When I was an undergraduate student of mathematics in the late 1970s I always wondered why am I learning what has been taught to me. Simply because I was good in Mathematics did not mean that I understood it. I was a student of science because I wanted to explore the mysteries of the universe, decode the method in the madness of the God.
I was driven to it with a fascination like a child to the Railway engine. I was told that it was impossible to understand modern Physics without Mathematics. Nevertheless, whenever I attended pure mathematics class whether it was about "Analysis" or "Point-set Topology", I could not get rid of the feeling that I am in a mental gymnasium working out my mind. I do not know whether it had something to do with me, the Mathematics, the manner in which it was taught or something of all this put together. However, the feeling of strange depression never really departed - it still has not.
One day I stumbled upon "The principles of Mathematics" by Bertrand Russell NL [1]. It was the first book about mathematics which made some sense out of this body of knowledge called Mathematics. The time when I read the book I was not really conversant with the raging debates between ‘formalists' and ‘intuitionist' schools of philosophies about Mathematics. Today, I realize, I may not have missed much. It was Russell's contention that "Mathematics is symbolic logic" is the one which survived the post Einstein era. [2]
After Einstein's arrival on the scientific horizon knowledge exploded like never before. We began to make progress in exponential proportion and developed aplomb, almost bordering on insolence, regarding our understanding of the universe. What Albert Einstein did was to crack the code of one of the hitherto unknown vaults of the God which now lay open before us to walk in and plunder. Physicists were the first to arrive and slowly every branch of modern science walked away with the wares which opened new vistas, avenues and methods before them.
One of the most fascinating things that happened during the modern era was the development of science of engineering. This has had huge social repercussions. It is engineering that gives modern science a special character. Engineering makes it possible, for the first time in the history of the ascent of man, to use knowledge without understanding it. Anyone today uses a cellular phone without knowing anything about Physics. This has given rise to a different kind of ‘faith'. Anything qualified by the words "scientifically established" is swallowed (a statement and a capsule alike) till it is scientifically contradicted and the coterie of thinkers who scientifically establish (or contradict) something are perhaps more alienated from the society than the God himself.
Once I was astounded, while initiating a group of students for advanced studies in Physics, when I realised that most of them believed that Albert Einstein said, "everything in the world was relative"!. Well, it had never occurred to them that if everything in the world is relative then there would be nothing for it to be relative to! [3].
Another fact which is philosophically far more worrisome than the other aspects of the development of the last century is that almost every field of science has become dependent on the body of knowledge called "Mathematics". In fact the whole knowledge seems to be incarcerated behind the solid impregnable bars of Mathematics which seems the only source of reasoning and at times, mathematics is the only way of understanding. This fact was disliked by Albert Einstein himself and was protested, albeit mildly, by Nobel Laureate Feynman in his speech in 1965 [4] Of course, the view hundred years before these gentlemen would have been completely different as reflected in Adolphe Quetelet's [1776-1874] writings of the early nineteenth century. He said, "The more progress physical sciences make the more they tend to enter the domain of mathematics which is a kind of centre to which they all converge. We may even judge the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation"
Let us try to understand the philosophic problems if we too, at the advent of 21st century, attempt to stick by the Quetelet's views.
II] Brief Definition of Mathematics
(i) Simply stated, the entire body of knowledge called the ‘Mathematics' is just two sets of statements, S1 and S2. Mathematics neither asserts that statements in set S1 are true or false, nor does it make any such claims about the statements in S2. All that it states is that IF S1 Then S2 follows from it. For example when we say, f (x, y, z...) implies y (x, y, z,) it merely says "implies" and such an implication is completely independent of x, y, &/or z being true, false, factual, imaginary or even absurd. Therefore, all of Mathematics is symbolic logic.
Let us take a simple ‘Arithmetical' statement 1+1 = 2, the actual statement made here is "if x is ‘1' and y is ‘1' and, x is not same as y; then x and y is ‘2' [5] However, things are not that simple. In the above example I have taken you away from the real problem by saying that the expression is ‘Arithmetical'. I have, therefore, led you into believing that symbol ‘1' always represents a number. In the body of Mathematical knowledge, these are just one class of propositions which are called ‘arithmetical propositions'. Mathematics deals with many more types and classes of propositions. What if x and y are not numbers? To illustrate, I have numbered the ‘endnotes' to this article as 1, 2 etc; Does the above expression holds good for the endnotes? Yes, in case of endnotes too the implication is ‘true' though the hypothesis and consequent [in case of endnotes], in the strict sense of concept of arithmetic addition that we have in our minds, may not be.
Let us look at it in a different way. When I am typing this article in ‘MS-Word' and instruct my computer to insert an endnote it just executes the above expression 1+1 = 2 and if I have previously entered an ‘endnote' it numbers the following ‘endnote' by a method which we call "addition". The problem that I presented does not occur to my computer because it too, like a true mathematician, is not concerned about truth and/or absurdness of the hypothesis or the consequent.
(ii)Now, consider the expression (I) below. When both ‘x' and ‘y' are algebraic variables we are all familiar with this expression. However, if in the expression (I) below I were to say that ‘x' is Aristotle and ‘y' is Socrates? Sure, the hypothesis and consequent, in this case, would be absurd. The fact is that the expression below forms a part of pure mathematics. The truth of the implication remains unhampered even after the above substitution despite the fact that hypothesis and consequent may appear absurd to us.
