Mathematical Philosophy Quotes

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

The science of government it is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take the place of, indeed exclude, in a manner, all other arts. I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.

Our species is the only creative species, and it has only one creative instrument, the individual mind and spirit of man. Nothing was ever created by two men. There are no good collaborations, whether in music, in art, in poetry, in mathematics, in philosophy. Once the miracle of creation has taken place, the group can build and extend it, but the group never invents anything. The preciousness lies in the lonely mind of a man.

What comes, is called.

Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being... This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont, to be called Lord God παντοκρατωρ or Universal Ruler.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Whereas I think: I’m lying here in a haystack... The tiny space I occupy is so infinitesimal in comparison with the rest of space, which I don’t occupy and which has no relation to me. And the period of time in which I’m fated to live is so insignificant beside the eternity in which I haven’t existed and won’t exist... And yet in this atom, this mathematical point, blood is circulating, a brain is working, desiring something... What chaos! What a farce!

Ideas that require people to reorganize their picture of the world provoke hostility.

Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

God without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and everywhere, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing.

Some people majored in English to prepare for law school. Others became journalists. The smartest guy in the honors program, Adam Vogel, a child of academics, was planning on getting a Ph.D. and becoming an academic himself. That left a large contingent of people majoring in English by default. Because they weren't left-brained enough for science, because history was too try, philosophy too difficult, geology too petroleum-oriented, and math too mathematical - because they weren't musical, artistic, financially motivated, or really all that smart, these people were pursuing university degrees doing something no different from what they'd done in first grade: reading stories. English was what people who didn't know what to major in majored in.

The person who wishes to attain human perfection should study logic first, next mathematics, then physics, and, lastly, metaphysics.

No code of ethics and no effort are justifiable a priori in the face of the cruel mathematics that command our condition.

I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture in order to give their children a right to study paintings, poetry, music, architecture, statuary, tapestry, and porcelain.

We find that at present the human race is divided into one wise man, nine knaves, and ninety fools out of every hundred. That is, by an optimistic observer. The nine knaves assemble themselves under the banner of the most knavish among them, and become 'politicians'; the wise man stands out, because he knows himself to be hopelessly outnumbered, and devotes himself to poetry, mathematics, or philosophy; while the ninety fools plod off under the banners of the nine villains, according to fancy, into the labyrinths of chicanery, malice and warfare. It is pleasant to have command, observes Sancho Panza, even over a flock of sheep, and that is why the politicians raise their banners. It is, moreover, the same thing for the sheep whatever the banner. If it is democracy, then the nine knaves will become members of parliament; if fascism, they will become party leaders; if communism, commissars. Nothing will be different, except the name. The fools will be still fools, the knaves still leaders, the results still exploitation. As for the wise man, his lot will be much the same under any ideology. Under democracy he will be encouraged to starve to death in a garret, under fascism he will be put in a concentration camp, under communism he will be liquidated.

I should attempt to treat human vice and folly geometrically... the passions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature... I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes, and solids.

In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, "The sole remaining task for philosophy is the analysis of language." What a comedown from the great tradition of philosophy from Aristotle to Kant!

When we are young, we spend much time and pains in filling our note-books with all definitions of Religion, Love, Poetry, Politics, Art, in the hope that, in the course of a few years, we shall have condensed into our encyclopaedia the net value of all the theories at which the world has yet arrived. But year after year our tables get no completeness, and at last we discover that our curve is a parabola, whose arcs will never meet.

Intuition is not a special source of ineffable insight: it is the womb of articulated understanding.

Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero – and an infinity.

Human thought, flying on the trapezes of the star-filled universe, with mathematics stretched beneath, was like an acrobat working with a net but suddenly noticing that in reality there is no net.

Academics are only useful when they try to be useless (say, as in mathematics and philosophy) and dangerous when they try to be useful.

It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers.

