Galileo Mathematics Quotes

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.

Mathematics is the language in which God has written the universe

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

Mathematics is the language with which God has written the universe.

he presented me with a mathematical conundrum,” he said. “It’s a famous one, the P = NP problem. Basically, it asks whether it’s more difficult to think of the solution to a problem yourself or to ascertain if someone else’s answer to the same problem is correct.

And, believe me, if I were again beginning my studies, I should follow the advice of Plato and start with mathematics.

You’re familiar with the P = NP problem, right?” Yukawa asked from behind him. Ishigami looked around. “You’re referring to the question of whether or not it is as easy to determine the accuracy of another person’s results as it is to solve the problem yourself—or, failing that, how the difference in difficulty compares. It’s one of the questions the Clay Mathematics Institute has offered a prize to solve.

The first man to understand the extraordinary magical power of applying mathematical calculation to things in nature was an Italian called Galileo Galilei.

. . . we come astonishingly close to the mystical beliefs of Pythagoras and his followers who attempted to submit all of life to the sovereignty of numbers. Many of our psychologists, sociologists, economists and other latter-day cabalists will have numbers to tell them the truth or they will have nothing. . . . We must remember that Galileo merely said that the language of nature is written in mathematics. He did not say that everything is. And even the truth about nature need not be expressed in mathematics. For most of human history, the language of nature has been the language of myth and ritual. These forms, one might add, had the virtues of leaving nature unthreatened and of encouraging the belief that human beings are part of it. It hardly befits a people who stand ready to blow up the planet to praise themselves too vigorously for having found the true way to talk about nature.

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.

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

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?

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics.

Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.

Physics was the first of the natural sciences to become fully modern and highly mathematical. Chemistry followed in the wake of physics, but biology, the retarded child, lagged far behind. Even in the time of Newton and Galileo, men knew more about the moon and other heavenly bodies than they did about their own.

But why has our physical world revealed such extreme mathematical regularity that astronomy superhero Galileo Galilei proclaimed nature to be “a book written in the language of mathematics,” and Nobel Laureate Eugene Wigner stressed the “unreasonable effectiveness of mathematics in the physical sciences” as a mystery demanding an explanation?

Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind.

Mathematics is the alphabet with which God has written the universe.

Whether we like it or not, if we are to pursue a career in science, eventually we have to learn the “language of nature”: mathematics. Without mathematics, we can only be passive observers to the dance of nature rather than active participants. As Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.” Let me offer an analogy. One may love French civilization and literature, but to truly understand the French mind, one must learn the French language and how to conjugate French verbs. The same is true of science and mathematics. Galileo once wrote, “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word.

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.

Generalizations in biology are almost invariably of a probabilistic nature. As one wit has formulated it, there is only one universal law in biology: 'All biological laws have exceptions.' This probabilistic conceptualization contrasts strikingly with the view during the early period of the scientific revolution that causation in nature is regulated by laws that can be stated in mathematical terms. Actually, this idea occurred apparently first to Pythagoras. It has remained a dominant idea, particularly in the physical sciences, up to the present day. Again and again it was made the basis of some comprehensive philosophy, but taking very different forms in the hands of various authors. With Plato it gave rise to essentialism, with Galileo to a mechanistic world picture, and with Descartes to the deductive method. All three philosophies had a fundamental impact on biology.

The Galileo saga is typically told as a conflict between science and religion. But in reality it was a conflict among Christians over the correct philosophy of nature. Was it Aristotle’s quality or Galileo’s quantity? Galileo’s victory was the triumph of the idea that the nature is constructed on a mathematical blueprint.

Mind-boggling, isn't it? Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.

The extreme mathematical weirdness of (infinity), which Galileo spends a lot of time in TNS giving examples of, is rather presciently attributed to epistemology instead of metaphysics. Paradoxes arise, according to G.G.'s mouthpiece, only "when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited.

Galileo showed that the same physical laws that govern the movements of bodies on earth apply aloft , to the celestial spheres; and our astronauts, as we have all now seen, have been transported by those earthly laws to the moon. They will soon be on Mars and beyond. Furthermore, we know that the mathematics of those outermost spaces will already have been computed here on earth by human minds. There are no laws out there that are not right here; no gods out there that are not right here, and not only here, but within us, in our minds. So what happens now to those childhood images of the ascent of Elijah, Assumption of the Virgin, Ascension of Christ - all bodily - into heaven?