Therefore, even if ‘x' is Aristotle and ‘y' is Socrates then;
.... I
is a valid mathematical expression [6]. If we have a difficulty in accepting this expression, then we shall also have considerable difficulty in accommodating even the following expression (II) in the body of knowledge called Mathematics. Anybody who has graduated out of high school knows that expression (II) below is the "theorem of Pythagoras"
If ABC is a right angled triangle formed by AB=x and BC= y then AC = h = hypotenuse then
..... II
We find nothing strange about the hypothesis and consequent of the above expression due to shear familiarity. Strictly speaking we do not know whether the above expression is true / false or fictitious. This is because, in accepting the theorem of Pythagoras as represented by (II) above we have accepted an implication under the assumption of the axioms of the "Euclidian Geometry". In short, we have accepted that:
"There exist a rigid body on which it is possible to have two points in such a manner that a unique straight line passes through them." ..... III [7]
Empirically, if there are two points P1 and P2 so that seeing through one eye it is possible that they look the same (or if P2 can be completely hidden behind P1) then we say that these two points are in a straight line and the length of a segment (P1,P2 ) is the shortest distance between them. Therefore, all set of propositions J(x,y,z) implies Z(x,y,z) based on the statement III above form a body of knowledge called the Euclidian Geometry and the body of Mathematics accepts it as its part. In doing so, as a mathematician, it is not my job to ever question the truth, validity &/or absurdness of the statement III above. It is also true that much of our everyday life depends on II & III above. These expressions and other expressions derived from them determine a lot of things from the stability of the overhead bridge to informing the pilot of the height of a particular thunder cloud above him. Unless one shrinks to the size of an electron or travels at speeds close to the electromagnetic waves the expression II & III above will continue to serve faithfully and even if one were to assert that these expressions are ‘true' nobody except the God will laugh.
However, it must be kept in the mind that we are a miniscule minority in the whole universe to whom the above expressions appear "real" or "true". For vast majority of things in the universe "a rigid body" does not exist.
At the same time if statement III above is replaced by some other proposition/s then Mathematics accepts new set of propositions e.g. F (x,y,z) implies W(x,y,z) as in case of the Riemannian Geometry or the Pseudo-Riemannian Geometry [which offers more convenient apparatus for the Theory of Relativity].
(iii) The discussion in paragraph (i) & (ii) above can be summed up using the words of a great mathematician Henri Poincare "Is mathematical analysis just a vain play of mind? It offers a convenient language to the physicists. Is this not a mediocre service which, strictly speaking, can be done without? Nevertheless, without this language of mathematics many beautiful things in the universe would have remained forever unknown to us ... " [8]
[III]Law of limitation of knowledge - Vedanta
On my part, as far as theory of knowledge is concerned, I prefer to call back Vyasa and Kapila from the fields of Kurukshetra rather than call upon Plato from the fields of Olympiad. It is in the Uttar Mimansa (Vedanta) of Vyasa and Sankhya of Kapila that one finds candid admission of bankruptcy of human intelligence to understand the "ultimate" and dispassionate analysis of the reasons thereof.
One verse in the Bhagvad Gita that always held me in fascination is the verse 46 in the Ch II. It's one of the distilled out principles of Kapilacharya's Sankhya Shastra. The verse reads as:
yAvAn artha udapAne sarvataH saMplutodake |
tAvAn sarveShu vedeShu brAhmansya vijAnataH ||
[As is the use of the pond in a place flooded with water; same is the use of all the Vedas for a knowledgeable man who understands] [9]
Maharishi Vyasa, through this verse, brings in the sharp focus the limitation of knowledge and incapability of seeking the ultimate truth by shear acquisition of knowledge that exists in the Vedas. The knowledge that can be obtained by the five senses, mind and intellect is in the Vedas that this is not sufficient for understanding the ultimate truth. It is necessary, therefore, to transcend the knowledge after acquiring it and carry on enquiry of the ultimate truth with a mind that is unadulterated by the knowledge. Vyasa clearly strengthens this argument in the Ch IX of Bhagvad Gita when Lord Krishna is described to have bestowed, albeit temporarily, divya-chakShu (the divine vision) to Arjuna to enable the latter to see (or perceive) Krishna's divine form as a "Narayana - the supreme self" saying that the ordinary mortal vision (mAMsa-chakShu) is incapable of perceiving the form. The idea of metaphor is inspired by the tenet of Sankhya Shastra: "what is unimaginable cannot be imagined". The law of limitation of knowledge is clearly laid down in the opening verses of the Ch. IX of Bhagvad Gita:
mayA tatamidaM sarvaM jagadavyaktamUrtinA |
matsthAni sarvabhUtAni na cAhaM teShvavasthiH || 9-4
na ca matsthAni bhUtAni pashya me yogamaishvaram* |
bhutabhRunna ca bhutastho mamAtmA bhUtabhAvanaH || 9-5
[Verbatim: (Krishna as supreme Godhead says): "By me all this universe is pervaded through my un-manifested form. All beings abide in me but I do not abide in them. And (yet) the beings do not dwell in me; behold my divine mystery. My spirit which is a source of all beings sustains the beings but does not abide in them". Translation © Dr S Radhakrishnan]
These two verses bring forth principles elucidated in Kena Upanishad and Sankhya-Pravachan Sutra. The most significant word in the verses is *yogam-aishvaram" which is translated by S Radhakrishnan as divine mystery, an alternate translation is: "wily trick played by the creator". The use of "accusative" form of the nouns here actually tells the Arjuna: "comprehend the wily trick that I (the God) am used to playing". What is this trick? The trick is: "That which makes matter behave under its influence is neither concerned about how the matter behaves under its influence nor does it share any of the properties with the matter "This could well be construed as a Hindu definition of the God and in its essence comes close to Bertrand Russell's concept of the God "Being impartial and impersonal, from being without and being fully aware" [10]. This is also the most logical explanation since if anything that makes matter behave were to be concerned about how the matter behaves (under its influence) then the very concern would dampen its influence, similarly if "it" had to share the properties with the matter the matter would somewhere create a field of resistance inhibiting it to establish complete control. Vyasa's metaphor of the Lord bestowing divine vision to Arjuna to grant him the temporary capacity to grasp the supreme form of Narayana is a splendid metaphor explaining the inherent systemic dyslexia that is built into our system.
