IST: A Set Theory for Scientists and Engineers

Engineers know that they can land a man on the moon without using the Lebesgue integral and they will never encounter Skolem paradox in their nuclear reactor design. Intuitive Set Theory (IST) defined here, de-emphasizes concepts that are not required by scientists in their practical work.

Axiom of combinatorial Sets. A set as important as the powerset of Cantor is what I call the combinatorial set of \aleph_0, which is defined as the set of all subsets of \aleph_0 with cardinality \aleph_0. Axiom of Combinatorial Sets (ACS) says that \aleph_1 is equal to the combinatorial set of \aleph_0. Even though, the combinatorial set is a subset of the powerset, it is easy to show that powerset and combinatorial set have the same cardinality. Click here for a simple proof. For a Principia Mathematica type proof, click here, for a musical version, click here. Cantor's continuum hypothesis immediately follows.

Axiom of Infinitesimals. First of all, let us note that corresponding to every real recursive number it is possible to visualize an infinitesimal attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. From this, we can say that an infinitesimal is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. The Axiom of Infinitesimals (AI) says that the unit interval is a set, with cardinality \aleph_0, of infinitesimals. We call an infinitesimal an relement and the elements in it figments, claiming that not even the axiom of choice can pick a figment from an relement.

intuitive Set Theory. We define IST as the theory we get when AI and ACS are added to ZF theory. For a statement of these axioms in mathematical notation, see Definition of Intuitive Set Theory. The discerning reader will easily recognize that the notion of a figment will not allow nonLebesgue measurable sets in IST. Also, the fact that \aleph_0 is the cardinality of the set of infinitesimals in a unit interval, provides us with a way to circumvent the Skolem paradox.

For an older version of these axioms, see The Essence of Intuitive Set Theory.

Infinitesimal Graph: Visualizing the Unit Interval

Briefly stated, infinitesimal graph is what we get when we consider an infinitesimal in the unit interval as a dedekind edge separating rational numbers above and below a number. The dedekind cut itself we imagine to be on the right side of the edge and call it a dedekind node.

Infinitesimal Graph. In representing the unit interval as a graph, we consider only the recursive numbers, that is, the numbers that are programmable on a computer. Since the programs in a computer can be listed, it follows that the cardinality of recursive numbers in a unit interval is \aleph_0. Thus the cardinality of the set of edges in the infinitesimal graph corresponding to a unit interval is \aleph_0.

Just as an infinite graph cannot be drawn in its entirety, we cannot draw a completed infinitesimal graph also. But we can imagine a limit process which will ultimately provide the infinitesimal graph in the limit, as described below.

Write the numbers 1,2,3,4,... in binary in the reverse order as 1,01,11,001,... and put a binary point in the beginning of each resulting in the infinite sequence .1,.01,.11,.001,... representing fractions in the unit interval. Mark the dedekind cut corresponding to these points and consider them as nodes. The graph we get as a limit is the infinitesimal graph for the unit interval (0,1].

The cardinality of the set of points in a dedekind edge can easily seen to be 2^\aleph_0. First of all, note that any fraction in the unit interval can be written as an infinite binary sequence. For example, we can write the sequence for 2/3 as 0.10101... , which we can use to have an infinite set representation for 2/3. If we take the positions at which the sequence takes the value 1, we get the set representation for 2/3 as the set of odd integers. Further, we can claim that the dedekind edge for 2/3 contains all those sets of positive integers which contain the odd integers. The cardinality of this set we can easily recognize as 2^\aleph_0.

The description above allows us to state the following facts. The unit interval is a set of dedekind edges, each edge representing a recursive number, and the cardinality of the set is \aleph_0. A dedekind node corresponding to a number cuts the unit interval into two pieces, each piece representing reals below and above the number. Each dedekind edge contains 2^\aleph_0 elements.

Real Cardinality. We may call an element of an edge a figment and the edge itself a relement (real element), and claim that not even the axiom of choice can pick a figment from a relement. We define real cardinality as the cardinality of a set when only relements in it are considered.