Neither agreeable nor disagreeable," I answered. "It just is." Istigkeit — wasn't that the word Meister Eckhart liked to use? "Is-ness." The Being of Platonic philosophy — except that Plato seems to have made the enormous, the grotesque mistake of separating Being from becoming and identifying it with the mathematical abstraction of the Idea. He could never, poor fellow, have seen a bunch of flowers shining with their own inner light and all but quivering under the pressure of the significance with which they were charged; could never have perceived that what rose and iris and carnation so intensely signified was nothing more, and nothing less, than what they were — a transience that was yet eternal life, a perpetual perishing that was at the same time pure Being, a bundle of minute, unique particulars in which, by some unspeakable and yet self-evident paradox, was to be seen the divine source of all existence.

The method of 'postulating' what we want has many advantages ; they are the same as the advantages of theft over honest toil.

Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought.

This most beautiful system [The Universe] could only proceed from the dominion of an intelligent and powerful Being.

The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we’d now call mathematics, things we’d now call religion, and things we’d now call mental illness.

‎In modern physics, there is no such thing as "nothing." Even in a perfect vacuum, pairs of virtual particles are constantly being created and destroyed. The existence of these particles is no mathematical fiction. Though they cannot be directly observed, the effects they create are quite real. The assumption that they exist leads to predictions that have been confirmed by experiment to a high degree of accuracy.

You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated. I mean, what is it in those little molecules and stuff that make one cat behave differently than another, or that make a cat? And how do you define a cat? I have no idea.

We do not know space. We do not see it, we do not hear it, we do not feel it. We are standing in the middle of it, we ourselves are part of it, but we know nothing about it.

If you bake a cupcake, the world has one more cupcake. If you become a circus clown, the world has one more squirt of seltzer down someone's pants. But if you win an Olympic gold medal, the world will not have one more Olympic gold medalist. It will just have you instead of someone else.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

Our craving for generality has [as one] source … our preoccupation with the method of science. I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is “purely descriptive.

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.

If you can't test it, it's not theorics -- it's metatheorics. A branch of philosophy. So, if you want to think of it this way, our test equipment is what defines the boundary separating theorics from philosophy.

Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

Surely, the gods' judgment is certain. But as for us, we must be satisfied to 'come close' to those things, for we are men, who speak according to what is likely, and whose lectures resemble fables.

Tom felt his darkness. His father was beautiful and clever, his mother was short and mathematically sure. Each of his brothers and sisters had looks or gifts or fortune. Tom loved all of them passionately, but he felt heavy and earth-bound. He climbed ecstatic mountains and floundered in the rocky darkness between the peaks. He had spurts of bravery but they were bracketed in battens of cowardice.

In existential mathematics that experience takes the form of two basic equations: The degree of slowness is directly proportional to the intensity of memory; the degree of speed is directly proportional to the intensity of forgetting.

A circle in a straight line is the mathematical symbol of miracle.

It might seem that the empirical philosopher is the slave of his material, but that the pure mathematician, like the musician, is a free creator of his world of ordered beauty.

India was the motherland of our race and Sanskrit the mother of Europe's languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity... of self-government and democracy. In many ways, Mother India is the mother of us all.

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centers of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One.

Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.

Hypotheses non fingo (Latin for "I feign no hypotheses", "I frame no hypotheses", or "I contrive no hypotheses")

At night I would return home, set out a lamp before me, and devote myself to reading and writing. Whenever sleep overcame me or I became conscious of weakening, I would turn aside to drink a cup of wine, so that my strength would return to me. Then I would return to reading. And whenever sleep seized me I would see those very problems in my dream; and many questions became clear to me in my sleep. I continued in this until all of the sciences were deeply rooted within me and I understood them as is humanly possible. Everything which I knew at the time is just as I know it now; I have not added anything to it to this day. Thus I mastered the logical, natural, and mathematical sciences, and I had now reached the science.

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.

Disbelief in God, openly admitted by a majority, is a recent event, not yet completed. Because this event is so recent, Non-Religious Ethics is at a very early stage. We cannot yet predict whether, as in Mathematics, we will all reach agreement. Since we cannot know how Ethics will develop, it is not irrational to have high hopes.

