Some Quotes About Mathematics
Problems worthy of attack prove their worth by fighting back.
A good stack of examples, as large as possible, is indispensable
for a thorough understanding of any concept, and when I want to
learn something new, I make it my first job to build one.
It is always the case, with mathematics, that a little direct
experience of thinking over things on your own can provide a much
deeper understanding than merely reading about them.
Perhaps I could best describe my experience of doing mathematics in
terms of entering a dark mansion. You go into the first room and it's
dark, completely dark. You stumble around, bumping into the
furniture. Gradually, you learn where each piece of furniture is. And
finally, after six months or so, you find the light switch and turn it
on. Suddenly, it's all illuminated and you can see exactly where you
were. Then you enter the next dark room…
I prided myself in reading quickly. I was really amazed by my first
encounters with serious mathematics textbooks. I was very interested
and impressed by the quality of the reasoning, but it was quite hard
to stay alert and focused. After a few experiences of reading a few
pages only to discover that I really had no idea what I'd just read, I
learned to drink lots of coffee, slow way down, and accept that I
needed to read these books at 1/10th or 1/50th standard reading speed,
pay attention to every single word and backtrack to look up all the
obscure numbers of equations and theorems in order to follow the
We need heuristic reasoning when we construct a strict proof as we
need scaffolding when we erect a building.
It is by logic that we prove, but by intuition that we discover. To
know how to criticize is good, to know how to create is better.
The heart of mathematics consists of concrete examples and concrete
problems. Big general theories are usually afterthoughts based on
small but profound insights; the insights themselves come from
concrete special cases.
The mathematician’s patterns, like the painter’s or the
poet’s, must be beautiful; the ideas, like the colours or the
words, must fit together in a harmonious way. Beauty is the
first test: there is no permanent place in the world for ugly
mathematics. … It may be very hard to define mathematical
beauty, but that is just as true of beauty of any kind — we
may not know quite what we mean by a beautiful poem, but that
does not prevent us from recognizing one when we read it.
G. H. Hardy
Mathematicians need proofs to keep them honest. All technical
areas of human activity need reality checks. It is not enough to
believe that something works, that it is a good way to proceed,
or even that it is true. We need to know why it's
true. Otherwise, we won't know anything at all.
It is an error to believe that rigor in proof is the enemy of
simplicity. On the contrary, we find it confirmed by numerous
examples that the rigorous method is at the same time the
simpler and the more easily comprehended. The very effort for
rigor forces us to discover simpler methods of proof. It also
frequently leads the way to methods which are more capable of
development than the old methods of less rigor.
If my false figures came near to the facts, this happened merely
by chance … These comments are not worth printing. Yet it gives
me pleasure to remember how many detours I had to make, along
how many walls I had to grope in the darkness of my ignorance
until I found the door which lets in the light of the truth … In
such manner did I dream of the truth.
The best way to learn is to do; the worst way to teach is to talk.
If you really wish to learn then you must mount the machine and
become acquainted with its tricks by actual trial.
It is the simple hypotheses of which one must be most wary; because
these are the ones that have the most chances of passing unnoticed.
I mean the word proof not in the sense of the lawyers, who set two
half proofs equal to a whole one, but in the sense of a mathematician,
where where ½ proof = 0, and it is demanded for proof that every
doubt becomes impossible.
Carl Friedrich Gauss
The best of ideas is hurt by uncritical acceptance and thrives on