23 April 2022

On Consistence (1925-1949)

"[…] the main object of physical science is not the provision of pictures, but in the formulation of laws governing phenomena and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance. In the case of atomic phenomena no picture can be expected to exist in the usual sense of the word ‘picture’, by which is meant to model functioning essentially on classical lines. One may extend the meaning of the word ‘picture’ to include any way of looking at the fundamental laws which make their self-consistency obvious. With this extension, one may acquire a picture of atomic phenomena by becoming familiar with the laws of quantum theory. (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics. (Samuel T Sanders, "Mathematics", National Mathematics Magazine, 1937)

"The ethos of science involves the functionally necessary demand that theories or generalizations be evaluated in [terms of] their logical consistency and consonance with facts." (Robert K Merton, "Science and the Social Order", Philosophy of Science Vol 5 (3), 1938)

"A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"Mathematicians deal with possible worlds, with an infinite number of logically consistent systems. Observers explore the one particular world we inhabit. Between the two stands the theorist. He studies possible worlds but only those which are compatible with the information furnished by observers. In other words, theory attempts to segregate the minimum number of possible worlds which must include the actual world we inhabit. Then the observer, with new factual information, attempts to reduce the list further. And so it goes, observation and theory advancing together toward the common goal of science, knowledge of the structure and observation of the universe." (Edwin P Hubble, "The Problem of the Expanding Universe", 1941)

"Mathematicians deal with possible worlds, with an infinite number of logically consistent systems. Observers explore the one particular world we inhabit. Between the two stands the theorist. He studies possible worlds but only those which are compatible with the information furnished by observers. In other words, theory attempts to segregate the minimum number of possible worlds which must include the actual world we inhabit. Then the observer, with new factual information, attempts to reduce the list further. And so it goes, observation and theory advancing together toward the common goal of science, knowledge of the structure and observation of the universe. (Edwin P Hubble, "The Problem of the Expanding Universe", 1941)

"The pictures we draw of nature show similar limitations; these are the price we pay for limiting our pictures of nature to the kinds that can be understood by our minds. As we cannot draw one perfect picture, we make two imperfect pictures and turn to one or the other according as we want one property or another to be accurately delineated. Our observations tell us which is the right picture to use for each particular purpose […] . Yet some properties of nature are so far-reaching and general that neither picture can depict them properly of itself. In such cases we must appeal to both pictures, and these sometimes give us different and inconsistent information. Where, then, shall we find the truth?" (James H Jeans,"Physics and Philosophy" 3rd Ed., 1943)

"It is likely then that the nervous system is in a fortunate position, as far as modelling physical processes is concerned, in that it has only to produce combinations of excited arcs, not physical objects; its ’answer’ need only be a combination of consistent patterns of excitation - not a new object that is physically and chemically stable." (Kenneth Craik, "The Nature of Explanation", 1943)

"Perhaps the extraordinary pervasiveness of number, and the multiplicity of operations which can be performed on number without leading to inconsistency, is not a proof of the ’real existence’ of numbers as such, but a proof of the extreme flexibility of the neural model or calculating machine. This flexibility renders a far greater number of operations possible for it than for any other single process or model. (Kenneth Craik, "The Nature of Explanation", 1943)

"The pictures we draw of nature show similar limitations; these are the price we pay for limiting our pictures of nature to the kinds that can be understood by our minds. As we cannot draw one perfect picture, we make two imperfect pictures and turn to one or the other according as we want one property or another to be accurately delineated. Our observations tell us which is the right picture to use for each particular purpose […] . Yet some properties of nature are so far-reaching and general that neither picture can depict them properly of itself. In such cases we must appeal to both pictures, and these sometimes give us different and inconsistent information. Where, then, shall we find the truth? (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"We have now to enquire how the neural mechanism, in producing numerical measurement and calculation, has managed to function in a way so much more universal and flexible than any other. Our question, to emphasize it once again, is not to ask what kind of thing a number is, but to think what kind of mechanism could represent so many physically possible or impossible, and yet self-consistent, processes as number does. (Kenneth Craik, "The Nature of Explanation", 1943)

"A model, like a novel, may resonate with nature, but it is not a ‘real’ thing. Like a novel, a model may be convincing – it may ‘ring true’ if it is consistent with our experience of the natural world. But just as we may wonder how much the characters in a novel are drawn from real life and how much is artifice, we might ask the same of a model: How much is based on observation and measurement of accessible phenomena, how much is convenience? Fundamentally, the reason for modeling is a lack of full access, either in time or space, to the phenomena of interest." (K. Belitz, Science, Vol. 263, 1944)

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