If your wish is to become really a man of science and not merely a petty experimentalist, I should advise you to apply to every branch of natural philosophy, including mathematics.

Russell's books should be bound in two colours, those dealing with mathematical logic in red — and all students of philosophy should read them; those dealing with ethics and politics in blue — and no one should be allowed to read them.

Our species is the only creative species, and it has only one creative instrument, the individual mind and spirit of a man. Nothing was ever created by two men. There are no good collaborations, whether in music, in art, in poetry, in mathematics, in philosophy. Once the miracle of creation has taken place, the group can build and extend it, but the group never invents anything. The preciousness lies in the lonely mind of a man. And now the forces marshaled around the concept of the group have declared a war of extermination on that preciousness, the mind of man. By disparagement, by starvation, by repressions, forced direction, and the stunning hammerblows of conditioning, the free, roving mind is being pursued, roped, blunted, drugged. It is a sad suicidal course our species seems to have taken. And this I believe: that the free, exploring mind of the individual human is the most valuable thing in the world. And this I would fight for: the freedom of the mind to take any direction it wishes, undirected. And this I must fight against: any idea, religion, or government which limits or destroys the individual. This is what I am and what I am about. I can understand why a system built on a pattern must try to destroy the free mind, for this is one thing which can by inspection destroy such a system. Surely I can understand this, and I hate it and I will fight against it to preserve the one thing that separates us from the uncreative beasts. If the glory can be killed, we are lost.

گروهى بدون آن كه به فن منطق مجهز باشند سراغ علوم عقلى آمده، نتوانسته اند حق را از باطل جدا كنند و پنداشته اند علوم، نسبى اند. چنين كسانى را بايد با آموزش قوانين منطق و علوم رياضى معالجه كرد.

The qualities of character can be arranged in triads, in each of which the first and last qualities will be extremes and vices, and the middle quality a virtue or an excellence. So between cowardice and rashness is courage; between stinginess and extravagance is liberality; between sloth and greed is ambition; between humility and pride is modesty; between secrecy and loquacity, honesty; between moroseness and buffoonery, good humor; between quarrelsomeness and flattery, friendship; between Hamlet’s indecisiveness and Quixote’s impulsiveness is self-control.49 “Right,” then, in ethics or conduct, is not different from “right” in mathematics or engineering; it means correct, fit, what works best to the best result. The

I knew that the languages which one learns there are necessary to understand the works of the ancients; and that the delicacy of fiction enlivens the mind; that famous deeds of history ennoble it and, if read with understanding, aid in maturing one's judgment; that the reading of all the great books is like conversing with the best people of earlier times; it is even studied conversation in which the authors show us only the best of their thoughts; that eloquence has incomparable powers and beauties; that poetry has enchanting delicacy and sweetness; that mathematics has very subtle processes which can serve as much to satisfy the inquiring mind as to aid all the arts and diminish man's labor; that treatises on morals contain very useful teachings and exhortations to virtue; that theology teaches us how to go to heaven; that philosophy teaches us to talk with appearance of truth about things, and to make ourselves admired by the less learned; that law, medicine, and the other sciences bring honors and wealth to those who pursue them; and finally, that it is desirable to have examined all of them, even to the most superstitious and false in order to recognize their real worth and avoid being deceived thereby

When I was young, most teachers of philosophy in British and American universities were Hegelians, so that, until I read Hegel, I supposed there must be some truth to his system; I was cured, however, by discovering that everything he said on the philosophy of mathematics was plain nonsense.

Absolute certainty is no more attainable in metaphysics than it is in any other field of rational inquiry and it is unfair to criticize metaphysics for failing to deliver what no other discipline - not even mathematics - is expected to deliver.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

As John Adams famously wrote during the American Revolution, “I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.” So maybe today they’re writing apps rather than studying poetry, but that’s an adjustment for the age.

Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

The notion of God, on the other hand, however inferior it may be in clearness to those mathematical notions so current in mechanical philosophy, has at least this practical superiority over them, that it guarantees an ideal order that shall be permanently preserved.

the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part.