[IV] Our understanding of infinity
Let us review our understanding of "infinity". We know that infinity is a very large magnitude; some may say it's the maximum magnitude beyond which there is nothing. Mathematically infinity is denoted by a symbol 
Mathematics also gives us ability to deal with this strange quantity. It tells you that + anything =, - anything = , 1/ = 0 , 1/0 = and so on, mathematics also tells you how to integrate a function from 0 to or add an infinite series etc . In physics, in statistics and in various different disciplines of science we, with mathematical apparatus, have been dealing with this abstract quantity with ease and comfort and we have been almost led into believing that we understand infinity. Yes, in parts we may be able to understand or imagine the abstraction but do we completely comprehend this abstract quantity? Readers are free to draw their own conclusions after going through the following example.
Here, I take the same example which Bertrand Russell uses [11] (though Russell's intention is different).
(i) For a moment suppose you are standing on a platform of a railway station and a long goods train is relatively at rest with you. You can neither see the engine of the train, nor the last compartment. Now, assume that the train begins to move. What would you see? You would first see an impulse travelling from one compartment to another (in form of a jerk) and then you will see train moving in the direction opposite to the direction in which the impulse travelled. In case of a normal goods train one can always suitably position oneself so that one can see the whole train and reconfirm that the train has indeed been set into motion.
(ii) In a second example assume that the goods train is infinite and there is one-to-one correspondence between the set of integers and the compartments of the train. Here too, as the train begins to move one would see an impulse travelling in one direction and compartments moving in another direction. From our experience of the example (i) above we will have no difficulty in concluding that the train has indeed been set in motion though in this case it is not possible to find a suitable place from where we can see the whole train. It's infinite in either direction no matter where you position yourself.
(iii) In example (i) above we can perceive, conclude and confirm. In example (ii) above we can perceive and infer the movement. Now consider another kind of infinity. All that I say here is that compartments of the train have one-to-one correspondence with "Real Numbers [R]". We know that if ‘a' is a real number and ‘b' is a real number the ‘a+b/2' is a real number, for all a & b [Mathematically: if a e R and b e R then (a+b)/2 e R for all a, b]. In plain language it means that between any two real numbers there exists another real number. Therefore, our train this time not only has infinite compartments but between any two compartments there is a compartment. Here, no matter how fast the train is in motion (relative to you) you will neither perceive the impulse (because between any two compartments, there is a compartment) nor, for the same reason, you will be able to see the motion of the compartments. In example (i) above the train is finite, in (ii) its infinite but you still have some clue of dealing with the situation. However, in example (iii) above though the train is as infinite as in (ii) but you are completely incapable of dealing with the situation. Take the process further, let us say that you are accidentally, for a moment, at rest relative to the train and manage to board it. Can you communicate with the engine driver? No. No matter how fast the electromagnetic waves travel they will never reach the engine driver because between any two compartments there is a compartment, in fact, this is same as saying that between any two compartments there are infinite number of compartments. Given the situation you would not be able to communicate with anybody even in the next compartment because given that between any two compartments there are infinite compartments and since there is no concept called > infinity velocity there is no way of communication. In this example as the reader would appreciate we have created a situation which can be manipulated mathematically, but can not be imagined or understood otherwise.
[V] Understanding of an Atom
(i)It was probably in early sixties (or even earlier) a mascot emerged symbolising what was called nuclear science or nuclear age. A symbol depicting a central nucleus with particles, supposedly the electrons, goings around it much like the planets in the solar system became popular and is still is(as depicted in the figure below).
The very fact that this model, though theoretically impossible, continues to be still a popular depiction of an atom suggests that we expect neat, clean and orderly system. This picture is something which we can easily imagine and seem to understand. We expect nature to be disciplined, stickler of rules and orderly. We look around and find nothing that contradicts this view. Every time we throw a ball in the air it falls down, we see the Sun rising in the East and setting in the West with great consistency. We too want to do things in a disciplined way. Whether it's about hanging suites in the wardrobe or social laws we seek discipline. This is the yogam-aishvaram - the wily trick played by the creator! If I were to say, "Every person has a freedom to behave the way he wants and the result would be perfect harmony" I would be laughed at but, the Mother Nature seems to work on this principle. The nature seems to create an illusion (!) of perfect order through complete chaos.
If one is not a student of Physics nothing much is lost if you imagine the atom to be like the one that is depicted in the above picture. However, if you are a student of advanced physics then you would know that it is impossible to sustain such a structure. Even if the electrons are assumed to be moving at a constant speed around the nucleus since they change their direction continuously, there is acceleration; if there is acceleration then they must emit energy (electromagnetic waves/light), if they do emit energy then there is loss of energy every time an electron makes a trip around the nucleus; and therefore, at some point in time the electron shall collapse on the nucleus.
Today, theoretical physicist knows that the electron does not orbit around the nucleus the way it is depicted in the above model but it resides in an abstract space in the neighbourhood of a nucleus moving in a chaotic fashion at fantastic speed and this space is called an ‘orbital', an unfortunate misnomer, which has nothing to do with a round or elliptical ‘orbit' that we are used to. We really do not know how an ‘orbital' looks but it is possible that it looks like the surface which is represented by mathematical apparatus that follows by manipulating the equation (III below) known as time-dependent Schrödinger equation (or popularly Schrödinger wave equation) [12]
III
Solution of mathematical apparatus emerging out of the equation (III) above we can simulate the shape of an atomic orbital. Adjoining figure gives a simulation of one of the atomic orbitals of the Oxygen atom. (Copyright credits see below[13]).
The figure tells us that given the nucleus of Oxygen atom there is a space surrounding it in a shape of lobes (as shown in the adjoining picture) where there is maximum possibility of finding an electron at a given time. We assume this probability to be 95%. Difficulty arises when we want to raise this probability to 100%, in which case the each lobe will have to be as big as the universe itself.
