The Faustian Infinite:
Western Mathematics and the Humanities of Endless Space
By Arthur Chandler
Give me a place to stand on, a fulcrum, and a lever, and I will move the world. –Archimedes
Give me extension and motion anI will construct the universe. --Descartes
• That sly or Arabian alchemist had brewed him such a henbane that Johannes Faustus, Doctor of Theology, could only gape into the waning embers of his fireplace and wonder:
Terrible: For it seemed
A void was made in Nature; all her bonds
Cracked; and he saw the flaring atom-streams
And torrents of her myriad universe,
Ruining along the illimitable inane,
Fly on to clash together again, and make
Another and another frame of things
The billion-winking heavens ceased to be the outer sphere pierced by the light of the Empyrean. Earth and planet and star receded to a point in an endless, changeless space and a sempiternal time of duration—eternal infinity. He wept at the thought of dying in such a vastness which not even a paradise could assuage. No sage, no saint, not Christ Himself could tell him how to live and die in the boundless prison. Only God…
Henceforth, Johannes Faustus resolved in his shadowed chamber, he would pattern himself after the Highest, whose sensorium was the very matrix of infinite space itself. Knowledge would be no vain quibbling (in the manner of the Schoolmen) over essences or the name of essences, no inquiry (in the manner of the Greeks) into the nature of static properties of bodies, but the study and use of forces in space. The Greeks only wanted to know. The Schoolmen only wanted to spar with their beliefs. But to know and imitate the act of God in endless space, to master the ceaseless reign of universal power…stunned with the splendor of the new ideal, Faust yearned for the power of God. All the resources of his soul would he thence forth our into his vaulting ambition. He would sacrifice this very soul for the universe.
But for all his resolve, for all his sudden and profound conviction in the absolute reality of infinity, it took Johannes Faustus centuries to break the spell of Greek beauty, the hypnotic vision of perfect bodies existing in an ordered proportion with all other bodies. The nostalgia for Helen, the loveliness of pure and present beauty, was strong enough to divert (but leave unchanged) his longing for ungraspable darkness. For the centuries of the Renaissance, Faust vainly attempted to conjure and actualize the Greek ideal. But his temple-forms in architecture were built to serve the worship of the infinite God; his drama enacted the prospect of salvation or damnation; his painting projected bodies and buildings into landscapes that receded forever into the infinitesimal vanishing point. Faust could never be anyone but Faust.
In his springtime, Faust cut a brave path through his studies to reversed majesty of the title "Doctor of Theology." But even before his confirmation as Man of Great Learning and Defender of the Faith, his inner eye had fastened on darker resolves. His own faith, in fact, had dwindled to mere forensics, to be picked over, analyzed, and argued about. His soul no longer belonged to those old Hebrews and Latins, though he could defend their ipsissima verba against all comers.
After he had done with "school-play" (for so low had his new doctorate sunk in his esteem), Faust began his search for a new certitude, one unassailable by doubt. In all his studies, only mathematics had struck him absolutely unquestionable. So, with the fresh confidence or youth embarking on a great journey, he opened Euclid’s Elements in search of The Answer.
No doctrine invented by man has ever held more universal currency than the faith of Euclid. Even in times when the Bible or the Koran were, officially, the sole receptacles of truth, Euclid’s beliefs went unchallenged. Good Christians, Moslems, and Brahmins who came into contact with Greek mathematics accepted its premises and conclusions without question. The Son might be equal or inferior to the Father; the actual body of Christ might or might not be present in the Eucharist; but when equals were added to equals, the sums were always and forever equal.
Even in a theological education, the ability to master Euclid’s rigorous method was the criterion for the basic intelligence. In fact, the entire "quadrivium" program of Medieval university study was essentially mathematical. When Faust studied arithmetic, geometry, astronomy, and music, he studied pure, stationary, moving and applied number, respectively. The mental prowess necessary to master Euclid was considered so essential to all further study that one of his theorems was used to divide the intelligent students from the dull ones. The famous "pons asinorum," or "bridge of asses," was the proof of the theorem that the base angles of an isosceles triangle are equal. If the student-ass could not cross this simple and elegant proof-bridge, he would advance no further in his studies.
