Mathematical Truth III

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts",  1989)

“Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics.” (Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty”, 1997)

“Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net.” (John Polkinghorne, “Belief in God in an Age of Science”, 1998)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, “Philosophy of Mathematics”, 1999)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it istrue, thre will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of theircreator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it istrue, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] Mathematical facts are not isolated, they are woven into a vast spider's web of interconnections." (Gregory Chaitin "Meta Math: The Quest for Omega", 2005)

"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", 2007)

"There is an absolute nature to truth in mathematics, which is unmatched in any other branch of knowledge. A theorem, once proven, requires independent checking but not repetition or independent derivation to be accepted as correct."  (James Glimm, "Reflections and Prospectives", 2009)

"Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity. Why does this matter, and why does it go beyond a cultural oddity of our profession? The answer is that mathematics is deeply embedded in the reasoning used within many branches of knowledge. That reasoning often involves conjectures, assumptions, intuition. But whatever aspect has been reduced to mathematics has an absolute validity. As in other subjects search for truth, the mathematical components embedded in their search are like the boulders in the stream, providing a solid footing on which to cross from one side to the other.” (James Glimm, "Reflections and Prospectives", 2009)

On Problem Solving VII: Mathematicians I

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves) 

"Finding the right answer is important, of course. But more important is developing the ability to see that problems have multiple solutions, that getting from X to Y demands basic skills and mental agility, imagination, persistence, patience." (Mary H Futrell)

"I knew nothing, except how to think, how to grapple with a problem and then go on grappling with it until you had solved it." (Sir Barnes Wallis) 

"It’s not that I’m so smart, it’s just that I stay with problems longer." (Albert Einstein)

"Man is not born to solve the problems of the universe, but to find out where the problems begin, and then to take his stand within the limits of the intelligible." (Johann Wolfgang von Goethe) 

"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice." (George Polya) 

"The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying." (Jean Dieudonne)

"The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems." (Charles W Churchman) 

"The real raison d’etre for the mathematician’s existence is simply to solve problems. So what mathematics really consists of is problems and solutions." (John Casti) 

"When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." (Buckminster Fuller) 

Previous <<||>> Next