Skolem Paradox. These definitions allow us to resolve some of the vexing problems of set theory. Cantor's theorem asserts that every model of Zermelo-Fraenkel set theory (ZF) has to have cardinality greater than \aleph_0. On the other hand, Lowenheim-Skolem theorem (LS) says that there is a model of ZF theory, whose cardinality is \aleph_0. These two statements together is called Skolem Paradox. The notion of relements gives us a reasonable way to circumvent the paradox. We merely take the LS theorem as stating that the real cardinality of a model of set theory need not be greater than \aleph_0. Cantor's theorem is satisfied when we consider figments also as elements.

Nonmeasurable Sets. With the introduction of the concept of relements, we are in a position to dispense with the notion of sets which are not Lebesgue measurable. This is because a nonmeasurable set cannot be constructed without using the axiom of choice.

Further reading: See the paper http://www.e-atheneum.net/science/ist_visualization.pdf for some more details. Note that the axioms given in this paper are dated, even though nothing has been found to be wrong with them.

The Book: Visualizing the Intellectual Universe

The Book is an invention of Paul Erdos, one of the most prolific mathematicians of the twentieth century. This book lists all the proofs of mathematics, written in the formal language of mathematical logic, according to length and in lexical order. With our knowledge of formal logic today, we know that a computer can be set up to start writing this book, and the computer will write as many pages as we desire, but we also know that the computer will never be able to complete the book. Fortunately for us, the author of The Book is not any computer, but the handiwork of The Almighty, and what we have is a totally finished book.

While the initial part of the book can be visualized without any difficulty, getting a mental picture of the last pages of the book does present a problem. The following is a visualization of the book for those who have no difficulty with the cardinals and ordinals of Cantor.

Anatomy of The Book. The salient features and the physical appearance of the book can be visualized as below.

• The front cover and the back cover are each one millimeter thick, and the entire book, including the covers, is three millimeters thick.
• The first sheet of paper is half-millimeter thick, the second sheet is half thick as the first, the third sheet is half thick as the second, and so on.
• On every odd page is written a full proof, and in the next even page is written the corresponding theorem.
• All the pages corresponding to the ordinals after \aleph_0 are stuck together so that not even the axiom of choice can be invoked to open them. For this reason, unfortunately, we will never know what His Last Theorem is, in fact, we will not even know whether there is a last theorem.

From the description of the book, we can infer that any formula which is a theorem can be found in the book, by sequentially going through the pages of the book. The only difficulty is that, if a formula is not a theorem, we will be eternally searching for it.

Note that the lexical order is with respect to proofs and not with respect to theorems, a cruel joke by The Almighty. David Hilbert, the high priest of formalism, once had high hopes of rewriting the book with theorems in the lexical order, but it did not take very long for logicians to find out that He is not in favor of the project.

A scientist can use The Book for taking his oath in the Ultimate Court of Nature. Years of cogitation on the limits of his knowledge makes the oath somewhat subdued: I solemnly swear and affirm that, if I am sane, I will tell the truth, nothing but the truth, however, not the whole truth.

Obviously, the following facts must have weighed heavily in the mind of the scientist:

• Mathematical logic is unable to decide the consistency of even the Elementary Arithmetic of Goedel. We believe our mathematics has no contradictions, just because it has worked well for us for the last two thousand years. Thus, the scientist is unable to categorically vouchsafe for the sanity of his logic.
• The exhilarating experience of the scientist is that whenever he writes his musico-logical composition strictly conforming to mathematical logic, he finds that nature is faithfully dancing to his tune. Earth dances around the sun, because Newton's laws forces it to do so, and electromagnetic wave radiates rhythmically through the universe because Maxwell's equation says, let there be light. Considering all this, the scientist feels that he is telling the truth and nothing but the truth, about all his observations of nature.
• The incompleteness theorems of Goedel and other developments in mathematical logic make it clear to the scientist that he will never be able to tell the whole truth regarding any subject matter. The talk of subquarks after the discovery of the final sixth quark reinforces this conviction of his.

