On Numbers (1925-1949)

"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well distinguishable individuals. Even we enlightened scientists can still feel it e.g. in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1927)

"For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Anschauliche Geometrie", 1932)

"Numbers constitute the only universal language." (Nathanael West, "Miss Lonelyhearts", 1933)

"[...] our knowledge of the external world must always consist of numbers, and our picture of the universe - the synthesis of our knowledge - must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols - they do not belong to Nature, but to the parables by which we try to make Nature comprehensible." (Sir James H Jeans, "The New World-Picture of Modern Physics", Supplement to Nature, Vol. 134 (3384), 1934)

"Mathematics is the science of number and space. It starts from a group of self-evident truths and by infallible deduction arrives at incontestable conclusions […] the facts of mathematics are absolute, unalterable, and eternal truths." (E Russell Stabler, "An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics", The Mathematics Teacher, Vol 26, 1935)

"[…] 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)

"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)