Mathematics In The Modern World Quotes

The mathematicians are the priests of the modern world.

scientific theories cannot be deduced by purely mathematical reasoning.

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.

The certainty of mathematics depends on its complete abstract generality.

Voltaire would mock the empire as “neither Holy, nor Roman, nor an Empire,

But science has given us new eyes that allow us to see down to the deeper roots of the world’s structure, and there all we see is order and symmetry of pristine mathematical purity.

Modern science was born through the Scientific Revolution in the 11th/17th century at a time when, as we saw earlier, European philosophy had itself rebelled against revelation and the religious world view. The background of modern science is a particular philosophical outlook which sees the parameters of the physical world, that is, space, time, matter and energy to be realities that are independent of higher orders of being and cut off from the power of God, at least during the unfolding of the history of the cosmos. It views the physical world as being primarily the subject of mathematicization and quatification and, in a sense, absolutizes the mathematical study of nature relegating the non-quantifiable aspects of physical existence to irrelevance.

In other words, if geometry was top-down mathematics, the method of indivisibles was bottom-up mathematics.

The sole purpose of terms such as spirit and immortal soul was to allow unscrupulous and corrupt clergymen to frighten men and subject them to their will.

Mathematics is the means by which we deduce the consequences of physical principles. More than that, it is the indispensable language in which the principles of physical science are expressed.

Our modern conception of the average person is not a mathematical truth but a human invention, created a century and a half ago by two European scientists to solve the social problems of their era.

Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God?

The modern science of nature is significant for many other reasons, beyond the obvious setting of conservation priorities and actions. Foremost in my mind being the fact that it is beautiful. Its wondrous mathematical synchronicities, the specifics of its chemical analyses, the complexity of its physics are beyond both the practical and intuitive knowledge of most lay naturalists (or mystics), no matter how seasoned. When mingled with the wildness of the natural world and the creativity of the human mind, good science reveals its center, its story, its deeper teaching. The science has its own poetic force.

Though resident much of his life in the city of Cnidus on the coast of Asia Minor, Eudoxus was a student at Plato’s Academy, and returned later to teach there. No writings of Eudoxus survive, but he is credited with solving a great number of difficult mathematical problems, such as showing that the volume of a cone is one-third the volume of the cylinder with the same base and height. (I have no idea how Eudoxus could have done this without calculus.) But his greatest contribution to mathematics was the introduction of a rigorous style, in which theorems are deduced from clearly stated axioms.

The Elements is arguably the most influential mathematical text in history.

in a land without a sovereign, Hobbes learned, every man lives in fear and makes war upon his neighbor.

The popular image of the lone (and possibly slight mad) genius-who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts-is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew....Actually, I find the reality of mathematical research today-in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck-to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of "geniuses.

What we call “the laws of nature” merely reflect the normal way in which God sustains or governs the natural world. Perhaps the most wicked concept that has captured the minds of modern people is the belief that the universe operates by chance. That is the nadir of foolishness. Elsewhere, I have written more extensively on the scientific impossibility of assigning power to chance, because chance is simply a word that describes mathematical possibilities.* Chance is not a thing. It has no power. It cannot do anything, and therefore it cannot influence anything, yet some have taken the word chance, which has no power, and diabolically used it as a replacement for the concept of God. But the truth, as the Bible makes clear, is that nothing happens by chance and that all things are under the sovereign government of God, which is exceedingly comforting to the Christian who understands it.

Grothendieck transformed modern mathematics. However, much of the credit for this transformation should go to a lesser-known forerunner of his, Emmy Noether. It was Noether, born in Bavaria in 1882, who largely created the abstract approach that inspired category theory.1 Yet as a woman in a male academic world, she was barred from holding a professorship in Göttingen, and the classicists and historians on the faculty even tried to block her from giving unpaid lectures—leading David Hilbert, the dean of German mathematics, to comment, “I see no reason why her sex should be an impediment to her appointment. After all, we are a university, not a bathhouse.” Noether, who was Jewish, fled to the United States when the Nazis took power, teaching at Bryn Mawr until her death from a sudden infection in 1935.

As Bertrand Russell, the famous British mathematical philosopher and Nobel laureate, famously lamented in an essay condemning the rise of Nazi Germany, “the fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.

There are some mysteries in this world," Yukawa said suddenly, "that cannot be unraveled with modern science. However, as science develops, we will one day be able to understand them. The question is, is there a limit to what science can know? If so, what creates that limit?" Kyohei looked at Yukawa. He couldn't figure out why the professor was telling him this, except he had a feeling it was very important. Yukawa pointed a finger at Kyohei's forehead. "People do." he said. "People's brains, to be more precise. For example, in mathematics, when somebody discovers a new theorem, they may have other mathematicians verify it to see if it's correct. The problem is, the theorems getting discovered are becoming more and more complex. That limits the number of mathematicians who can properly verify them. What happens when someone comes up with a theorem so hard to understand that there isn't anyone else who can understand it? In order for that theorem to be accepted as fact, they have to wait until another genius comes along. That's the limit the human brain imposes on the progress of scientific knowledge. You understand?" Kyohei nodded, still having no idea where he was going with this. "Every problem has a solution," Yukawa said, staring straight at Kyohei through his glasses. "But there's no guarantee that the solution will be found immediately. The same holds true in our lives. We encounter several problems to which the solutions are not immediately apparent in life. There is value to be had in worrying about those problems when you get to them. But never feel rushed. Often, in order to find the answer, you need time to grow first. That's why we apply ourselves, and learn as we go." Kyohei chewed on that for a moment, then his mouth opened a little and he looked up with sudden understanding. "You have questions now, I know, and until you find your answers, I'll be working on those questions too, and worrying with you. So don't forget, you're never alone.

Before we criticize Gerbert and his compatriots for their foolish adherence to ancient Greek and Hebrew authority, consider this: if someone asked you today to demonstrate that the earth orbits the sun, you almost certainly could not do it. You could show them every book and ask every expert, but you could not provide them with direct evidence without a telescope, a lot of time, and a lot of mathematics. Gerbert lacked the telescope and the math, so we cannot blame him for believing his books when they so clearly echoed common sense. The idea that the earth moves was absurd, and it would take a great deal of careful thought before people realized that it was even possible.

This parallelism between physics and psychology should occasion no great surprise. The human nervous system, after all - the "mind" in pre-scientific language - created modern science, including physics and quantum mathematics. One should expect to find the genius, and the defects, of the human mind in its creations, as one always finds the autobiography of the artist in the artwork.

Yes, it is indeed by way of the mathematical forms that the physicist gains knowledge of the external world; Eddington's point, however, is that the forms in question have been artificially imposed: "The mathematics is not there until we put it there." And it is for this reason, and in this sense, that our knowledge of mathematical structures—our knowledge of the physical world!—is said to be subjective.

