual love triangle (love triangle) does not exists.
Proof :

Proof directly follows from Property 3. For being a love triangle, transitivity must hold and this is not the case. Without a loss of generality, We can assume that A \in M. Now if A \heartsuit B is true then B \in F. Further if A \heartsuit C then, C \in F. In this case A \in M while B \in F, C \in F. If A \in F then, using the same argument it can be proved that, B, C \in M. In either case two person belongs to a same gender. A contradiction!

A more general result which can be found in many situations is that if there is a continuous switching between two states then any three consequent states comprises of two same states. e.g. Take this state machine, in which, at every event, a odd number changes to even number or even number to odd number, like, 1,2,3,4,5,6,7,8,9,10 … Take any three consequent numbers you’ll get either 2 odd or 2 even numbers.

If above three properties are true then the relation is called equivalence relation. For \heartsuit, none of them are true in most of the cases. But still that much gives me to conclude that, “Since the equivalence relations are way to strong, one may try to bring these properties into his/her \heartsuit for forming a nice relationship. Property 3 (i.e. transitivity) could be seen differently by different people, and in my opinion can be relaxed to accommodate your love-partner reservations about it. Nonetheless, first two properties are must for a strong and ever lasting relationship. One more interesting point is that property 1 is person dependent. It takes only you to build it. Property 2 is like “It takes two to tango’ and Property 3 is dependent on social context. All three environments, personal, bi-personal (or bi-sexual) and social environments are important and ought to be dealt with.

Then one may ask, why bring mathematics in. By common consent, this is true that in Math, cheating is not allowed. This is the most honest subject around known to humanity. Second, its the curiosity which makes a human different from donkeys.

If you have no girl/boy to kiss of today (Valentine day), you would like to do this exercise. Denote binary relation ‘hate’  by \spadesuit and explore its properties. Does it have stronger mathematical properties? Does Hate Triangle exist? Send your answers by email to this author. Also prove/disprove that, IITians, Times of India and United States form a love triangle under ‘nonsexual love’. Email me/comment your solutions.

Dilawar 

Unique phone numbers in India

When I posted mine and other teaching assistant’s mobile numbers on IITB Moodle and mentioned that my number has two prime factor, in case you wants to verify; and others should be right since I copy and pasted them, Srihari  replied back,

Sunny got 3 prime factors and you 2 as you said..while achin had 6..
now i started to wonder how many prime factors on avg for 10 digit telephone number (starting with 9).. any clues?
sorry for spamming.

Well, unfortunately these kind of emails are considered spam in Indian Universities. But anyway, here is what I am able to reply,

Achin number is then probably worth envy. One can do the simulation. But theorem proving will be good. The maximal is 2x3x5x7x11x13x19x23x29 = 6469693230. It has ten distinct prime factors. But it does not start with 9. And another multiplication will make it 11 digit. If we drop one of the prime in this list, we can check whether there exists another prime which can be used as replacement. In fact, 41 is a prime which can do that. Drop 29, and multiply by 41 we’ll get 9146807670 and 9592993410 with 43. What about other numbers with 10 prime factors? If we drop two primes e.g. 23 and 29, then product of rest becomes 9699690. Then we need two distinct prime such that their product is somewhere between 928 and 1030 (else the total product will not have 9 as its first digit). There aren’t any! SO THERE ARE ONLY TWO NUMBERS WITH TEN PRIME FACTORS. I called them up. First one is in Maharashtra (it was switched off) and second one is in Haryana (He did not know what the hell is a prime number so he gave the phone to his brother, he also did not know and I did not explain.)

One can find many numbers with 9 prime factors by dropping 23 and multiplying by others (this makes 223092870 one of the mothers of all such numbers). This mother have, in addition to the above two, following children,

9369900540
9592993410
9816086280

One sure can take clues from it. I have dropped 29 and multiplied with other numbers (41, 42, etc). You can drop others and figure out. They should not be more than 4 children for each mother (proof?). I can establish upper bound by 36 (surely less than that). Now since the problem is yours, you deserve the joy of solving it fully. This was a nice one by the way!! smile

So there could be at max 36 such numbers (with at least 9 prime factors, and they are surely less than 36). I wonder who are the lucky bastards possessing them?

