"Mathematical truth is like any other truth. A statement is true if our embodied understanding of the statement accords with our embodied understanding of the subject matter and the situation at hand. Truth, including mathematical truth, is thus dependent on embodied human cognition." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is an objective feature of the universe; mathematical objects are real; mathematical truth is universal, absolute, and certain." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)
"That a proof must be convincing is part of the anthropology of mathematics, providing the key to understanding mathematics as a human activity. We invoke the logic of mathematics when we demand that every informal proof be capable of being formalized within the confines of a definite formal system. Finally, the epistemology of mathematics comes into play with the requirement that a proof be surveyable. We can't really say that we have created a genuine piece of knowledge unless it can be examined and verified by others; there are no private truths in mathematics." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"Traditionally, mathematical truths have been considered to be a priori truths, either in the sense that they are truths that would be true in any possible universe, or in the sense that they are truths whose validity is independent of our sensory impressions." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)
"In essence, mathematicians wanted to prove two things: 1.Mathematics is consistent: Mathematics contains no internal contradictions. There are no slips of reason or ambiguities. No matter from what direction we approach the edifice of mathematics, it will always display the same rigor and truth. 2.Mathematics is complete: No mathematical truths are left hanging. Nothing needs adding to the system. Mathematicians can prove every theorem with total rigor so that nothing is excluded from the overall system." (F David Peat, "From Certainty to Uncertainty", 2002)
"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws, which will be revealed in the coming chapters. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"Where we find certainty and truth in mathematics we also find beauty. Great mathematics is characterized by its aesthetics. Mathematicians delight in the elegance, economy of means, and logical inevitability of proof. It is as if the great mathematical truths can be no other way. This light of logic is also reflected back to us in the underlying structures of the physical world through the mathematics of theoretical physics." (F David Peat, "From Certainty to Uncertainty", 2002)
"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, 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. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)
"We do not discover mathematical truths; we remember them from our passages through this world outside our own." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others." (William Byers, "How Mathematicians Think", 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. […] 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" (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)
"What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?" (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)
"Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built." (David Perkins, "Calculus and Its Origins", 2012)
"There are thousands of apparent mathematical truths out there that we humans have discovered and believe to be true but have so far been unable to prove. They are called conjectures. A conjecture is simply a statement about mathematical reality that you believe to be true [..]" (Paul Lockhart, "Measurement", 2012)
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