But even Euclid’s "timeless" method could bear strange fruit in the minds of men whose inner eye, ever sought spiritual light, and not the satisfying stability of static, corporeal forms. In the Optus Tertium Roger Bacon (himself an Urfaust) assures Pope Clement that mathematics "should aid us in ascertaining the position paradise and hell." In Opus Maius, Bacon, irrefutably demonstrates the relevance of Euclid to Christian doctrine, since "the first proposition of Euclid"—that on any given line an equilateral triangle may be constructed—shows "that if the person of God the Father be granted, a Trinity of equal persons presents itself."
In the fourteenth century came the first stirring if a new birth in mathematics. Thinking like Euclid, but feeling like Faust, Bishop Nicholas Oresme developed the first, inchoate form of co-ordinate geometry. In this new space-construction, Oresme used a horizontal and vertical axis—the X- and Y-axes of Cartesian geometry—to locate points in space. In place of the closed curves and triangles of Greek geometry, Oresme predicated a vast, two-dimensional matrix of space, which extended indefinitely in all directions away from the origin. His purpose remained Euclidean: to describe mathematical bodies. But by placing all Euclidean figures in an infinite plane of co-ordinated points, Oresme had already unconsciously subsumed the whole Euclid into a new, Western vision of space:
With Descartes, the co-ordinate geometry reaches maturity. In fact, it is so removed from its Greek progenitors that the common word "geometry" only conceals a deep division between the two. The genius of Descartes was to associate the algebraic equation with the geometrical curve. Greek geometry, Arabian algebra, and a burgeoning feeling for the infinite resolved into a unit that would mesh exactly with the Western feeling fort the world as a planet around a sun in deep space.
If there can be a two-axis (X and Y) plane for organizing flat figures in space, there clearly can be added a Z-axis for describing three-dimensional bodies. A circle with a five-unit radius can be mapped with the equation x2+y2 =25; and a sphere of the same unit dimension can be placed in an X-Y-Z axis with the equation x2+y2+z2=25. But is there a figure describable by the equation x2+y2+z2+t2=25? To the Greeks, the impossibility of visualizing such a figure meant not only that such a figure could not exist, but even that the sum of four numbers was inconceivable, because no figure could be drawn to represent it. The area of a circle is a function of the radius squared (A= _r2), and with a sphere, the volume is a function of the radius cubed (V= 4/3_r3). For the Greek, a figure whose final dimension would be a function of a radius to the fourth power did not and could not exist.
But the post-Cartesian mathematician no longer depended on his senses for his mathematics. With the aid of algebra he could construct a purely abstract idea of a four-dimensional figure, and discover the properties of such hyperspheres (whose hypervolume 1/2_2r4), tesseracts (hypercubes), and a whole fourth dimension with distinctive properties all its own.
And why should he stop at four dimensions? With the Cartesian co-ordinate system and the breathtaking revelation of N-dimensional spaces, Faust had given root to a purely Western mathematic that would first overshadow, then replace the Greek. From the seventeenth to the twentieth centuries, Faust formulated and energized the motif-method he had craved. He patterned his life more and more to the existence of an endless expanse, not of three-dimensions, but N-dimensions of length, width, depth, time, velocity, acceleration, rate of change of acceleration…°
Before Faust would turn his new mathematic into power, he would express its force in art. The method of fastening points and bodies in space brought forth a music of function a painting of projective geometry. In the time span between the fourteenth century of Oresme and the seventeenth century of Descartes, the rules for both counterpoint and perspective would be fully developed. But the first great triumphs were reserved for painting from the hands of the Flemish and Italian masters of the Renaissance. Through them, Faust transformed his vision from the spaceless eternity of Medieval frescoes and manuscript illumination into an infinitesimal instant held fast in the depths of space.
There is distance in Gothic art, but it is the distance of time. The "vanishing point" on a Medieval portrayal of Christ, the Madonna, or the saints lies in their gaze, which projects past or through the viewer into eternity. But as early as the trecento, Duccio is attempting to place sacred events and people in a contemporary landscape. The world of his Christ Entering Jerusalem and The Temptation of Christ is a walled city of Medieval Italy, complete with bell towers and parapets, spires and Gothic windows. Acts of sacred, eternal significance have been translated into contemporary drama.