Epilogue. Essentially, what The Book teaches us is humility, and consistent with this, it is not surprising that Bertrand Russell has the following advice for us:

United with his fellow-men by the strongest of all ties, the tie of a common doom,.... Be it ours to shed sunshine on their path, to lighten their sorrows by the balm of sympathy, to give them the pure joy of a never-tiring affection, to strengthen failing courage, to instil faith in hours of despair. Let us not weigh in grudging scales their merits and demerits, but let us think only of their need, of the sorrows, the difficulties, perhaps the blindnesses, that make the misery of their lives; let us remember that they are fellow-sufferers in the same darkness, actors in the same tragedy with ourselves.

White Hole and Black Whole: Visualizing the Physical Universe

Here, white hole is the terminology we use when we cram a set of points of cardinality 2^\aleph_0 into an interval of infinitesimal length and black whole is the terminology for the entire space beyond the finite space.

We will consider only the real line in its entirety, infinite on both sides, with the understanding that we can have a visualization of the three-dimensional space, if we have a clear mental picture of the one-dimensional line.

White hole. First of all, let us note that corresponding to every real number it is possible to visualize a white hole attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. Simply stated, we can say that a white hole is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. From this description it should be clear that the term white hole does introduce us to the notion of an infinitesimal.

It is easy to see that any point in a unit interval can be represented by an infinite subset of natural numbers. For example, the representation 0.101010... shows that 2/3 can be represented by the set of odd integers, the places where the 1's occur in the sequence. Recognizing this, we can define the white hole or infinitesimal corresponding to 2/3 as the set of all those subsets of \aleph_0 which contain the odd integers. We will take it for granted that the cardinality of this set is 2^\aleph_0.

Black Whole. When we flip the real number xxx...xxx.xxxxx... around the binary point, the resulting ...xxxxx.xxx...xxx we define as a supernatural number and from symmetry considerations we assert that a transfinite stretch comes attached with every supernatural number. Using the notion of point at infinity of complex analysis, a transfinite stretch may also be called a black hole, without violating the conventional sense of the word. If we define the black whole as the set of all black holes and the white whole as the set of all white holes, we will have the universe neatly divided into the two halves, white whole and black whole.

Epilogue. According to our visualization, the infinitesimal corresponding to 2/3 is a dedekind edge between the rational numbers less than 2/3 and greater than 2/3. We do not call it a Dedekind Cut because of its nonzero length.

If we call the elements of a white hole, figments, and consider it as axiomatic that not even the axiom of choice can pick up a figment from a white hole, scientists will have the pleasant situation where they will not have to deal with sets which are not Lebesgue measurable. This follows from the fact that the axiom of choice is crucial for the creation of nonmeasurable sets. As a corollary, we can also claim that we will never be
able to walk along a transfinite stretch.

For some more details, see an earlier blog, Infinitude: the Infinite and the Infinitesimal.

Mysticism and Logicism: Reality is Unrealizable

We want to show that the part of reality that can be understood from set theory can be represented by a specific point in a unit interval, and further, we want to claim that this point is illusive and beyond our grasp.

Since set theory forms the foundations of mathematics, it will not be unreasonable for us to assume that a substantial part of our knowledge of reality can be derived from set-theoretic concepts.

We need the following classification of formulas to proceed further. Originally it was thought that we will be able to show that every formula in set theory is either a theorem or a falsehood, until Goedel showed that there are formulas in set theory which are neither true nor false. Creating a specific formula, he showed that the assumption of the formula itself or its negation will create contradictions in set theory. Self-reference is a crucial concept used by Goedel in the generation of his formula and for that reason we will classify this kind of formulas as introversions. Investigating Cantor's Continuum Hypothesis (CH), Cohen and Goedel have shown that neither CH nor its negation can generate a contradiction in set theory. Accepting that there can be more formulas of this kind, we will classify these formulas as profundities. From all these facts, we conclude that there are four kinds of formulas possible in set theory, namely, theorems, falsehoods, introversions, and profundities. If we are able to classify every formula in set theory in one of these categories, we can claim that we have some understanding of reality, at least that part of reality understandable through set theory.

