A companion volume to 'The Destiny of the Body,' this explores man as a species, his past beginnings, present achievements & failures, his evolutionary future.
IV
Introduction
"Perhaps it is the scientific world that will first get transformed; because that demands a very great sincerity and a great perseverance in effort, and these are indeed the qualities that open the door to a higher life."
The Mother
It is a curious but significant fact that some of the greatest philosophical thinkers, notably Pythagoras, Descartes, Pascal and Leibnitz, were great mathematicians as well. Some others like Thales, Democritus, Plato, Saint Augustine, Condorcet, Kant, Auguste Comte and Husserl were not professional mathematicians, but their acquaintance with 'the Queen of the Sciences' was certainly not negligible. One remembers in this connection a pregnant remark made by Leibnitz: "Mathematicians have as much need of being philosophers as the philosophers have of being mathematicians." In fact mathematics, that wonderful "distilling alembic in which is received the quintessence of abstraction'1 has been styled the bridge linking philosophic meditation to scientific experimentation. It deals with the most abstract of ideas, remaining at the same time the most potent weapon in the arsenal of technology. This double aspect of 'the Queen of the Sciences' confers on one who approaches her in a right spirit the rare boon of enjoying the beneficial fruits both of philosophy and of science, but at the same time helps him to steer clear of the limitations of either. For a mathematician learns by experience that advancing Knowledge should "correct her errors sometimes by a return to the restraint of sensible fact, the concrete realities of the physical world."2 On the other hand, he equally realises that "To refuse to enquire upon any general ground preconceived and a priori is an obscurantism as prejudicial to the extension of knowledge as the
1.F. Le Lionnais.
2.Sri Aurobindo, The Life Divine, p. 11.
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religious obscurantism which opposed in Europe the extension of scientific discovery."3 This dual realisation helps to create in him a harmoniously balanced psychological frame.
If we pass from philosophy to Yoga, we note that a student of mathematics discovers to his agreeable surprise that his previous initiation in the Temple of Mathematics has already given him a psychological poise which is much conducive to the proper appreciation of the philosophy and practice of the Integral Yoga. For, by blending harmoniously an austere exercise of pure reason with the concomitant realisation of the inadequacy of Reason ever to reach the absolute truth, also by forcing its votary to acquire a certain number of essential qualities such as an unalloyed intellectual integrity, a boundless patience and an unflagging perseverance, devotion to truth above everything else, readiness to meet problems in the spirit of an adventurer who is fired with an ardent enthusiasm to march ever forward, mathematics prepares one to approach the path of Yoga in a spirit of heroic humility.
We propose to treat in the present chapter some of the more salient features of a mathematical culture, which stand a mathematician in good stead if he ever turns to Yoga as a sincere aspirant. But in order to dispel a possible misunderstanding, let us add at the very outset that the conclusions regarding mathematics (e.g., all creative mathematicians rely more on intuition than on reason; mathematicians have realised the impotency of Reason to be the arbiter of Truth, etc.) advanced in the course of this essay are all derived from the utterances of eminent mathematicians themselves. In fact, to substantiate these conclusions we shall be drawing freely upon the writings of Henri Poincaré, David Hilbert, Nicolas Bourbaki, Andre Weil, Arnaud Denjoy, René Dugas, etc.
"Mysteria Infiniti"
"The one becomes Many, but all these Many are That which was already and is always itself and in becoming the Many remains the One."4
3.The Life Divine, p. 650.
4.Ibid., p. 339.
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One of the principal difficulties experienced by man's intellect in the path of Yoga is the apparently paradoxical nature of certain truths revealed to spiritual vision. For, the besetting sin of an arrogant intellect is to try to judge spiritual ideas with the help of a logic based on the finite, but the spirit's logic is essentially a logic of the infinite. This unwarranted extrapolation of the finite logic into the domain of the infinite is bound to create obstacles in the way of comprehending spiritual truths.
To a student of mathematics this miscarriage of finite logic in the sphere of the infinite is not at all a surprise. In fact, the instructive history of his encounter with the infinite in his own domain, accompanied by all sorts of puzzles and paradoxes, invariably engenders in him a sense of caution and humility before the infinite.