(ii)Let us try to summarise what we know of this basic entity of particle physics called the ‘electron', a word which has become a household name today. We postulate that there exists an entity bearing a charge opposite to that of a nucleus of an atom (we call this negative charge) which in all probability resides in a space surrounding the nucleus of an atom. This entity we know at times behaves like a particle and at times like a wave. We can draw an inference from a particular perceptible phenomenon as to when it behaved as a particle and when like a wave; however, no realistic method exists to 100% predict that behaviour. We know that at a given instant of time we cannot simultaneously know the position and direction of the movement of this entity. We can at the most know one of the two.(Heisenberg's Uncertainty Principle) [14]. It is impossible to see this entity because any attempt to see this entity will interfere with it (our seeing is dependent on the reflection of photons from a surface). There are four lobes in the adjoining figure and each one of them is disjoint, i.e. there is a space between them where probability of finding an electron at any time is ‘Zero'. How does an electron (if at all) passes from one lobe to another? One of the theories is that it does so partly by reflection and partly by representation; the other, known as Born's hypothesis, says that the above figure indicates a space where there is maximum probability of finding an electron and that's all there is to know [15] The latter theory is regarded more acceptable because there is no empirical evidence of splitting of an electron and this aspect of partly by reflection and partly by representation is not palatable at least today.
(iii) Understanding - elaborated:
What has been said in the paragraph (ii) above is been based on our experiences of certain perceptible phenomena which we observed at the beginning of the 19th Century in the laboratories around the world. In our quest to form explanations to these perceptible phenomena we developed the above postulates and finally found a mathematical apparatus which seem to give us an ability to predict few more properties of an entity which we can never see by our own eyes. If one is endowed with well honed skill set that is required to be a good theoretical physicist one would be able to manipulate the mathematical apparatus little further and make logical guesses about few more properties of this entity. The important question is: Do we understand this entity? Do we really understand what it means by being a particle and wave at same time? Again, I am talking about understanding and not saying whether the deductions that we have reached are correct or otherwise. Given the set of experiences of scientists in the laboratories in the early part of the 19th Century this was the most rational, logical and only feasible conclusion/s that we could come to [16].
This needs some elaboration. If I were to make prediction about my death all that I can say with certainty is this: Probability that I will eventually die is 100% however, probability that I shall die now is very small (almost negligible). This is not saying much. However, there is another way of saying the same thing in the language of Mathematics. If ‘x' is the population and ‘l' is a very small number and ‘t' is the time then the probability function of my death follows the following distribution:
IV [17]
If I were an actuary then the statement in English language does not help me at all. However, equation IV gives me an infrastructure on which I can build an apparatus to predict ‘life expectancy' of a various sections of population and determine the quantum of premium that a life insurance company can charge at a different age/s. An actuary is not wrong. He is correct so far and so forth he understands the difference between correctness and truth. He understands nothing more about death than anybody else who does not know to use the equation IV above. However, given the context of estimating ‘chargeable premium' he is correct. Truth is the same in every context; or, the truth has no context at all.
This is what a theoretical physicist accomplishes by knowing to handle the apparatus emerging out of Schrödinger wave equation. In a given context a physicist can correctly estimates the properties of microscopic particles like an electron. That's still being far away from understanding the truth about an electron.
Lord Krishna in Mahabharata - limits of understanding
Let us look at this differently. If one has reasonably well developed linguistic skills of English language then one can understand the anguish in Shelley's writings, one can understand the melancholia of Shakespearian characters, one can appreciate the beauty of words in Keats and almost feel the breath of fresh air while reading Wordsworth. Let us move to Mahabharata of Vyasa. We can understand Duryodhana and his ambitions, we can understand Yudhishthira, we have no difficulty in appreciating Arjuna's Valour and courage and strength of Bhīma, we can feel the agony of the leading lady Draupadi feel her anger about injustice done to her and understand her ardent desire to seek revenge. However, as we reach the character of the chief protagonist the Lord Krishna we begin to feel uneasy, as we probe deeper into Krishna's character we realise that we cannot understand it, we quickly idolize him and say that's God not us. We begin our difficulties in understanding his relationships, his legendary relationship with Radha - we postulate it to be pure friendship to platonic love to anything that suites our convenience. We meet Krishna in Mahabharata in adulthood as a "Yogeshwar - Sthitaprajnya"; a master of Yoga and who has mastered all his senses transcended pain and pleasure. Yet, we see him living like a householder with two wives (Satyabhama & Rukhmini). Mahabharata describes him as a great foodie with a sweet tooth and a connoisseur of good clothes and jewels; yet a Yogeshwar-Sthitaprajnya!
At the Mahabharata war his behaviour is more perplexing; unlike the ordinary mortals, Krishna does not incite Arjuna into the war by reminding him of the insults and humiliations inflicted by the opposite party but chooses to motivate him through highly esoteric lecture on Sankhya, Yoga and Bhakti by reciting Bhagvad Gita. We don't understand him as he is the only character who seems to be in the battle with full heart and mind without having any personal stake while others fumble and stumble in their resolve despite huge personal stake. In creating the character of Lord Krishna Vyasa takes you to the boundaries of human ‘understanding' hinting at possible systemic dyslexia that lies further. Reams of paper have been written about Lord Krishna some have called him a mere mortal who lived at the time of Mahabharata, some consider him as a one of the Rishis who contributed to Rigveda, others theorise him as a "boy-god" who was worshiped by an aboriginal tribe in the planes of Ganges glorified by Vyasa's poetical imagination [18] and by far most regard him as a avatar [19] of Narayana - the almighty himself. All theories are fascinating but they all stop at the assertion of an implication insouciant of the truth and/or absurdity of hypothesis and the consequent.