Gradually, as the Renaissance approaches, eternity is transformed into its spatial equivalent: the infinitely far-away reach of the vanishing point. Here, in the metamorphosis of eternity into infinity, is the fusion of the Western painting and geometry. From the Renaissance down to the Impressionists, virtually every picture painted is the West is an application of projective, co-ordinate geometry to painting. In this new perspective art, the eye directed, not across the surface of bodies (as in Greek art), nor up to a higher world of symbolic religious significance (as in Medieval art), but into a pictorial space through which man, building, and nature have been projected. The vanishing point leads—commands—the eye to follow one and only one path through space. The force of this compulsion becomes clear when we see an "exposed" perspective drawing, like Uccello’s famous perspective rendering of a goblet, or Jan de Vrie’s remarkable series of perspective constructions. The primary and secondary vanishing points thrust us down this corridor or across that broad mountain range, or even tantalize us with a vanishing point that converges outside the picture plane, leaving our mind’s eye to search outside the canvas for the resting point on the horizon.
From the time of the van Eycks and Alberti’s Della Pittura, a confident mastery of perspective was sine qua non for every painter, regardless of his subject matter. But the preoccupation with Christian and classical themes prevented artists from rendering space as their predominant subject matter until the late Renaissance. As long as people were present, their facial features and bodily attitudes ruled the content of painting, and pictorial space remained subservient to the vignette, some drama of human life caught in its significant moment. But in the High Renaissance, with Altdorfer’s Danube Landscape (the first pure landscape painting in the West) and The Battle of Alexander, space itself becomes a theme in painting.
With the Baroque, space painting reaches its peak. In the great landscapes of Ruysdael, Rembrandt, and Lorrain, characterization is reduced or even eliminated in favor of an immense volume of atmosphere. Still-life painting, which is landscape painting in miniature, could take seeming disarray of objects and project them into a cavernous darkness that receded into an umber chasm only a few feet behind the picture plane. In the still lives of Claesz and de Heem, as in landscapes of Turner (the last Baroque and first Impressionist painter), all things, people, and events take their place and significance only in relation to the enveloping vastness. There is a strong suggestion of a Chinese feeling for nature in these works; and there is also something of Medieval eternity in them. But Baroque spacescapes are vast and remote in a Copernican spirit. Man and his works are dwarfed in the cosmic perspective of a co-ordinated space whose theme, which is identical with the theme of the Baroque calculus of Newton and Leibnitz, is the capturing of an instantaneous moment of an endlessly continuous space.
When thus I hail the moment flying:
"Ah, still delay, thou art so fair!"
Then bind me in thy chains undying,
My final ruin then declare!
Thus had Faust bound his soul to Mephistopheles. If he ever begged for a stop to his perpetual motion, he would be Faust no more.
The desire for peace and stillness had come upon him when he first loved a woman and longed for the contentment of home and children; again when he conjured the beauty of Hellenism from the past, and ached for a perfection never to be his own; and once again, when he knew the phosphorescent glow of good works performed in the spirit of charity. But he never cried out for time to have a stop, because he knew a peace in restlessness, a harmony in motion. His soul was saved by music, where he could delay in the very moment flying, and find his own kind of peace in the intervals of counterpoint.
As painting designed a geometry for the eye, music sounded a geometry for the ear. In the early Renaissance, when Oresme was creating co-ordinate geometry and Alberti systematized perspective, the ars nova movement in music developed the rough beginnings of vocal polyphonic melody into a methodical instrumental polyphony. From the parallel movement of the third and sixth intervals (note that the very notion of a moving interval implies dynamic soundspace) to the labyrinthine complexity of nineteenth-century orchestral theme development, European composers transformed this simple polyphony into a grand architecture of sounds in which the functional relation of each note to its enveloping chord progression is of primary importance.
With the development of polyphony came staff notation (probably from Guido of Arezzo in the eleventh century) as the equivalent of the co-ordinate plane in Western geometry. Instead of a vertical Y-axis there is the staff grid, in which harmonies are built; instead of the horizontal X-axis, there is the progression of measures in which the development of the piece proceeds. Conceived in this way, it is apparent that the basis of Western music, strict counterpoint, is function music. The art that draws together many lines of sound into one progressive harmony is the precise counterpart if the mathematic that generates families of curves which display algebraically equivalent properties. No matter how lovely the melody, it exists, in Western music, inside a vertical grid of horizontal chord progressions which imply certain harmonies and exclude others.
