"[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost." (Mark Kac, "Mathematics: Tensions", 1992)
"Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston, "On Proof and Progress in Mathematics", 1994)
"The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design." (Freeman J Dyson, [interview] 1994)
"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers", 1995)
"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood […]" (Andrew Granville, 1997)
"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)
"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)
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