To tackle the infinite has been one of the hardest tasks for the mathematician. As Tobias Dantzig aptly remarks, "From the very threshold of mathematics we come up against the dilemma of the infinite blocking like the mythical dragon all entrance to the Magic Garden."5 Since far past times when Zeno of Elea (c. 450 B.C.) formulated his famous arguments (Dichotomy, Achilles and the Tortoise, the Arrow, the Stadium) which implied that all change and movement are illusory and which came into conflict with the traditional conception concerning the infinitely small and the infinitely large, up to this day which has seen in the shrine of the Cantorian School the birth of the Theory of Aggregates, mathematicians have been ever busy exploring and unravelling the mys-teria infiniti. And each time they have endeavoured to apply the logic of the finite, they have discovered to their utter dismay that all sorts of paradoxes crop up in the process. That bizarre statement: "Infinity is where things happen that don't", claimed by W. W. Sawyer to have been made by a perplexed schoolboy, epitomises as well the perplexities of mathematicians before the problem of the infinite appearing in a new guise. Did not De Morgan exclaim that he could believe anything of a function which became infinite?
To give only a few instances showing the utter confusion of mathematicians faced with the failure of the logic of their day, every time they have encountered infinity anew, we may mention the case of the discovery of irrational numbers, which disconcerted
5. Le Nombre, Langage de la Science, p.66.
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the Pythagoreans so much so that they took a solemn vow not to divulge this intriguing discovery for fear of incurring the wrath of the Supreme Architect; for, to their view, the existence of irrational numbers in the continuum of rational numbers shows a sure sign of imperfection in the creation of the Supreme Being. Remember, too, in this connection that now-famous statement of Leopold Kronecker made at a meeting in Berlin in 1886: "Die ganzen Zahlen hat der liebe Gott gemacht, alles undere ist Menschenwerk." ("The integer numbers have been made by God, everything else is the work of man.")
Consider next the case of fluxions and infinitesimals introduced by Newton and Leibnitz in the building up of the Differential and the Integral Calculus, of which the initial logical difficulties provoked the opposition of George Berkeley and Bernard Nieuwen-tijt. Berkeley derided the infinitesimals as "ghosts of departed quantities" and exclaimed: "He who can digest a second or a third Fluxion, a second or a third Difference, need not, methinks, be squeamish about any Point in Divinity."6
Now come to the case of the infinite divergent series which seemed in the beginning almost to be a house of miracles, disconcerting even great spirits like Leibnitz and Euler. Remember in this connection the quasi-magical summation of Guido Grandi series which purported to prove that the final result of the summation might equally be zero or a round sum. In fact, logical paradoxes surging up in the process of tackling the infinite divergent series became so great that Abel, that young genius who discovered elliptic integrals, exclaimed in one of his letters, dated 16th January 1826: "The divergent series are the devil's creation, and it is a shame to base on them any mathematical demonstration whatsoever; for while using them, one can draw at will any conclusion one likes."
Let us come down to the present times and consider the case-history of Moritz Cantor's transfinite numbers, those strange mathematical entities which, as the name suggests, are situated on the other side of infinity! The Theory of Aggregates ("Mengen-lehre") first developed by Cantor in the course of an essay (A.D. 1883) was a bold and distinct step away from all traditions of the past. With this, Cantor created an entirely new field of mathematical
6. Dirk J. Struik, A Concise History of Mathematics, p. 178.
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research; he developed a theory of transfinite cardinal numbers based on a systematic mathematical treatment of the actually infinite. Cantor also defined transfinite ordinal numbers, expressing the way in which infinite sets are ordered. Cantor's ideas were so novel and shocking to the supposed inviolability of all traditional concepts concerning the infinite that they initiated a fierce intellectual battle waged for years in the field of mathematics. Logical difficulties were pointed out in the theory of transfinite numbers, various paradoxes were proposed, notably by Burali Forti in Italy, Bertrand Russell in England, Koenig in Germany and Richard in France. The subject of transfinite numbers is so enthralling that one is tempted to treat it in some detail. But space forbidding, we content ourselves with a bare mention of the fact that faced with the dilemma of the Theory of Aggregates, mathematicians divided themselves into two opposite camps, the "formalists" represented among others by Hilbert, Russell and Zermelo, and the "intuitionists" by Kronecker, Poincaré, Brower and Weyl.
