繁星客栈 - 望月殿 (数学逻辑论坛)

那一剑的寂寞

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Mathematicians are born, not made

I entered an omnibus to go to some place or other. At that moment when I put
my foot on the step the idea came to me, without anything in my former thou
ghts seeming to have paved the way for it, that the transformations I had us
ed to define the Fuchsian functions were identical with non-Euclidean geomet
ry.

In the old days when people invented a new function they had something usefu
l in mind. Now, they invent them deliberately just to invalidate our ancesto
rs' reasoning, and that is all they are ever going to get out of them.

How is an error possible in mathematics? A sane mind should not be guilty of
a logical fallacy, yet there are very fine minds incapable of following mat
hematical demonstrations. Need we add that mathematicians themselves are not
infallible?

Point set topology is a disease from which the human race will soon recover.
Quoted in D MacHale, Comic Sections (Dublin 1993)

Later generations will regard Mengenlehre (set theory) as a disease from whi
ch one has recovered.
[Whether or not he actually said this is a matter of debate amongst historia
ns of mathematics.]
The Mathematical Intelligencer 13 (1991).

Mathematics is the art of giving the same name to different things.
[As opposed to the quotation: Poetry is the art of giving different names to
the same thing].

What is it indeed that gives us the feeling of elegance in a solution, in a
demonstration? It is the harmony of the diverse parts, their symmetry, their
happy balance; in a word it is all that introduces order, all that gives un
ity, that permits us to see clearly and to comprehend at once both the ensem
ble and the details.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).

Thus, be it understood, to demonstrate a theorem, it is neither necessary no
r even advantageous to know what it means. The geometer might be replaced by
the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine
might be imagined where the assumptions were put in at one end, while the th
eorems came out at the other, like the legendary Chicago machine where the p
igs go in alive and come out transformed into hams and sausages. No more tha
n these machines need the mathematician know what he does.
Quoted in J R Newman, The World of Mathematics (New York 1956).

Talk with M. Hermite. He never evokes a concrete image, yet you soon perceiv
e that the more abstract entities are to him like living creatures.
Quoted in G Simmons Calculus Gems (New York 1992).

Science is built up with facts, as a house is with stones. But a collection
of facts is no more a science than a heap of stones is a house.
La Science et l'hypothèse.

A scientist worthy of his name, about all a mathematician, experiences in hi
s work the same impression as an artist; his pleasure is as great and of the
same nature.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).

The mathematical facts worthy of being studied are those which, by their ana
logy with other facts, are capable of leading us to the knowledge of a physi
cal law. They reveal the kinship between other facts, long known, but wrongl
y believed to be strangers to one another.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).

Mathematicians do not study objects, but relations between objects. Thus, th
ey are free to replace some objects by others so long as the relations remai
n unchanged. Content to them is irrelevant: they are interested in form only
.

Thought is only a flash between two long nights, but this flash is everythin
g.
Quoted in J R Newman, The World of Mathematics (New York 1956).

The mind uses its faculty for creativity only when experience forces it to d
o so.

Mathematical discoveries, small or great are never born of spontaneous gener
ation They always presuppose a soil seeded with preliminary knowledge and we
ll prepared by labour, both conscious and subconscious.

Absolute space, that is to say, the mark to which it would be necessary to r
efer the earth to know whether it really moves, has no objective existence..
.. The two propositions: "The earth turns round" and "it is more convenient
to suppose the earth turns round" have the same meaning; there is nothing mo
re in the one than in the other.
La Science et l'hypothèse.

...by natural selection our mind has adapted itself to the conditions of the
external world. It has adopted the geometry most advantageous to the specie
s or, in other words, the most convenient. Geometry is not true, it is advan
tageous.
Science and Method.

One would have to have completely forgotten the history of science so as to
not remember that the desire to know nature has had the most constant and th
e happiest influence on the development of mathematics.
Les faits ne parlent pas
Facts do not speak.

If one looks at the different problems of the integral calculus which arise
naturally when one wishes to go deep into the different parts of physics, it
is impossible not to be struck by the analogies existing. Whether it be ele
ctrostatics or electrodynamics, the propogation of heat, optics, elasticity,
or hydrodynamics, we are led always to differential equations of the same f
amily.
American Journal of Physics 12 (1890) 211.

When one tries to depict the figure formed by these two curves and their inf
inity of intersections, each of which corresponds to a doubly asymptotic sol
ution, these intersections form a kind of net, web or infinitely tight mesh
. . . . One is struck by the complexity of this figure that I am not even at
tempting to draw.

If geometry were an experimental science, it would not be an exact science.
it would be subject to continual revision ... the geometrical axioms are the
refore neither synthetic a priori intuitions nor experimental facts. They ar
e conventions. Our choice among all possible conventions is guided by experi
mental facts; but it remains free, and is only limited by the necessity of a
voiding every contradiction, and thus it is that postulates may remain rigor
ously true even when the experimental laws which have determined their adopt
ion are only approximate. In other words the axioms of geometry (I do not sp
eak of those of arithmetic) are only definitions in disguise. What then are
we to think of the question: Is Euclidean geometry true? It has no meaning.
We might as well ask if the metric system is true and if the old weights and
measures are false; if Cartesian coordinates are true and polar coordinates
are false. One geometry cannot be more true than another; it can only be mo
re convenient.
Quoted in M J Greenberg, Euclidean and non-Euclidean geometries: Development
and history (San Fransisco, 1980).

The scientist does not study nature because it is useful; he studies it beca
use he delights in it, and he delights in it because it is beautiful. If nat
ure were not beautiful, it would not be worth knowing, and if nature were no
t worth knowing, life would not be worth living. Of course I do not here spe
ak of that beauty that strikes the senses, the beauty of qualities and appea
rances; not that I undervalue such beauty, far from it, but it has nothing t
o do with science; I mean that profounder beauty which comes from the harmon
ious order of the parts, and which a pure intelligence can grasp.

... it may happen that small differences in the initial conditions produce v
ery great ones in the final phenomena.

Zero is the number of objects that satisfy a condition that is never satisfi
ed. But as never means "in no case", I do not see that any progress has been
made.

发表时间：2005-10-18, 23:25:34