Merlin smiled. "Thank you, sir! It was just Game Theory." "It was just – is this a Maths thing?" "Well, Maths and Philosophy, and..." "Stop! Stop right there – I really have no wish to hear you expound upon either Mathematics or Philosophy, Merlin. Just take the compliment

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.

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

It is impossible to derive happiness from the company of those whom we deprive of happiness. To be happy in old age it is necessary that we accustom ourselves to objects than can accompany the mind all the way through life, and that we take the rest as good in their day. The mere man of pleasure is miserable in old age; and the mere drudge in business is but little better: whereas, natural philosophy, mathematical and mechanical science, are a continual source of tranquil pleasure, and in spite of the gloomy study of the true theology; it teaches man to know and to admire the Creator, for the principles of science are in the creation, and are unchangeable, and of divine origin.

In mathematics or physics, infinity is greater than one or two or any number countable. In how many ways can the world be destroyed based on ordered knowledge? You may be able to count this. But the truth is, you “really” don’t know. These possibilities in your mind hold a set of unpredictable orders. One effect may be causative of another of another. It could be a culmination of effects you know as events where events are sets and subsets of potential possibilities. In how many ways can the world be destroyed based on unordered possibilities?

We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.

From all this we concluded that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of the matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry (tr. Toomer, p. 6).

Ever since Plato most philosophers have considered it part of their business to produce ‘proofs’ of immortality and the existence of God. They have found fault with the proofs of their predecessors — Saint Thomas rejected Saint Anselm's proofs, and Kant rejected Descartes' — but they have supplied new ones of their own. In order to make their proofs seem valid, they have had to falsify logic, to make mathematics mystical, and to pretend that deepseated prejudices were heaven-sent intuitions.

The conclusion is that the physical theory and the mathematical theory of science are valid methods but not valid philosophies. Facts need interpretation the physical theory forgets that it has no such principles of interpretation with its own bosom.

Everything we learn—economics, philosophy, biology, mathematics—has to be understood in light of the overarching reality of the character of God. That is why, in the Middle Ages, theology was called “the queen of the sciences” and philosophy “her handmaiden.” Today the queen has been deposed from her throne and, in many cases, driven into exile, and a supplanter now reigns. We have replaced theology with religion.

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth. —Galileo Galilei, The Assayer, 1623

It's with such profound happiness. Such a hallelujah. Hallelujah, I shout, hallelujah merging with the darkest human howl of the pain of separation but a shout of diabolic joy. Because no one can hold me back now. I can still reason - I studied mathematics, which is the madness of reason - but now I want the plasma - I want to eat straight from the placenta. I am a little scared: scared of surrendering completely because the next instant is unknown. The next instant, do I make it? or does it make itself? We make it together with our breath. And with the flair of the bullfighter in the ring.

Kepler's laws, although not rigidly true, are sufficiently near to the truth to have led to the discovery of the law of attraction of the bodies of the solar system. The deviation from complete accuracy is due to the facts, that the planets are not of inappreciable mass, that, in consequence, they disturb each other's orbits about the Sun, and, by their action on the Sun itself, cause the periodic time of each to be shorter than if the Sun were a fixed body, in the subduplicate ratio of the mass of the Sun to the sum of the masses of the Sun and Planet; these errors are appreciable although very small, since the mass of the largest of the planets, Jupiter, is less than 1/1000th of the Sun's mass.

The assumption that numbers and mathematical or logical laws are mental is due to the even more widespread notion that only particular sensible entities exist in nature, and that relations abstractions, or universals cannot have any such objective existence - hence they are given a shadowy existence in the mind.