It is a known fact that the formulas of set theory can be enumerated, thereby, assigning a unique positive integer number to every formula. Consider the point in a unit interval, determined by the following rules. The point is specified by a quaternary number, the n-th digit in the number after the quaternary point getting the value 0,1,2, or 3 depending upon whether the n-th formula is a falsehood, theorem, introversion, or profundity respectively. With all that we know about mathematical logic today, it will not be unreasonable to say that we will never be able to precisely locate this point in the unit interval. If we call the point, Reality, we can reinforce the opinion of many mystics and formulate our predicament as a thesis.

Incomprehensibility Thesis: Reality is Unrealizable.

What is Reality?

Reality is the name used by philosophers for God, after the name has been depleted of all fanatism of all organized religions, so that the highly emotional content that goes with the mention of God can be avoided during a rational discussion. Many scientists believe in what is called pantheism, the view that the only part of reality that we can rationally investigate is the enormously complex working of nature that is visible to us. These scientists carry out their investigations using mathematics as their tool. They fabricate theories like electromagnetic theory, group theory, and derive all their theorems from a set of axioms pertinent to the theory.

Metamathematics. Since the mathematical derivations are known to be mechanical, it becomes clear that the most important part of any theory is the set of axioms that define it. When we investigate the axioms and the derivation rules of a theory, the study is called metamathematics.

From all this one might conclude that the best that humans can do is to restrict themselves to metamathematics, however, this is easier said than done. Unfortunately, we have been attempting to do this at least for the last two millennia, but our violent history shows that we are unable to do it. There seem to be something in the human soul which pines for a realm even beyond metamathematics. Obviously, we cannot rationally comment about it here.

We are not out of trouble even if we restrict ourselves to metamathematics. If you ask a mathematician to show that 2+3=5, he we will ask you to count five apples and give two to Tom and three to Dick and notice that there is none left. If you ask him to justify commutative law of multiplication, he will tell you that you will need a dozen apples whether you distribute them between four children three each or three children four each. The situation becomes more complicated when he tells you that a four dimensional cube has 32 edges. When pressed to explain he will ask you to take the projection of a four dimensional cube on plain paper and count the number of edges in the projection. The situation becomes impossible when he tells you that the volume of a four dimensional sphere is ((\pi^2)/2)r^4. When pressed he will explain that you need to know quite a few fundamental concepts of mathematics to understand and accept this formula.

Mathematics as an Axiomatic Theory. So, what do we make out of all this? The basic facts here can be explained without introducing complications. The whole of mathematics can be considered as a theory with its set of axioms and derivation rules. It is believed that these set of axioms do not have hidden contradictions in it, since it has worked well for us for the last two thousand years. Any derivation we make without violating rules of mathematical logic is supposed to give us a theorem which we can trust. The volume of the four dimensional sphere given above is such a derivation.

With all that we have said so far, we are still not out of trouble. Even though the natural numbers form an infinite set, it turns out that they are just not enough to count the number of points in a unit interval or explain the space beyond the stars. Defining the concept of powersets and cardinals we can generate bigger and bigger infinities indefinitely, eventually finding out that there is no biggest cardinal. With the proliferation of cardinals in set theory it has not been possible to order them in a sensible fashion. The continuum hypothesis of Cantor attempts to solve this problem in a reasonable way.