A student of mathematics draws two valuable morals from these episodes concerning the mathematicians' encounter with the infinite: (i) never to extrapolate into the field of the infinite a logic essentially based on the finite; (ii) if our existing intellectual resources get baffled by the apparently paradoxical nature of truths touching infinity, not to cry out in utter despair nor to indulge in a priori condemnation, but rather to accept them in all humility, at least provisionally before one can realise them himself in practice.
And this helps him immensely to appreciate the truth of the following statement: "But what appear as contradictions to a reason based on the finite may not be contradictions to a vision or a larger reason based on the infinite. What our mind sees as contraries may be to the infinite consciousness not contraries but complementaries.... All the intellectual problem and difficulty is raised by the finite reason cutting, separating, opposing the power of the Infinite to its being, its kinesis to its status, its natural multiplicity to its essential oneness, segmenting self, opposing Spirit to Nature."7
This incompetence of the finite reason for the task of comprehending the infinite leads us to an enquiry of the status of Reason in that great architecture otherwise called mathematics.
7. The Life Divine, pp. 474-75.
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Reason and Mathematics
"It is necessary, therefore, that advancing Knowledge should base herself on a clear, pure and disciplined intellect."8
Mathematics is the training ground par exellence for developing a clear and disciplined intellect, because a successful pursuit of this Science of relations demands above everything else an intense mental concentration which ruthlessly eliminates all idle wandering of mind. A peu près is banished for ever from the kingdom of mathematics, and the very nature of the symbolic language used in this realm forces one to practise an arduous intellectual gymnastic. Here, all scope for self-illusion is immediately rooted out. In fact, an absolute intellectual integrity and a dispassionate exercise of reason are the two basic lessons one has to learn at each step on entering the temple of mathematics. Here, knowledge has to be sought for the sake of knowledge without any ulterior motive, with every consideration put away except the rule of keeping the eye on the subject under enquiry and finding out its truth, its process, its law.
Crisis of Mathematical Logic
"Whether the intellect is a help or a hindrance depends upon the person and upon the way in which it is used. There is a true movement of the intellect and there is a wrong movement; one helps, the other hinders. The intellect that believes too much in its own importance and wants satisfaction for its own sake, is an obstacle to the higher realisation.... Any part of the being that keeps to its proper place and plays its appointed role is helpful; but directly it steps beyond its sphere, it becomes twisted and perverted and therefore false.'9
We have stated above that the study of mathematics invariably creates an extremely alert and keen intellect: logical rigour is
8.The Life Divine, p. 11.
9.Words of the Mother (First Series), p. 52.
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almost another name for mathematics. But if it happens to be a fact that an absolute reliance on the potency of Reason is the only character of mathematics, the study of this Queen of the Sciences would surely prove to be a great hindrance, and not at all an asset, for the aspirant taking up the path of Yoga. For reason is in its very nature only "an imperfect light with a large but still restricted mission"10, and "it must be exceeded, put away to a distance and its insistences often denied if we are to arrive at more adequate conceptions of the truth of things."11
Fortunately for those who drink deep in the fount of mathematics, this problem does not arise at all; for the study of this particular science, if undertaken in a right spirit, creates that happy psychological frame in which, while toning up a keen intellect, one becomes simultaneously aware of the limitations and inadequacy of reason. Mathematicians have long since discarded the arrogant claim of Logic and Reason to be the sole arbiters of all Reality. They are no longer hesitant in admitting the fact that the search of the absolute within the four corners of logic is futile and condemned to an ignoble failure. Modern mathematics is faced with its own uncertainties, it is even haunted by glaring inconsistencies. The attitude of the fanatic who would declare his idea to contain absolute truth has lost ground here.12 Poincaré, that towering giant of modern mathematics, never got tired of insisting on this fact that even in mathematics there exists something which is other than logic. For, reason unfertilised by intuition is sterile by the very definition of the term. If it knows how to separate the wheat from the tares, it never knows how to create.13
But does it even know how to separate the wheat from the tares? If not a creator, can it be adjudged at least a supreme debater who is absolutely certain of his own position? To this question modern mathematics answers with a categorical 'No'. In fact, the advent of the theory of transfinite numbers with all its logical surprises has shattered the previous assumption of the inviolability and the absolute nature of mathematical concepts. Traditional logic has definitely failed before the problems raised by the Cantorian transfinite. "This confusion of reasoning in that
10.Sri Aurobindo, The Human Cycle, p. 111.