REF :
[1] http://primes.utm.edu/howmany.shtml

Lobbying for logarithms

The trouble with spending time on a ‘good university’ (or having educated parents) is that it might make you incapable of thinking about certain ideas. If you are a historian trained in Harvard University, than like New York Times, you may not be able to think beyond the nations of Atlantis. An economist, who have spend a fair amount of his/her time in ‘Chicago-Stockholm Express’ can not digest the criticism of ‘free-market’. A typical students of Jawaharlal National University find it hard to listen to what neoliberals has to say. Indian journalists trained in Western privy league universities can not believe that poor do exists in India e.g. according to Times of India, we are living in a golden age. A typical engineer or scientist can not believe that there could be some soocial problems which can not solved by implementing some sort of technology. Fortunately, on the campus of IITB (or on any campuses which takes pride in producing ‘Whiz Kids’ rather than Bose of Raman) we do not have these kind of problems. Actually no one can accuse us of having any radical idea. Outside India, engineers have shown tremendous diversity of thoughts. From Ludwig Wittgenstein to Osama Bin Laden, we have touched upon many facets of ideas. Indeed, seeking a career in tangentially related fields have now become a norm rather than an exception on most of the engineering campuses in India. I have more to say about it, see [Appendix]

It not just universities which can blind us. We loose many faculties as we move ahead in our school education system. Here, I am going to reflect on logarithms (in fact, going to lobby for them) and their superiority over linear scales generally used in academia. In passing we’ll touch upon many seeming unrelated ideas. Our central argument is going to be based on evolution. And social construct which might have played their roles building up our predilection for a liner scale at the cost of logarithms.

Necessity is NOT a mother of all inventions? Though necessity is mother of most of the inventions in Industrial age. Most of the early inventions might have been done out of curiosity only. Their necessity only became apparent later e.g. cooking food on fire was not a necessity, some chap might have got curious about it. But invention of scales might have been a necessity. Comparison is what we do when we start seeing. To compare we need a scale. Early humans might have liked to compare few things for sure. Size of their potential hunt and meat it can provide them. Size of their social group and of adversaries etc. Comparing is an old habit. Nothing has changed in this regard. In modern times, on schools or on any universities, we are always compared. Although the methods are changed. First we were compared whether we are born in a certain class or caste, now we are compared, along with these primitive social construct, how well we perform in certain competitions and how much we earn. Linear scales are mostly employed. Lets consider something we are all concerned about, marks! On a linear scale, anyone who scored 70 in one test and 100 in the next has a growth of 30; another who improved from 20 to 50 also has a growth of 30. Should both progress be termed as ‘same’? On a linear scale, yes! Perhaps the the range of the marks is not quite large in which we can produce a visible contrast. Besides linear scales become more relevant when we compare seemingly ‘same’ entities. A class can be compared on a linear scale but a high-school student and a graduate student surely can not. How about when we have to compare different entities? I can run half a kilometer and and take Mr. X who can run 10 kilometer in one stretch. If somehow I am able to run 4 kilometer; will my willpower is ‘same’ as Mr. X should he run 14 kilometers. If yes, then this is unfair! My improvement by a factor of ‘4’ make Mr. X run 10×4 = 40 kilometer to claim the same level of ‘willpower’. What about frequency response of two amplifiers? Would you measure it in linear scale and then compare? Or pH of two solutions, magnitude of two earthquakes, level of two sounds. Then question arises which scale is more natural. Linear of Logarithm?

