Steve Landsburg alerted me to an amazing set of interactions in Mathematics this week. A very distinguished Princeton professor, Ed Nelson, announced what may have been the most profound mathematical result of the century (right up there with Godel’s Theorem last century): that the Peano axioms in mathematics were inconsistent. Nelson announced his finding on a mailing list:

I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent. This is posted as a Work in Progress at http://www.math.princeton.edu/~nelson/books.html

A short outline of the book is at:

http://www.math.princeton.edu/~nelson/papers/outline.pdf

The outline begins with a formalist critique of finitism, making the case that there are tacit infinitary assumptions underlying finitism. Then the outline describes how inconsistency will be proved. It concludes with remarks on how to do modern mathematics within a consistent theory.

Here is a discussion.

But what happened next says much about how the Internet has changed science. At Google+ (in a discussion from another mathematician John Baez), mathematician Terry Tao raised some concerns:

I suppose it is possible that this obstruction could be evaded by a suitably clever trick, but personally I think that the FTL neutrino confirmation will arrive first.

That quote hoped back to John Baez’s blog on Sept 27 where Ed Nelson himself chose to respond provoking Terry Tao to expand on his thoughts. After a couple of rounds from Sept 28 onwards, on October 1, Nelson posted a reply comment:

You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Wow. To be clear, we have a famous mathematician working on a path breaking finding — enough to have written a yet-to-be-published book about it, announcing his claim and then, within days, partly using a platform for communication that was not available when Nelson was writing up his results has led to an interaction that has caused the claim to be withdrawn. To be sure, this could have occurred at a conference in the days of old but this time around occurred based on a wider set of interactions. And this time, all this happened quicker than Wikipedia could be revised and reverted.

There are several things that are worth noting. First, as Michael Nielsen documents in his forthcoming book, Reinventing Discovery: The New Era of Networked Science, science is being transformed by new online tools — particular of peer review but also for collaboration. Mathematics is leading the charge and it is no coincidence that Terry Tao is right at the heart of that.

Second, this says as much about secrecy as it does about communication in science. Prior to his announcement, Nelson appears not to have divulged what he was doing. That, as it turns out, was to his own peril. To his credit, science progressed as it should — challenge, response, agreement — but had Terry Tao been at Princeton and the two had caught each other in the hallway, some months of — at the very least — write-up would have been saved. What led Nelson to work in this way?

Third, I have had my doubts about Google+ and, indeed, this interaction could have taken place in a number of places. But Google+ is great for conversations and that is how this started. Had it been slightly harder to use, perhaps Tao wouldn’t have thrown in his concerns into another conversation. There is something to be said for that.

On that score, and slightly off topic, Terry Tao occasionally veers into economics and last year, on Google Buzz of all places, he and I had an interaction on some economics of environmental policy. You can read about that here (from my perspective at least).