"All knowledge, the sociologist could say, is conjectural and theoretical. Nothing is absolute and final. Therefore all knowledge is relative to the local situation of the thinkers who produce it: the ideas and conjectures that they are capable of producing: the problems that bother them; the interplay of assumptions and criticism in their milieu; their purposes and aims; the experiences they have and the standards and meanings they apply." (David Bloor, "Knowledge and Social Imagery", 1976)
"The essential function of a hypothesis consists in the guidance it affords to new observations and experiments, by which our conjecture is either confirmed or refuted." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1976)
"The verb 'to theorize' is now conjugated as follows: 'I built a model; you formulated a hypothesis; he made a conjecture.'" (John M Ziman, "Reliable Knowledge", 1978)
"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case."
"Three shifts can be detected over time in the understanding of mathematics itself. One is a shift from completeness to incompleteness, another from certainty to conjecture, and a third from absolutism to relativity." (Leone Burton, "Femmes et Mathematiques: Y a–t–il une?", Association for Women in Mathematics Newsletter, Intersection 18, 1988)
"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)
"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof - which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lan, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)
"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)
"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed [...] were made explicit when logic was formalized early in the this century [...] These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a [...] conjecture. [...] Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. [...] Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)
"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)
"The everyday usage of 'theory' is for an idea whose outcome is as yet undetermined, a conjecture, or for an idea contrary to evidence. But scientists use the word in exactly the opposite sense. [In science] 'theory' [...] refers only to a collection of hypotheses and predictions that is amenable to experimental test, preferably one that has been successfully tested. It has everything to do with the facts." (Tony Rothman & George Sudarshan, "Doubt and Certainty: The Celebrated Academy: Debates on Science, Mysticism, Reality, in General on the Knowable and Unknowable", 1998)
"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, 1999)
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