It is quite astonishing to note that how frequently we encounter logarithms (also pi) in natural phenomenons. One wonders where our presupposition with linear scale is misguided. Whether it is an equation representing diode characteristics, sum of an infinite series, frequency response of an amplifier, no of animals in animal kingdom (save humans) at various level of food chain, distribution or errors, definite pattern in seemingly chaos, transmission of energy in food chain, distribution of prime numbers (Prime Number Theorem), no of applicants universities can accommodate (10% for others, 1% for IIT’s), distribution of most search keywords on the Internet http://terrytao.files.wordpress.com/2011/01/universality.pdfSee (or on IITB DC, you can get the data using tcpdump utility); topic of most shared article on Times of India [2] website, pH of solutions, strength of an earthquake, level of sound, decline in interest in reading and writing (and possibly in thinking also), distribution of letters in a write-up, frequency of location of wars, neural response to pain etc. If logarithmic scale is really so natural then why we have come to be possessed by linear scale?

There is very little evidence to pass a judgment on this. Written evidence, firstly very few exists, does not say very concrete about these tendencies. However there are certain patterns which give some idea about existence of some logarithmic scales in ancient civilizations. Take for example, in old Chinese and Egypt number system they have used power of 10 [3] to represent their numbers. Oldest language of this world, language of Indus valley [4], is still an enigma and has not been decrypted. It contains in itself a treasure trove about early human civilization.


Before we use evolution to justify our stand, we have to say something about evolution. In fact evolution deserve a separate note. Its has been a major force in last century. Few ideas have influenced humanity in such a great extent. Evolution has been used and misused [5] by many. Recently many weakness of evolution has been confirmed. It is not able to explain everything. It does not explain why chicks of a hen who was put under stressful condition are also stressful and shows a tremendous decline in their learning capabilities. This disability continued in their successive generation even if their children were raised under normal condition. A small stretch of distress of mother can make her offspring capabilities limited. Take Sweden for example. Sweden used to have a pattern in which after every 20 years there was a severe lack of food. People born in these times have shown life expectancy well below from those who were born in normal times. Evolution has no clue about it. You may be in IIT not because it is in your genes but because your grandfather ate well or your grandmother of your great grandfather had a happy life. Indeed, there is a high correlation between your social standing and your achievements [Bet83] . On top of it, it is a truism that a person born in socially backward family cannot achieve the same level of success despite of having same level of natural abilities. It is quite visible among the backwards and socially deprived cast group. The stigma and inhumane practice of discrimination can make low of their self esteem. Anyone who is born in high social group but in a poor family only has to strive for resources. For example, the great Indian cricketer Palwankar Baloo (eldest and most talented among the four Palwanker brothers who were supremely gifted cricketer) had to go through in their career [Guh]. Evolution does not give any credible explanation how culture and social practices will influence an individual or a society in next 100 years.

Evolution is quite successful in explaining behaviors which have ‘evolved’ due to ‘necessity’ and ‘interests’ over a long period of time. For example take the universal phenomenon of choosing mating partners (or life partners). Robert Trivers [rob72] believes that among the mammals a ‘sex with the greater minimal investment in offspring is selected to be more choosy; the sex with the lesser investment is selected to be more promiscuous and competitive.’ ‘In the human species’, argues Steven Pinker, author of Blank Slate, ‘our mammalian physiology makes women the greater-investing sex’, though the fact that our males also invest in their offspring ‘blunts the asymmetry, and makes both sexes compete and choose,’. 6.Though using different criteria: fertility for men choosing women, ability and willingness to invest for women choosing men. It explains why men ‘have liking for female breasts’. Just due to the fact that it symbolizes how well a female can feed their offspring should he have one with her. Woman craving for ‘gossips’ can also be explained in similar terms 7 . This was the only way to gather information regarding their potential mates (foes and friends also). Their tendencies to pick a ‘dark hero’ (dark dashing outlaws know as ‘clads’ in academic circle) for short term affair and ‘romantic and proper hero’ (caring and well meaning, known as ‘dads’) for marriage and relationships can also be explained. Former can provide better genes while the later can provide nurture and a stable future to their offspring. Steven Pinker [Pin07], author of the Blank Slate has more to say on it. He has also tried to figure out why humanity crave for fiction; but be advised, he is not admired in certain academic circle for really good reasons [you] .