The so-called physical universe—"the world so described"—turns out to be constituted by mathematical structures which we ourselves have imposed; in a word, it proves to be "man-made." Yet this way of putting it is also misleading; for inasmuch as physical knowledge is partly objective, "the world so described" must be "partly objective" as well. One is left with a curiously equivocal conception, which may enlighten the wise but is bound to deceive the unwary

The “Muslim speech,” as we took to calling the second major address, was trickier. Beyond the negative portrayals of terrorists and oil sheikhs found on news broadcasts or in the movies, most Americans knew little about Islam. Meanwhile, surveys showed that Muslims around the world believed the United States was hostile toward their religion, and that our Middle East policy was based not on an interest in improving people’s lives but rather on maintaining oil supplies, killing terrorists, and protecting Israel. Given this divide, I told Ben that the focus of our speech had to be less about outlining new policies and more geared toward helping the two sides understand each other. That meant recognizing the extraordinary contributions of Islamic civilizations in the advancement of mathematics, science, and art and acknowledging the role colonialism had played in some of the Middle East’s ongoing struggles. It meant admitting past U.S. indifference toward corruption and repression in the region, and our complicity in the overthrow of Iran’s democratically elected government during the Cold War, as well as acknowledging the searing humiliations endured by Palestinians living in occupied territory. Hearing such basic history from the mouth of a U.S. president would catch many people off guard, I figured, and perhaps open their minds to other hard truths: that the Islamic fundamentalism that had come to dominate so much of the Muslim world was incompatible with the openness and tolerance that fueled modern progress; that too often Muslim leaders ginned up grievances against the West in order to distract from their own failures; that a Palestinian state would be delivered only through negotiation and compromise rather than incitements to violence and anti-Semitism; and that no society could truly succeed while systematically repressing its women. —

Alan Turing appears to be becoming a symbol of the shift towards computing, not least because of his attitude of open-minded defiance of convention and conventional thinking. Not only did he conceptualise the modern computer – imagining a simple machine that could use different programmes – but he put his thinking into practice in the great code breaking struggle with the Nazis in World War II, and followed it up with pioneering early work in the mathematics of biology and chaos.

... not only did perspective elevate art to a "science"... the subjective visual impression was indeed so far rationalized that this very impression could itself become the foundation for a solidly grounded and yet, in an entirely modern sense, "infinite" experiential world. One could even compare the function of Renaissance perspective with that of critical philosophy... The result was a translation of psychophysiological space into mathematical space; in other words, an objectification of the subjective.

Because much of the content of education is not cognitively natural, the process of mastering it may not always be easy and pleasant, notwithstanding the mantra that learning is fun. Children may be innately motivated to make friends, acquire status, hone motor skills, and explore the physical world, but they are not necessarily motivated to adapt their cognitive faculties to unnatural tasks like formal mathematics. A family, peer group, and culture that ascribe high status to school achievement may be needed to give a child the motive to persevere toward effortful feats of learning whose rewards are apparent only over the long term.

Sighing, he rose from his desk and walked to the windows to stare out at the Vatican through the rain. What a burden men like Sandoz carried into the field. Over four hundred of Ours to set the standard, he thought, and remembered his days as a novice, studying the lives of sainted, blessed and venerated Jesuits. What was that wonderful line? "Men astutely trained in letters and in fortitude." Enduring hardship, loneliness, exhaustion and sickness with courage and resourcefulness. Meeting torture and death with a joy that defies easy understanding, even by those who share their religion, if not their faith. So many Homeric stories. So many martyrs like Isaac Jogues. Trekking eight hundred miles into the interior of the New World—a land as alien to a European in 1637 as Rakhat is to us now, Giuliani suddenly realized. Feared as a witch, ridiculed, reviled for his mildness by the Indians he'd hoped to gain for Christ. Beaten regularly, his fingers cut off joint by joint with clamshell blades—no wonder Jogues had come to Emilio's mind. Rescued, after years of abuse and deprivation, by Dutch traders who arranged for his return to France, where he recovered, against all odds. Astonishing, really: Jogues went back. He must have known what would happen but he sailed back to work among the Mohawks, as soon as he was able. And in the end, they killed him. Horribly. How are we to understand men like that? Giuliani had once wondered. How could a sane man have returned to such a life, knowing such a fate was likely? Was he psychotic, driven by voices? A masochist who sought degradation and pain? The questions were inescapable for a modern historian, even a Jesuit historian. Jogues was only one of many. Were men like Jogues mad? No, Giuliani had decided at last. Not madness but the mathematics of eternity drove them. To save souls from perpetual torment and estrangement from God, to bring souls to imperishable joy and nearness to God, no burden was too heavy, no price too steep.

121. George Bernard Shaw – Plays and Prefaces 122. Max Planck – Origin and Development of the Quantum Theory; Where Is Science Going?; Scientific Autobiography 123. Henri Bergson – Time and Free Will; Matter and Memory; Creative Evolution; The Two Sources of Morality and Religion 124. John Dewey – How We Think; Democracy and Education; Experience and Nature; Logic; the Theory of Inquiry 125. Alfred North Whitehead – An Introduction to Mathematics; Science and the Modern World; The Aims of Education and Other Essays; Adventures of Ideas 126. George Santayana – The Life of Reason; Skepticism and Animal Faith; Persons and Places 127. Vladimir Lenin – The State and Revo

The focus of history and philosophy of science scholar Arthur Miller’s (2010) "137: Jung and Pauli and the Pursuit of Scientific Obsession" is Jung and Pauli’s mutual effort to discover the cosmic number or fine structure constant, which is a fundamental physical constant dealing with electromagnetism, or, from a different perspective, could be considered the philosopher’s stone of the mathematical universe. This was indeed one of Pauli and Jung’s collaborative passions, but it was not the only concentration of their relationship. Quantum physics could be seen as the natural progression from ancient alchemy, through chemistry, culminating in the abstract world of subatomic particles, wave functions, and mathematics. [Ancient Egypt and Modern Psychotherapy]

But that there is a simple relation between literary and other fictions seems, if one attends to it, more obvious than has appeared. If we think first of modern fictions, it can hardly be an accident that ever since Nietzsche generalized and developed the Kantian insights, literature has increasingly asserted its right to an arbitrary and private choice of fictional norms, just as historiography has become a discipline more devious and dubious because of our recognition that its methods depend to an unsuspected degree on myths and fictions. After Nietzsche it was possible to say, as Stevens did, that 'the final belief must be in a fiction.' This poet, to whom the whole question was of perpetual interest, saw that to think in this way was to postpone the End--when the fiction might be said to coincide with reality--for ever; to make of it a fiction, an imaginary moment when 'at last' the world of fact and the mundo of fiction shall be one. Such a fiction--the last section of Notes toward a Supreme Fiction is, appropriately, the place where Stevens gives it his fullest attention--such a fiction of the end is like infinity plus one and imaginary numbers in mathematics, something we know does not exist, but which helps us to make sense of and to move in the world. Mundo is itself such a fiction. I think Stevens, who certainly thought we have to make our sense out of whatever materials we find to hand, borrowed it from Ortega. His general doctrine of fictions he took from Vaihinger, from Nietzsche, perhaps also from American pragmatism.

Common sense requires modern man’s recognition of the scientific method as a spectacularly useful instrumentality for transforming our environment. Respect and gratitude are indeed due the scientist for many comforts and conveniences furnished to modern living, often as the fruit of painstakingly sacrificial research and experimentation, although in recent times not often without financial reward. This practical success of science inclines many persons to a tacit acceptance of the scientific world-picture of external reality as a realm merely of impersonal processes and mathematically connectible sequences. Charles H. Malik observes rightly that all too often the highly merited prestige of scientists in their own fields of competence is transferred to areas of publicly expressed opinion in which they are novices.

But in connection with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from what appeared self-evident, not inductively from what had been observed. Its amazing successes in the employment of this method misled not only the ancient world, but the greater part of the modern world also. It has only been very slowly that scientific method, which seeks to reach principles inductively from observations of particular facts, has replaced the Hellenic belief in deduction from luminous axioms derived from the mind of the philosopher. For this reason, apart from others, it is a mistake to treat the Greeks with superstitious reverence. Scientific method, though some few among them were the first men who had an inkling of it, is, on the whole, alien to their temper of mind, and the attempt to glorify them by belittling the intellectual progress of the last four centuries has a cramping effect upon modern thought.