By the eighteenth century, the melody had been translated into the motif, which existed, from Haydn’s time thenceforward, as development material which could be worked out in endless series. A true melody has a beginning and an end; but a developmental motif may be transformed contrapuntally as long as the composer likes, using not only the original harmonies of the subject, but an indefinite number of implied harmonies drawn from the original chord structure. This method of developing the higher harmonies inherent (for the Western ear) in any sequence of notes was so well in hand that by the 1940’s Charlie Parker could improvise, at incredibly rapid tempos, coherent solos patterned on the implied harmonies of a complex tune.
By the eighteenth century, the West had finalized its musical and artistic vocabulary for the next two centuries. With the division of the octave into twelve equal semi-tones by Bach’s contemporary, Andreas Werckmeister, the resources of the composer were both arithmetically exact in their foundation and limitless in possibilities for development. The rules of composition were so purely intellectual (as opposed to merely auditory) that Beethoven could compose his final works while deaf. Music had achieved the ideal of theoretical base, the Western composer could boast of a consistency equal to that of the mathematician; and indeed, at the bottom, they were identical in their thought processes.
In the Enlightenment, Faust could survey his achievements with justifiable self-satisfaction. His own accomplishments in the mathematics, painting, and music—to say nothing of architecture, sculpture and literature—were considerable. And if, on occasion, he had failed to live up to the Ancients in the practice of some of the arts, he had nowhere violated their precedent standards of excellence.
His science, though, had not dealt kindly with the Ancient. Ptolemy’s geocentric spheres had been replaced by the Copernican universe; Aristotle’s physics of the "Why" had been discarded in favor of the Galilean "How"; and the authority of Galen in medical matters had long since given way to the labors of Vesalius, Harvey, and their successors. But in spite of the revolutions in the sciences and in mathematics itself, the authority of Greek geometry held fast. No revolution, not even a slight modification, had touched Euclid’s axioms for two thousand years.
It bothered Faust, though, that there was one axiom which, while undoubtedly true, did not seem as perfect as the rest. Faust loved complete perfection as he yearned for the Infinite (not yet realizing that his goals were incompatible), and so he set about to erase his only doubt concerning Euclid’s flawless domain.
The crucial axiom stated that parallel lines never meet, no matter how far they are extended. More precisely, Euclid asserts that through any point P not on line A, there can be drawn one and only one line that is parallel to A, assuming that A and B are in the same plane.
This axiom to many of Euclid’s successors looked more like an assumption than an axiom; but they regarded it as true, both because it seemed true (though not quite self-evident), and because it was needed to prove a good many theorems. In 1733, Father Girolamo Saccheri, professor of mathematics at the University of Pavia, stepped forward to announce that he had cleared up the difficulty. His Euclid Vindicated from All Defects is both a creditable achievement in mathematical reasoning, and a remarkable testament to man’s ability to make a revolutionary discovery without having any idea of its implications.
Saccheri attacked the problem of the parallel axiom with a time-honored method, the reductio ad absurdum. To prove the necessary correctness of the parallel axiom, Saccheri proposed to assume that either (a) no parallels of (b) at least two parallels could be drawn through P parallel to A. If the assumptions produced contradictory theorems when applied with other axioms, then it could be safely assumed that there could be only one parallel through point P, since this assumption had never produced contradictions.
The first part of the proof went along well. The assumption that no parallels could be drawn immediately gave rise to contradictory theorems. But the second stage of the proof proved mind-bending to the unfortunate Saccheri. The assumption that at least two parallels could be drawn through P gave birth to some bizarre geometry, as he thought it would—but there were no contradictions. The entire system stood intact with the newly fabricated axiom, though it gave a warped image of a geometry of the fun-house mirrors and Tanguy nightmares.
Saccheri had discovered non-Euclidean geometry. He had found that other mutually consistent systems of mathematical thought could be constructed with a different set of consistent assumptions as axioms. Not even Euclid was absolutely true.
Father Saccheri looked at his astounding conclusions—and refused to believe them. He closed his book with a passionate plea to reject the grotesqueries generated by his novel axiom, calling it "absolutely false" and "repugnant." The eternal truth of Euclid’s geometry had been undermined by a man who had hoped to save it. But the feelings of the same man could not face what his intellect had done, and thereby spared the absolute validity of Euclidean geometry for another century.