11.Ibid., p. 146.
12.Adolphe Buhl, Esthétinque Scientifique.
13.René Dugas, La Mathématique, Object de Culture.
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very science which is universally considered to be the surest of all, thanks to its essentially and purely deductive character, shows pointedly the fragility of our strongest certitudes, or rather the relativity of all our logical apparatus and the impossibility of ever affirming that all demonstration vigorously controlled and validated by logic will perforce be free from any error, at least from all serious errors."14 Modern mathematics has been rudely awakened to the realisation that all our thoughts, even the most rigorous of them, suffer from "an organic and ineluctable frailty."15 It has faced a situation where
"A doubt corroded even the means to think,
Distrust was thrown upon Mind's instruments;
All that it takes for reality's shining coin,
Proved fact, fixed inference, deduction clear,
Firm theory, assured significance,
Appeared as frauds upon Time's credit bank
Or assets valueless in Truth's treasury."16
Place of Intuition in Mathematics
"The scientist who gets an inspiration revealing to him a new truth, receives it from the intuitive mind. The knowledge comes as a direct perception in the higher mental plane illumined by some other light still farther above."17
Mathematicians have not only discarded the vain hope of enthroning reason as the sole arbiter of Reality and Truth; as an instrument of discovery, too, they rely more on intuition than on a sequential process of ratiocination. Whosoever has studied the history of the growth of mathematics must have unmistakably noticed that all the great turning-points in the progress of this marvellous science are marked with the appearance of brilliant flashes of intuition, this mathematical intuition being "some sort of
14.Arnaud Denjoy, L'lnnéité du Transfini.
15.Ibid.
16.Savitri, Bk. II, Canto 13, pp. 284-85.
17.Words of the Mother, p. 140.
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a direct divination anterior to all reasoning."18 And the wonder is that this intuition invariably brings with it appropriate symbols and language for its expression. That is why M. Andre Weil, the leading light of the polycephalous Bourbaki School, declares: "In mathematics more than in any other branch of knowledge, an idea springs up all armed out of the brain of its creator."19 We may cite as instances the geometrical representation of complex numbers, the introduction of p-adic numbers by Hensel, the Haar measure and the Hilbertian space.
While speaking of the important role played by intuition in the field of mathematical researches, we should not forget to mention a highly remarkable fact. The history of mathematics furnishes many instances in which a great discovery has been made by two different persons independently and almost simultaneously. Take the case of the discovery of descriptive geometry by Pascal and Desargues or of the principles of probability by Pascal and Fermat. Coming to the 18th century we see the birth of Calculus at the hands of Newton and Leibnitz. The 19th century offers the example of almost simultaneous discovery of an interpretation of complex numbers made independently by Wessel, Argand and Gauss. Similarly Hamilton and Grassmann wrote at the same time those papers which were to become the foundation of modern vector analysis. The first two non-euclidean geometries, one by the Hungarian Janos Bolyai, and the other by the Russian Nikolai Lobachevsky "were so nearly alike that they seem like different drafts of the same composition."20 Finally, we witness in the closing years of the last century the independent formulation of the principle of continuum both by Richard Dedekind and Moritz Cantor.
We leave it to our readers to offer an adequate explanation of this striking fact and pass on to the consideration of an ideal which is almost the life-breath of all mathematical research.
Unity in Diversity
"What we see everywhere is an infinitely variable fundamental
18.Nicolas Bourbaki, L'Architecture des Mathématiques.
19.Andre Weil, L'Avenir des Mathématiques.
20.Gilbert N. Lewis, The Anatomy of Science.
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oneness which seems the very principle of Nature. The basic Force is one, but it manifests from itself innumerable forces; the basic substance is one, but it develops many different substances and millions of unlike objects."21
A diversity of isolated facts showing no essential unity fails to satisfy a mathematician. His whole attempt is never to get lost in appearances but rather to penetrate deeper and deeper into the mystery of things until he gets at their inmost heart which is Unity. He always seeks for hidden connexions and an underlying unity in all things. "If it were possible to weld together the whole of knowledge into two general laws, a mathematician would not be satisfied. He would not be happy until he had shown that these two laws were rooted in a single principle."22
The classical example of conic sections reveals in an ample way how mathematicians proceed step by step to discover the essential unity holding together disparate entities showing no apparent link.