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

In this book, you will encounter various interesting geometries that have been thought to hold the keys to the universe. Galileo Galilei (1564-1642) suggested that "Nature's great book is written in mathematical symbols." Johannes Kepler (1571-1630) modeled the solar system with Platonic solids such as the dodecahedron. In the 1960s, physicist Eugene Wigner (1902-1995) was impressed with the "unreasonable effectiveness of mathematics in the natural sciences." Large Lie groups, like E8-which is discussed in the entry "The Quest for Lie Group E8 (2007)"- may someday help us create a unified theory of physics. in 2007, Swedish American cosmologist Max Tegmark published both scientific and popular articles on the mathematical universe hypothesis, which states that our physical reality is a mathematical structure-in other words, our universe in not just described by mathematics-it is mathematics.

What we know today, if we know anything at all, is that every individual is unique and that the laws of his life will not be those of any other on this earth. We also know that if divinity is to be found anywhere, it will not be “out there,” among or beyond the planets. Galileo showed that the same physical laws that govern the movements of bodies on earth apply aloft, to the celestial spheres; and our astronauts, as we have all now seen, have been transported by those earthly laws to the moon. They will soon be on Mars and beyond. Furthermore, we know that the mathematics of those outermost spaces will already have been computed here on earth by human minds. There are no laws out there that are not right here; no gods out there that are not right here, and not only here, but within us, in our minds. So what happens now to those childhood images of the ascent of Elijah, Assumption of the Virgin, Ascension of Christ - all bodily - into heaven?

Johannes Kepler, who was one of the first to apply mathematics to the motion of the planets, was an imperial adviser to Emperor Rudolf Il and perhaps escaped persecution by piously including religious elements in his scientific work. The former monk Giordano Bruno was not so lucky. In 1600, he was tried and sentenced to death for heresy. He was gagged, paraded naked in the streets of Rome, and finally burned at the stake. His chief crime? Declaring that life may exist on planets circling other stars. The great Galileo, the father of experimental science, almost met the same fate. But unlike Bruno, Galileo recanted his theories on pain of death. Nonetheless, he left a lasting legacy with his telescope, perhaps the most revolutionary and seditious invention in all of science. With a telescope, you could see with your own eyes that the moon was pockmarked with craters; that Venus had phases consistent with its orbiting the sun; that Jupiter had moons, all of which were heretical ideas. Sadly, he was placed under house arrest, isolated from visitors, and eventually went blind. (It was said because he once looked directly at the sun with his telescope.) Galileo died a broken man. But the very year that he died, a baby was born in England who would grow up to complete Galileo's and Kepler's unfinished theories, giving us a unified theory of the heavens.

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.

SALV. I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to (37) eternity there would still remain finite parts which were undivided. Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is, in my opinion, getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows [83] that, since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri].

Queen Anne of England established the Longitude Act in 1714, and offered a monetary prize of over a million in today’s dollars to anyone who invented a method to accurately calculate longitude at sea. Longitude is about determining one’s point in space. So one might ask what it has to do with clocks? Mathematically speaking, space (distance) is the child of time and speed (distance equals time multiplied by speed). Thus, anything that moves at a constant speed can be used to calculate distance, provided one knows for how long it has been moving. Many things have constant speeds, including light, sound, and the rotation of the Earth. Your brain uses the near constancy of the speed of sound to calculate where sounds are coming from. As we have seen, you know someone is to your left or right because the sound of her voice takes approximately 0.6 milliseconds to travel from your left to your right ear. Using the delays it takes any given sound to arrive to your left and right ears allows the brain to figure out if the voice is coming directly from the left, the right, or somewhere in between. The Earth is rotating at a constant speed—one that results in a full rotation (360 degrees) every 24 hours. Thus there is a direct correspondence between degrees of longitude and time. Knowing how much time has elapsed is equivalent to knowing how much the Earth has turned: if you sit and read this book for one hour (1/24 of a day), the Earth has rotated 15 degrees (360/24). Thus, if you are sitting in the middle of the ocean at local noon, and you know it is 16:00 in Greenwich, then you are “4 hours from Greenwich”—exactly 60 degrees longitude from Greenwich. Problem solved. All one needs is a really good marine chronometer. The greatest minds of the seventeenth and eighteenth centuries could not overlook the longitude problem: Galileo Galilei, Blaise Pascal, Robert Hooke, Christiaan Huygens, Gottfried Leibniz, and Isaac Newton all devoted their attention to it. In the end, however, it was not a great scientist but one of the world’s foremost craftsman who ultimately was awarded the Longitude Prize. John Harrison (1693–1776) was a self-educated clockmaker who took obsessive dedication to the extreme.