Pythagoras, as everyone knows, said that 'all things are numbers'. This statement, interpreted in a modern way, is logically nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music, and the connection which he established between music and arithmetic survives in the mathematical terms 'harmonic mean' and 'harmonic progression'. He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares and cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or, as we should more naturally say, shot) required to make the shapes in question. He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.

Perhaps the most influential person ever associated with Samos was Pythagoras,* a contemporary of Polycrates in the sixth century B.C. According to local tradition, he lived for a time in a cave on the Samian Mount Kerkis, and was the first person in the history of the world to deduce that the Earth is a sphere. Perhaps he argued by analogy with the Moon and the Sun, or noticed the curved shadow of the Earth on the Moon during a lunar eclipse, or recognized that when ships leave Samos and recede over the horizon, their masts disappear last. He or his disciples discovered the Pythagorean theorem: the sum of the squares of the shorter sides of a right triangle equals the square of the longer side. Pythagoras did not simply enumerate examples of this theorem; he developed a method of mathematical deduction to prove the thing generally. The modern tradition of mathematical argument, essential to all of science, owes much to Pythagoras. It was he who first used the word Cosmos to denote a well-ordered and harmonious universe, a world amenable to human understanding.

As Descartes said to his close correspondent, Marin Mersenne, “I may tell you, between ourselves, that these six Meditations contain all the foundations of my physics. But please do not tell people, for that might make it harder for supporters of Aristotle to approve of them. I hope that readers will gradually get used to my principles, and recognize their truth, before they notice that they destroy the principles of Aristotle” (18 January 1641). The Meditations attempts a complete intellectual revolution: the replacement of Aristotelian philosophy with a new philosophy in order to replace Aristotelian science with a new science. For a 17th-century Aristotelian, a body is matter informed by substantial and accidental forms, and change is explained by the gain or loss of such forms: in mutation by the acquisition of a substantial form, and in what Aristotelians would call true motion (that is, augmentation and diminution, alteration, or local motion) by the successive acquisition of places or of qualitative or quantitative forms. The mechanist program consisted in doing away with qualitative forms and reducing all changes to something mathematically quantifiable: matter in motion. As Descartes said in The World, not only the four qualities called heat, cold, moistness, and dryness, “but also all the others

Isaac Newton is perhaps the greatest scientist who ever lived. In a world obsessed with witchcraft and sorcery, he dared to write down the universal laws of the heavens and apply a new mathematics he invented to study forces, called the calculus. As physicist Steven Weinberg has written, 'It is with Isaac Newton that the modern dream of a final theory really begins.' In its time, it was considered to be the theory of everything-that is, the theory that described all motion. It all began when he was twenty-three years old. Cambridge University was closed because of the black plague. One day in 1666, while walking around his country estate, he saw an apple fall. Then he asked himself a question that would alter the course of human history. If an apple falls, then does the moon also fall? Before Newton, the church taught that there were two kinds of laws. The first were the laws found on Earth, which were corrupted by the sin of mortals. The second were the pure, perfect, and harmonious laws of the heavens. The essence of Newton's idea was to propose a unified theory that encompassed the heavens and the Earth. In his notebook, he drew a fateful picture (see figure 1). If a cannonball is fired from a mountaintop, it goes a certain distance before hitting the ground. But if you fire the cannonball at increasing velocities, it travels farther and farther before coming back to Earth, until it eventually completely circles the Earth and returns to the mountaintop. He concluded that the natural law that governs apples and cannonballs, gravity, also grips the moon in its orbit around the Earth. Terrestrial and heavenly physics were the same.

A Code of Nature must accommodate a mixture of individually different behavioral tendencies. The human race plays a mixed strategy in the game of life. People are not molecules, all alike and behaving differently only because of random interactions. People just differ, dancing to their own personal drummer. The merger of economic game theory with neuroscience promises more precise understanding of those individual differences and how they contribute to the totality of human social interactions. It's understanding those differences, Camerer says, that will make such a break with old schools of economic thought. "A lot of economic theory uses what is called the representative agent model," Camerer told me. In an economy with millions of people, everybody is clearly not going to be completely alike in behavior. Maybe 10 percent will be of some type, 14 percent another type, 6 percent something else. A real mix. "It's often really hard, mathematically, to add all that up," he said. "It's much easier to say that there's one kind of person and there's a million of them. And you can add things up rather easily." So for the sake of computational simplicity, economists would operate as though the world was populated by millions of one generic type of person, using assumptions about how that generic person would behave. "It's not that we don't think people are different—of course they are, but that wasn't the focus of analysis," Camerer said. "It was, well, let's just stick to one type of person. But I think the brain evidence, as well as genetics, is just going to force us to think about individual differences." And in a way, that is a very natural thing for economists to want to do.

In 1935, three years before his death, Edmund Husserl gave his celebrated lectures in Vienna and Prague on the crisis of European humanity. For Husserl, the adjective "European" meant the spiritual identity that extends beyond geographical Europe (to America, for instance) and that was born with ancient Greek philosophy. In his view, this philosophy, for the first time in History, apprehended the world (the world as a whole) as a question to be answered. It interrogated the world not in order to satisfy this or that practical need but because "the passion to know had seized mankind." The crisis Husserl spoke of seemed to him so profound that he wondered whether Europe was still able to survive it. The roots of the crisis lay for him at the beginning of the Modern Era, in Galileo and Descartes, in the one-sided nature of the European sciences, which reduced the world to a mere object of technical and mathematical investigation and put the concrete world of life, die Lebenswelt as he called it, beyond their horizon. The rise of the sciences propelled man into the tunnels of the specialized disciplines. The more he advanced in knowledge, the less clearly could he see either the world as a whole or his own self, and he plunged further into what Husserl's pupil Heidegger called, in a beautiful and almost magical phrase, "the forgetting of being." Once elevated by Descartes to "master and proprietor of nature," man has now become a mere thing to the forces (of technology, of politics, of history) that bypass him, surpass him, possess him. To those forces, man's concrete being, his "world of life" (die Lebenswelt), has neither value nor interest: it is eclipsed, forgotten from the start.

Those who are condemned to see the world in shadows will be forced to formulate a description of reality based on insufficient information and, therefore, contrivance. Human history verifies the axiom, “In the absence of knowledge, superstition prevails.” Impaired perception impedes comprehension and breeds fabrication. We find the gaps in our knowledge irritating and uncomfortable, like a road riddled with potholes, and so we fill them in with fabrications to make the ride smoother. The rutted highway of human knowledge is mended with all sorts of contrivances concerning the nature of the universe, a query that for many centuries was beyond investigation. The tools of modern science have enabled us to repave the road, in a manner of speaking, and to upend, one by one, the falsities of our former ignorance. But the road is long, and the work is slow. We must concede that our current conception of the universe is still infantile. Like a child staring bewilderedly at a blackboard chalked from end to end with the esoteric figures of a complex mathematical formula, we are able to recognize some of the numbers and symbols but cannot hope to comprehend the equation, much less solve it. But rather than accept the irreducible complexity before us, many Christians have endeavored to reduce what they cannot comprehend into facile religious concepts that they can. This “Sunday school reductionism” tends to transform profound truths into coloring book illustrations and connect-the-dot puzzles. Instead of illuminating the problem with the lamp of logic and admitting our ignorance, we tend to obscure the problem beneath a canopy of nebulous abstractions, commending ourselves with the false satisfaction of having “solved” it.

