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The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination.
‘in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since’.
A MATHEMATICIAN, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
A mathematician, on the other hand, has no material to work with but ideas,
most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
that if a chess problem is, in the crude sense, ‘useless’, then that is equally true of most of the best mathematics; that very little of mathematics is useful practically, and that that little is comparatively dull.
The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.
The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains.
Each is as fresh and significant as when it was discovered—two thousand years have not written a wrinkle on either of them.
Pythagoras’s discovery exposed the unsoundness of this foundation, and led to the construction of the much more profound theory of Eudoxus which is set out in the fifth book of the Elements, and which is regarded by many modern mathematicians as the finest achievement of Greek mathematics.
This theory is astonishingly modern in spirit, and may be regarded as the beginning of the modern theory of irrational number, which has revolutionized mathematical analysis and had much influence on recent philosophy.
extreme speciality of both the enunciations and the proofs, which are not capable of any significant generalization.
It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics; and this, I suppose, all astronomers would admit when pressed, however many eclipses they may have predicted correctly.
It makes no difference to a chess problem whether the pieces are white and black, or red and green, or whether there are physical ‘pieces’ at all; it is the same problem which an expert carries easily in his head and which we have to reconstruct laboriously with the aid of the board.
The board and the pieces are mere devices to stimulate our sluggish imaginations, and are no more essential to the problem than the blackboard and the chalk are to the theorems in a mathematical lecture.
Some measure of generality must be present in any high-class theorem, but too much tends inevitably to insipidity.
It is undeniable that a good deal of elementary mathematics—and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility.
For me, and I suppose for most mathematicians, there is another reality, which I will call ‘mathematical reality’; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers.
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.
This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards,
A reader who does not like the philosophy can alt...
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Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern.
Do the theorems which I have proved become false?
A mathematician, on the other hand, is working with his own mathematical reality.
The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers.
Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
The Whitehead mathematics may affect astronomy or physics profoundly, philosophy very appreciably—high thinking of one kind is always likely to affect high thinking of another