Vyasa uses two significant words in the Sanskrit language to describe Krishna in different circumstances. The first one is "Vasudev" meaning (greatest amongst) the rays of light [20] and the second one is paradoxical in its very character, its "Bhagvana". Bhaga is a well defined adjective in Sanskrit language [21] Bhaga is an adjective applied to one who, due to complete detachment from material objects (Vairagya) has acquired Success (yasha), Greatness (shree), Fame (kirti), Valour (virya) and Wealth (aishvarya). Lord Krishna of Mahabharata emerges out of this paradox. Vyasa does not talk of Krishna like Bhagavata, he attributes no miracles or the folklore of killing giant snakes or lifting mountains but creates enough mystery in the character to leave comfortable space in his character for all these miracles to be contained and sustained.
In Mahabharata Vyasa stops at dropping hints; aware that his intent here is sociological, his focus is to provide (through Bhagvad Gita) a single philosophical umbrella to more than 1,300 folk traditions that existed in the subcontinent and his audience is the common man.
Causes of systemic dyslexia (Vedanta)
(Attempting to conclude)
Vyasa's rendition in Shanti-Parva of Mahabharata (Sec 290: Verses 102 -04) [22] clearly explains immense respect that he has for Kapilacharya's Sankhya shastra and that's the reason the entire second chapter of Bhagvad Gita has been devoted to the basics of this school of thought. Though Bhagvad Gita treats Sankhya-shastra, Yoga-shastra and Bhaktivada with equal respect it is hugely influenced by the epistemology of Sankhya. Vyasa in Bhagvad Gita has indeed modified Sankhya in a manner that can be called "seshvara" (believing in the existence of the God) whereas, the ‘adhikR^ita' (originally ruled) Sankhya doctrines propounded by Kapilacharcya and sage Panchashikha may be ‘nirIshvarvAdi' [23] i.e. does not regard God as an essential postulate. Considering the fact that Bhagvad Gita had an onerous social responsibility as stated above it is impossible to reason whether Vyasa's version of Sankhya (if I may call it that) was borne out of the need to play to the gallery or out of philosophical necessity or both. The point is that in the chain of effect & cause, infinite regression is no logic at all and therefore, this regression must end at some place. The choice is either to accept Vyasa's doctrine and in turn accept the God as an essential postulate or like Kapilacharcya end infinite regression by saying, "it is like a lame man with help of a blind man reaching desired destination by pure chance - where is the place for God in this?" Fortunately, in the context of the current discussion, this debate is not significant.
There is an unanimous agreement amongst all commentators on the Sankhya that life in the manifested form comes into being out of desire on part of the ‘purusha' (or the atma or soul) to individualise the experience of the universe [24]. It is with this desire of individualising the experience that ‘purusha' separates from Brahma and plants the seed of intelligence (mahata), intelligence in turn brings forth the mind, senses are needed to be able to experience the universe and lastly to sustain senses limbs are born; consequently, the whole package that comes into existence is what we term as a living being. The very purpose of the manifesting in this universe is to individualise the experience of the universe which is essentially an effect of the cause and not to understand the cause. The Mother Nature [25] is not known to splurge or waste, it provides neither a penny less nor a penny more. In other words, what we are provided for is an apparatus to experience the universe and individualise that experience. We are completely devoid of the apparatus which can understand the cause of that effect which we term as the universe at least till such time that we continue in the manifested state. What is unimaginable cannot be imagined or in the context of the above discussion capacity to imagine only that which is essential to fulfil the purpose of our manifestation is what we are provided for.
In the last century as scientific knowledge exploded it became fashionable to assign explanations in terms of scientific jargon for certain concepts like the atman etc. This is far more confusing. For example there is over simplification to the point of trivialisation of the very concept when we try to compare atman by calling it "energy". Then there is a desire of comparing it with electricity or the like. This is the very pitfall. Electricity, as we must understand, is a manifested form which is understandable and therefore can be subject to manipulations by us which we have done with amazing success whereas the atman is un-manifested. As Uttar Mimansa (Vedanta) of Vyasa argues, that which is un-manifested does not share any of its properties with that which is manifested and therefore two way dialogue between the two is impossible and this impossibility is not due to lack of knowledge on our part but due to the very nature of the things. The lack of understanding in imbedded in the very manner in which we seek to understand things. It is, therefore, only natural to be not able to understand the cause of the perceptible universe or, the God if you may want to give it a name. Our knowledge, as expounded in the Vedas or in the modern Scientific Books, makes us capable of ascertaining the implication and we have no real means to assert the truth and or absurdity of the hypothesis or the consequent. As Bhagvad Gita II 47 (above) seem to suggest that any endeavour to understand the Brahma must begin with the understanding that it is not possible to understand it with knowledge.
©Surin Usgaonkar, 2009
Researched and written by Surin Usgaonkar, any unauthorised publication is prohibited. If the text, or a part of it is quoted in any work for scholarly/research purposes, due credit must be given to the author in accordance with established academic norms
References:
[1] Published in 1902, revised edition 1937. Later in the year 1910-14 all volumes of the famous "Principia Mathematica" by Bertrand Russell NL and Professor A N Whitehead were published which included a great deal of metaphysical discussion which was left out in the Principles
[2] Ef. Couturat, La Logique de Leibniz, 1901
[3] ABC of Relativity , Bertrand Russell
[4] "The Development of the Space-Time View of Quantum Electrodynamics" - A Nobel Banquet lecture by Dr Feynman during Nobel Award (physics) acceptance 1965 © Archives of Nobel foundation
[5] The example is from the Principles of Mathematics B Russell pp 7
[6]Elaborate arguments of the case have been presented by Bertrand Russell in "Principles of Mathematics" beginning Ch I Sec 8 onwards. There is no need to repeat the whole set of arguments here.
[7] Detailed discussion ref: "Relativity" by Albert Einstein © 1937English Translation by Dr Robert W Lawson
[8] Translation by W Kaplan - Advanced Calculus
[9] translation © Dr S Radhakrishnan - Bhagvad Gita 22nd impression(2005)
[10]Analysis of Matter , by Nobel Laureate B. Russell
[11] Principia Mathematica Vol II , B Russell , A N Whitehead (Oxford 1914)
[12]Discovered by Dr Erwin Schrödinger (1887-1961) in 1926 Awarded Nobel Prize for the discovery in 1929.