That such a surprisingly powerful philosophical method was taken seriously can be only partially explained by the backwardness of German natural science in those days. For the truth is, I think, that it was not at first taken really seriously by serious men (such as Schopenhauer, or J. F. Fries), not at any rate by those scientists who, like Democritus2, ‘would rather find a single causal law than be the king of Persia’. Hegel’s fame was made by those who prefer a quick initiation into the deeper secrets of this world to the laborious technicalities of a science which, after all, may only disappoint them by its lack of power to unveil all mysteries. For they soon found out that nothing could be applied with such ease to any problem whatsoever, and at the same time with such impressive (though only apparent) difficulty, and with such quick and sure but imposing success, nothing could be used as cheaply and with so little scientific training and knowledge, and nothing would give such a spectacular scientific air, as did Hegelian dialectics, the mystery method that replaced ‘barren formal logic’. Hegel’s success was the beginning of the ‘age of dishonesty’ (as Schopenhauer3 described the period of German Idealism) and of the ‘age of irresponsibility’ (as K. Heiden characterizes the age of modern totalitarianism); first of intellectual, and later, as one of its consequences, of moral irresponsibility; of a new age controlled by the magic of high-sounding words, and by the power of jargon. In order to discourage the reader beforehand from taking Hegel’s bombastic and mystifying cant too seriously, I shall quote some of the amazing details which he discovered about sound, and especially about the relations between sound and heat. I have tried hard to translate this gibberish from Hegel’s Philosophy of Nature4 as faithfully as possible; he writes: ‘§302. Sound is the change in the specific condition of segregation of the material parts, and in the negation of this condition;—merely an abstract or an ideal ideality, as it were, of that specification. But this change, accordingly, is itself immediately the negation of the material specific subsistence; which is, therefore, real ideality of specific gravity and cohesion, i.e.—heat. The heating up of sounding bodies, just as of beaten or rubbed ones, is the appearance of heat, originating conceptually together with sound.’ There are some who still believe in Hegel’s sincerity, or who still doubt whether his secret might not be profundity, fullness of thought, rather than emptiness. I should like them to read carefully the last sentence—the only intelligible one—of this quotation, because in this sentence, Hegel gives himself away. For clearly it means nothing but: ‘The heating up of sounding bodies … is heat … together with sound.’ The question arises whether Hegel deceived himself, hypnotized by his own inspiring jargon, or whether he boldly set out to deceive and bewitch others. I am satisfied that the latter was the case, especially in view of what Hegel wrote in one of his letters. In this letter, dated a few years before the publication of his Philosophy of Nature, Hegel referred to another Philosophy of Nature, written by his former friend Schelling: ‘I have had too much to do … with mathematics … differential calculus, chemistry’, Hegel boasts in this letter (but this is just bluff), ‘to let myself be taken in by the humbug of the Philosophy of Nature, by this philosophizing without knowledge of fact … and by the treatment of mere fancies, even imbecile fancies, as ideas.’ This is a very fair characterization of Schelling’s method, that is to say, of that audacious way of bluffing which Hegel himself copied, or rather aggravated, as soon as he realized that, if it reached its proper audience, it meant success.

This makes a mockery of real science, and its consequences are invariably ridiculous. Quite a few otherwise intelligent men and women take it as an established principle that we can know as true only what can be verified by empirical methods of experimentation and observation. This is, for one thing, a notoriously self-refuting claim, inasmuch as it cannot itself be demonstrated to be true by any application of empirical method. More to the point, though, it is transparent nonsense: most of the things we know to be true, often quite indubitably, do not fall within the realm of what can be tested by empirical methods; they are by their nature episodic, experiential, local, personal, intuitive, or purely logical. The sciences concern certain facts as organized by certain theories, and certain theories as constrained by certain facts; they accumulate evidence and enucleate hypotheses within very strictly limited paradigms; but they do not provide proofs of where reality begins or ends, or of what the dimensions of truth are. They cannot even establish their own working premises—the real existence of the phenomenal world, the power of the human intellect accurately to reflect that reality, the perfect lawfulness of nature, its interpretability, its mathematical regularity, and so forth—and should not seek to do so, but should confine themselves to the truths to which their methods give them access. They should also recognize what the boundaries of the scientific rescript are. There are, in fact, truths of reason that are far surer than even the most amply supported findings of empirical science because such truths are not, as those findings must always be, susceptible of later theoretical revision; and then there are truths of mathematics that are subject to proof in the most proper sense and so are more irrefutable still. And there is no one single discourse of truth as such, no single path to the knowledge of reality, no single method that can exhaustively define what knowledge is, no useful answers whose range has not been limited in advance by the kind of questions that prompted them. The failure to realize this can lead only to delusions of the kind expressed in, for example, G. G. Simpson’s self-parodying assertion that all attempts to define the meaning of life or the nature of humanity made before 1859 are now entirely worthless, or in Peter Atkins’s ebulliently absurd claims that modern science can “deal with every aspect of existence” and that it has in fact “never encountered a barrier.” Not only do sentiments of this sort verge upon the deranged, they are nothing less than violent assaults upon the true dignity of science (which lies entirely in its severely self-limiting rigor).

The textbooks of history prepared for the public schools are marked by a rather naive parochialism and chauvinism. There is no need to dwell on such futilities. But it must be admitted that even for the most conscientious historian abstention from judgments of value may offer certain difficulties. As a man and as a citizen the historian takes sides in many feuds and controversies of his age. It is not easy to combine scientific aloofness in historical studies with partisanship in mundane interests. But that can and has been achieved by outstanding historians. The historian's world view may color his work. His representation of events may be interlarded with remarks that betray his feelings and wishes and divulge his party affiliation. However, the postulate of scientific history's abstention from value judgments is not infringed by occasional remarks expressing the preferences of the historian if the general purport of the study is not affected. If the writer, speaking of an inept commander of the forces of his own nation or party, says "unfortunately" the general was not equal to his task, he has not failed in his duty as a historian. The historian is free to lament the destruction of the masterpieces of Greek art provided his regret does not influence his report of the events that brought about this destruction. The problem of Wertfreíheit must also be clearly distinguished from that of the choice of theories resorted to for the interpretation of facts. In dealing with the data available, the historian needs ali the knowledge provided by the other disciplines, by logic, mathematics, praxeology, and the natural sciences. If what these disciplines teach is insufficient or if the historian chooses an erroneous theory out of several conflicting theories held by the specialists, his effort is misled and his performance is abortive. It may be that he chose an untenable theory because he was biased and this theory best suited his party spirit. But the acceptance of a faulty doctrine may often be merely the outcome of ignorance or of the fact that it enjoys greater popularity than more correct doctrines. The main source of dissent among historians is divergence in regard to the teachings of ali the other branches of knowledge upon which they base their presentation. To a historian of earlier days who believed in witchcraft, magic, and the devil's interference with human affairs, things hàd a different aspect than they have for an agnostic historian. The neomercantilist doctrines of the balance of payments and of the dollar shortage give an image of presentday world conditions very different from that provided by an examination of the situation from the point of view of modern subjectivist economics.

