"The final truth about phenomena resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge is complete. We go beyond the mathematical formula at our own risk; we may find a [nonmathematical] model or picture that helps us to understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault." (James Jeans, "The Mysterious Universe", 1930)
"In the study of ideas, it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions." (Alfred N Whitehead, "Adventures of Ideas", 1933)
"Our reasonings grasp at straws for premises and float on gossamers for deductions." (Alfred N Whitehead, "Adventures of Ideas", 1933)
"Science is not a system of certain, or -established, statements; nor is it a system which steadily advances towards a state of finality […] And our guesses are guided by the unscientific, the metaphysical" (though biologically explicable) faith in laws, in regularities which we can uncover - discover. Like Bacon, we might describe our own contemporary science - 'the method of reasoning which men now ordinarily apply to nature' - as consisting of 'anticipations, rash and premature' and as 'prejudices'." (Karl R Popper, "The Logic of Scientific Discovery", 1934)
"Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and" (in fortunate cases) making it possible to predict future occurrences." (Bertrand Russell, "Religion and Science, Grounds of Conflict", 1935)
"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1937)
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct" (leaving aside the question as to what is meant by 'correct'). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. One may for instance judge that all positive integers are uniquely factorable into primes; a detailed mathematical argument leads to the same result. It will also involve intuitive judgments, but they will be ones less open to criticism than the original judgment about factorization." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45" (2), 1939)
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