On Arithmetic (1925 - 1949)

"[…] extensions beyond the complex number domain are possible only at the expense of the principle of permanence. The complex number domain is the last frontier of this principle. Beyond this either the commutativity of the operations or the rôle which zero plays in arithmetic must be sacrificed." (Tobias Dantzig, "Number: The Language of Science", 1930)

"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)

"Will anyone seriously assert that the existence of negative numbers is guaranteed by the fact that there exist in the world hot assets and cold, and debts? Shall we refer to these things in the structure of arithmetic? Who does not see that thereby an entirely foreign element enters into arithmetic, which endangers the pureness and clarity of its concepts?" (Friedrich Waismann, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1936)

"If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant - but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error." (Clive S Lewis, "Bulverism", 1941)

"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)

"If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability." (Vannevar Bush, "As We May Think", Atlantic Monthly, 1945)

"With a literature much vaster than those of algebra and arithmetic combined, and as least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." (Eric T Bell, "The Development of Mathematics", 1945)