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Infinity: Beyond the Beyond the Beyond

Lillian R. Lieber
Paul Dry Books
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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I must have been around twelve years old. Newly arrived in Junior High, I rejoiced in the privilege of using the big kids' library, and spent a lot of time rummaging in the stacks looking for interesting things. At some point, I found a treasure-trove of mathematics books aimed at the mythical "general reader." Many of the early volumes in what is now the Anneli Lax New Mathematical Library were there; I have a blurry memory of reading and enjoying The Lore of Large Numbers and of despairing quickly of the Contest Problem Book. On that shelf as well were several of Lillian R. Lieber's books, and the one that caught my eye was Infinity.

I had probably already made my first acquaintance with transfinite numbers through Kasner and Newman's Mathematics and the Imagination, so Lieber's book was an opportunity to learn a little more. And learn I did, though of course there was much I didn't quite understand. I loved Lieber's idiosyncratic style, laid out on the page in short lines with lots of words in ALL CAPS. I even enjoyed the very strange illustrations.

That was many years ago, and I had never seen Infinity since. When a review copy of this new edition arrived, I knew I had to read it again.

Infinity was first published in 1953; the new edition has been edited by Barry Mazur, who explains in his preface that he has deleted some chapters of the original. Most of these comprised a lengthy section on calculus, which Mazur argues has been treated more effectively in other "popular math" books.

Mazur has also removed the preface and first chapter, where SAM is introduced (see below) and Lieber's preachiness must have been at high intensity. He comments that all sorts of typical concerns of the early 1950s show up in the text: the perils of nuclear weapons, the promise of nuclear energy, the role and effectiveness of the United Nations are all discussed at several points. I was, in fact, somewhat surprised at Lieber's sermonizing; clearly when I was young it had washed right past me.

The main focus of the book is on how mathematicians have studied the notion of infinity. Lieber starts with a discussion of "potential infinity," including the "points at infinity" of projective geometry, then launches into her real theme: Cantor's theory of infinite sets. The story proceeds on familiar paths for a while, but then goes on to discuss topics that aren't usually included in books like this: Cantor's transfinite ordinals, for example, and Russell's paradox and its resolution.

Lieber is eager to argue that these ideas are useful, and that leads to rather far-fetched claims. I suspect that this is why the chapters on calculus were there. Lieber is certainly right that one cannot do modern real analysis without some sort of underlying set-theoretical foundation. Claiming this as an "application" of the theory of transfinite ordinals and cardinals is, however, something of an overstatement.

I was surprised at some things. Lieber reproduces Cantor's incorrect "interleaving" proof that the unit square and the unit interval have the same cardinality without any comment whatsoever. Surely she realized it didn't quite work? Her discussion of the ordinals never talks about order-preserving bijections; instead, she uses Cantor's language about "abstracting only the nature of the elements" versus "abstracting both the nature and the order of the elements." This may be part of the reason I found that section of the book impossible to understand oh-so-many years ago.

Lieber also says that the question of whether all sets can be well-ordered is "still under discussion," as is the Continuum Hypothesis. In 1953? By that time one knew that both the Axiom of Choice and the Continuum Hypothesis were consistent with standard set theory, and surely it was also quite clear that AC implied the existence of well-orderings. But AC and CH seem to have a very different status, at least today. Were there still mathematicians prepared to refuse the Axiom of Choice in the 1950s?

For today's readers, it is probably the preachiness that will stand in the way of enjoyment. For example, here she is after a discussion of non-Euclidean geometry:

…all this is another example of
as displayed in mathematics,
and, as you will soon see,
it is not just
empty useless bragging about courage,
but the genuine article with
tremendously useful consequences!

That is actually fairly tame by comparison with other passages. There's lots of talk about the United Nations, about atomic bombs, about the desirability of applying the strictures of logic to political discourse. Much of this is summarized in references to SAM, a kind of "spirit guide" (as Mazur describes him). SAM stands for Science, Art, and Mathematics; Lieber personifies in him what she thinks are the three components of human thought: observation and experiment (S), intuition and expression (A), and rational thought (M). So we get lots of passages like

In short,
can we not learn,
from mathematics,
how to use our SAM
effectively and constructively
in dealing with
ourselves and our environment?

I doubt it, Prof. Lieber, I really do.

This kind of thing does date the book. I still feel a nostalgic affection for it, and one can still learn quite a bit of mathematics from it. I hope it can still seduce twelve-year-olds into loving mathematics.

Fernando Q. Gouvêa thanks the Escola Graduada de São Paulo for having such a good library.

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