But Faust could not yet, in the eighteenth century, turn atheist to Euclid. Immanuel Kant, unaware that Euclid’s cosmos was already in doubt, posited an entire a priori perceptual state of mind based on the validity of Euclidean space. We accept, Kant argued, the proposition that the straight line is the shortest distance between two points because there exists in the mind of every human being an innate idea of the unchangeable structure of reality, despite the flux of experience. "How far we can advance independently of experience, in a priori knowledge," Kant asserted in the Critique of Pure Reason, "is shown by the brilliant example in mathematics." The axioms of mathematics are true because our minds, which are themselves a part of reality, immediately fathom and affirm that which is intuitively apparent and unquestionably true. Though the mind can never know reality directly, never perceive the Ding an Sich without the aid of the senses, universal truth does exist in the mind as an a priory apprehension of real (Euclidean) space.
It was a heroic attempt to save the search for knowledge from an unmoored relativism—the implications inherent in Hume’s rejection of casuality and in Berkeley’s anti-materialism. Given the apparently unshakeable authority of Euclid, it is easy to comprehend why Kant chose to rest the heart of his case on the validity of mathematics—and easy to understand why his case would vanish with the appearance of other equally valid systems of organizing mathematical spaces.
Kant was doing little more than what centuries of scientific thinkers before him had attempted to establish. "No human enquiry," said Leonardo, "can be called true knowledge unless it proceeds through mathematical demonstrations." Western thought had already taken the decisive turn when thinkers began to predicate their belief in universals on a mathematical model of the structure of reality. So, too, did Pythagoras. But the Westerners succeeded, as no other culture had ever done, in transforming the very base of their culture into a technological plenum that depended, through and through, on mathematical reasoning for virtually every phrase of it life. From farming machines to electric lights to the fine point of intercontinental war to the conquest of outer space, mathematical though has led the edge of Western culture. And having staked so much on numbers, its entire foundation was sure to change qualitatively when its mathematics underwent a drastic conceptual revolution.
In the nineteenth century, Euclid fell. First Gauss, then Bolayi, Lobatchevsky, and Riemann invented new geometries that all had the consistency of Euclid’s, but posited different kinds of space. On the surface of a sphere, for example, Euclid’s geometry simply does not hold true. The shortest distance between two points is not a straight line, but an arc segment know as a geodesic. Furthermore, in Riemannian geometry (the geometry of a spherical surface), the sum of the angles of any triangle is always greater than 180°--a flat contradiction of Euclid’s plane geometry, in which the sum is always exactly 180°.
Euclid’s geometry, it became evident, was only one of an infinite number of possible constructions. If one conceived of space as a plane, then Euclid would do. But if space were positively curved, like the surface of earth, then Rimannian geometry would fit the case more aptly. The world that the Greeks had known was close at hand; hence their geo-metry (measure of the earth) was founded on a sense experience that took the flat surface as the ideal space. But a culture that progressively dealt with a vaster universe would be certain, sooner or later, to discover that Euclid was not the final answer. So far had Euclid fallen that by the twentieth century Bertrand Russell could exclaim that "it is nothing less than a scandal that he should still be taught to boys in England."
Greek mathematics was an idealization of the senses. That which had no corporeal counterpart was quite inconceivable to them—even Plato’s world of ideal forms is Euclidean-visible. The Greeks had no notion of zero as a number, no negative numbers, no notion of infinity (Anaximander’s apeiros had to do with what lies beyond corporeal limits, not a positive conception of infinite space), no working theory of equations. The Greek mathematic was, in the deepest sense, closed and finite, and irrevocably bound to geometry. The foundations of the straight line (triangle, square, etc.) and the conic section (ellipse, parabola, hyperbola, and circle) practically exhaust the categories of Greek mathematical thought. They had no mature theory of numbers, and never considered the properties of a class of numbers (like the class of all positive integers), or the types of operations (such as division) as uniform transformations.
The mathematics of the West has evolved into an infinite system of functions. And in the process of constructing this new universe of number, the entirety of Greek mathematics has been subsumed as a special case in the infinitely richer and more extensive theories of Western thought. All Greek geometry of plane and solid surfaces, for example, has been taken in as an extremely specific subdivision of topology (which deals with the properties of surfaces and volumes under continuous transformation)—and even here, the Greek original may be transformed almost beyond recognition. The circle, for instance, which was such a bright figure for Greek contemplation and ecstasy, emerges in topology as a specific instance of a general class of diagrams know as turbines, which consist of an infinite number of points radiating, with the same degree of inclination, from a center with a constant radius:
The transformation of the circle into a turbine is representative of the transvaluation of a closed and finite Greek figure into an open and infinite Western class of curves that evince common properties. And when, at the close of the nineteenth century, mathematicians began to create the theory groups, the entire number system was reformulated. Even 1+1=2 could take on endless variable values, depending on the properties of the group to which the number belong.