Moreover, we look in vain to philosophy for the answer to the great riddle. Despite its noble purpose and history, pure philosophy long ago abandoned the foundational questions about human existence. The question itself is a reputation killer. It has become a Gorgon for philosophers, upon whose visage even the best thinkers fear to gaze. They have good reason for their aversion. Most of the history of philosophy consists of failed models of the mind. The field of discourse is strewn with the wreckage of theories of consciousness. After the decline of logical positivism in the middle of the twentieth century, and the attempt of this movement to blend science and logic into a closed system, professional philosophers dispersed in an intellectual diaspora. They emigrated into the more tractable disciplines not yet colonized by science – intellectual history, semantics, logic, foundational mathematics, ethics, theology, and, most lucratively, problems of personal life adjustment. Philosophers flourish in these various endeavors, but for the time being, at least, and by a process of elimination, the solution of the riddle has been left to science. What science promises, and has already supplied in part, is the following. There is a real creation story of humanity, and one only, and it is not a myth. It is being worked out and tested, and enriched and strengthened, step by step. (9-10)

Russell is reputed at a dinner party once to have said, ‘Oh, it is useless talking about inconsistent things, from an inconsistent proposition you can prove anything you like.’ Well, it is very easy to show this by mathematical means. But, as usual, Russell was much cleverer than this. Somebody at the dinner table said, 'Oh, come on!’ He said, 'Well, name an inconsistent proposition,’ and the man said, 'Well, what shall we say, 2 = 1.’ 'All right,’ said Russell, 'what do you want me to prove?’ The man said, 'I want you to prove that you’re the pope.’ 'Why,’ said Russell, 'the pope and I are two, but two equals one, therefore the pope and I are one.

Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy. Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.

Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? And who created him? Up to now, most scientists have been too occupied with the development of new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, 'The sole remaining task for philosophy is the analysis of language.' What a comedown from the great tradition of philosophy from Aristotle to Kant! However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason--for then we would know the mind of God.

The peculiarity of the evidence of mathematical truths is, that all the argument is on one side. There are no objections, and no answers to objections. But on every subject on which difference of opinion is possible, the truth depends on a balance to be struck between two sets of confliting reasons. Even in natural philosophy, there is always some other explanation possible of the same facts; some geocentric theory instead of heliocentric, some phlogiston instead of oxygen; and it has to be shown why that other theory cannot be the true on: and until this is shown, and until we know how it is shown, we do not understand the grounds of our opinion. But when we turn to subjects infinitely more complicated, to morals, religion, politics, social relations, and the business of life, three-fourths of the arguments for every disputed opinion consist in dispelling the appearances which favour some opinion different from it.

The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm, everything that makes any difference to the value of life and the world. According to our temperaments, we shall prefer the contemplation of the one or of the other.

In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Kepler's rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

In France at least, the history of science and thought gives pride of place to mathematics, cosmology, and physics – noble sciences, rigorous sciences, sciences of the necessary, all close to philosophy: one can observe in their history the almost uninterrupted emergence of truth and pure reason. The other disciplines, however – those, for example, that concern living beings, languages, or economic facts – are considered too tinged with empirical thought, too exposed to the vagaries of chance or imagery, to age-old traditions and external events, for it to be supposed that their history could be anything other than irregular.

And yet sometimes she worried about what those musty old books were doing to her. Some people majored in English to prepare for law school. Others became journalists. The smartest guy in the honors program, Adam Vogel, a child of academics, was planning on getting a Ph.D. and becoming an academic himself. That left a large contingent of people majoring in English by default. Because they weren't left-brained enough for science, because history was too dry, philosophy too difficult, geology too petroleum-oriented, and math too mathematical--because they weren't musical, artistic, financially motivated, or really all that smart, these people were pursuing university degrees doing something no different from what they'd done in first grade: reading stories. English was what people who didn't know what to major in majored in.

Ultimately Russell himself admitted that he made his greatest efforts in the field of traditional philosophy – in epistemology, the search for the ultimate grounds of our knowledge about the world. How can we be certain that what we claim to know is true? Where lies the certainty in our experience of the world? Can even the most precise knowledge – such as mathematics – be said to rest on any sure logical foundation? These were the questions that Russell sought to answer during the periods of his most profound philosophical thinking. They have remained the perennial questions of philosophy from Plato and Aristotle through Descartes, Hume, and Kant, to Russell and Wittgenstein.