The introduction of the notion of focus brings an ellipse and a hyperbola under the same roof and a little reflexion shows that a circle is nothing but a special case of an ellipse. With the introduction of the directrix along with the notion of eccentricity, the parabola is brought into close relation with the other two curves. Then comes Projective Geometry to throw much more light on the essential likeness of the properties possessed by the conic sections. Finally, Analytical Geometry demonstrates that the general equation of the second degree represents all the conic curves with all their rich diversity and apparent irreconcilability on the surface.
In this context let us consider the case of the Geometry of Transformations. (Note the remarkable word transformation which has a special significance for those who practise the Integral Yoga.) Here is a branch of mathematics which helps the student to appreciate in a profound way the truth of the following statement: "The Self becomes insect and bird and beast and man, but it is always the same Self through these mutations because it is the One who manifests himself infinitely in endless diversity."23 For this is a frequent phenomenon in the domain of the geometry of transformations
21.The Life Divine, p. 339.
22.W. W. Sawyer, Prelude to Mathematics.
23.The Life Divine, p. 340.
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that two figures showing absolutely no similarity in appearance conserve in them an essential oneness. In fact, starting from the one one can very easily pass to the other through a suitable transformation, and for a particular group of transformations one discovers a particular group of properties which remain absolutely invariable however diverse may be the modifications undergone by the figures. To give only a few examples in which an extreme apparent divergence conceals within it a basic oneness revealed by transformation geometry:
A strophoid and a rectangular hyperbola (inversion); the MacLaurin Trisectrix and a folium of Descartes (orthogonal projection); the conchoid of Cappa and a Maschéroni curve; etc.
In conclusion, let us add that this incessant search for unity in diversity should normally create in a student of mathematics a great reverence for the path of Yoga which declares:
"All contraries are aspects of God's face.
The Many are the innumerable One,
The One carries the multitude in his breast."24
Mathematical Knowledge and Scientific Truth
"...No law is absolute, because only the infinite is absolute, and everything contains within itself endless potentialities quite beyond its determined form and course, which are only determined through a self-limitation by Idea proceeding from an infinite liberty within."25
Another psychological benefit which an aspirant derives in full from his previous study of mathematics is his enthusiastic readiness to accept the truth of transformation so important in the philosophy and practice of Sri Aurobindo's Yoga. The revolutionary conception that there is nothing sacrosanct and absolute about the so-called laws of nature purported to be discovered by the Experimental Sciences, and that there may equally be other possible 'rules of the game' implying a different type of universe
24.Savitri, Bk. X, Canto 4, p. 656.
25.The Life Divine, p. 267.
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with a different set of laws, does not appear strange to a mathematician; it may well be so to the so-called experimental scientist who, basing himself on the supposed inviolability of all physical laws, may dub as a chimera all hope of transforming this earth-life.
But, after all, what is a law if not "the habits of the world"?26 And, while considering "the possible relation between the divine life and the divine mind of the perfected human soul and the very gross and seemingly undivine body or formula of physical being in which we actually dwell,"27 we should never forget that "that formula is the result of a certain fixed relation between sense and substance from which the material universe has started. But as this relation is not the only possible relation, so that formula is not the only possible formula. Life and mind may manifest themselves in another relation to substance and work out different physical laws, other and larger habits, even a different substance of body with a freer action of the sense, a freer action of the life, a freer action of the mind."28
Here we may digress a little and discuss the role of mathematics in the domain of the exact sciences.
No science is ever called an exact science unless and until it can mathematise itself. In fact in its relation to mathematics every science passes through four distinct phases of its growth: (i) empirical, when it collects facts; (ii) experimental, when it starts measurements; (iii) analytical, when it begins to calculate; and (iv) axiomatical, when it finally endeavours to become a deductive science basing itself on mathematical certainty.29 Mathematics is the model for all exact sciences and Physics is given the most honoured place only because it has succeeded most in mathematising itself.