Symbol Manipulation: the toy of homo sapiens. I consider symbol manipulation as the highest achievement of our civilization. To answer the question, what is reality, I would say that it is the set of derivations we make with well chosen symbols and axioms of a theory. If the symbols have been created with insight and the axioms have been chosen carefully, we can be sure that our derivations will be meaningful and represent some facts of reality. A conspicuous example of symbol manipulation is provided by the prediction of radiation by Maxwell. Another example is provided by the derivation of bending of light rays by Einstein. The developments in mathematical logic shows that we will never be able to comprehend reality in toto. Philosophers get uptight and uneasy when they find that they have reached this impossible situation after all their painstaking studies. To shift his problem to the common man, Bertrand Russell asks the question: What should a true democrat do, when the majority insists that they do not want democracy? Attempting to find a solution to the problem, vedanta philosophy represents the inexplicable Brahman (Reality) by the symbol OM, and asks the vedantist to make a resounding utterance of OM in helpless supplication. Recently, this method of investigation seems to have gained some popularity around the world.

For more of my opinions regarding reality, see the presentation The Mathematical Universe in a Nutshell.

Infinitude: the Infinite and the Infinitesimal

Prologue. Stated simply, we want to claim that corresponding to every infinitesimal on the real line, there is a transfinite stretch representing the space beyond the finite space.

Infinitude. Infinity has been always a difficult and intriguing subject for mathematicians, but they find themselves embedded in a universe which is infinite and hence forced to deal with it. Here, we want to talk about supernatural numbers which, hopefully, would make visualization of the space beyond the stars easier. We claim that corresponding to every real number, there is a supernatural number representing the infinite space.

Real Numbers. Note that one way to uniquely represent a number within the unit interval (0,1] is by an infinite binary string of the form .xxxxx... where the x's after the initial binary point represent either 0 or 1. As examples we have for the rational numbers 1, 2/3, 3/4, the unique representations .111111...., .101010...., .101111.... respectively.

Supernatural Numbers. The question we want to investigate here is that whether it is possible to give some meaning to these strings if we flip them around the binary point. Fortunately for us, computer science tell us that the appropriate meaning for ....111111. is -1 (minus one). The daring wrong argument used by the computer engineer to reach the right result is that the sequence here is the power series expansion of 1/(1-x), where x=2. Ignoring these arguments, we will accept the fact that the flipped sequences can have meaning and suggest that the flipped sequences ...xxxxxx.xxx...xxx corresponding to the real numbers xxx...xxx.xxxxx... should be called supernatural numbers.

Transfinite Stretches. Just as there are infinitesimals attached with the real numbers, we can claim that there are transfinite stretches attached with supernatural numbers. A little investigation shows that infinitesimals and transfinite stretches can be considered as duals of each other, reminiscent of the point at infinity of complex analysis. Also, it should be clear that corresponding to every transcendental number there is a supernatural number. Click here for some details. If your computer can download files fast, click here.

Epilogue. Tolerating some abuse of language, we can state that an infinitesimal is what we get when we compress and fuse a set of points of cardinality 2^\aleph_0. Similarly, we get a transfinite stretch when we keep the points of 2^\aleph_0 a finite distance apart from each other. Because of the duality between the infinitesimal and the transfinite stretch, it should be clear we need to study only one of them. In short, studying the reachable fused infinitesimal is as good as studying the unreachable transfinite stretch.

Proving Incompleteness Theorems Formally

Prologue. We are told that we should not trust our natural language arguments when it comes to mathematics, and we should follow the formal derivation rules of predicate calculus, if we want to prove any thing rigorously. Yet, when it comes to the most serious matter of incompleteness of Elementary Arithmetic (EA) of Goedel, what we do is to talk in plain English and conclude for certain that the theory cannot be complete.

Adding three derivation rules. The issue we want to consider is whether it is possible to rectify this flaw and make all our logical arguments totally formal. A little investigation suggests that this can be done by adding three more derivation rules to Predicate Calculus. For details of these derivation rules, click here. Of course, adding derivation rules in a theory is a risky enterprise, and to make it worse, Goedel tells us that there is no logical way to prove the consistency of a significant axiomatic theory. The time tested method is to wait and see whether a given theory eventually turns out to be inconsistent. Recall, Bertrand Russell's devastating letter to Gottlob Frege about set theory.