Now we have argued more than enough in defense of the suitability of ‘evolution’ to study language and numbers evolution. If this is right than babies, children and uneducated (or unaffected) adults should show a liking towards logarithmic scale. In fact, if you try to teach children in nearby primary school, they seem to follow linear scale up to few tens. They can give some examples if you ask them. When you insist to use abstraction e.g. ask them to use their hands to represent quantities by stretching them away from each other; they do not follow linear scale. Its not only the kids in Primary School Nichalpur who do that. Few months back a study has also confirmed it [AS10]. Apparently children in the West also tend to do the same.But adults instead use a linear scaling, in which the distance between each number is the same irrespective of their magnitude. Philip Ball, a consultant editor with Nature, reviewed this study. ‘This could be because adults are taught that is how numbers are ‘really’ distributed, or it could be that some intrinsic aspect of brain development creates a greater predisposition to linear scaling as we mature. To distinguish between these possibilities’, Seigler and his colleagues ‘tested an adult population that was ‘uncontaminated’ by schooling’. ‘The implication of their finding’, they say, is that “the concept of a linear number line seems to be a cultural invention that fails to develop in the absence of formal education”. [AS10]


Numbers and languages8 are deeply related. One can locate same parallels there also. Untouched tribal cultures of Amazon have surprised us many times.9 It was surprising enough for anthropologists to learn that there exists a tribal language called http://pib.socioambiental.org/en/povo/piraha/803Piraha in which natural numbers are totally absent. For them it was inconceivable to think a language without having natural numbers in it. They have sounds of one, few and many which they encode by souds of ‘hoi’, ‘hoye’, ‘hoyeeeiiii’ [Gor04]. Indeed, their hoi’s have an indication of thinking in logarithmic scale. For them 105 is more akin to 100, while 1 and 6 surely are different. Another well studied tribal language is of Mundurucus which does not have any exposure to linear counting scale of the industrialized world, just magnitude on log scale. 10

One could argue that real quantities are linear. 1 kilometer is a kilometer whether you have traveled 1 km or 100 km. Well yes and no, as Philip Ball has written, ‘Many creatures, execute random walks or the curious punctuated random walks called Levy flights, in which migrations over a fixed increment in distance takes an ever longer time. Besides, we can usually assume that an animal capable of covering 100 kilometres could manage 101, but not necessarily that one capable of 1 kilometre could manage 2 kilometres (try the latter case with a young child).’

END NOTES :END NOTES :

[2] It can well be named Times of Soft-Porn or Times of Idiots

[3] Most of ancient civilization have used 5, 10 or 20 as their base for number system. The reason is quite intuitive. When you have to count, the simplest device is your fingers. You can use either one hand (5) or both (10). Some might have liked to use their toes also. After all not everyone was fond of wearing shoes. There is one major exception of Babylonian. They have used 60! Why one would do that? Freaks!

[4] Asca Parpola and I. Mahadevan are two most prominent scholar of this language. The difficulty in its decoding is due to the fact that only few samples, that to are of very short length are available. History tells it and present proves it that Indians have never been great writers. Though our exceptional cases are truly exceptional. Despite of spending 3000 years not writing a damn thing, some of them wrote ‘Vedas’, and multitude of stories including longest epic ‘Mahabharta’.

[5] For example, in their anti reservation stand, some students and some section of media used this aporism survival of the fittest. If this is relevent for social situation then a robber shold not be charged for any robbery he does. He is fit to rob you and it is also a survival of fittest. Other major victims of this ‘selfish’ misuse of great man are Gandhi and Adam Smith.

[6] One comment is in order. Pinker is from Harvard and as I have argued in the opening of this article, can not think beyond the nations of Atlantis. In India our social construct of marriages defies this logic. In fact India has defied almost all of the social studies ever done in the West. This hardly holds in a ‘system of arranged marriages’. However if we consider a small and certainly growing percentage of teenagers who prefer to choose their mates by themselves, this logic that females also compete does not hold up to the level it holds in West. The reason for this is simple. During world war, there were fewer men left in the Europe since its natural and females there are more competitive (simple supply and demand). In India, males outnumbered females. They have many options to choose from. Unfortunately, larger the choices more confused one get and ended up with someone they tend to avoid. See this remarkable study done by Sheena Iyenger on Art of Choosing’ [Iye]. On this rather a new phenomenon of ‘love marriages’ among Indians. See Vijay Nagaswamy’s article.

