"Although the prime numbers are rigidly determined, they somehow feel like experimental data." Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"What a mathematical proof actually does is show that certain conclusions, such as the irrationality of, follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"Compactness is a powerful property of spaces, and it is used in many ways in many different areas of mathematics. One is via appeal to local-to-global principles: one establishes local control on a function, or on some other quantity, and then uses compactness to boost the local control to global control." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)
"The curious switch, from initially perceiving an obstruction to a problem to eventually embodying this obstruction as a number or an algebraic object of some sort that we can effectively study, is repeated over and over again, in different contexts, throughout mathematics." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)
"Topology allows the possibility of making qualitative predictions when quantitative ones are impossible." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)
"A discovery tends to be more notable than an observation and
less easy to verify afterward. And inventions tend to be more general than
creations."
"A straightforward use of the word “invention” in mathematics
is to refer to the way general theories and techniques come into being. This way
of coming into being certainly covers the example of calculus, which is not an
object, or a single fact, but rather a large collection of facts and methods
that greatly increase your mathematical power when you are familiar with them."
"Another reason for our ambivalence about the complex numbers
is that they feel less real than real numbers. [...] We can directly relate the
real numbers to quantities such as time, mass, length, temperature, and so on
(though for this usage, we never need the infinite precision of the real number
system), so it feels as though they have an independent existence that we
observe. But we do not run into the complex numbers in that way. Rather, we
play what feels like a sort of game - imagine what would happen if -1 did have a
square root."
"Nevertheless, there does seem to be a spectrum of
possibilities, with some parts of mathematics feeling more like discoveries and
others more like inventions. It is not always easy to say which are which, but there
does seem to be one feature that correlates strongly with whether we prefer to
use a discovery- type word or an invention- type word."
"The fact that some parts of mathematics are unexpected and others not, that some solutions are unique and others multiple, that some proofs are obvious and others take a huge amount of work to produce - all these have a bearing on how we describe the process of mathematical production, and all of them are entirely independent of one’s philosophical position." (Timothy Gowers, "Is Mathematics Discovered or Invented?", ["The Best Writing of Mathematics: 2012"] 2012)
"It is efficient to look for beautiful solutions first and settle for ugly ones only as a last resort. [...] It is a good rule of thumb that the more beautiful the guess, the more likely it is to survive." (Timothy Gowers)
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