Introversion and profundity. If what we are suggesting is indeed a solution, every formula in a theory would fall in one of four categories: theorem, falsehood, introversion, profundity. To give a rough idea of the classification, an example of an introversion is the Consistency of EA, and an example of a profundity is the Continuum Hypothesis of Zermelo-Fraenkel theory. Introversions arise when we attempt to use the theory to investigate itself, and profundities are the profound concepts which we are not sure to choose or not to choose as axioms.

Elevating Goedel's Incompleteness Theorem. It is worthwhile to note that what Goedel has shown is that there are introversions in EA. A more significant result would be to show that there will always be profundities in any significant theory. Whether this is what Goedel really wanted to show is a moot question. Shankara, a philosopher of eighth century, in what he calls "maya" and what I call Shankara's Incomprehensibility Thesis, claims that humanity will have profound questions facing them perennially and our understanding of nature will always be imperfect.

Inescapable Question. By allowing four categories of statements, we are in essence giving freedom of expression to a theory. Throughout civilization humans have indulged in introversion, mostly with disastrous results. Wise men have always been preoccupied with profundities and have enjoyed the consequent agony and ecstasy. A conspicuous example is provided by Mahatma Gandhi's autobiography, The Story of my Experiments with Truth. What if, we decide to whole-heartedly accept and live contented with Goedel's incompleteness theorem and Shankara's incomprehensibility thesis. The fact is that we will still have no peace of mind when the following question about religion, in the best sense of the word, crops up. It is as though God wants us to be permanently on tenterhooks when it comes to important issues.
Humanity's Inescapable Question: Is religion, an introversion or a profundity?

Epilogue. Those who may consider this as a frivolous question should look into some facts of history. It is easy to see that, if an introversion is accepted as an axiom in a theory, the theory will be destroyed by contradictions. From the gruesome violence prevalent in the world today in the name of religion, it is tempting to classify religion as an introversion. But on the other hand Mahatma Gandhi, deeply religious, spent all his life experimenting with Truth, which he considered the same as Reality or God. His autobiography, makes it clear that he considers religion as a profundity.

An Axiom to Derive Continuum Hypothesis

Prerequisite. No more than a passing acquaintance with the cardinals of Cantor is required to read what follows. However, the notations here are a little clumsy since Internet still lacks math symbols.

Axiom of Combinatorial Sets. This blog entry is about the most fascinating problem I know in metamathematics, the Continuum Hypothesis (CH) of Cantor. CH states as a guess that \aleph_1, the cardinal next to \aleph_0, is equal to 2^\aleph_0. It is known that neither CH, nor its negation, can be derived from Zermelo-Fraenkel theory (ZF) and hence it is necessary to introduce an axiom into ZF, if we want to derive CH. The suggestion we want to make here is that the The Axiom of Combinatorial Sets (ACS) can serve as an appropriate axiom for the purpose. Here, by combinatorial set, \aleph_0 \choose \aleph_0, is meant the set of all subsets of \aleph_0 of cardinality \aleph_0. Click here for details.

Can we hope for a better axiom? It is easy to see that ACS can accomplish the job, but the real issue is whether we can have any hope of getting a simpler or more elegant axiom for the same purpose. In other words, is it possible to generate an infinite subset of 2^\aleph_0 substantially different from the combinatorial set which has cardinality less than or equal to that of 2^\aleph_0? Or, is there a simpler way to generate a superset of \aleph_0 from \aleph_0, other than the combinatorial set, whose cardinality is less than or equal to that of 2^\aleph_0?

A fact that may be of some interest here is that the value of, \aleph_0 \choose k, is respectively 1, \aleph_0, 2^\aleph_0, for k=0, finite, \aleph_0.

Internet Principia Mathematica. For a Principia Mathematica type proof of the equivalence of combinatorial set and power set of \aleph_0, see Metamath Proof Explorer (MPE). In my opinion, this MPE is indeed a spectacular achievement of mathematical logic since Principia Mathematica. For more of my comments, see my review.