[13] Simulation is through software called Gaussian 03, Revision D.01, M. J. Frisch et al, with all rights exclusively reserved with Gaussian, Inc., Wallingford CT © 2004
[14] Mathematical proof of Heisenberg's Uncertainty principle is found in any Advanced text book of Quantum Mechanics
[15] It is the Born's hypothesis which is currently assumed to be more appropriate by the physicists
[16]Lucid explanations of most of these experiments and conclusions drawn from them are easily available on the Internet
[17] This is known as Poisson distribution in Statistics and is generally used to estimate number of deaths by heart attack at time ‘t' etc.
[18] See Indian Philosophy Vol I by Dr S Radhakrishnan ; Oxford Univ. Press 25th Ed © 2005
[19] The word ‘avatar' in Sanskrit meaning' coming down from up' did not exist during the time of Mahabharata or Vedas, it is found in the Sanskrit literature little before the Middle Ages (cf.: Dr P V Kane et al ) therefore it unlikely that Vyasa could have described him as avataara of Narayana.
[20] Etymology by Dr. Monier Williams
[21] Defined in Vishnu Purana, also ref: etymological discussions by Bharatratna Dr P V Kane, et al, and Dr D. D. Kossambi et al
[22] "Mahabharata" Pune researched edition © Bhandarkar Oriental Research Institute, Pune , India ; references in other edition may change
[23] Unfortunately, the original document supposedly scripted by sage Panchashikha known as ‘Shashtitantra' which is presumed to have had about 60,000 verses is completely lost to us in the passage of time. Currently, the only authentic sources of Sankhya-shastra are Bhagvad Gita, Mahabharata and Ishwarkrishna's Sankhya Karika and the references in Vyasa's Uttar Mimansa, Yoga Bhashya and Sankhya Pravachan-Sutra and many commentaries made by hundreds of writers on the subject for last couple of thousand years.
[24] Suggested References : Kane et al History of Dharmashastra Vol V part II, S Radhakrishnan Indian Philosophy , Sankhya Aphorisms of Kapila (Ballyntine) , Sankhya Karika, Yoga Bhashya , Sankhya Pravachan sutra
[25] The word used here is in the literally sense and not in the philosophical sense
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written by Riverine, 2009-05-13 17:36:20
This work is, I am sure, quite a struggle filled sincere attempt in presenting in as simple words as possible the corelation between philosophy and mathematics. And I feel we benefit quite a lot! Thanks.
While I quite agree that 20th century advancements in theories of mathematics and science are historically remarkable, it seems still lacking in many an area. For example, the dimensional aspects of science do not answer many a question yet (the expansion of the Universe for eg. which is interrogated with 'into what?' by the layman). Time is yet another aspect. Like you expressed 'relative to everything else' would fall short tremendously without the absolute.
Again, when you mentioned the ruthlessly tough Bhagavad Gita within the immensely interesting Mahabharata, one feels ascertained that the interest component has just been touched upon by the field of Mathamatics and science. The Gita has not yet happened!
Thanks once again!
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written by Surin Usgaonkar, 2009-05-06 01:51:11
Thanks for stopping by. I appreciate your comments. I submit the following:
Comment No 1
Sir, at most times the mathematical proof is taken with a degree of finality and as “Truth”. So far there is awareness regarding the nature of mathematics and the asserted facts it may not be worrisome but to assume the assertions made by Mathematical apparatus is the ultimate truth is definitely worrisome. While reading some of the works of Einstein I did develop a feeling that he was not comfortable with the fact that large part of the physics was being explainable only in the language of Mathematics. Of course, this is an opinion and I understand and respect a rival opinion.
Comment No 2
Arithmetical “addition” is an abstraction of “counting” where one-to-one correspondence between numbers and objects is presumed. We say there are two apples only when they are two separate entities. Adding an apple to itself does not amount to two apples. Therefore, an arithmetical expression, in logic, is written as: if x is 1 and y is 1 and x is not same as y then x and y are two. (If there is an apple and if there is another apple and these two apples are not the same then there are two apples). This is a standard custom in logic.
Comment No 3
The example here is used to show that a valid mathematical expression can be built up insouciant of the truth/validity/absurdity of the hypothesis or the consequent. The fact that this is a valid mathematical expression has been elaborated by Bertrand Russell in “The Principles of Mathematics” Ch I Sec 8 onwards (as my footnote says) and the metaphysical discussions appear in “The Principia Mathematica” by B. Russell and A. N. Whitehead ©1937.
Comment No 4
I am not sure whether I have understood your comment. Hindu scriptures do speak about all that is manifested and “the cause” whose “effect” is this manifestation and assert that the cause can neither be concerned about the effect nor can it share its properties with the effect. I agree Hindu scriptures do not talk about “matter” in a manner Greeks do. Lastly I have said this could be construed – I am not dogmatic about it.
Comment No 5
You are right that the probabilities cannot be arbitrarily assumed, they need to be calculated. I would have been wrong had I been talking about the Physics of the Oxygen atom here. I have used the figure and the example only to facilitate imagination of a non-physicist reader who still thinks ‘orbit’ and ‘orbital’ is the same. Mentioning of probability was only to indicate that it is not 100%. In the context of this article I think, I can be excused of this inaccuracy.
Comment No 6
I have indeed stated that Born’s hypothesis is currently the only accepted hypothesis.
The purpose of this article is to trigger a chain reaction of thoughts amongst various readers and not to make any dogmatic assertions. I shall be happy as far as that objective is fulfilled.
Regards
Surin
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written by Raju, 2009-04-29 15:38:57
God works like a mathematical operator; being detached yet necessary for all changes.
