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.
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.
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.
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.
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.
A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
If it were true [that there are mathematical problems undecidable by the human mind] it would mean that human reason is utterly irrational in asking questions it cannot answer, while asserting emphatically that only reason can answer them. Human reason would then be very imperfect and, in some sense, even inconsistent, in glaring contradiction to the fact that those parts of mathematics which have been systematically and completely developed show an amazing degree of beauty and perfection. In these fields, by entirely unexpected laws and procedures, means are provided not only for solving all relevant problems, but also solving them in a most beautiful and perfectly feasible manner.
If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies
Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
God used beautiful mathematics in creating the world.
I must say that I am very dissatisfied with the situation, because this so called good theory does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small - not neglecting it just because it is infinitely great and you do not want it!
If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics lead me by the hand. It has lead me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.
It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.