The Greeks sought definiteness and absolute surety in their mathematics. When we read of the Pythagorean’s despair at the discovery of irrational numbers, we get a glimpse of the purpose of their mathematics for their world feeling. A non-repeating decimal was an augury of chaos, a number that tumbled on indefinitely without repetition, and therefore without order. What the study of geometry was supposed to supply was not mental exercise, but an insight into the real structure of the Absolute with its unchanging harmonies and proportions.
Western mathematicians, too, have craved certitude; but they have been willing to deal in approximations. The whole of calculus is nothing but a converging series of approximations. The very basis of Western mathematics is variable and fully abstract to a degree that no Greek would have found tolerable. The willingness to deal with incertitude finds its fullest expression in probability theory, when even error comes under mathematician’s scrutiny. The "norm" of the bell-curve is a world away from the Greek notion of the Ideal, but it has enabled the West to deal effectively with values in a universe of chance.
As the implications of the new mathematics were worked out during the nineteenth and twentieth centuries, it became more and more apparent that the mature mathematics if Western culture bears few fundamental affinities with Greek mathematics. For Euclid, a point is "that which has no parts." For the Westerner, a point is an infinitesimal something/nothing that has no length, width, of depth—pure position. After the appearance of Georg Cantor’s epochal work on infinite sets, it was not great conceptual difficulty to grasp the notion that a line of one unit length has the same number of points as a line of 10, 100, or 1,000,000 units length, or that the same one-unit line contains as many points in its modest length as a sphere whose volume would fill the Milky Way. To a Greek, such a proposition would seem utter madness. To the Faustian mathematician, it simply exemplifies one of the unusual properties of infinite sets—namely, that the part may be equal to the whole. Archimedes might reckon up all the grains of sand in the world; but only a Westerner mathematician would prove that there is infinity in a grain of sand!
In youth, Faust loved to dance; but now, he thought about dancing—the erotic push and pull to it, the controlled abandon of the whole body to a rhythm greater than itself, yet a part of itself. The music had no power to lift him to his feet; but his mind could interpenetrate the labyrinths of polyrhythms and dense textures of sound with such fleet assurance that his thoughts began to dance anew to the understanding of what his music was about. Why was the dance just so, and not otherwise?
When Euclid’s geometry lost its absolute authority, mathematicians began to search for a genuinely unmovable foundation, a system whose axioms could be demonstrated to be thoroughly consistent (as the parallel axiom was not) and complete—that is, a system of axioms that generated no contradictory theorems, and which could account for all known truths of mathematics.
The search began when David Hilbert attempted an exhaustive formalization of all mathematical systems. Formalization, for Hilbert, meant removing all meaning from the signs and operations of mathematics, leaving only empty shells of symbols. Then the rules for manipulating these signs were to be postulated in a rigorously precise set of rules, so that any combination within the system would be tautologously true. Furthermore, Hilbert hoped, this reconstituted mathematic could generate all the possible truths inherent in mathematical reasoning, because the axioms would be complete. Finally, the axioms should be finite in number, since an infinite number of axioms is no axioms at all—if everything is red, nothing is red.
The result of Hilbert’s proposed reconstruction would be the mathematical counterpart to a Mondrian composition. For Mondrian, the two primary positions of line (horizontal and vertical) are combined with the three primary colors plus black and white to represent the equivalent opposition of all forces in the universe. The principle, in spite of the difference of media, is the same as Hilbert’s: reduce all basic properties of the language to absolute fundamentals of symbol, then recombine them in "theorem mosaics" that will consistently carry out the tautologous truths of the primary operations. But for such matters as placement and hue, Mondrian took the subjective liberties always granted to a painter, but never to a mathematician. Hilbert himself could never prove that even elementary arithmetic could be fully formalized. His program was, in the end, "the substance of things hoped for."