Science proceeds by inference, rather than by the deduction of mathematical proof. A series of observations is accumulated, forcing the deeper question: What must be true if we are to explain what is observed? What "big picture" of reality offers the best fit to what is actually observed in our experience? American scientist and philosopher Charles S. Peirce used the term "abduction" to refer to the way in which scientists generate theories that might offer the best explanation of things. The method is now more often referred to as "inference to the best explanation." It is now widely agreed to be the philosophy of investigation of the world characteristic of the natural sciences.

What if one were to want to hunt for these hidden presences? You can’t just rummage around like you’re at a yard sale. You have to listen. You have to pay attention. There are certain things you can’t look at directly. You need to trick them into revealing themselves. That’s what we’re doing with Walter, Jaz. We’re juxtaposing things, listening for echoes. It’s not some silly cybernetic dream of command and control, modeling the whole world so you can predict the outcome. It’s certainly not a theory of everything. I don’t have a theory of any kind. What I have is far more profound.’ ‘What’s that?’ ‘A sense of humor.’ Jaz looked at him, trying to find a clue in his gaunt face, in the clear gray eyes watching him with such - what? Amusement? Condescension? There was something about the man which brought on a sort of hermeneutic despair. He was a forest of signs. ‘We’re hunting for jokes.’ Bachman spoke slowly, as if to a child. ‘Parapraxes. Cosmic slips of the tongue. They’re the key to the locked door. They’ll help us discover it.’ ‘Discover what?’ ‘The face of God. What else would we be looking for?

Among this bewildering multiplicity of ideals which shall we choose? The answer is that we shall choose none. For it is clear that each one of these contradictory ideals is the fruit of particular social circumstances. To some extent, of course, this is true of every thought and aspiration that has ever been formulated. Some thoughts and aspirations, however, are manifestly less dependent on particular social circumstances than others. And here a significant fact emerges: all the ideals of human behaviour formulated by those who have been most successful in freeing themselves from the prejudices of their time and place are singularly alike. Liberation from prevailing conventions of thought, feeling and behaviour is accomplished most effectively by the practice of disinterested virtues and through direct insight into the real nature of ultimate reality. (Such insight is a gift, inherent in the individual; but, though inherent, it cannot manifest itself completely except where certain conditions are fulfilled. The principal pre-condition of insight is, precisely, the practice of disinterested virtues.) To some extent critical intellect is also a liberating force. But the way in which intellect is used depends upon the will. Where the will is not disinterested, the intellect tends to be used (outside the non-human fields of technology, science or pure mathematics) merely as an instrument for the rationalization of passion and prejudice, the justification of self-interest. That is why so few even of die acutest philosophers have succeeded in liberating themselves completely from the narrow prison of their age and country. It is seldom indeed that they achieve as much freedom as the mystics and the founders of religion. The most nearly free men have always been those who combined virtue with insight. Now, among these freest of human beings there has been, for the last eighty or ninety generations, substantial agreement in regard to the ideal individual. The enslaved have held up for admiration now this model of a man, now that; but at all times and in all places, the free have spoken with only one voice. It is difficult to find a single word that will adequately describe the ideal man of the free philosophers, the mystics, the founders of religions. 'Non-attached* is perhaps the best. The ideal man is the non-attached man. Non-attached to his bodily sensations and lusts. Non-attached to his craving for power and possessions. Non-attached to the objects of these various desires. Non-attached to his anger and hatred; non-attached to his exclusive loves. Non-attached to wealth, fame, social position. Non-attached even to science, art, speculation, philanthropy. Yes, non-attached even to these. For, like patriotism, in Nurse Cavel's phrase, 'they are not enough, Non-attachment to self and to what are called 'the things of this world' has always been associated in the teachings of the philosophers and the founders of religions with attachment to an ultimate reality greater and more significant than the self. Greater and more significant than even the best things that this world has to offer. Of the nature of this ultimate reality I shall speak in the last chapters of this book. All that I need do in this place is to point out that the ethic of non-attachment has always been correlated with cosmologies that affirm the existence of a spiritual reality underlying the phenomenal world and imparting to it whatever value or significance it possesses.

