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Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.
What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze.
Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood.
... there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.
As soon as a thought or word becomes a tool, one can dispense with actually 'thinking' it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of mathematics—the model of all neo-positivistic thinking—lies in just this 'intellectual economy.' Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced.
The distinction between the mathematical method and the scientific is seen in the agonies of very young children to do what their teachers sometimes tell them is mathematics. Anyone who was subjected to elementary geometry when his infantile brain was as unripe as a green walnut will recall the protracted misery he endured.