But in this process of mathematisation, a developing science erroneously considers the mathematical terms offered by the mathematicians, not as so many symbols or pure ideas, but as images of reality.30 And as a result, it makes use of that much of mathematics as it considers to reflect adequately the supposedly
26.Savitri, Bk. II, Canto 10.
27.The Life Divine, p. 254.
28.Ibid., p. 254.
29.Raymond Queneau, La Place des Mathématiqués dans la Classification des Sciences.-
30.Tobias Dantzig, Le Nombre, p. 127.
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true reality. Note the irony of the situation when it tries to sermonize and pass its wise (!) verdict that other branches of the mathematical edifice raised by the mathematicians' creative zeal violate reality and are therefore false. But the mathematician enjoys the fun and goes on as ever indulging in the creation of all possible mathematical universes, each logically self-consistent.
For, after all, what is mathematics? It is that particular science which studies the relations between certain abstract entities defined in an arbitrary way; the only restriction imposed on a mathematician's creative imagination is that there should not exist any logical inconsistency and self-contradiction inherent in the agreed initial 'rules of the game', we mean, in the definitions and axioms with which he starts. The building up of the revolutionary non-euclidean geometries illustrates this point in a striking way. Before 1800, it was universally believed that the Euclidean geometry is the one true geometry, something certain and proved. But the great Gauss, that "lonely giant, austere and integral", was the first to believe in the independent nature of the 'parallel postulate', which implied that other geometries based on another choice of axioms were logically possible. But Gauss never published his thoughts on this subject. Then came the epoch-making event in the history of mathematics when Nikolai Lobachevsky and Janos Bolyai openly challenged the authority of two millennia and constructed a non-euclidean geometry which was at first totally ignored because of the influence of the prevailing Kantian philosophy. Finally, Bernhard Reimann arrived on the scene, recognised the importance of this revolutionary idea and himself created, through his general theory of manifolds, many other, so-called Reimannian, geometries.31
Desperate attempts were made by the antagonists to prove the absurdity and logical contradictions of these non-euclidean geometries but all in vain. "Mathematically it is certain that geometries other than that of Euclid are equally possible, physically it may well be that Euclid's geometry is not exactly true of this universe."32
So we see that for a mathematician there is nothing sacrosanct in a particular set of laws as distinguished from another set. He is solely concerned with things which could not be otherwise without
31.Dirk J. Struik, A Concise History of Mathematics, p. 251.
32.W.W. Sawyer, Prelude to Mathematics, p. 66.
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logical contradiction. And this is why the following utterance of Sri Aurobindo comes as a supreme assurance to him:
"But the being and action of the Infinite must not be therefore regarded as if it were a magic void of all reason; there is, on the contrary, a greater reason in all the operations of the Infinite, but it is not a mental or intellectual, it is a spiritual and supramental reason: there is a logic in it, because there are relations and connections infallibly seen and executed; what is magic to our finite reason is the logic of the infinite."33
Now we pass on to a close examination of some of the essential qualities invariably created by a study of mathematics. The important thing to note is that these are the very qualities needed in abundance by an aspirant on the path of the Integral Yoga.
Aspects of Mathematical Culture
"The method of scientific work is a marvellous discipline. Those who follow it in all sincerity truly prepare themselves for Yoga. It requires but a slight turn, somewhere in their being, which will enable them to come out of their a little too narrow point of view and enter into an integrality which will surely lead them towards the Truth and the supreme mastery."
The study of mathematics, 'the Queen of the Sciences', pursued in a right spirit is an exceptionally potent training ground for the-impulsive and emotional nature as well as for the mind. For none can ever expect to fare well in this realm of mathematics unless he possesses an all-consuming zeal for knowledge, a profound intellectual integrity, a spirit of heroic venturesomeness and a total consecration to the task in hand.