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 first overt clerical attack took place in December 1614. Standing in the pulpit of the Dominican Church of Santa Maria Novella in Florence, Father Thomas Caccini delivered a sermon that denounced mathematics as inconsistent with the Bible and detrimental to the State.

The doctrine of creation of the kind that the Abrahamic faiths profess is such that it encourages the expectation that there will be a deep order in the world, expressive of the Mind and Purpose of that world’s Creator. It also asserts that the character of this order has been freely chosen by God, since it was not determined beforehand by some kind of pre-existing blueprint (as, for example, Platonic thinking had supposed to be the case). As a consequence, the nature of cosmic order cannot be discovered just by taking thought, as if humans could themselves explore a noetic realm of rational constraint to whichthe Creator had had to submit, but the pattern of the world has to be discerned through the observations and experiments that are necessary in order to determine what form the divine choice has actually taken. What is needed, therefore, for successful science is the union of the mathematical expression of order with the empirical investigation of the actual properties of nature, a methodological synthesis of a kind that was pioneered with great skill and fruitfulness by Galileo.

claiming publicly that comets follow natural law and not God’s whim was a gutsy thing to do, especially given that the prior year—almost fifty years after Galileo’s condemnation—the professor of mathematics at the University of Basel, Peter Megerlin, had been roundly attacked by theologians for accepting the Copernican system and had been banned from teaching it at the university.

Thomson particularly admired Fourier’s agnostic theoretical method, based on mathematical models that were useful but at the same time noncommittal on the difficult question of the nature of heat.

He had always thought of mathematics as a treasure hunt. First, one had to decide where to dig; then one had to determine the proper excavation route that led to the answer. Once you had a plan, you could make formulas to fit it, and they would give you clues. If you wound up empty-handed, you had to go back to the beginning and choose another route. Only by doing this over and over, patiently, yet boldly, could you hope to find the treasure—a solution no one else had ever found.

Look, I know this theory of general relativity sounds wacky, but that’s what they said about Galileo!”)

Unfortunately, Galileo never managed to derive this rule mathematically. It was an empirical pattern crying out for a theoretical explanation. He worked at it for years but failed to solve it. In retrospect,

When Galileo divided experienced reality into two spheres, a subjective sphere, which he chose to exclude from science, and an objective sphere, freed theoretically from man's visible presence, but known through rigorous mathematical analysis, he was dismissing as unsubstantial and unreal the cultural accretions of meaning that had made mathematics-itself a purely subjective distillation-possible.

Mathematics is the language with which God wrote the universe. Galileo Galilei said

As science has become more abstract and remote from everyday experience, the role of metaphor in our descriptions of the world has become more central. The language that nature speaks, as Galileo long pointed out, is mathematics. The language that ordinary human beings speak, especially those of us who are not fluent in mathematics, is metaphor. Lightman ends his discussion with another metaphor: "We are blind men, imagining what we don't see." That is a good description of theoretical physics.

There remained a main melody, or a path through a maze—a maze that was like the delta of the Po. He seemed to look down on it as he sang it. A great number of channels were weaving down a slightly tilted plain. Each channel was a mathematical specialty—some of them shallow and disappearing into the sand, but most making their loop and reconnecting to other flows. A few were the kind of deep channels that ships would use. Upstream they coalesced until there were fewer, scattered streams. Fewer tributaries rather than more, leading up in different directions to sources, often at springs. Water out of the rock. This was, he saw, an image of mathematics in time. Or maybe it was all time, or humanity in time; but it was the mathematics that sprang out at him. The fewer channels upstream, in the distant past, well before his time, were where Aurora’s tutorial now led him. Then he was flying over the time stream, or in it, sometimes returning upstream to view a contemporaneous discipline. Mainly he had a general sense of flying downstream, over or occasionally inside some eternal landscape, the nature of which could not be discerned. He inhabited an image he had heard some time before, of history as a river, in which people were water, eroding the banks and depositing soil elsewhere downstream, so that the banks slowly changed and the river ran otherwise than it had, without the water ever noticing the changed courses of the braiding stream.

The Pythagoreans' discovery that there was a relationship between musical intervals and numerical ratios led to the belief that the study of mathematics was the key to the understanding of the structure and order of the universe. Astronomy and harmony, they said, were sister sciences, one for the eyes and one for the ears). However, it was not until two millennia later that Galileo and his successors showed the sense in which it is true that the book of the universe is written in numbers.

