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Un insegnamento e' pessimo se presenta una processione senza fine di segni, parole e regole senza significato, e non riesce ad arrivare all'immaginazione. W. W. Sawyer |
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Niente appaga un matematico piu' che
scoprire che due cose, prima considerate come completamente distinte, sono
matematicamente identiche. " La matematica ", diceva Poincare', "
e' 1'arte di
dare lo stesso nome a cose differenti. "
W. W. Sawyer |
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Mathematics possesses not only truth but supreme beauty, a beauty cold and austere, like that of sculpture, sublimely pure and capable of a stern perfection, such as only the greatest art can show. Bertrand Russell |
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Mathematics takes us into the region of absolute necessity, to which not
only the actual word, but every possible word, must conform. Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988). |
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Although this may seem a paradox, all exact science is dominated by the
idea of approximation. Quoted in W H Auden and L Kronenberger, The Viking Book of Aphorisms (New York 1966). |
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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 Autobiography of Bertrand Russell . |
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Ordinary language is totally unsuited for expressing what physics really
asserts, since the words of everyday life are not sufficiently abstract.
Only mathematics and mathematical logic can say as little as the physicist
means to say. With equal passion I have sought knowledge. I have wished to understand
the hearts of men. I have wished to know why the stars shine. And I have
tried to apprehend the Pythagorean power by which number holds sway about
the flux. A little of this, but not much, I have achieved. At first it seems obvious, but the more you think about it the stranger
the deductions from this axiom seem to become; in the end you cease to
understand what is meant by it. Calculus required continuity, and continuity was supposed to require the
infinitely little; but nobody could discover what the infinitely little
might be. The fact that all Mathematics is Symbolic Logic is one of the greatest
discoveries of our age; and when this fact has been established, the
remainder of the principles of mathematics consists in the analysis of
Symbolic Logic itself. A habit of basing convictions upon evidence, and of giving to them only
that degree or certainty which the evidence warrants, would, if it became
general, cure most of the ills from which the world suffers. The method of "postulating" what we want has many advantages; they are
the same as the advantages of theft over honest toil. [Upon hearing via Littlewood an exposition on the theory of relativity:] "But," you might say, "none of this shakes my belief that 2 and 2 are 4."
You are quite right, except in marginal cases -- and it is only in marginal
cases that you are doubtful whether a certain animal is a dog or a certain
length is less than a meter. Two must be two of something, and the
proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs
and two dogs are certainly four dogs, but cases arise in which you are
doubtful whether two of them are dogs. "Well, at any rate there are four
animals," you may say. But there are microorganisms concerning which it is
doubtful whether they are animals or plants. "Well, then living organisms,"
you say. But there are things of which it is doubtful whether they are
living organisms or not. You will be driven into saying: "Two entities and
two entities are four entities." When you have told me what you mean by
"entity," we will resume the argument. I wanted certainty in the kind of way in which people want religious
faith. I thought that certainty is more likely to be found in mathematics
than elsewhere. But I discovered that many mathematical demonstrations,
which my teachers expected me to accept, were full of fallacies, and that,
if certainty were indeed discoverable in mathematics, it would be in a new
field of mathematics, with more solid foundations than those that had
hitherto been thought secure. But as the work proceeded, I was continually
reminded of the fable about the elephant and the tortoise. having
constructed an elephant upon which the mathematical world could rest, I
found the elephant tottering, and proceeded to construct a tortoise to keep
the elephant from falling. But the tortoise was no more secure than the
elephant, and after some twenty years of very arduous toil, I came to the
conclusion that there was nothing more that I could do in the way of making
mathematical knowledge indubitable. Men who are unhappy, like men who sleep badly, are always proud of the
fact. Work is of two kinds: first, altering the position of matter at or near the earth's surface relatively to other such matter; second, telling other people to do so. The first kind is unpleasant and ill paid; the second is pleasant and highly paid. A sense of duty is useful in work but offensive in personal relations.
Certain characteristics of the subject are clear. To begin with, we do not,
in this subject, deal with particular things or particular properties: we
deal formally with what can be said about "any" thing or "any" property. We
are prepared to say that one and one are two, but not that Socrates and
Plato are two, because, in our capacity of logicians or pure mathematicians,
we have never heard of Socrates or Plato. A world in which there were no
such individuals would still be a world in which one and one are two. It is
not open to us, as pure mathematicians or logicians, to mention anything at
all, because, if we do so we introduce something irrelevant and not formal.
It can be shown that a mathematical web of some kind can be woven about
any universe containing several objects. The fact that our universe lends
itself to mathematical treatment is not a fact of any great philosophical
significance. Almost everything that distinguishes the modern world from earlier
centuries is attibutable to science, which achieved its most spectacular
triumphs in the seventeenth century. The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. Boredom is a vital problem for the moralist, since at least half the sins
of mankind are caused by the fear of it. Bertrand Russell |
A cura di L.T.