"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist Vol. 87 (5), 1999)
"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject." (Steven G Krantz, "The History and Concept of Mathematical Proof", 2007)
"There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements […]." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"An introverted mathematician is one who looks at his shoes when he talks to you. An extroverted mathematician is one who looks at your shoes when he talks to you." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)
"Definitions are part of the bedrock of mathematical writing and thinking. Mathematics is almost unique among the sciences - not to mention other disciplines - in insisting on strictly rigorous definitions of terminology and concepts. Thus we must state our definitions as succinctly and comprehensibly as possible. Definitions should not hang the reader up, but should instead provide a helping hand as well as encouragement for the reader to push on." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)
"One fault that all mathematicians have is this: we think that when we have said something once clearly then that is the end of it; nothing further need be said. This observation explains why mathematicians so often lose arguments. You must repeat." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)
"That is the trouble with facts: they sometimes force you to conclusions that differ with your intuition." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)
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