For some: proof is music to their ears. Did you ever feel that there is something musical about mathematical proofs? If you did, you are absolutely right. To hear a musical version of the proof mentioned above click here (courtesy: Norm Megill).

Adi Shankara and His Incomprehensibility Thesis

Shankara (788-820) was born in a small village called Kaladi in Kerala (God's Own Country, a "must see" place of National Geographic), India, and died at the young age of 32. In this short span of life he traveled the four corners of the Indian subcontinent and established the four most revered places of worship which are thriving even today. He wrote Atma Bodha, and commentaries on Brahma Sutra, Bhagvad Gita, and the most important Upanishads. Atma Bodha is about the awakening of the mind and Brahma Sutra is about the structure of Reality. When Ervin Schrodinger says that Atman and Brahman are the same, we can see from where the idea originates. The celestial song of India, Bhagvad Gita, has dictated the sense of ethics for India for the last two millennia and also it seems to have given consolation to many who had to face the harsh realities of life. Shah Jahan, the Mughal emperor of Taj Mahal fame, who had to suffer the atrocities of his son Aurangazeb, was the one who got it translated into Persian. Warren Hastings, the first Viceroy of India, who had to face the impeachment of his Parliament, ordered it to be translated into English, saying that it is a document that will survive in civilization long after the British Empire is gone from India. Mahatma Gandhi has said that he had turned to Gita whenever he had not even a ray of hope in his predicament with the British. Suffice it to say that Shankara did not leave any of the important literature of the time uncommented.

Real Estate and Intellectual Properties. Unlike these days, in olden days it was unnecessary to say "don't be evil" or "don't kill unborn babies", people were born pious and mostly remained pious. Totally egoless, Shankara never wrote his name in any of his voluminous writings. In his time, owning real estate property was frowned upon and intellectual property right was considered disgraceful. He wrote many hymns which are popular even today. He is the originator of Advaita Philosophy, a sophisticated form of Vedanta.

Shankara's Incomprehensibility Thesis. Stated simply, Shankara's contention is that even metamathematics (or any other form of rational thinking) is not enough for a complete understanding of Reality. An important concept initiated by Shankara is called "Maya", loosely and perhaps wrongly, translated in English as "illusion". I look at it differently and call it Shankara's Incomprehensibility Thesis (SIT): a kind of extension of Goedel's Incompleteness Theorem (GIT). My view is that every formula in any axiomatic theory can be put in one of four categories: theorem, falsehood, introversion, profundity. Introversions arise when we attempt to use the theory to investigate itself, and profundities are the profound concepts which we are not sure to choose or not to choose as axioms. While GIT says that there are introversions in any significant theory, SIT says that there will always be profundities in any branch of knowledge, no matter how much we advance in our understanding. Of course, Shankara did not say this in bland logic, but with religious fervor in the form of hymns.

The following quote from Will Durant gives a rough idea of the line of thinking adopted by Shankara to reach his conclusion:

Sankara establishes the source of his philosophy at a remote and subtle point never quite clearly visioned again until a thousand years later. Immaunel Kant wrote his Critique of Pure Reason. How, he asks, is knowledge possible? Apparently, all our knowledge comes from the senses, and reveals not the external itself, but our sensory adaptation-perhaps transformation of that reality. By sense, then, we can never quite know the "real"; we can know it only in the garb of space, time and cause which may be a web created by our organs of sense and understanding, designed or evolved to catch and hold that fluent and elusive reality whose existence we can surmise, but whose character we never objectively describe; our way of perceiving will forever be inextricable mingled with the thing perceived.

Note that it is only in the twentieth century that quantum physicists grappled with the idea that the observer and the observed cannot be seperated from an observation.

Further reading. To get a gist of Shankara's philosophy, see Will Durant. To know more about Adi Shankara, type "shankara" without quotes in google search box.