The (in)consistency of PA and consensus in mathematics

(Cross-posted at NewAPPS)

Last week, the foundations of mathematics community was shaken by yet another claim of the inconsistency of Peano Arithmetic (PA). This time, it was put forward by Edward Nelson, professor of mathematics in Princeton, who claimed to have found a proof of the inconsistency of PA. A few months ago, quite some stir had been caused when Fields medalist V. Voevodsky seemed to be saying that the consistency of PA was an open question; but Nelson’s claim was much more radical; he claimed to have proved that PA was outright inconsistent! (Here is a great post by Jeff Ketland with a crash-course on PA and a discussion of ways in which it might be inconsistent.)

Nelson announced his results on the FOM mailing list on September 26^th 2011, providing links to two formulations of the proof: one in book form and one in short-summary form. Very quickly, a few math-oriented blogs had posts up on the topic; we all wanted to understand the outlines of Nelson’s purported proof, and most of us bet all our money on the possibility that there must be a mistake in the proof. External evidence strongly suggests that PA is consistent, in particular in that so many robust mathematical results would have to be revised if PA were inconsistent (not to mention several proofs of the consistency of arithmetic in alternative systems, such as Gentzen’s proofs -- see here).

Indeed, it did not take long for someone to find an apparent loophole in Nelson’s purported proof, and not just someone: math prodigy and Fields medalist Terence Tao (UCLA), who is considered by many as the most brilliant mathematician currently in activity. The venue in which Tao first formulated his reservations was somewhat original: on the G+ thread opened by John Baez on the topic. (So those who dismiss social networks as a pure waste of time have at least one occurrence of actual top-notch science being done in a social network to worry about!) At the same time, Daniel Tausk, a professor of mathematics at the University of São Paulo, had identified the same mistake in Nelson’s argument, and alerted Nelson of the problem in private communication. Judging from his replies (which can be read in the FOM archive as well as in The n-Category Café post), Nelson did not immediately understand the objection, but within a few days consensus had emerged that the mistake identified by Tao and Tausk was irreparable. Then, on October 1^st, Nelson graciously acknowledged the correctness of the objections and withdrew the claim of having a proof of the inconsistency of PA. As has been remarked by many commentators, it is certainly a sign of mathematical greatness to acknowledge one’s own mistakes so promptly!

While it must have taken true mathematical insight to see what was wrong with the purported proof, once the mistake was spotted, it was actually not overly difficult to understand – if not in all its technical details, which presuppose a good understanding of Chaitin’s theorem, at least in its basic idea: it was based on the unwarranted assumption that the hierarchy of sub-theories built for the proof is going to map neatly into the hierarchy of the Kolmogorov complexity of each of these sub-theories. In the game of mathematics, tacitly relying on assumptions is not a good move, even less so when it turns out that the assumption is false.

From the point of view of the social practice of mathematics, this episode raises a number of interesting questions, which can be framed from the point of view of Jody Azzouni’s insightful analysis of how and why mathematics is unique as a social practice. Azzouni argues that mathematics is one of the most homogeneous groups of human practices around, and in particular one which seems particularly favorable for the emergence of consensus. Indeed, mathematicians virtually always agree on whether a purported proof is or is not a valid mathematical proof; it may take a while for enough members of the community to understand its details (as with Deolalikar’s purported proof of $P \not= NP$ last year, but this was mostly because the latter was rather poorly formulated), but past this initial period, consensus virtually always emerges. (To be clear, there can be disagreement on whether a proof is elegant, interesting etc., but not as to its correctness and incorrectness.) This remains a very puzzling feature of mathematical practice, which at least to some people suggests that Platonism is the only plausible philosophical account of mathematical truth.

The swiftness with which consensus emerged as to what was wrong with Nelson’s purported proof is yet another example of the strong tendency towards consensus in mathematical practice – and arguably, it cannot be fully explained as a merely socially imposed kind of consensus, due to homogeneous ‘indoctrination’ by means of mathematical education. That it only took a few days for consensus to emerge naturally also depends crucially on the technologies of information dissemination currently at our disposal – mailing lists, home-pages, blogs, social networks; but more generally, the episode has been a dramatic but classical occurrence of a well known pattern in mathematics as a social practice.

The comparison with another dramatic claim, put forward for the first time more than 80 years ago, may prove to be informative. Indeed, the reception of Gödel’s incompleteness results (which has been extensively studied by Dawson and Mancosu, among others) also illustrates the emergence of a consensus, but this time it favored the highly surprising claim being put forward; in last week’s episode, by contrast, the highly surprising claim has been refuted.

In September 1930, Gödel made the first public announcement of his incompleteness result. In first instance, he had proved only the first incompleteness theorem, and as is well known, from the first theorem the second incompleteness theorem was proved independently by Gödel himself and by John von Neumann. But von Neumann was one of the few who immediately understood the result (Bernays and Carnap seem to have struggled); in fact, he may have been the first to see the far-reaching consequences it had. In particular, while Gödel was still rather cautions regarding the impact of his results for Hilbert’s program, von Neumann immediately drew the right conclusions. However, quite a few mathematicians were so taken aback by Gödel’s results that they were convinced that there must have been a mistake somewhere in the proof; just as in last week’s episode, many of them set out to find out the ‘loophole’ in Gödel’s proof, but on that occasion to no avail. As narrated in this wonderful paper by P. Mancosu, the news of Gödel’s results quickly spread, but in the absence of the WWW or similar platforms, many people did not have immediate access to off-prints of the paper where the results were presented. For example, before seeing the actual results in detail, mathematician Leon Chwistek suspected that Gödel’s proof would be based on misunderstandings, but once he read the actual paper, he stood corrected:

In my last letter I have raised some doubts concerning Dr Gödel’ s work that have completely disappeared after a more attentive study of the problem. I had at first thought that there was a tacit introduction of non-predicative functions which makes the use of a rigorous symbolical procedure impossible and I even feared the possibility of an antinomy. Now I see that this is out of the question. The method is truly wonderful and it fits pure type theory thoroughly. (Letter to Kaufmann, in Mancosu 1999, 44)

Both the consensus that eventually emerged concerning the correctness of Gödel’s surprising results, and the consensus swiftly established as on what was wrong with Nelson’s purported proof last week are examples of the amazing and quite unique tendency to converge towards consensus in mathematical practice. In many senses, the most pressing task for the philosophy of mathematics, practice-based or otherwise, is to formulate a satisfactory account of this almost eerie fact about mathematics.