Clarification
There are some similarities or parallels between Hindu version/description of Brahman and mathematics & concepts discovered by modern science (quantum mechanism, relativity, genetics, astronomy etc). These parallels have been proved to very useful for understanding Brahman for modern man. In my view these parallels should not be considered as either knowledge or proof of Brahman.
Thanks,
Rajendra
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written by Dr. Pradip Gangopadh, 2009-04-29 13:10:30
1. Another fact which is philosophically far more worrisome than the other
aspects of the development of the last century is that almost every field of
science has become dependent on the body of knowledge called "Mathematics". In
fact the whole knowledge seems to be incarcerated behind the solid impregnable
bars of Mathematics which seems the only source of reasoning and at times,
mathematics is the only way of understanding. This fact was disliked by Albert
Einstein himself and was protested, albeit mildly, by Nobel Laureate Feynman
in his speech in 1965 [4]
My Comment:
I do not see why it should be worrisome that every field of science has become
dependent on the body of knowledge called "Mathematics". It is puzzling but
not worrisome. What Einstein or Feynman or Schroedinger found worrisome was
quantum mechanics.
2. Let us take a simple 'Arithmetical' statement 1+1 = 2, the actual statement
made here is "if x is '1' and y is '1' and, x is not same as y; then x and y
is '2'
My Comment:
If x is 1 and y is 1 then x and y are in fact same. So why does the author
say that 'x is not same as y;'. Moreover what does 'then x and y is '2'' mean?
3. Therefore, even if 'x' is Aristotle and 'y' is Socrates then;
(x+y)^2=x^2 + 2xy + Y^2
is a valid mathematical expression [6].
My Comment:
How can this be true? What does Aristotle squared mean?
4. The trick is: "That which makes matter behave under its influence is
neither concerned about how the matter behaves under its influence nor does it
share any of the properties with the matter "This could well be construed as a
Hindu definition of the God and in its essence comes close to Bertrand
Russell's concept of the God "Being impartial and impersonal, from being
without and being fully aware" [10].
My Comment:
I do not agree that it 'could well be construed as a Hindu definition of the
God'. Hindu scripture does not say anything about whether God is concerned
about how matter behaves under its influence.
5. The figure tells us that given the nucleus of Oxygen atom there is a space
surrounding it in a shape of lobes (as shown in the adjoining picture) where
there is maximum possibility of finding an electron at a given time. We assume
this probability to be 95%. Difficulty arises when we want to raise this
probability to 100%, in which case the each lobe will have to be as big as
the universe itself.
My Comment:
One can't arbitrarily assume a percentage. It has to be calculated!
6. There are four lobes in the adjoining figure and each one of them is
disjoint, i.e. there is a space between them where probability of finding an
electron at any time is 'Zero'. How does an electron (if at all) passes from
one lobe to another? One of the theories is that it does so partly by
reflection and partly by representation; the other, known as Born's
hypothesis, says that the above figure indicates a space where there is
maximum probability of finding an electron and that's all there is to know
[15] The latter theory is regarded more acceptable because there is no
empirical evidence of splitting of an electron and this aspect of partly by
reflection and partly by representation is not palatable at least today.
My Comment:
Born's assumption has been the standard approach for the last 80 years. An
electron does not pass from one lobe to another lobe.
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written by P. Desikan, 2009-04-28 16:41:14
Like the human being himself,which is a system with known and unknown assets and limitations
and unlike his Atma, a system whose assets could be unlimited and need either to be searched and sought more or experienced in ultimate stillness,
human science has defined assets and visible or palpable limitations.
Therefore the Heisenberg Uncertainty Principle, which must have been first revealed and justified and understood following the revelation, shall live on. Someday a continuum may be understood which merges causality with complementarity and a yet to be defined unifying property of everything.
Surin's article had to cover the field to help different kinds of readers to focus on familiar ground and step into his track. Hence it appears longer than what you may find necessary.
Regards. Partha
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written by Raju, 2009-04-28 11:22:35
I enjoyed reading your article.
Here are my thoughts.
1.Science is methodical study of nature i.e. manifested things. It is for this reason field of science keeps changing as more things are known. Science cannot answer questions about origin of laws of nature and cannot account for existence of facts. Science is about mechanism and not meaning even though engineering can create useful/meaningful products for us.
2.Gödels Incompleteness theorems points us the problems of reason i.e. basis of science and mathematics.
3.Heisenbergss uncertainty principle - One consequence of the uncertainty principle is that the macroscopic principle of causality cannot apply at the atomic level. Complementarity principle -is a useful concept but does not provide single unifying view.
4.In my view details of science and math given in the articles are superfluous with respect to main point of knowing Brahman or unmanifested entity.
5.I agree that Vedas is not sufficient for knowing ultimate truth, but Vedas and Upanishads are the best guides that are available to us.
6.God works like a mathematical operator; being detached yet necessary for all changes.
7.Fundamentally knowledge is self-revealing and Atman is same as Brahman otherwise we will never know the truth.
8.Relationship between Krishna and Radha is not known in older Puranas but found in later puranas. It is said that Devaki putra Krishna may same as the one mentioned in Chandyga Upanishad. ( Ref: Did Krsna have a Companion named Radha?
Retrieval of History From Puranic Myths By P.L. Bhargava )
Erotic relationship between Radha and Krishna is found in literature of various Vaisnava Hindu saints in middle age like Jayadeva, Vidyapti, Candidas, Vallabha, and Surdas etc.
Thanks,
Rajendra Rajput
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written by Surin Usgaonkar, 2009-04-27 03:37:57
Regards
Surin
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written by P. Desikan, 2009-04-25 20:55:44
To your beautiful, masterly presentation appropriately ending with
'any endeavour to understand Brahmam must begin with the understanding that it is not possible to understand it with knowledge'
I humbly add my faith which seems capable of letting me understand somewhat some aspects of it some of the time, and find that it does not alter at all your wonderful conclusion.
regards. Partha
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written by Surin Usgaonkar, 2009-04-23 03:24:20
Regards
Surin
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written by TheBigThinkg, 2009-04-23 01:52:46
That was a good analysis.
