I see a certain order in the universe and math is one way of making it visible.
You see, the chemists have a complicated way of counting: instead of saying "one, two, three, four, five protons", they say, "hydrogen, helium, lithium, beryllium, boron."
All of mathematics can be deduced from the sole notion of an integer; here we have a fact universally acknowledged today.
More and more I'm aware that the permutations are not unlimited.
We have not begun to understand the relationship between combinatorics and conceptual mathematics.
'Without the help of mathematics,' the wise man continued, 'the art could not advance and all the sciences would perish.'
All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).
Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.
Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.
Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.
The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).
It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers.
Moreover, if one selects a problem, works on it in isolation for a few years and finally solves it, there is a danger, unless the problem is very famous, that it will no longer be regarded as all that significant.
At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied.
If we have no idea why a statement is true, we can still prove it by induction.
It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.
To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.
Today, it is not only that our kings do not know mathematics, but our philosophers do not know mathematics and - to go a step further - our mathematicians do not know mathematics.
It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.
At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.
The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes.
There is something in statistics that makes it very similar to astrology.
If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do.
When I give this talk to a physics audience, I remove the quotes from my 'Theorem'.
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