But when great and ingenious artists behold their so inept performances,
not undeservedly do they
ridicule the blindness of such men; since sane judgment abhors nothing
so much as a picture
perpetrated with no technical knowledge, although with plenty of care
and diligence. Now the sole
reason why painters of this sort are not aware of their own error is
that they have not learnt
Geometry, without which no one can either be or become an absolute
artist; but the blame for this
should be laid upon their masters, who are themselves ignorant of this
art.
Albrecht Dürer, The Art of Measurement. 1525.
And since geometry is the right foundation of all painting, I have decided
to teach its rudiments and
principles to all youngsters eager for art...
Albrecht Dürer, Course in the Art of Measurement
Welcher aber ... durch die Geometria sein Ding beweist und die gründliche
Wahrheit anzeigt,
dem soll alle Welt glauben. Denn da ist man gefangen.
Whoever ... proves his point and demonstrates the prime truth geometrically
should be believed by
all the world, for there we are captured
Albrecht Dürer, Von menschlicher Proportion
The traditional mathematics professor of the popular legend is absentminded.
He usually appears in
public with a lost umbrella in each hand. He prefers to face a blackboard
and to turn his back on the
class. He writes a, he says b, he means c, but it should be d. Some
of his sayings are handed down
from generation to generation.
George Pólya
In order to solve this differential equation you look at it till a solution
occurs to you.
George Pólya
My method to overcome a difficulty is to go round it.
George Pólya
What is the difference between method and device? A method is a device
which you used twice.
George Pólya
Even fairly good students, when they have obtained the solution of the
problem and written down
neatly the argument, shut their books and look for something else.
Doing so, they miss an important
and instructive phase of the work. ... A good teacher should understand
and impress on his students
the view that no problem whatever is completely exhausted.
George Pólya
One of the first and foremost duties of the teacher is not to give his
students the impression that
mathematical problems have little connection with each other, and no
connection at all with anything
else. We have a natural opportunity to investigate the connections
of a problem when looking back
at its solution.
George Pólya
If there is a problem you can't solve, then there is an easier problem
you can't solve: find it.
George Pólya
A GREAT discovery solves a great problem but there is a grain of discovery
in any problem.
George Pólya
The first rule of discovery is to have brains and good luck. The second
rule of discovery is to sit tight
and wait till you get a bright idea.
George Pólya
If you have to prove a theorem, do not rush. First of all, understand
fully what the theorem says, try
to see clearly what it means. Then check the theorem; it could be false.
Examine the consequences,
verify as many particular instances as are needed to convince yourself
of the truth. When you have
satisfied yourself that the theorem is true, you can start proving
it.
George Pólya
Mathematics consists of proving the most obvious thing in the least
obvious way.
George Pólya,Quoted in N Rose Mathematical Maxims and Minims
(Raleigh N C 1988).
Mathematics is the cheapest science. Unlike physics or chemistry, it
does not require any expensive
equipment. All one needs for mathematics is a pencil and paper.
George Pólya, Quoted in D J Albers, G L Alexanderson and C Reid,
Mathematical People (Boston 1985).
To teach effectively a teacher must develop a feeling for his subject;
he cannot make his students
sense its vitality if he does not sense it himself. He cannot share
his enthusiasm when he has no
enthusiasm to share. how he makes his point may be as important as
the point he makes; he must
personally feel it to be important.
George Pólya, Mathematical discovery (New York, 1981).
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 unity, that permits us to see clearly and to
comprehend at once both the
ensemble and the details.
Henri Poincaré, Quoted in N Rose Mathematical Maxims and Minims
(Raleigh N C 1988).
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.
Henri Poincaré, La Science et l'hypothèse.
A scientist worthy of his name, about all a mathematician, experiences
in his work the same
impression as an artist; his pleasure is as great and of the same nature.
Henri Poincaré, Quoted in N Rose Mathematical Maxims and Minims
(Raleigh N C 1988).
Mathematical discoveries, small or great are never born of spontaneous
generation They always
presuppose a soil seeded with preliminary knowledge and well prepared
by labour, both conscious
and subconscious.
Henri Poincaré
Mathematics takes us into the region of absolute necessity, to which
not only the actual word, but
every possible word, must conform.
Bertrand Russell, Quoted in N Rose Mathematical Maxims and Minims (Raleigh
N C 1988).
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 the notes of our observations.
G H Hardy, A Mathematician's Apology (London 1941).
Archimedes will be remembered when Aeschylus is forgotten, because languages
die and
mathematical ideas do not. "Immortality" may be a silly word, but probably
a mathematician has the
best chance of whatever it may mean.
G H Hardy, A Mathematician's Apology (London 1941).
In great mathematics there is a very high degree of unexpectedness,
combined with inevitability and
economy.
G H Hardy, A Mathematician's Apology (London 1941).
The mathematician's patterns, like the painter's or the poet's must
be beautiful; the ideas, like the
colors or the words must fit together in a harmonious way. Beauty is
the first test: there is no
permanent place in this world for ugly mathematics.
G H Hardy, A Mathematician's Apology (London 1941).
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's
finest weapons. It is
a far finer gambit than any chess play: a chess player may offer the
sacrifice of a pawn or even a
piece, but a mathematician offers the game.
G H Hardy, A Mathematician's Apology (London 1941).
I am interested in mathematics only as a creative art.
G H Hardy, A Mathematician's Apology (London 1941).
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.
Eric Temple Bell