33. The Life Divine, p. 329.
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a) "Merchants and Soldiers of Light"
"Near it, retreated; far, it called him still. "34
"A warrior in the dateless duel's strife."35
A ceaseless search, unalloyed and absolute, for necessary implications and a veritable disdain for all à peu près helps the mathematician to rise far above all sense of utilitarian consideration. He becomes ready to sacrifice all his time and all his energy in an attempt to raise, even if a little, the veil covering the face of truth. He is a "merchant of light" whose voice peals forth: "We maintain a trade, not for gold, silver or jewels, nor for silks, nor for spices, nor any other commodity of matter; but only for God's first creature, which was light."36
Consider only the case-history of the Last Theorem of Fermat. A search for the solution of this problem has been haunting the mathematician for the last three hundred years. And what a heroic creature he is! His patience knows no bounds, his perseverance is almost endless, and that too towards solving a problem pertaining to the Theory of Numbers, which has apparently no application in the field of the practical sciences; at least it is not the pure mathematician who concerns himself with this aspect of the question. (Incidentally, the Theory of Numbers and Topology are the most difficult branches of mathematics.) One after another, mathematicians have come forward, taken up the standard from their fallen predecessors and marched forward in their zeal to "wrest from the Sphinx the secret of his enigma."37 And what an array of great names do we not find in this struggle against the unknown: Fermat, Euler, Legendre, Sophie Germain, Lejeune Dirichlet, Le-besgue, Liouville, Cauchy and Kummer! But no issue is in sight and the battle continues in full earnest against "the problem that seems to have been thrown as an eternal challenge to man's intelligence."38
In fact, a spirit of heroic adventure animates the whole fabric of
34.Savitri, Bk. III, Canto I, p. 305.
35.Ibid., p. 227.
36.Bacon.
37.Théophile Got, le Dernier Théorème de Fermat.
38.Edouard Lucas.
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a true mathematician. He never shuns problems, rather welcomes them with zest in order only to conquer them. In fact, "problems are the daily bread for the sustenance of a mathematician."39 According to David Hilbert, the greatest mathematical genius of the present century, "an absence of problems is the sure sign of death". The attitude of a true mathematician has been mirrored in that popular saying (quoted by W.W. Sawyer): "You have only to show that a thing is impossible and some mathematician will go and do it!" This spirit of meeting the unknown and the uncon-quered marks out a mathematician who never "shrinks from adventure", nor "blinks at glorious hope", nor again tends towards
"Preferring a safe foothold upon things
To the dangerous joy of wideness and of height"40,
and it proves to be a great psychological asset for him, if he ever enters the path of the Integral Yoga.
b) Place of Imagination in Mathematics
"Imagination called her shining squads
That venture into undiscovered scenes
Where all the marvels lurk none has yet known."41
Karl Weierstrass uttered a great truth when he declared: "No mathematician can be a complete mathematician unless he is also something of a poet." To seize in full the importance of this statement, we should remember that it comes from the pen of a great mathematician whose fame has been based on his extremely careful reasoning, on "Weierstrassian rigour" as other mathematicians style it. "Weierstrass was the mathematical conscience par excellence, methodological and logical."42
In fact mathematics possesses the alchemic virtue of marrying opposites. While not sacrificing even a bit of extreme logical rigour, it encourages the development of a great play of imagination. He who is not imaginative will never succeed as a mathematician.
39.Andre Weil.
40.Savitri, Bk. II, Canto 10, p. 246.
41.Ibid., p. 242.
42.Dirk J. Struik.
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c) Beauty in Mathematics
"The All-Beautiful is a miracle in each shape."43
While being an austere intellectual culture, mathematics creates at the same time a high sense of harmony and beauty. All true mathematicians are greatly responsive to harmony of forms and beauty of thoughts. The logical validation of a particular result is not sufficient to content a mathematician; his aesthetic sense must be fully satisfied before he can declare a demonstration truly deserving its name. It has been said that the mathematical works of Abel, that young genius who invented elliptical integrals, are veritable lyrics of sublime beauty.
"As in the ease of pure music, great art or great poetry, the emotions evoked by beauty in mathematics are most often of another world, which can never be explained to one who has not felt the illumination in himself."44
d) Faith in Mathematics
"Faith is indispensable to man, for without it he could not proceed forward in his journey through the Unknown; but it ought not to be imposed, it shouldcome as a free perception or an imperative direction from the inner spirit. "45
Contrary to all belief, the study of mathematics does not stifle faith; it is rather the fact that "a new discovery in the field of mathematics is nearly always a matter of faith in the first instance; later, of course, when one has seen that it does work, one has to find a logical justification that will satisfy the most cautious critics."46 The mathematician is essentially a creative creature who is ever eager to strike out new pathways, open up new vistas and explore new avenues to the unknown. And in this dan for creation he is not bound by a priori prejudices. Not without significance did D'Alembert, the leading mathematician of the Encyclopedists,
43.Savitri, Bk. X, Canto 4, p. 663.