We are living now, not in the delicious intoxication induced by the early successes of science, but in a rather grisly morning-after, when it has become apparent that what triumphant science has done hitherto is to improve the means for achieving unimproved or actually deteriorated ends. In this condition of apprehensive sobriety we are able to see that the contents of literature, art, music—even in some measure of divinity and school metaphysics—are not sophistry and illusion, but simply those elements of experience which scientists chose to leave out of account, for the good reason that they had no intellectual methods for dealing with them. In the arts, in philosophy, in religion men are trying—doubtless, without complete success—to describe and explain the non-measurable, purely qualitative aspects of reality. Since the time of Galileo, scientists have admitted, sometimes explicitly but much more often by implication, that they are incompetent to discuss such matters. The scientific picture of the world is what it is because men of science combine this incompetence with certain special competences. They have no right to claim that this product of incompetence and specialization is a complete picture of reality. As a matter of historical fact, however, this claim has constantly been made. The successive steps in the process of identifying an arbitrary abstraction from reality with reality itself have been described, very fully and lucidly, in Burtt’s excellent “Metaphysical Foundations of Modern Science"; and it is therefore unnecessary for me to develop the theme any further. All that I need add is the fact that, in recent years, many men of science have come to realize that the scientific picture of the world is a partial one—the product of their special competence in mathematics and their special incompetence to deal systematically with aesthetic and moral values, religious experiences and intuitions of significance. Unhappily, novel ideas become acceptable to the less intelligent members of society only with a very considerable time-lag. Sixty or seventy years ago the majority of scientists believed—and the belief often caused them considerable distress—that the product of their special incompetence was identical with reality as a whole. Today this belief has begun to give way, in scientific circles, to a different and obviously truer conception of the relation between science and total experience. The masses, on the contrary, have just reached the point where the ancestors of today’s scientists were standing two generations back. They are convinced that the scientific picture of an arbitrary abstraction from reality is a picture of reality as a whole and that therefore the world is without meaning or value. But nobody likes living in such a world. To satisfy their hunger for meaning and value, they turn to such doctrines as nationalism, fascism and revolutionary communism. Philosophically and scientifically, these doctrines are absurd; but for the masses in every community, they have this great merit: they attribute the meaning and value that have been taken away from the world as a whole to the particular part of the world in which the believers happen to be living.

In many fields—literature, music, architecture—the label ‘Modern’ stretches back to the early 20th century. Philosophy is odd in starting its Modern period almost 400 years earlier. This oddity is explained in large measure by a radical 16th century shift in our understanding of nature, a shift that also transformed our understanding of knowledge itself. On our Modern side of this line, thinkers as far back as Galileo Galilei (1564–1642) are engaged in research projects recognizably similar to our own. If we look back to the Pre-Modern era, we see something alien: this era features very different ways of thinking about how nature worked, and how it could be known. To sample the strange flavour of pre-Modern thinking, try the following passage from the Renaissance thinker Paracelsus (1493–1541): The whole world surrounds man as a circle surrounds one point. From this it follows that all things are related to this one point, no differently from an apple seed which is surrounded and preserved by the fruit … Everything that astronomical theory has profoundly fathomed by studying the planetary aspects and the stars … can also be applied to the firmament of the body. Thinkers in this tradition took the universe to revolve around humanity, and sought to gain knowledge of nature by finding parallels between us and the heavens, seeing reality as a symbolic work of art composed with us in mind (see Figure 3). By the 16th century, the idea that everything revolved around and reflected humanity was in danger, threatened by a number of unsettling discoveries, not least the proposal, advanced by Nicolaus Copernicus (1473–1543), that the earth was not actually at the centre of the universe. The old tradition struggled against the rise of the new. Faced with the news that Galileo’s telescopes had detected moons orbiting Jupiter, the traditionally minded scholar Francesco Sizzi argued that such observations were obviously mistaken. According to Sizzi, there could not possibly be more than seven ‘roving planets’ (or heavenly bodies other than the stars), given that there are seven holes in an animal’s head (two eyes, two ears, two nostrils and a mouth), seven metals, and seven days in a week. Sizzi didn’t win that battle. It’s not just that we agree with Galileo that there are more than seven things moving around in the solar system. More fundamentally, we have a different way of thinking about nature and knowledge. We no longer expect there to be any special human significance to natural facts (‘Why seven planets as opposed to eight or 15?’) and we think knowledge will be gained by systematic and open-minded observations of nature rather than the sorts of analogies and patterns to which Sizzi appeals. However, the transition into the Modern era was not an easy one. The pattern-oriented ways of thinking characteristic of pre-Modern thought naturally appeal to meaning-hungry creatures like us. These ways of thinking are found in a great variety of cultures: in classical Chinese thought, for example, the five traditional elements (wood, water, fire, earth, and metal) are matched up with the five senses in a similar correspondence between the inner and the outer. As a further attraction, pre-Modern views often fit more smoothly with our everyday sense experience: naively, the earth looks to be stable and fixed while the sun moves across the sky, and it takes some serious discipline to convince oneself that the mathematically more simple models (like the sun-centred model of the solar system) are right.

The Soviet system had left many valuable legacies—a huge network of large industrial enterprises (though stranded in the 1960s in terms of technology); a vast military machine; and an extraordinary reservoir of scientific, mathematical, and technical talent, although disconnected from a commercial economy. The highly capable oil industry was burdened with an ageing infrastructure. Below ground lay all the enormous riches in the form of petroleum and other raw materials that Gorbachev had cited in his farewell address

With over 5,000 years of continuous history, the subcontinent known as India has flourished. Its culture, people, and history have added a crucial, colorful chapter to the history of humankind as a whole. India has participated in many events that shaped the progress and future of mankind, and its art, philosophy, literature, and culture have influenced billions. From the culture's inception in the Indus Valley or Harappan Civilization, the people of the Indian subcontinent have acted as the fulcrum between the east and west. Their civilization once flourished as a trading titan and provided the ancient world with a rich and varied society, unlike its contemporaries it did so without succumbing to the horrors of war. This tradition of economic and philosophic focus would be transmitted throughout the ages through each of the different eras in Indian history. In the ancient world, the Indus Valley civilization provided the backbone of what would become Indian culture. As the society eventually collapsed, it left behind traces of its existence to be found and adopted by the Vedic peoples that sprung from their demise. In the Vedic period, Indian culture and history were shaped and transformed into literary masterpieces that survive today as a lynchpin of Hindu philosophy. It also saw the birth of Buddhism, the ascension of the Buddha and the spread of a counter culture that has expanded far across the globe, influencing the lives of millions. This very formative era in Indian history gives modern-day society an idea of what the structure of Indian history and society would become. This feudal period in India was one of ideological development in both the Vedic or Hindu ways and the ways of the Sramana traditions that arose as a countercultural movement. These two ideologies would go on to influence the various empires that would begin to form after the Vedic Age. In the Age of Empires, the Indian subcontinent would witness the birth of empires like that of Cyrus the Great in Persia and Alexander the Great of Macedonia. The disunity of the Indian kingdoms would allow foreign invaders to influence this era, but although the smaller Indian kingdoms were defeated in many ways, India remained unconquered as a whole. From this disunity and vulnerability, the first Indian empires would begin taking shape. From the Mauryan to the Gupta and beyond, the first Indian empires would shape the history of India in ways that are hard to fathom. Science, mathematics, art, architecture, and literature would flourish in this age. This period would provide India with a national identity that hangs on to this day. In the Age of Muslim Expansion, India was introduced to yet another vital part of its history and culture. Though many wars were fought between the Indian kingdoms and the Muslim sultanates, the people of the Indian subcontinent adopted an attitude of religious tolerance that persists to this day. In modern-day India, you can see the influence of the Muslim cultures that put down roots in India during this time, most notably in the Taj Mahal. In the Age of Exploration, the expansion of European power across the globe would shape the history of India under the Portuguese, Dutch, and eventually the British. This period, although known for exploitation, can also be attributed with the birth of Indian democracy and republican values that we would see born in the modern age. Though the modern age is but a minuscule fraction of the gravitas of Indian history, it maintains itself as a colorful portrait of the Indian soul. If one truly wants to understand Indian history, one but has to look at the astounding culture of modern-day India. The 50 events chosen to be illustrated in this book are but a few of the thousands if not millions of crucial events that shaped and built the extravagance of the country we now call India.