Galileo also wanted to avoid the error of his colleague Johannes Kepler (1571–1630). For all his brilliant work, Kepler had insisted on making the five Platonic solids the basis for his model of the solar system, rendering it useless for further empirical research. It was a bizarre example of carrying the faith in the perfection of mathematics too far.35 By contrast, Galileo understood that even a divinely ordered cosmos could not be perfect. There were craters on the moon and spots on the sun. Kepler himself had shown that the planetary orbits were not perfect circles as Aristotle and even Copernicus had assumed, but ellipses.

Galileo’s science managed to fuse the Platonist’s faith in mathematics with the Aristotelian faith in experience as the basis of discovery. All his work on mechanics, optics, and astronomy was deeply rooted in experiment and empirical research. When experience proved ambiguous or unreliable, however, Galileo realized then that mathematics must take over.

The universe, Galileo wrote, “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 world of it.” Without mathematics, he concluded, “one wanders about in a dark labyrinth”—or what Plato might have called a cave.

This was Galileo’s most important contribution to the future of science. Galileo knew that if he could measure it mathematically, then it must exist even if he could not see it. This was whether one was talking about his own concept of velocity or, later, Newton’s concept of gravity. His “mathematization of Nature” allowed scientists for the first time to anticipate discoveries and work out scientific theories, including Einstein’s relativity three centuries later, long before the means of testing them existed.

Elliptical orbits. Langdon recalled that much of Galileo’s legal trouble had begun when he described planetary motion as elliptical. The Vatican exalted the perfection of the circle and insisted heavenly motion must be only circular. Galileo’s Illuminati, however, saw perfection in the ellipse as well, revering the mathematical duality of its twin foci. The Illuminati’s ellipse was prominent even today in modern Masonic tracing boards and footing inlays.

In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world — to show, that is, in how far culture in the "become" state can express or portray an idea of human existence — I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. [...] Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. [...] But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks, speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the "alien," in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. [...] Nature is the numerable, while History, on the other hand, is the aggregate of that which has no relation to mathematics hence the mathematical certainty of the laws of Nature, the astounding Tightness of Galileo's saying that Nature is "written in mathematical language," and the fact, emphasized by Kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach.

When it comes to the basic causal workings of the universe, scientists provide mathematical laws which describe with great accuracy how matter behaves, but they provide no explanation of why matter behaves in that way.

As William Bouwsma pointed out, the late medieval and early Renaissance crises of representation did not stall out at their skepticism of the old systems but rather progressed to an even more urgent defense of objective boundaries and quantifiable truths.27 In “The Secularization of Language in the Seventeenth Century,” Margreta de Grazia has shown how this pursuit of certainty led to a skepticism about language itself that dissociated words from God and deverbalized God’s message, prompting thinkers from Thomas Sprat of the Royal Society to Hobbes, Robert Hooke, Galileo, and Newton to seek cer- tainty in mathematical knowledge; quantifiable, identifiable substances; and trial, experiment, and experience.28 As Puritan propagandist Vavasor Powell put it in the middle of the seventeenth century, “Experience is like 42 Rituals of Spontaneity steel to an edged tool, or like salt to fresh meat, it seasons brain- knowledge, and settles a shaking unsetled soule.” Paralleling more sec- ular quests for certainty, the Puritan quest for grounding religious knowledge in a literalist reading of Scripture focused ever more intensely on manifest, genuine experience confirming salvation and the personal application of scriptural truth. The spontaneous “pouring out of the heart” in prayer was just such an evidentiary experience.

As William Bouwsma pointed out, the late medieval and early Renaissance crises of representation did not stall out at their skepticism of the old systems but rather progressed to an even more urgent defense of objective boundaries and quantifiable truths. In “The Secularization of Language in the Seventeenth Century,” Margreta de Grazia has shown how this pursuit of certainty led to a skepticism about language itself that dissociated words from God and deverbalized God’s message, prompting thinkers from Thomas Sprat of the Royal Society to Hobbes, Robert Hooke, Galileo, and Newton to seek certainty in mathematical knowledge; quantifiable, identifiable substances; and trial, experiment, and experience. As Puritan propagandist Vavasor Powell put it in the middle of the seventeenth century, “Experience is like steel to an edged tool, or like salt to fresh meat, it seasons brain- knowledge, and settles a shaking unsetled soule.” Paralleling more secular quests for certainty, the Puritan quest for grounding religious knowledge in a literalist reading of Scripture focused ever more intensely on manifest, genuine experience confirming salvation and the personal application of scriptural truth. The spontaneous “pouring out of the heart” in prayer was just such an evidentiary experience.