44.FranÇois Le Lionnais, La Beauté en Mathématiques.
45.The Life Divine, p. 864.
46.W.W. Sawyer, Prelude to Mathematics.
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declare: "Allez en avant, et la foi vous viendra." ("March forward, and you will get the faith").
Mathematics and Yoga
"Above mind's twilight and life's star-led night There gleamed the dawn of a spiritual day."47
We have made a rapid survey of some of the more salient aspects of modern mathematical thought and have tried to point out their psychological advantage for a student of mathematics, if he ever sincerely enters the path of Yoga.
And, we think, if a mathematician is truly sincere in his own domain and if he follows faithfully the curve of his development, he should not find it at all difficult to come one day to this supreme decision: to enter the path of Yoga. For, if it is true, as has been claimed on their behalf, that the mathematicians are not for the proximate, they seek the ultimate, then they should realise that
"...not by Reason was creation made
And not by Reason can the Truth be seen".48
For
"In her high works of pure intelligence,
In her withdrawal from the senses' trap,
There comes no breaking of the walls of mind,
There leaps no rending flash of absolute power,
There dawns no light of heavenly certitude."49
But have not the mathematicians known by experience, as pointed out by one of their greatest leaders, Henri Poincaré, that "there is no such thing as a solved problem, there are only problems more or less solved," For although
47.Savitri, Bk. I, Canto 3, p. 26.
48.Ibid., Bk. II, Canto 10, p. 256.
49.Ibid., p. 251.
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"Each brief idea, a structure perishable,
Publishes the immortality of its rule,"50
it has been amply demonstrated many a time in the course of the development of mathematics that
"There is no last certitude in which thought can pause."51
"It reasons from the half-known to the unknown,
Ever constructing its frail house of thought,
Ever undoing the web that it has spun."52
In fact,
"If Mind is all, renounce the hope of Truth.
For Mind can never touch the body of Truth
And Mind can never see the soul of God."53
Because
"On the ocean surface of vast Consciousness
Small thoughts in shoals are fished up into a net
But the great truths escape her narrow cast."54
But
"Mind is not all his tireless climb can reach,
There is a fire on the apex of the worlds,
There is a house of the Eternal's Light."55
And so we say that if a mathematician is really sincere in his pursuit of truth, if he does not want to "turn in a worn circle of
50.Ibid., Bk. II, Canto 6, p. 198.
51.Ibid., Bk. I, Canto 4, p. 69.
52.Ibid., Bk. II, Canto 10, p. 240.
53.Ibid., Bk. X, Canto 4, p. 645.
54.Ibid., Bk. X, Canto 3, p. 626.
55.Ibid., Bk. XI, Canto 1, p. 704.
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ideas"56 ever repeating "a circuit ending where it first began",57 he should be prepared to break asunder the walls of mind and take a leap into the splendours of the Spirit. It is true that "by constant enlargement, purification, openness the reason of man is bound to arrive at an intelligent sense even of that which is hidden from it, a power of passive yet sympathetic reflection of the Light that surpasses it".58 But this cannot be the final goal, and a sincere seeker should not falter at the last step. He should clearly see that
"Its [reason's] limit is reached, its function is finished when it can say to man, 'There is a Soul, a Self, a God in the world and in man who works concealed and all is his self-concealing and gradual self-unfolding. His minister I have been, slowly to unseal your eyes, remove the thick integuments of your vision until there is only my own luminous veil between you and him. Remove that and make the soul of man one in fact and nature with this divine; then you will know yourself, discover the highest and widest law of your being, become the. possessors or at least the receivers and instruments of a higher will and knowledge than mine and lay hold at last on the true secret and the whole sense of a human and yet divine living."59
When the 'thick integuments' are removed and the 'luminous veil' is lifted, what is it that replaces our conceptual knowledge with a much more surer and authentic way to the Truth?
The next three chapters attempt to give an answer to this question.
56.Savitri, Bk. II, Canto 10 p. 245.
57.Ibid., Bk. II, Canto 6, p. 198.
58.Sri Aurobindo, The Human Cycle, p. 114.
59.Ibid., p. 114.
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