“There is no rigorous definition of rigor”

The quotation of this post’s title is from Morris Kline, one of the great writers about mathematics. Kline, who died in 1992 and taught at New York University from 1938 to 1975, published a number of books — e.g. Mathematics: The Loss of Certainty (from which the quote comes); Mathematics: A Cultural Approach, and Mathematics in Western Culture — that continue to be worth reading.

Recently I returned to Kline’s Mathematics for the Nonmathematician with an idea that it would help me brush up my rude understanding of non-Euclidean geometry. Instead I realized, and with great pleasure, that the book was in some sense only masquerading as a mathematics text, while being in fact an intellectual history of Western thought. Thus, reading, I was reminded again of what we might call the essential tensions: e.g. between appearance and this something else we call reality. Or is the tension between the reality of appearances and our idealizations, which we (or our scientists) often try to pass off as such things as basic physical principles, or as the underlying reality?

It would take a very different kind of piece or post to reprise Kline’s history or to review the several dichotomies that have dominated our Western thinking. I will close the present text with a quotation from Bertrand Russell (as quoted by Kline). To my mind it speaks both to the wonders of Western thinking and to the extent to which it has been psychologically motivated (a search for order) and has, in a sense, sacrificed truth in order to reduce anxiety.

Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.


— William Eaton, Executive Editor


The photograph of Kline is from a Web post on Mathematics: The Loss of Certainty.

The Russell quotation is from his own The Study of Mathematics.

The phrase “essential tension” may well have come to me from the title of another admired book, Thomas Kuhn’s The Essential Tension: Selected Studies in Scientific Tradition and Change. I have not sought, however, to harmonize my use of the phrase with Kuhn’s.

One comment

  1. Daniel D'Arezzo

    Thank you for this post. It reminds me that William Kuhn is essential reading that I haven’t done. And it reminds me, again, that I am like someone who has dwelt all his life in the same house but is entirely ignorant of when it was made, how it was made, of what it is made. I feel that scientific thought is something I desperately need in order to understand where I live. And yet, as you point out, the most subtle thinkers among the scientists question the “reality” that science purports to describe. That’s an interesting proposition in a time when many Americans think that science is a vast left-wing conspiracy to distort reality.

    Today’s luddites would gleefully seize on Lord Russell’s observation and say that scientists, when they’re being candid, admit that science is a hoax, “remote even from the . . . facts of nature.” (There are always convenient elisions.) But for me, science remains the best idea anyone has ever had: Let us test our ideas of the world against what we can observe and measure in the world. There is plenty of room for error in observation and measurement, so we do it again and again and we try to think and rethink what our observations and measurements may mean. “Rigor” is not just difficult to define, it’s even harder to do. Despite the lack of a rigorous definition of rigor, it’s a worthy ideal.


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