On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities

Abstract

The full text appears in tBull. Inst. Math. Applic. (1968). Modern mathematics could be interpreted in the sense of modern art (an experimental abandonment of traditional skills) or in the sense of modern languages (a vehicle for communication in everyday affairs). At present the interpretation is too much in the former sense and not enough in the latter. There is of course no reason why pure mathematicians should not experiment with contemporary mathematical art forms, provided that what they do within university circles is not automatically taken as a pattern for the mathematical needs of the community or as the basis for school syllabuses. For prospective scientists, engineers, etc. it is very important to teach the skill of solving problems, which involves not only the construction of mathematical models but also manipulative ability in deducing conclusions from the latter. There is some evidence that a preoccupation with abstract mathematics inhibits manipulative ingenuity. The following problem defeated mathematicians and was first solved by physicists: you may care to assess your own problem-solving potential as a mathematician by attempting it. Let f a, b be the number of ways of filling a rectangle (whose sides are 2a, b) with ab small rectangles (whose sides are 2, 1). Prove that $$f_{a,b}^{1/2ab}$$ tends to a limit as a and b»∞. Show that the value of this limit is $$e^{G/\pi }$$ (where G=1-2−3-2+5-2−... is Catalan's constant). What can you say about the corresponding three-dimensional problem (a box of sides 2a, b, c filled with abc small boxes of sides 2, 1, 1)? Another problem at a much simpler level (suitable for school children) is this: is the pattern observed in successive columns of the following table a particular property of the entries in the first column or not? $$\begin{gathered} 16 77 8 42 0 16 0 0 ..... \hfill \\ 93 69 50 42 16 16 0 0 ..... \hfill \\ 24 19 8 58 0 16 0 0 ..... \hfill \\ 5 11 66 58 16 16 0 0 ..... \hfill \\ \end{gathered}$$