Organization of a Field by Mathematizing

Abstract

Up to now our didactical analysis has been mainly local. No global structure of mathematics to be taught was visible — it would have been otherwise if mathematics were supposed to be taught as a pre-established deductive system, as an inverse pyramid as it were, but it is now obvious that this would never lit the didactics of re-invention. Earlier analysis, however, shows how the global structure of mathematics to be taught should be understood: it is not a rigid skeleton, but it rises and perishes with the mathematics that develops in the learning process. Is it not the same with the adult mathematician’s mathematics? Its structure is not exhibited on a bookshelf by a collection of Bourbaki volumes that he has never read, nor by any other work written by other authors or by himself; it is changing every day. Why should students learn a mummified mathematics?

In the history of forestry science one knew the period of the mathematicians, yet today foresters are glad to know that this era of aberration has gone… These people would calculate the most incredible things, and the problems of forestry they could not force into the mathematical jacket were simply omitted as “not fit for scientific treatment”.

P. von Lossow, Zeitschrift d. Ver. d. Ing. 43 (1899), 360

He says “mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the the same general nature — authority and tradition furnish the data, and the mental operations are deductive.” It would seem from the above somewhat singularly juxtaposed paragraphs that, according to Prof. Huxley, the business of the mathematical student is from a limited number of propositions (bottled up and labelled ready for future use) to deduce any required result by a process of the same general nature as a student of language employs in declining and conjugating his nouns and verbs — that to make out a mathematical proposition and to construe or parse a sentence are equivalent or identical mental operations. Such an opinion scarcely seems to need serious refutation. The passage is taken from an article in Macmillan’s Magazine for June last, entitled “Scientific Education — Notes of an After-dinner Speech”, and I cannot but think would have been couched in more guarded terms by my distinguished friend had his speech been made before dinner instead of after.

The notion that mathematical truth rests on the narrow basis of a limited number of elementary propositions from which all others arc to be derived by a process of logical inference and verbal deduction, has been stated still more strongly and explicitly by the same eminent writer in an article of even date with the preceding in the Fortnightly Review, where we are told that “Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.” I think no statement could have been made more opposite to the undoubted facts of the case, that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand in somewhat the same general relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention.

J. J. Sylvester, The collected math, papers, II, 654