But when the agricultural villages of the Neolithic expanded into larger towns that grew to more than two thousand inhabitants, the capacity of the human brain to know and recognize all of the members of a single community was stretched beyond its natural limits. Nevertheless, the tribal cultures that had evolved during the Upper Paleolithic with the emergence of symbolic communication enabled people who might have been strangers to feel a collective sense of belonging and solidarity. It was the formation of tribes and ethnicities that enabled the strangers of the large Neolithic towns to trust each other and interact comfortably with each other, even if they were not all personally acquainted. The transformation of human society into urban civilizations, however, involved a great fusion of people and societies into groups so large that there was no possibility of having personal relationships with more than a tiny fraction of them. Yet the human capacity for tribal solidarity meant that there was literally no upper limit on the size that a human group could attain. And if we mark the year 3000 BC as the approximate time when all the elements of urban civilization came together to trigger this new transformation, it has taken only five thousand years for all of humanity to be swallowed up by the immense nation-states that have now taken possession of every square inch of the inhabited world. The new urban civilizations produced the study of mathematics, astronomy, philosophy, history, biology, and medicine. They greatly advanced and refined the technologies of metallurgy, masonry, architecture, carpentry, shipbuilding, and weaponry. They invented the art of writing and the practical science of engineering. They developed the modern forms of drama, poetry, music, painting, and sculpture. They built canals, roads, bridges, aqueducts, pyramids, tombs, temples, shrines, castles, and fortresses by the thousands all over the world. They built ocean-going ships that sailed the high seas and eventually circumnavigated the globe. From their cultures emerged the great universal religions of Christianity, Buddhism, Confucianism, Islam, and Hinduism. And they invented every form of state government and political system we know, from hereditary monarchies to representative democracies. The new urban civilizations turned out to be dynamic engines of innovation, and in the course of just a few thousand years, they freed humanity from the limitations it had inherited from the hunting and gathering cultures of the past.

Modern culture has disenchanted the world by disenchanting numbers. For us, numbers are about quantity and control, not quality and contemplation. After Bacon, knowledge of numbers is a key to manipulation, not meditation. Numbers are only meaningful (like all raw materials that comprise the natural world) when we can do something with them. When we read of twelve tribes and twelve apostles and twelve gates and twelve angels, we typically perceive something spreadsheet-able. By contrast, in one of Caldecott’s most radical claims, he insists, “It is not simply that numbers can be used as symbols. Numbers have meaning—they are symbols. The symbolism is not always merely projected onto them by us; much of it is inherent in their nature” (p. 75). Numbers convey to well-ordered imaginations something of (in Joseph Cardinal Ratzinger’s metaphor) the inner design of the fabric of creation. The fact that the words “God said” appear ten times in the account of creation and that there are ten “words” in the Decalogue is not a random coincidence. The beautiful meaningfulness of a numberly world is most evident in the perception of harmony, whether in music, architecture, or physics. Called into being by a three-personed God, creation’s essential relationality is often evident in complex patterns that can be described mathematically. Sadly, as Caldecott laments, “our present education tends to eliminate the contemplative or qualitative dimension of mathematics altogether” (p. 55). The sense of transcendence that many (including mathematicians and musicians) experience when encountering beauty is often explained away by materialists as an illusion. Caldecott offers an explanation rooted in Christology. Since the Logos is love, and since all things are created through him and for him and are held together in him, we should expect the logic, the rationality, the intelligibility of the world to usher in the delight that beauty bestows. One

This parallelism between physics and psychology should occasion no great surprise. The human nervous system, after all — the "mind" in pre-scientific language — created modern science, including physics and quantum mathematics. One should expect to find the genius, and the defects, of the human mind in its creations, as one always finds the autobiography of the artist in the art-work.

Though mathematics is used in the formulation of physical theories and in working out their consequences, science is not a branch of mathematics,

Democritus wrote books on ethics, natural science, mathematics, and music, of which many fragments survive. One of these fragments expresses the view that all matter consists of tiny indivisible particles called atoms (from the Greek for “uncuttable”), moving in empty space: “Sweet exists by convention, bitter by convention; atoms and Void [alone] exist in reality.

Most early scientists were compelled to study the natural world because of their Christian worldview. In Science and the Modern World, British mathematician and philosopher Alfred North Whitehead concludes that modern science developed primarily from “the medieval insistence on the rationality of God.” Modern science did not develop in a vacuum, but from forces largely propelled by Christianity. Not surprisingly, most early scientists were theists, including pioneers such as Francis Bacon (1561–1626), Johannes Kepler (1571– 1630), Blaise Pascal (1623–62), Robert Boyle (1627–91), Isaac Newton (1642–1727), and Louis Pasteur (1822–95). For many of them, belief in God was the prime motivation for their investigation of the natural world. Bacon believed the natural world was full of mysteries God intended for us to explore. Kepler described his motivation for science: “The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God, and which he revealed to us in the language of mathematics.

Richard Feynman had to say this about energy in his 1961 lecture: “There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.” All significant philosophers and scientists throughout history were in their own right, right if we consider the context, time, and place, the point from which they observed the world by the means available to them. If we understand this context, we know how much harder it was for them to decipher the world previously unknown, except as an experience without fundamental and deeper understanding. In this sense, all these philosophers and scientists were, in a way, “right,” even when they were “not” right. Correctness or wrongness of their ideas and opinions shall be measured more by how they helped our understanding and ideas developed directly from their thoughts. Even if they were in some way wrong, great ideas helped our ideas develop and allowed the formation and formulations of great ideas that will follow. Quality and potential of insights and ideas are more important than strict correctness without any potential. Progress in human history would not be possible without following the traces of long-bygone giants (as Newton understood them). We can hardly produce any new important question that ancient Greek philosophers did not pose. The whole idea of Western philosophy, as it is, would not be possible without the ancient Greeks. This statement holds even when we talk about the modern era’s greatest philosophers, starting with Descartes and culminating in the works of the great German philosophers Leibnitz, Kant, Hegel, Schopenhauer, the Dutch Spinoza, and others. Almost all central questions or problems treated by these philosophers were already postulated, discussed, or touched, directly or indirectly, by the great ancient philosophers who paved the way for the others.

is a Civill Warr with the Pen, which pulls out the sorde soon afterwards.

