Gregory Chaitin has done seminal work on the foundations of mathematics, especially the meaning of randomness and undecidability. In Meta Maths, he offers his ideas in a new popular version, which has a special interest because it comes directly from their originator. Long associated with the IBM Watson Research Center, Chaitin comes across as a kind of mathematical Richard Feynman, intuitive and high-spirited, irreverent and plain-spoken. Through this jovial persona, he presents many serious ideas in an engagingly dishevelled way, mixing autobiography and reminiscence with exposition and example. He presents his insights so agreeably that it all seems a grand lark.
Chaitin was something of a child prodigy, who glimpsed one of his essential insights at the age of fifteen. Trying to understand popular accounts of Gödel's Proof about the incompleteness of the axioms of ordinary arithmetic, the young Chaitin found himself puzzled by the abstract formal language of such systems, preferring to think in terms of Alan Turing's idealized calculating machine. Whatever the failures of mathematical formalism, Chaitin argues that it gave the essential grounding for modern computers; what Turing anticipated, our laptops deliver. By considering whether any given, self-contained program will eventually stop (by reaching some self-identified point of completion) or go on forever (perhaps in an endless loop), Turing proved that this ``halting problem'' could never be decided in advance. To find out whether a given program halts or not, we have actually to run it, which is Turing's version of Gödel's result: there are more things in heaven and earth than are dreamt of in any given set of axioms. Here the young Chaitin already divined his own seminal question: If we devise a program to generate a certain output, can that program be made smaller in size than its output? If so, that smaller program makes that output more intelligible, ``reducing'' it by showing how it flows from a simpler set of instructions. If not, the output data is essentially random because it is irreducible, not capable of being compressed into anything less complex than itself. For instance, the laws of physics presumably show the order of nature by condensing many observations into a smaller number of pregnant principles. A truly random universe could not be represented by any set of ``laws'' less complex than nature itself. It remains undecided whether nature, in the end, will prove to be infinitely complex or not.
Chaitin then turns his question on Turing's halting problem, arguing that the impossibility of deciding in advance whether a program halts means that the probability of halting is essentially random. Chaitin uses this to construct an ingenious number he calls Ω, a sum over the results of such attempts to record the halting of ever more complex programs. The star of his book, this ``halting probability'' Ω, is a radically random quantity that Chaitin discerns at the heart of pure mathematics, not resultant from physical causes (such as random quantum processes, for instance, or statistical fluctuations).
Such insights about the nature and definition of randomness are the crux of Meta Maths; they are deeply thought-provoking. Chaitin presents them not only in their broad sweep but also with significant detail and elaboration, in order to give his reader a thoughtful understanding of their underpinnings. He also includes an interesting and helpful variety of historical examples en route, especially showing his enthusiasm for G. W. Leibniz. This does Chaitin great credit in an era when many scientists scarcely look beyond the past few years in their field, nor care much about the roots and sources of their ideas. Chaitin's interest is not mere historical respectfulness; he is really struck by the ways and contexts in which Leibniz, Émile Borel and others already entertained ``his'' ideas, showing something of the ``irreducible'' character of human thought itself. Thus, his book is well worth reading for anyone who has wondered how to understand Gödel's work in a larger context. Though informal and engaging, it both requires and repays the reader's concentration; his idiomatic informality at certain points does need some puzzling out, compared to a less imaginative, idiosyncratic exposition. Trying to offer alternative versions, he also includes two other separate talks he has given that present his ideas in slightly different ways. In order to bring out the sequential coherence of his ideas, a single, more continuous line of exposition would have been an easier choice, I think. I do also wish he had included more references and notes giving fuller help for readers desiring to pursue these matters further; Chaitin's friendly excitement encourages us to think further and take up these questions for ourselves.
Mark Ronan is also a working mathematician whose first-hand knowledge adds a great deal to his account of the heroic ``Classification program'', which explored more and more complex groups of symmetries in the last decades of the twentieth century. A professor at the University of Illinois who also taught at Berlin, London and Birmingham, Ronan describes himself as having worked only on the ``fringes'' of this classification program, but his inside knowledge of the protagonists and, more importantly, of the full scope and details of their work is crucial. It may also help that Professor Ronan reads Babylonian cuneiform, has acted in a dozen operas at the Lyric Opera of Chicago, and danced in The Nutcracker; no narrow specialist, he.
In Symmetry and the Monster, Ronan looks back to Greek geometry and its beautiful polyhedra but really begins with Niels Henrik Abel and Évariste Galois applying the general concept of symmetry to the solvability of algebraic equations. This, as Ronan neatly explains, disclosed certain ``atoms'' of symmetry, basic elements and structures Galois called ``groups''; finding the ``simple groups'' corresponds to the search for a ``periodic table'' of such elemental symmetries. The heart of Ronan's story is the unfolding saga of these ever more complex ``simple'' symmetries, at first ramifying along lines suggested by symmetries realizable in three-dimensional space, then going far beyond more familiar low-dimensional terrain. Ronan unfolds this story with admirable verve and clarity; his clever examples often evince his interest in dance. This dizzying cavalcade of ever-larger groups culminates in the so-called ``Monster'', an amazing symmetry in 196,884 dimensions.
Ronan's exposition includes entertaining glimpses of the personalities involved in this extraordinary quest but best of all gives an admirable amount of detail concerning the actual substance of their work, especially through the bizarre and wonderful examples that emerged. He also includes avenues still under exploration, such as the audacious ``moonshine'' conjectures relating the Monster to other, seemingly quite unrelated problems in mathematics and even physics (string theory, of course, that impudent changeling). Here Ronan's book seems to confirm Chaitin's overarching argument that mathematics increasingly looks like a kind of imaginary physics, exploring an ever-expanding and seemingly inexhaustible array of mathematical ``phenomena'' that call for new fundamental principles and questions. This surprising sense of emergence may also explain the recent florescence of general interest in mathematics, viewed not merely as the dry elaboration of consequences from given assumptions but as the discovery of whole new realms of thought that call for commensurate --- and exciting --- leaps of human imagination.