[7]A noted English writer Oscar Wilde once commented in one of his work of fiction, ‘A man loves by his eyes, a woman by her ears’

[8]It is believed that languages are evolved to influence others rather than to tell the truth. Chimpanzee do have basic tool to build up a language. They can suffix and prefix the sound to make another but they have yet not come up with a language. This man says just because they do not know that others have thoughts and they can influence it. Using language effectively might give them certain advantages in group like saying something cool to impress a girl. See their is necessity of language for them but they do not realize it. So I refuse to belive that ‘necessity is the mother of all invention’. Possibly this loose aphorism is perpetuated by industries.

[9]Fortunately, in these times ‘anthropologists’ are not narrow minded as they used to be. Whenever they studies ‘tribal’ they called them ‘savages’ and likewise names. These days they used euphemism like ‘other culture’ or ‘primitive culture’. More importantly, they have come to accept that ‘tribal culture’ is least adulterated and hence can be used as a reliable source for studying our ancestors.

[10] The researchers concluded that numerical approximation ability is a basic cognitive ability that is common to all human beings, which could be independent of language. Their findings moreover run counter to the conclusions of research by American scientist Peter Gordon[Gor04] conducted on the Piraha’s, a population that is geographically near the Mundurucus. These conclusions hold that the Indians’ capacities are “immeasurably” different from our own.

Appendix :
The doyen of Indian anthropologist M. N. Srinivas has reflected on this phenomemon many times. See  [Sri96]. About educated parents who ‘do not pay any attention to the talents and inclinations of their children in choosing careers’ and force them to pick up engineering and medicine so that they can boost ‘their image in thier own social circle’, he notices ‘A surer and more expensive way of producing misfits can not imagined and the country has been doing this for over 50 years’. Talink of professional edcation and IIT does not figure is not possible. He continues, ‘The IITs enjoy great prestige as Institutions and they are highly subsidised by the government. Admission to them is eagerly sought after and highly competetive, but how many of those who graduate from them stay in India? Indeed, the question need be asked, how many of those who benefit from studying in the elite institutes of higher education in India seek careers in the county? Our most successful exports are our best and brightest men and women’. Since 90’s, though the situation has improved a lot still what he had poited out about the U-turn Indian media took in 90’s on the question of brain-drain is still relevent, ‘Until recently, some concern was expressed about the “brain drain” from India to the developed countries, and need to “reverse” the flow. But such concerns seem to have evaporated quietly, and the expatriat have become NRIs, whose dollars are more important to the country than their skills, qualification and expericance. The irony of this situation seems to be lost on everyone. Greenbacks are preferred to grey cells’ .

Bibliography

AS10
Clarissa A.Thompson1 and Robert S. Siegler.  Linear numerical-magnitude representations aid children’s memory for numbers.  2010.
Bet83
Andre Beteille.  The Idea of Natural Inequalities and Other Essays.  Oxford India, 1983.  Available in IITB Library.
Gor04
Peter Gordan.  Numerical cognition without words: Evidence from amazonia.  Oct 2004.
Guh
Ramachandra Guha.  The Corner of a Foreign Field.
Iye
Sheena Iyengar.  The art of choosing.
Pin07
Steven Pinker.  Toward a consilient study of literature.  2007.
rob72
Parental investment and sexual selection.  1972.  Available on google books.
Sri96
M N Srinivas.  Social Change in Modern India.  2008 edition, 1996.  It will cost you Rs. 195.
you
Rebecca Jordan young.  Brainstorm.

The Unified Mathematics..