To coin a Uexküllian-Heideggerian neologism, Jews were to Uexküll the epitome of Umweltvergessenheit or the “forgetfulness of Umwelt”—an inability to grasp and experience one’s own preordained environment that is both brought about and glossed over by vague appeals to universal liberty and justice. But this was nothing specifically or uniquely Jewish; historical circumstances conspired to make the Jews the avant-garde of modern decline universal, a portent of what was to come if the world succumbed to newfangled notions of absolute time, absolute space, absolute symbolic exchange in the shape of money and mathematics, and the abstractions of modern science. This “regrettable laying-waste of the worlds-as-sensed [that] has arisen from the superstition started by the physicists”38 could be averted if people—or rather, the elites—were to accept his new biology, but while Uexküll could pass on the knowledge of what it means to inhabit and shape one’s own Umwelt next to all the myriads of other human and animal Umwelten, he was not able to impart the experience. That is the business of artists.

seemed almost certain to the mathematicians that since the general first, second, third, and fourth degree equations can be solved by means of the usual algebraic operations such as addition, subtraction, and roots, then the general fifth degree equation and still higher degree equations could also be solved. For three hundred years this problem was a classic one. Hundreds of mature and expert mathematicians sought the solution, but a little boy found the full answer. The Frenchman Évariste Galois (1811— 1832), who refused to conform to school examinations but worked brilliantly and furiously on his own, showed that general equations of degree higher than the fourth cannot be solved by algebraic operations. To establish this result Galois created the theory of groups, a subject that is now at the base of modern abstract algebra and that transformed algebra from a a series of elementary techniques to a broad, abstract, and basic branch of mathematics.

The distinction between mathematics and science is pretty well settled. It remains mysterious to us why mathematics that is invented for reasons having nothing to do with nature often turns out to be useful in physical theories. In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

in the absence of political order, human life is “solitary, poor, nasty, brutish, and short.

one person, of whose acts a great multitude … have made themselves every one the author” in order that peace will prevail.

No longer a critic, he was now a rebel,

What up to that point could have been viewed as an internal rebellion within the Church now became a schism in which two rival faiths confronted each other in open hostility.

I think, therefore I am.

Of all the tragedies wrought by this collective amnesia in economics, the greatest loss to the world is the eclipse of the “Austrian School.” Founded in the 1870s and 1880s, and still barely alive, the Austrian School has had to suffer far more neglect than the other schools of economics for a variety of powerful reasons. First, of course, it was founded a century ago, which, in the current scientific age, is in itself suspicious. Second, the Austrian School has from the beginning been self-consciously philosophic rather than “scientistic”; far more concerned with methodology and epistemology than other modern economists, the Austrians arrived early at a principled opposition to the use of mathematics or of statistical “testing” in economic theory. By doing so, they set themselves in opposition to all the positivistic, natural-science-imitating trends of this century. It meant, furthermore, that Austrians continued to write fundamental treatises while other economists were setting their sights on narrow, mathematically oriented articles. And third, by stressing the individual and his choices, both methodologically and politically, Austrians were setting themselves against the holism and statism of this century as well.

In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that … a huge proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility…. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. —Alfred North Whitehead

Strings possess one defining property which is their tension. This quantity plays a crucial role in the overall picture of how strings can be reconciled with the miraculous experimental success of the point-like Quantum field theories in explaining the observed features of the world at lower Energies. For the strings possess a tension that varies with the energy of the environment, so that at low energies, the tension is high and pulls the strings taut into points and we recover the favorable features of a world of Point like Elementary particles. At high energies where the string tension is low, their essential stringiness becomes evident and creates Behavior that is qualitatively different from that of the point particle theories. Unfortunately at present the mathematical expertise required to reveal these properties is somewhat Beyond us. For the first time modern physicists have found that off-the-shelf mathematics is insufficient to extract the physical content of their theories. But, in time, suitable techniques will no doubt emerge, or perhaps a better way to look at the theory will be found: one that is conceptually and technically simpler.

Literacy, usually in Arabic, was spread through the teaching of the Qur’an. Thus the mosques became centres of learning. In this way, the peoples of northern and western Africa were exposed to and contributed to the intellectual achievements of the Muslim world. These achievements were considerable, especially in the fields of mathematics and science; it was people from this vast Muslim-Arab world who developed our modern numeral system based on counting from 1 to 10. They invented algebra, the use of the decimal point and the number zero – a mathematical concept missed by the Ancient Greeks. They developed physics and astronomy. They studied chemistry and were the first people to separate medicine from religion and develop it as a secular science. As we shall see later in this chapter, the peoples of the western Sudan became part of this Muslim intellectual tradition.

The most decisive argument against democracy can be summed up in a few words: the higher cannot proceed from the lower, because the greater cannot proceed from the lesser; this is an absolute mathematical certainty that nothing can gainsay. And it should be remarked that this same argument, applied to a different order of things, can also be invoked against materialism; there is nothing fortuitous in this, for these two attitudes are much more closely linked than might at first sight appear. It is abundantly clear that the people cannot confer a power that they do not themselves possess; true power can only come from above, and this is why-be it said in passing-it can be legitimized only by the sanction of something standing above the social order, that is to say by a spiritual authority, for otherwise it is a mere counterfeit of power, unjustifiable through lack of any principle, and in which there can be nothing but disorder and confusion.

They believed in angles and alchemy and the devil, and they believed that the universe followed precise, mathematical laws.

Magic is a practical science,' he began quickly. He talked to the wall, as if dictating. 'There is all the difference in the world between a formula in physics and a formula in magic, although they have the same name. The former describes, in terse mathematical symbol, cause-effect relationships of wide generality. But a formula in magic is a way of getting or accomplishing something. It always takes into account the motivation or desire of the person invoking the formula—be it greed, love, revenge, or what not. Whereas the experiment in physics is essentially independent of the experimenter. In short, there has been little or no pure magic, comparable to pure science. 'This distinction between physics and magic is only an accident of history. Physics started out as a kind of magic, too—witness alchemy and the mystical mathematics of Pythagoras. And modern physics is ultimately as practical as magic, but it possesses a superstructure of theory that magic lacks. Magic could be given such a superstructure by research in pure magic and by the investigation and correlation of the magic formulas which could be expressed in mathematical symbols and which would have a wide application. Most persons practicing magic have been too interested in immediate results to bother about theory. But just as research in pure science has ultimately led, seemingly by accident, to results of vast practical importance, so research in pure magic might be expected to yield similar results.

Physicists are more opportunistic, demanding only enough precision and certainty to give them a good chance of avoiding serious mistakes. In the preface of my own treatise on the quantum theory of fields, I admit that “there are parts of this book that will bring tears to the eyes of the mathematically inclined reader.

Mistakes like the ones in the following pages aren’t just amusing; they’re revealing. They briefly pull back the curtain to reveal the mathematics that is normally working unnoticed behind the scenes. It’s as if, behind our modern wizardry, Oz is revealed working overtime with an abacus and a slide rule.

The trouble with the world today is not that we have too much Mathematics, but that we do not yet have enough.

All my knowledge of the world, even my scientific knowledge, is gained from my own particular point of view. The whole universe of science is built upon the world as directly experienced. It is unfortunate that modern science, and in particular mathematics, would like to forget this. Scientific points of view, according to which my existence is a moment of the world’s, are always both naive and at the same time dishonest, because they take for granted, without explicitly mentioning it, the other point of view, namely that of consciousness, through which from the outset a world forms itself around me and begins to exist for me. To return to things themselves is to return to that world which precedes knowledge, of which knowledge always speaks, and in relation to which every scientific schematization is an abstract and derivative sign language, as is geography in relation to the country-side in which we have to learn beforehand what a forest, a prairie or a river is.

