Whenever, for any reason, we wish to think of the world, not as it appears to common sense, but as a continuum, we find that our traditional syntax and vocabulary are quite inadequate. Mathematicians have therefore been compelled to invent radically new symbol-systems for this express purpose. But the divine Ground of all existence is not merely a continuum, it is also out of time, and different, not merely in degree, but in kind from the worlds to which traditional language and the languages of mathematics are adequate.
There exists a passion for comprehension, just as there exists a passion for music. That passion is rather common in children but gets lost in most people later on. Without this passion, there would be neither mathematics nor natural science.
All the commandments follow three ways: faith, word, and deed. As one is basic in mathematics, so the essence of every commandment, whether it depends on speech or action, is faith of heart. If it has not that , all else is meaningless.
Music has always had its own syntax, its own vocabulary and symbolic means. Indeed, it is with mathematics the principal language of the mind when the mind is in a condition of non-verbal feeling.
That the truths of the Bible have the power of awakening an intense moral feeling in every human being; that they make bad men good, and send a pulse of healthful feeling through all the domestic, civil, and social relations; that they teach men to love right, and hate wrong, and seek each other's welfare as children of a common parent; that they control the baleful passions of the heart, and thus make men proficient in self-government; and finally that they teach man to aspire after conformity to a being of infinite holiness, and fill him with hopes more purifying, exalted, and suited to his nature than any other book the world has ever known - these are facts as incontrovertible as the laws of philosophy, or the demonstrations of mathematics.
Evolution is not necessarily a reductive theory: it does not explain away or reduce meaningfulness and value, any more than it explains away or reduces mathematics, economics, or even sociobiology itself. It aims to provide a naturalistic explanation of biological characteristics, including the capacities that enable us to recognize value and meaning. Giving a causal explanation of the origin of capacities is not the same as giving an account of the relevant meaning or content.
Computational irreducibility tends to make infinite questions undecidable. The presence of universality implies that there must at some level be computational irreducibility… This means that today’s mathematics will be viewed as small and surprisingly uncharacteristic sample of what is possible. If a system is computationally irreducible this means that there is in effect a tangible separation between the underlying rules for the system and its overall behavior associated with the irreducible amount of computational work needed to go from one to the other. And it is this separation that the basic origin of the apparent freedom we see in all sorts of system lie – whether those systems are abstract cellular automata or actual living brains.
The overall similarity between mathematics and nature must have a deeper origin, both involving processes that can be thought of as computations.
When it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature.
If there should be any [persons], who though ignorant in Mathematics, yet pretending a skill in those Learnings, should dare, upon the authority of some place of Scripture wrested to their purpose, to condemn and censure my Hypothesis, I value them not, but shall slight their inconsiderate judgment.
Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
To see what students learn in school, look at how they leave school. If they leave thinking that reading and writing are difficult and pointless, that mathematics is confusing, that history is irrelevant, and that art is a bore, then that is what they have been taught. People learn what is demonstrated to them, and this reality will not change to suit the convenience of politicians and educations administrators.
Mathematics... possesses not only truth, but supreme beauty - a beauty cold and austere, like that of a 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.
Reading maketh a full man; conference a ready man; histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral philosophy, grace; logic and rhetoric, able to contend.
Mathematics is the language with which God has written the universe.
If you help others, you will be helped, perhaps tomorrow, perhaps in one hundred years, but you will be helped. Nature must pay off the debt... It is a mathematical law and all life is mathematics.
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.
I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.
I must study politics and war, that my sons may have liberty 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.
Science and mathematics run parallel to reality, they symbolize it, they squint at it, they never touch it: consider what an explosion would rock the bones of men into little white fragments and unsky the world if any mind for a moment touch truth.