International Congress of Mathematics was held at Hyderabad in India this year. Unfortunately, these kinds of events generally lost in the glamor of awards and laudations of few individuals. Mathematicians are weired creatures, at least when it comes to awards. They try their best to keep the novice out of their domain. Engineers are diagonally opposite. Despite of all the claims that their awards are targeted on young people so they can be brought in to research, their rewards do not motivate anyone who gets it. How many awards they give anyway. Two are the most prominent of them which are given by Mathematicians to Mathematicians are Fields and Nevanlinna Awards. Once in every four years! How many they did motivate in a decade? 2 and a half men! Kid (below 40 years of age) who have got it is already super-motivated and hardly  need any support. Already all door are open to him – already a professor in some university!. Funny people.

Dan Spielman. Smooth Operator.

Well, for journalism and literature etc these awards do matter. They make their recipients hugely popular and lot of people start consuming their works. But in the case of Mathematics, this is also not the case. There is no chance I can read and understand their works. But wait, this year was slightly different. They awarded Nevanlinna prize to Daniel Spielman.. Now his work is written so well that even you and me can read it.  First time in the history of Mathematics, seems like readability matters.

The other Mathematician whom I can read is Terrence Tao. His blogs are very informative and his books are just awesome. Though, I also romanticize Manjul Bhargava, but you know, these Princeton guys find a lot of pleasure in making their work unreadable for lesser mortals.

Now since the time of G. H. Hardy’s A Mathematician Apology, a lot of things have changed for good in this world of Mathematics.  The boundaries are fast disappearing between pure and applied mathematics. Look at the work profile of all seven prize winners. Even historically, these boundaries drawn out between pure and applied mathematics were ill conceived (Sorry, Hardy!). People who were trying to solve everyday problems came up with theories which later laid the foundation of pure mathematics. Problem of Heat Conduction gave birth to Fourier Series. Euclid and Archimedes were fascinated with their own everyday problems. Even the innocent looking axioms did not reveal themselves. They were found while finding a solution to some problem. I am happy that these boundaries are disappearing. But there will always be a need to protect Mathematics from utilitarian school of thoughts.  

The Hindu newspaper, with a history of understanding science, had this to say about this year ICM,  

Indeed, some very important talks in the Hyderabad Congress are in areas of pure mathematics inspired by problems in applied mathematics. It is this wonderful duality of mathematics — the joy of pursuit of pure mathematics for its intrinsic aesthetic experience, and its increasing relevance to real-life problems — that must be projected in greater measure to school and college students.

INDIA RISING AGAIN?

There was bit euphoria in Indian Mathematical circles since it was the first time ICM was held in India, possible to acknowledge that current Indian contribution to the world of Mathematics is reasonable good and increasing. Possibly that is why they also they named one of their award after Leelavati.

Now Indian contribution to Number Theory is perhaps well known. A lot of Indian Mathematicians, except Ramanujan, were spoilt by Hardy to work on his life long fascination of series and convergence  – as Andrew Weil ( Professor at Aligarh Muslim University, of Barbauki fame) once said. Beyond Ramanujan, we had K. Ananda Rau, S.S. Pillai, S. Chawla, T. Vijayaraghvan, K. Chandrashekhran, S.  Minakshisundram.

A very prominent Indian Mathematician M. S. Raghunathan, wrote in 2007, “Our nurture of the potential has been less than satisfactory. But there is room for cautious optimism about the future.” After three years in 2010, he made a claim that, “While the numbers in India are small and on the average Indian mathematicians may not have a good standing, the peaks are world class.” Now the media, which are bitten by the super-power syndrome did not hesitate to headline it, “India a world mathematics power, says Professor Raghunathan.”


On some positive side, my Institute IIT Bombay and TIFR Mumbai are comping up National Center of Mathematics at IITB campus. And just a note, the committee which awarded Spielman award was headed by ex-EE IIT Bombay, Ravi Kannan.

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[1] This blogger follows R. Ramachandran, The Hindu for his daily science updates. He holds a Ph.D. from TIFR who has the history of producing some of the brightest scientist.