Bertrand Russell and Alfred North Whitehead made a heroic effort to find the common generative procedure of both logic and mathematics, and advance the thesis that since the two disciplines were at root identical, if the axioms of logic could be shown to be consistent, then the same would hold true for mathematics. But in 1931, a twenty-five-year-old German mathematician, Kurt Gödel, produced an incontrovertible proof that, first of all, it is impossible to prove internal consistency of arithmetic (or of any other deductive system); and that furthermore, even if arithmetic is consistent (which we can never prove), it is necessarily incomplete—that is, there is an indefinite number of mathematical truths that cannot be derived from the axioms.
Gödel’s depressing conclusions have not brought mathematics to a halt, any more than the Heisenberg Uncertainty Principle has stopped physicists from measuring electron velocity. But in both cases, an indeniable element of agnosticism has been introduced. In choosing axioms, you can never be sure that they are consistent; but you can always be sure that they are incomplete. Gödel proved that even in the very substructure of the mathematical enterprise itself, there is endemic uncertainty and incompleteness. Axioms remain absolutes in mathematics, but they are relative absolutes in the Einsteinian sense—absolutes relative to their respective systems. A set of axioms remains absolute only for those specific problems which it generates and solves. There is no guarantee that an inconsistency will never arise, not that any mathematical problem is soluble in terms of its own axioms. Indeed, there is a growing suspicion that some of the perennially unsolved problems of mathematics (such as notorious "Fermat’s Last Theorem") may belong to that class of mathematical truths not implicit in its axioms.
No certitude, even in mathematics….And there were other portents—
In the nineteenth century, Darwin gave the West its strongest assurance of a Faustian paradise: never-ending progress through never-ending adaptation. Though Tennysons might whine over "nature red in tooth and claw," strong people exulted over the cosmic justification of "The Struggle." As the Americas, China, India, and Japan were forcibly converted to Western forms, from political style to arithmetic, the universal validity of our beliefs seemed undeniable. Darwin’s vision succeeded, not because of the abundance of his evidence, but because he gave meaning and purpose to the Western idea of ceaseless human action on a Copernican world in a Newtonian space. It was a severe and kinetic ethic—even Darwin himself was too kindly to espouse it—but it fit the world of Carlyle, Nietzsche, Napoleon and John D. Rockefeller.
Genesis comes in the beginning, but Revelation closes the book. In the twentieth century, science began to take some alarmingly absurd (in the Samuel Beckett sense) notions about space and time and matter. Space is curved; there is no gravitational action at a distance—only geodesics of space warped around bodies; the bodies themselves are, in the subatomic last analysis, "pure shape, nothing but shape," according to Irwin Schrödinger; electrons (if such things be) can move from one space to another without traversing the distance between (like St. Thomas Aquinas’ angels moving in quantum jumps through the Empyrean); and, most devastating of all, Heisenberg proves that there are inherent and inescapable limitations to physical measurement.
Then come the words of the century’s leading cybernetics oracle, Norbert Wiener:
Sooner or later we shall die, and it is highly probable that the whole universe around us will die the heat death, in which the world shall be reduced to one vast temperature equilibrium in which nothing really happens. There will be nothing left but a drab uniformity out of which we can expect only minor and insignificant local fluctuations….
As entropy increases, the universe, and all closed systems in the universe, tend naturally to deteriorate and lose their distinctiveness, to move from the least to the most probable state, from a state of organization and differentiation in which distinctions and from exist, to a state of chaos and sameness. In this universe, order is least probable, chaos most probable. But while the universe as a whole, if indeed there is a whole universe, tends to run down, there are local enclaves whose direction seems opposed to that of the universe at large and in which there is a limited and temporary tendency for organization to increase. Life finds its home in some of these enclaves. (From The Human Use of Human Beings)
This is the tragic prophecy of the union of the mathematics, physics, computer technology, and astronomy. The long-range probabilities are always against order and life. There is no "light, nor certitude, nor peace, nor help for pain." Perhaps sooner, and certainly later, by all that we now know of the universe, even time must have a stop in infinite space. It is the Night of Judgement of rational thought—
— Unless, urges Old Belief, there lives a regenerative force that is Itself inexhaustible, which Faust, as a young Doctor of Theology, once called "God."
As the mind-fathoming drug began to loosen its dominion over the mind of Faust, he raised his eyes to the morning light. Around him were scattered ancient Authorities, his diagrams of knowledge, his paintings and his musical instruments. Before last night he had pursued each one with a separate passion, unaware of the unity that pervaded them all, a unity that was identical with himself. He knew now, with the certitude that needs no proof and brooks no contradiction, that the problem, and the solution, is bringing things to bear.