On Numbers (1940-1949)

"Our conception of space is, in a fashion similar to that of natural numbers, depending on a constructive grip on all possible places. Let us consider a metallic disk in a plane E. Places on the disk can be marked in concreto by scratching little crosses on the plate. But relatively to two axes of coordinates and a standard length scratched into the plate we can also put ideal marks in the plane outside the disk by giving the numerical values of their two coordinates. Each coordinate varies over the a priori constructed range of real numbers. ln this way astronomy uses our solid earth as a base for plumbing the sidereal spaces." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)

"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)

"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

"[…] statistical literacy. That is, the ability to read diagrams and maps; a 'consumer' understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number." (Douglas Scates,"Statistics: The Mathematics for Social Problems", 1943)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"Good physics is made a priori. Theory precedes fact. Experience is useless because before any experience we are already in possession of the knowledge we are seeking for. Fundamental laws of motion" (and of rest), laws that determine the spatio-temporal behavior of material bodies, are laws of a mathematical nature. Of the same nature as those which govern relations and laws of figures and numbers. We find and discover them not in Nature, but in ourselves, in our mind, in our memory, as Plato long ago has taught us." (Alexander Koyre, "Galileo and the Scientific Revolution of the Seventeenth Century", The Philosophical Review Vol. 52" (3), 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)

"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) 

"And nobody can get far without at least an acquaintance with the mathematics of probability, not to the extent of making its calculations and filling examination papers with typical equations, but enough to know when they can be trusted, and when they are cooked. For when their imaginary numbers correspond to exact quantities of hard coins unalterably stamped with heads and tails, they are safe within certain limits; for here we have solid certainty [...] but when the calculation is one of no constant and several very capricious variables, guesswork, personal bias, and pecuniary interests, come in so strong that those who began by ignorantly imagining that statistics cannot lie end by imagining, equally ignorantly, that they never do anything else." (George B Shaw, "The Vice of Gambling and the Virtue of Insurance", 1944)

"In other words, without a theory, a plan, the mere mechanical manipulation of the numbers in a problem does not necessarily make sense just because you are using Arithmetic!" (Lillian R Lieber, "The Education of T.C. MITS", 1944)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The straight line of the geometers does not exist in the material universe. It is a pure abstraction, an invention of the imagination or, if one prefers, an idea of the Eternal Mind." (Eric T Bell, "The Magic of Numbers", 1946)

"[…] the number of available analogies is a determining factor in the growth and progress of science." (Morris R Cohen, "The Meaning of Human History", 1947)