Pure Mathematics II

"In pure mathematics the maximum of detachment appears to be reached: the mind moves in an infinitely complicated pattern, which is absolutely free from temporal considerations. Yet this very freedom - the essential condition of the mathematician’s activity - perhaps gives him an unfair advantage. He can only be wrong – he cannot cheat." (Kytton Strachey, "Portraits in Miniature", 1931)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.  In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Albert Einstein, [Obituary for Emmy Noether], 1935)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, "Mathematics in Western Culture", 1953)

"General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the 'real' world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge." (Kenneth E Boulding, "General Systems Theory - The Skeleton of Science", Management Science Vol. 2 (3), 1956)

"The confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated." (Michael Polanyi, "Personal Knowledge", 1958)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists." (Imre Lakatos,"Mathematics, Science and Epistemology" Vol. 2, 1978)

"Theoretical physicists are like pure mathematicians, in that they are often interested in the hypothetical behaviour of entirely imaginary objects, such as parallel universes, or particles traveling faster than light, whose actual existence is not being seriously proposed at all." (John Ziman," Real Science: What it Is, and what it Means", 2000)

"Pure mathematics was characterized by an obsession with proof, rigor, beauty, and elegance, and sought its foundations in the disembodied worlds of logic or intuition. Far from being coextensive with physics, pure mathematics could be ‘applied’ only after it had been made foundationally secure by the purists." (Andrew Warwick, "Masters of Theory: Cambridge and the rise of mathematical physics", 2003)

"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature Vol. 475 (7355), 2011)

"We tend to think of maths as being an 'exact' discipline, where answers are right or wrong. And it's true that there is a huge part of maths that is about exactness. But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definite, 'right' answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest." (Rob Eastaway, "Maths on the Back of an Envelope", 2019)