Self-government" is a marriage of two terms, expressible mathematically as a ratio (self-government). The first thing we should observe about this relationship is that the first term is always static while the second is potentially infinite. The smaller the second term, which is to say, the fewer are the "others" that go to make up the apparatus of government, which within democracy is theoretically everyone, the more tolerable we find the arrangement. But as the second term approaches infinity, the more we feel our isolated "self' dissolving into insignificance. The wider the circumference of the "self-government," the smaller the share of each self in the governing of the selves which comprise it. We begin to understand that what was flattering in theory can become terrifying in practice.

On the other hand, the elimination of the qualitative aspects in favour of a tighter and tighter mathematical definition of atomic structure must necessarily reach a limit, beyond which precision gives way to the indeterminate. This is exactly what is happening with modern atomist science, in which mathematical reflection is being more and more replaced by statistics and calculations of probability, and in which the very laws of causality seem to be facing bankruptcy. If the 'forms' of things are 'lights', as Boethius said, the reduction of the qualitative to the quantitative can be compared to the action of a man who puts out all the lights the better to scrutinize the nature of darkness. Modern science can never reach that matter that is at the basis of this world. But between the qualitiatively differentiated world and undifferentiated matter lies something like an intermediate zoneand this is chaos. The sinister dangers of atomic fission are but one signpost indicating the frontier of chaos and dissolution.

Thus, the spirit of objective inquiry in understanding physical realities was very much there in the works of Muslim scientists. The seminal work on Algebra comes from Al-Khwarizmī and Fibonacci (Leonardo of Pisa) has quoted him. Al-Khwarizmī, the pioneer of Algebra, wrote that given an equation, collecting the unknowns on one side of the equation is called 'al-Jabr'. The word Algebra comes from that. He developed sine, cosine and trigonometric tables, which were later translated in the West. He developed algorithms, which are the building blocks of modern computers. In mathematics, several Muslim scientists like Al-Battani, Al-Beruni and Abul-Wafa contributed to trigonometry. Furthermore, Omar Khayyam worked on Binomial Theorem. He found geometric solutions to all 13 forms of cubic equations.

The Ultimate Creator has to be uncreated since it is necessary for the universe to be created in the first place. We find cause or creator for something that is created and that begins to exist at some finite point in time like the universe which came into existence 13.7 billion years ago. The Ultimate Creator did not come into existence at some finite point in time. It is ever-existing. This God is not the ‘scientific conjecture of god of the gaps’ which fits in the novel for pages that are not found in the novel. This God is the author of the whole novel and the programmer of nested loops within loops. He is not the pixel of the painting or a brush or a colour or the painting itself. It is the painter. It is not the laws of physics or theorems of mathematics alone. It is the source of these laws and theorems. Isaac Newton aptly said that gravity explains the motions of the planets, but it cannot explain who sets the planets in motion.

Once the relation of science and metaphysics with “intellectual intuition” is misunderstood, Kant has no difficulty in showing that our science is entirely relative and our metaphysics wholly artificial. Because he strained the independence of the understanding in both cases, because he relieved metaphysics and science of the “intellectual intuition” which gave them their inner weight, science with its relations presents to him only an outer wrapping of form, and metaphysics with its things, an outer wrapping of matter. Is it surprising, then, that the first shows him only frameworks within frameworks, and the second phantoms pursuing phantoms? He struck our science and metaphysics such rude blows that they have not yet entirely recovered from their shock. Our mind would willingly resign itself to see in science a wholly relative knowledge and in metaphysics an empty speculation. It seems to us even today that Kantian criticism applies to all metaphysics and to all science. In reality it applies especially to the philosophy of the ancients, as well as to the form—still ancient—that the moderns have given most often to their thought. It is valid against a metaphysics which claims to give us a unique and ready-made system of things, against a science which would be a unique system of relations, finally against a science and a metaphysics which present themselves with the architectural simplicity of the Platonic theory of Ideas, or of a Greek temple. If metaphysics claims to be made up of concepts we possessed prior to it, if it consists in an ingenious arrangement of pre-existing ideas which we utilize like the materials of construction for a building, in short, if it is something other than the constant dilation of our mind, the constantly renewed effort to go beyond our actual ideas and perhaps our simple logic as well, it is too evident that it becomes artificial like all works of pure understanding. And if science is wholly the work of analysis or of conceptual representation, if experience is only to serve as the verification of “clear ideas,” if instead of starting from multiple and varied intuitions inserted into the movement proper to each reality but not always fitting into one another, it claims to be an immense mathematics, a single system of relations which imprisons the totality of the real in a mesh prepared for it, it becomes a knowledge purely relative to the human understanding. A close reading of the Critique of Pure Reason will show that for Kant this kind of universal mathematics is science, and this barely modified Platonism, metaphysics. To tell the truth, the dream of a universal mathematics is itself only a survival of Platonism. Universal mathematics is what the world of Ideas becomes when one assumes that the Idea consists in a relation or a law, and no longer in a thing.

Chesterton's topic is nothing less than the fundamental contrast between deductive logic, true of all possible worlds, and inductive logic, capable only of telling us how we may reasonably expect this world to behave. Let us hasten to add that Chesterton's analysis is in full agreement with the views of modern logicians. Perhaps his "test of the imagination" is not strictly accurate--who can "imagine" the four-dimensional constructions of relativity?-but in essence his position is unassailable. Logical and mathematical statements are true by definition. They are "empty tautologies," to use a current phrase, like the impressive maxim that there are always six eggs in half a dozen. Nature, on the other hand, is under no similar constraints. Fortunately, her "weird repetitions," as GK calls them, often conform to surprisingly low-order equations. But as Hume and others before Hume made clear, there is no logical reason why she should behave so politely.

Authenticity and respect go to those who stick to their own specific field of work. For example, I am a Biologist and my work is the understanding of human nature - that's where I place all my attention. I know nothing revelatory about modern physics - I know nothing revelatory about mathematics - I know nothing revelatory about architecture - I know nothing revelatory about any field of understanding except for the ones directly related to biology. It doesn't mean that I cannot learn about other fields - I can, but every human has his or her own distinct knack, and mine is understanding humans - understanding how and why they think, what they think - how and why they feel, what they feel - how and why they behave, the way they behave - how and why they perceive, what they perceive.

A like-minded group coalesced with the common mission of bringing the study of living organisms in line with existing research into the inanimate world. In modern terms, they wanted to show that living organisms obeyed the same mathematical, physical, and chemical laws as everything else. However, this approach put Helmholtz and his network in conflict with a large section of the European scientific community who felt such a synthesis of the animate and inanimate worlds was not possible. Many scientists of the day believed in vitalism, the idea that living organisms, in addition to the sustenance they received from food, water, air, and so on, also possessed a “vital,” life-giving force. While an organism was alive, this vital force controlled the physical and chemical processes that took place within it. Logically, therefore, when it died, that vital force disappeared, leaving the dead organism to decay as if it were inanimate. Helmholtz and his friends opposed this “vitalist” view and felt disproving it was a crucial step to putting biology on the same footing as physics and chemistry.

Roger Bacon had complained that the calendar was “intolerable to all the wise, horrible to all astronomers, and ridiculed by all computists.

...general theories emerge from consideration of the specific, and they are meaningless if they do not serve to clarify and order the more particularized substance below. The interplay between generality and individuality, deduction and construction, logic and imagination -- this is the profound essence of live mathematics. Anyone or another of these aspects of mathematics can be at the center of a given achievement. In a far-reaching development all of them will be involved. Generally speaking, such a development will start from the "concrete" ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual "reality". In brief, the flight into abstract generality must start from and return to the concrete and specific. - Mathematics in the Modern World. Scientific American, Volume 211, No. 3, September 1964.