Unclad emperors: A case of mistaken identity

Conclusion

I am ready to rest my case. In summary, I will simply provide the two paragraphs of the conclusion of my “Trivial Mathematics” paper that Koblitz did not quote.

In the social science literature, much reasoning takes place about magnitudes without any help from mathematical formalisms, but solely in terms of ordinary language. As the illustrations [in “Trivial Mathematics”] suggest, a good deal of this reasoning makes implicit use of the properties of ordinal variables and monotonic transformations. In both the social science and natural science literatures, much reasoning is also done from diagrams, without concern for the exact forms of the functions depicted or the cardinal values of variables. This reasoning appears also to make use of only the ordinal properties of the magnitudes depicted in the diagrams. One can argue that a great deal of everyday “commonsense” reasoning, when it is concerned with magnitudes, is, implicitly, reasoning about ordinal relations.

To the extent that this is true, a strong case can be made for introducing students in elementary courses in mathematics to ordinal magnitudes and the kinds of inferences that can be drawn with their help. This could be done in courses on discrete mathematics and on the calculus. In these pages, I have provided some illustrative materials that I hope will help make that case.

I will be glad to send copies of the full text of the “Trivial Mathematics” manuscript to those who wish to read it. I would be delighted if it were to stimulate (or even to irritate) some mathematicians to undertake the constructive task of creating effective pedagogical materials for teaching the simple theory of ordinal relations. That would make a far more useful contribution to the social sciences than lecturing them about their supposed pretensions, or anguishing about an imaginary conflict between them and the humanities.