Regarding ur III
1. My understanding (from Nasadiya Suktam, http://www.medhajournal.com/in...Itemid=267)is that we, our ego, our intelligence, our actions all are result of the desire of HIM to realize himself through the process of evolution of intelligence.
So good, bad, punyam, pavam are all arising out of his desire to achieve higher and higher mode of intelligence in the process of realisation of his SELF. So all are same in a way.
2. As the evolution of intelligence happens the rule with which it is guided is by sequence of action - reaction. As I wrote in a post titled "Existentially Non-Existent" http://www.medhajournal.com/in...Itemid=281
we evolve continuously based on this rule and the HIM which makes us up is a witness to our struggles, though the very struggles are the result of HIS desire to realize himself.
Hence Kapila's Sankhya makes perfect sense with Bhakti Marga. True Bhakti/Devotion can establish the reference point for us more clearly. And establishing this reference is important for our brain.
3. Regarding infinity, in our terms it called "Poornam", which means complete.
While westerners see it from the human side and term it as "Infinite", our ancestors saw it from the other side and termed it 'Complete' or "Poornam".
To Poornam, if we add or subtract poornam, it will be still poornam.
Poornam is the reference point available everywhere for everything.
Since our brain understands everything wrt to reference and lacks the
understanding of absolute, Poornam cannot be comprehended, except by
visualizing it as poornam, the reference.
4. As I go through my understanding of Aitareya Upanishad, I see that there are several forces that are talked about there. These are forces or fields that manifest themself at different places as quanta or particles (like a dance of a stretched string, stretching extending everywhre). Though I have not completely blogged about it, I think the visualization could be useful to modern science. I am not sure. It may be my understanding of it. It could be wrong.
5. Regarding the usage of word Atman in Vedas, Bhagavad Gita etc, I think the word has been used with respect to a logical aspect of a physical entity, ie the logical seed of this universe.
6. I started my blog of Gita but cannot complete it due to paucity of time.
here is a link of it: http://www.gitaforuandme.blogspot.com/
7. Just some random thoughts, as your blog triggered these.
-TBT
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written by Dwai Lahiri, 2009-04-22 11:49:25
As usual a fantastic follow up to your previous masterful articles.
I have to make a point to mimi, who refers to the "Existence of God being both true as well as untrue" per her analysis. There is a famous technique called prasanga (I think Friend Mogambo introduced in the Bamboo/Flute discussion) which deals specifically with this kind of dichotomy (latin I believe is called reductio ad absurdum). The problem of rationality/irrationality of the God paradigm comes because God paradigm transcends reason and logic (paramartha satya)
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written by Mita Das, 2009-04-22 09:32:46
My comment below is not directly related to your excellent article but an effort to start a discussion and bring the mathematic, scientific and non-scientific minds together. There is a distinct difference between the mathematic mind and the scientific mind. In my opinion, Mathematical thinking is extreme rationality with extreme creativity. However, mathematics leave no room for doubt. The mathematical definition of truth is ‘conclusion logically deduced from postulates’. Once you decide that postulates are true and you agree on the rules of logic, nobody can ever cast a doubt on a theorem: proven once, true forever. However, in Science, fundamental facts that one holds true can always be questioned by new experiments. I believe scientific thinking is mathematics with doubt. Too many so-called scientists only have a rational mind, but I believe true science only comes with doubt.
I’m married to a Swiss Mathematician and I’m myself a Scientist. We have many debates about the proof of existence of God started by Euler’s famous joke:
"e^(i*pi)+1=0; therefore God exists"
Of course, Mathematics does not discover truth - it only packages bundles of logically consistent constructs. I was introduced to undecidable (or independent) propositions by my husband – that is even in well-built theories, there are propositions that can neither be demonstrated as true, nor false. And that is when the fun started. If I find an undecidable proposition, I can add it to the axioms, and build a new theory. The new theory extends the initial one. If the initial theory is consistent, then the new one does not have internal contradictions either. In this new theory, the undecidable proposition is now true, as it is now an axiom.
Conversely, I can add the negation of the undecidable proposition (which is also undecidable) to the axioms and make another new theory. This second new theory also extends the initial one. In this theory, the undecidable proposition is now false, as I have added its negation to the axioms. Clearly, both new theories are mutually incompatible, but they are both equally valid.
So now, if the existence of God is an undecidable proposition, I can choose to consider it as true or false with equal validity. I choose to consider it to be true. I can make a new theory where I have no doubt, where I am sure that God truly exists, because it is now one of my axioms. Of course, the converse holds true too to those who have chosen it to be false. So, I can say that I have a partial proof. A lot of scientists are atheists because they think religious belief is irrational. I have just proved in a very rational way that it can be justified as a logical construct.
As a scientist, I had rather see the laws of physics as the precious womb in which life can grow. It gives an overall consistency to my life, which I find more satisfactory, even though I cannot prove it is not culturally biased.
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I did not manage to read your complete article as I founds flaws at the very beginning. From classical logic, it is known that FALSE can IMPLY anything (TRUE or FALSE). Just with the basic idea that the story proceeds from top to bottom, and time limitations made me stop.
I have read Russell but his treatment not really the same as yours -- Mathematics is so flawed. Mathematical representation of one of the formulas of squares is like this:
For every x, y in R (Real numbers): (x + y)^2 = x^2 + 2x*y + y^2
Where "(", ")", "square (2)", "+", and "=" have their "usual" meanings.
Where are you? Aristotle or Gandhi do not belong to the real or any other such space where the statement is valid.
But if you want to really take x = Aristotle, y = Socrates, then to find the TRUTH value of x + y, you have do define "+" here, and .... Remember: FALSE can IMPLY anything (TRUE or FALSE).