"Architectural conjectures are mathematically precise assertions, as well milled as minted coins, provisionally usable in the commerce of logical arguments; less than ‘coins’ and more aptly, promissory notes to be paid in full by some future demonstration, or to be contradicted. These conjectures are expected to turn out to be true, as, of course, are all conjectures; their formulation is often away of "formally" packaging, or at least acknowledging, an otherwise shapeless body of mathematical experience that points to their truth." (Barry Mazur, "Conjecture", Synthese 111, 1997)
"A seemingly modest change of notation may suggest a radical shift in viewpoint. Any new notation may ask new questions." (Barry Mazur, "Imagining Numbers", 2003)
"If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer." (Barry Mazur, "Imagining Numbers", 2003)
“[…] history of mathematics. It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining." (Barry Mazur, "Imagining Numbers", 2003)
"If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer." (Barry Mazur, "Imagining Numbers", 2003)
“[…] history of mathematics. It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining." (Barry Mazur, "Imagining Numbers", 2003)
"The wonderful thing about mathematics is that, in the end as well as in the beginning, it can depend upon no authority other than one’s own (your own) mind; its verification comes from thinking alone, an activity open to anyone. If we have no theoretical equipment, we use the mathematical eyes and ears with which we were born and just experiment with our guesses to see whether we have faith in them." (Barry Mazur, "Imagining Numbers", 2003)
"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)
"At some point in his or her life every working mathematician
has to explain to someone, usually a relative, that mathematics is hardly a finished
project. Mathematicians know, of course, that it is far too soon to put the
glorious achievements of their trade into a big museum and just become happy
curators. In many respects, the study of mathematics has hardly begun."
"It is a hard balancing act: to explain important and beautiful mathematical ideas - to truly explain them - to people with a general cultural background but no technical training in math, and yet not to slip away from the full seriousness and ambitious goals of the subject being explained." (Barry Mazur, [foreword] 2006)
"Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible", 2014)"In mathematics, we often depend on the proof of a statement to offer not only a justification of its truth, but also a way of understanding its implications, its connections to other established truths - a way, in short of explaining the statement. But sometimes even though a proof does its job of showing the truth of a result it still leaves us with the nagging question of why.’ It may be elusive - given a specific proof - to describe in useful terms the type of explanation the proof actually offers. It would be good to have an adequate vocabulary to help us think about the explanatory features of mathematics (and, more generally, of science)." (Barry Mazur, "On the word ‘because’ in mathematics, and elsewhere", 2017)
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