But what is the use of the humanities as such? Admittedly they are not practical, and admittedly they concern themselves with the past. Why, it may be asked, should we engage in impractical investigations, and why should we be interested in the past? The answer to the first question is: because we are interested in reality. Both the humanities and the natural sciences, as well as mathematics and philosophy, have the impractical outlook of what the ancients called vita contemplativa as opposed to vita activa. But is the contemplative life less real or, to be more precise, is its contribution to what we call reality less important, than that of the active life? The man who takes a paper dollar in exchange for twenty-five apples commits an act of faith, and subjects himself to a theoretical doctrine, as did the mediaeval man who paid for indulgence. The man who is run over by an automobile is run over by mathematics, physics and chemistry. For he who leads the contemplative life cannot help influencing the active, just as he cannot prevent the active life from influencing his thought. Philosophical and psychological theories, historical doctrines and all sorts of speculations and discoveries, have changed, and keep changing, the lives of countless millions. Even he who merely transmits knowledge or learning participates, in his modest way, in the process of shaping reality - of which fact the enemies of humanism are perhaps more keenly aware than its friends. It is impossible to conceive of our world in terms of action alone. Only in God is there a "Coincidence of Act and Thought" as the scholastics put it. Our reality can only be understood as an interpenetration of these two.

Whether or not we subscribe to any particular religion or philosophy, it would be hard to deny that knowing our cosmic destiny must have some impact on how we think about our existence, or even how we live our lives. If we want to know whether what we do here ultimately matters, the first thing we ask is: how will it come out in the end? If we find the answer to that question, it leads immediately to the next: what does this mean for us now? Do we still have to take the trash out next Tuesday if the universe is going to die someday? I’ve done my own scouring of theological and philosophical texts, and while I learned many fascinating things from my studies, unfortunately the meaning of existence wasn’t one of them. I may just not have been cut out for it. The questions and answers that have always drawn me in most strongly are the ones that can be answered with scientific observation, mathematics, and physical evidence. As appealing as it sometimes seemed to have the whole story and meaning of life written down for me once and for all in a book, I knew I would only ever really be able to accept the kind of truth I could rederive mathematically.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

In Tibet, we have a traditional image, the windhorse, which represents a balanced relationship between the wind and the mind. The horse represents wind and movement. On its saddle rides a precious jewel. That jewel is our mind. A jewel is a stone that is clear and reflects light. There is a solid, earthly element to it. You can pick it up in your hand, and at the same time you can see through it. These qualities represent the mind: it is both tangible and translucent. The mind is capable of the highest wisdom. It can experience love and compassion, as well as anger. It can understand history, philosophy, and mathematics—and also remember what’s on the grocery list. The mind is truly like a wish-fulfilling jewel. With an untrained mind, the thought process is said to be like a wild and blind horse: erratic and out of control. We experience the mind as moving all the time—suddenly darting off, thinking about one thing and another, being happy, being sad. If we haven’t trained our mind, the wild horse takes us wherever it wants to go. It’s not carrying a jewel on its back—it’s carrying an impaired rider. The horse itself is crazy, so it is quite a bizarre scene. By observing our own mind in meditation, we can see this dynamic at work. Especially in the beginning stages of meditation, we find it extremely challenging to control our mind. Even if we wish to control it, we have very little power to do so, like the infirm rider. We want to focus on the breathing, but the mind keeps darting off unexpectedly. That is the wild horse. The process of meditation is taming the horse so that it is in our control, while making the mind an expert rider.

