Excerpt from The University of New Mexico Bulletin: An Introduction to Mathematics; A Study of the Nature of Mathematical Knowledge
For several years the author has had the privilege of giving lectures upon the nature of mathematics to students in the University of New Mexico. One set of lectures has been given to certain classes of beginning students in college mathematics as it is the belief of many mathematicians that little actual mathematical thinking can be indulged in without some comprehension of the meaning of mathematics. An additional advanced course of lectures of the same fundamental nature has been made available to advanced students in mathematics with the hope that they would appreciate many of the problems which face the creative mathematician. As a result of this experience it would seem that a short monograph of popular appeal upon the nature of mathematics might be welcomed. Of course, there are a few excellent books available to the student who cares to work seriously in the field. However, these books contain difficult vocabularies or symbolic languages which hide difficult concepts. So it is the purpose of this short work to present a simple introduction to modern mathematical thought, omitting completely many of the important extensions which are occupying the attention of mathematicians.
One must realize that the decision as to what is mathematics or what is not mathematics is, to a great extent, arbitrary. Hence not all mathematicians agree. Some mathematicians prefer to be called "pure mathematicians," and others prefer to be classified as "applied mathematicians." The purest of "pure mathematicians" refuses to soil his hands in the laboratory, and his meditations are the envy of modern philosophers to such an extent that philosophers are now studying mathematics. The extremist among the "applied mathematicians" goes to sleep while a theory is expounded; he wants a formula; and he wants to know when and how such a formula is used.
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