Galileo held that the laws of nature are written by the hand of God in the “language of mathematics”7 and that the “human mind is a work of God’s and one of the most excellent.”8

If human beings could communicate in mathematical language -- which is more poetical than any other language -- the world would be an easier place to live. Perhaps one day we will follow Nature's example, for her immense book is written in mathematical language, as Galileo has affirmed.

God makes the world using mathematics, and He has given us minds that can see it. We can discover the laws He used! It is a most beautiful thing to witness and understand. It's prayer. It's more than prayer, it's a sacrament, a kind of communion. An apprehension - an epiphany - it's seeing God, while still in this body and in this world! How blessed we are, to be able to experience God like that. Who would not devote their time to understanding more, to seeing deeper into God's manner of thinking about these things?

Christiaan Huygens was the first to use Galileo’s insights to build the first high-quality pendulum clocks. A better mathematician than Galileo, he was able to truly comprehend the intricacies of the dynamics of a weight swinging back and forth on a string. Thanks to his mathematical skills and a number of technical innovations, the clock he designed in 1657 represented a quantum leap in timekeeping technology. Before Huygens, the best clocks were off by approximately 15 minutes a day; his clock lost a mere 10 seconds a day.8 Ten seconds amounts to approximately 0.01 percent of a 24-hour day. This level of accuracy marked a milestone in the history of timekeeping: these were the first clocks designed by the human brain that were better than the clocks within the human brain.

Reality is mathematical, as long as you understand that uncertainty and contingency can be mathematically described, without them becoming any more certain.

The immediate catalyst for the emergence of the Enlightenment in the eighteenth century was the scientific revolution of the sixteenth and seventeenth centuries, which included three momentous discoveries in astronomy: Johannes Kepler delineated the rules that govern the movement of the planets, Galileo Galilei placed the sun at the center of the universe, and Isaac Newton discovered the force of gravity, invented calculus (Gottfried Wilhelm Leibniz independently discovered it at the same time), and used it to describe the three laws of motion. In so doing, Newton joined physics and astronomy and illustrated that even the deepest truths in the universe could be revealed by the methods of science. These contributions were celebrated in 1660 with the formation of the first scientific society in the world: the Royal Society of London for Improving Natural Knowledge, which elected Isaac Newton as its president in 1703. The founders of the Royal Society thought of God as a mathematician who had designed the universe to function according to logical and mathematical principles. The role of the scientist—the natural philosopher—was to employ the scientific method to discover the physical principles underlying the universe and thereby decipher the codebook that God had used in creating the cosmos. Success in the realm of science led eighteenth-century thinkers to assume that other aspects of human action, including political behavior, creativity, and art, could be improved by the application of reason, leading ultimately to an improved society and better conditions for all humankind. This confidence in reason and science affected all aspects of political and social life in Europe and soon spread to the North American colonies. There, the Enlightenment ideas that society can be improved through reason and that rational people have a natural right to the pursuit of happiness are thought to have contributed to the Jeffersonian democracy that we enjoy today in the United States.

But as the seventeenth century wore on, precision observations greatly improved due to the invention of the telescope and an increasingly mature application of mathematics to describe the data, and led a host of astronomers and mathematicians – including Johannes Kepler, Galileo and ultimately Isaac Newton – towards an understanding of the workings of the solar system. This theory is good enough even today to send space probes to the outer planets with absolute precision.

By apparently purging material reality of subjective experience, Galileo cleared the ground and Descartes laid the foundation for the construction of the objective or “disinterested” sciences, which by their feverish and forceful investigations have yielded so much of the knowledge and so many of the technologies that have today become commonplace in the West. The chemical table of the elements, automobiles, smallpox vaccines, “close-up” images of the outer planets—so much that we have come to assume and depend upon has emerged from the bold experimentalization of the world by the objective sciences. Yet these sciences consistently overlook our ordinary, everyday experience of the world around us. Our direct experience is necessarily subjective, necessarily relative to our own position or place in the midst of things, to our particular desires, tastes, and concerns. The everyday world in which we hunger and make love is hardly the mathematically determined “object” toward which the sciences direct themselves. Despite all the mechanical artifacts that now surround us, the world in which we find ourselves before we set out to calculate and measure it is not an inert or mechanical object but a living field, an open and dynamic landscape subject to its own moods and metamorphoses.