Our species is the only creative species, and it has only one creative instrument, the individual mind and spirit of a man. Nothing was ever created by two men. There are no good collaborations, whether in music, in art, in poetry, in mathematics, in philosophy. Once the miracle of creation has taken place, the group can build and extend it, but the group never invents anything. The preciousness lies in the lonely mind of a man. And now the forces marshaled around the concept of the group have declared a war of extermination on that preciousness, the mind of man. By disparagement, by starvation, by repressions, forced direction, and the stunning hammerblows of conditioning, the free, roving mind is being pursued, roped, blunted, drugged. It is a sad suicidal course our species seems to have taken. And this I believe: that the free, exploring mind of the individual human is the most valuable thing in the world. And this I would fight for: the freedom of the mind to take any direction it wishes, undirected. And this I must fight against: any idea, religion, or government which limits or destroys the individual. This is what I am and what I am about. I can understand why a system built on a pattern must try to destroy the free mind, for that is one thing which can by inspection destroy such a system. Surely I can understand this, and I hate it and I will fight against it to preserve the one thing that separates us from the uncreative beasts. If the glory can be killed, we are lost.

Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read, but not curiously; and some few to be read wholly, and with diligence and attention. Some books also may be read by deputy, and extracts made of them by others; but that would be only in the less important arguments, and the meaner sort of books, else distilled books are like common distilled waters, flashy things. Reading maketh a full man; conference a ready man; and writing an exact man. And therefore, if a man write little, he had need have a great memory; if he confer little, he had need have a present wit: and if he read little, he had need have much cunning, to seem to know, that he doth not. Histories make men wise; poets witty; the mathematics subtile; natural philosophy deep; moral grave; logic and rhetoric able to contend. ---- Alcuni libri devono essere gustati, altri masticati e digeriti, vale a dire che alcuni libri vanno letti solo in parte, altri senza curiosità, e altri per intero, con diligenza ed attenzione. Alcuni libri possono essere letti da altri e se ne possono fare degli estratti, ma ciò riguarderebbe solo argomenti di scarsa importanza o di libri secondari perché altrimenti i libri sintetizzati sono come l’acqua distillata, evanescente. La lettura completa la formazione di un uomo; il parlare lo fa abile, e la scrittura lo trasforma in un uomo preciso. E, pertanto, se un uomo scrive poco, deve avere una grande memoria, se parla poco ha bisogno di uno spirito arguto; se legge poco deve avere bisogno di molta astuzia in modo da far sembrare di sapere quello che non sa. Le storie fanno gli uomini saggi; i poeti arguti; la matematica sottile; la filosofia naturale profondi; la logica e la retorica abili nella discussione.

Wars and chaoses and paradoxes ago, two mathematicians between them ended an age d began another for our hosts, our ghosts called Man. One was Einstein, who with his Theory of Relativity defined the limits of man's perception by expressing mathematically just how far the condition of the observer influences the thing he perceives. ... The other was Goedel, a contemporary of Eintstein, who was the first to bring back a mathematically precise statement about the vaster realm beyond the limits Einstein had defined: In any closed mathematical system--you may read 'the real world with its immutable laws of logic'--there are an infinite number of true theorems--you may read 'perceivable, measurable phenomena'--which, though contained in the original system, can not be deduced from it--read 'proven with ordinary or extraordinary logic.' Which is to say, there are more things in heaven and Earth than are dreamed of in your philosophy, Horatio. There are an infinite number of true things in the world with no way of ascertaining their truth. Einstein defined the extent of the rational. Goedel stuck a pin into the irrational and fixed it to the wall of the universe so that it held still long enough for people to know it was there. ... The visible effects of Einstein's theory leaped up on a convex curve, its production huge in the first century after its discovery, then leveling off. The production of Goedel's law crept up on a concave curve, microscopic at first, then leaping to equal the Einsteinian curve, cross it, outstrip it. At the point of intersection, humanity was able to reach the limits of the known universe... ... And when the line of Goedel's law eagled over Einstein's, its shadow fell on a dewerted Earth. The humans had gone somewhere else, to no world in this continuum. We came, took their bodies, their souls--both husks abandoned here for any wanderer's taking. The Cities, once bustling centers of interstellar commerce, were crumbled to the sands you see today.