Time for a re-post. This first appeared in these pages on 18 August 2010.

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A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's *Elements*, David Hilbert's *Foundations of Geometry*, Richard Dedekind's *Continuity and Irrational Numbers*, Cantor's *Contributions to the Founding of the Theory of Transfinite Numbers,* etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of constructing the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to *supplant* a textbook-driven approach, but that the latter ought to be *supplemented* by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it.

So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

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