The title is provocative. In so many ways, it is at the same time interesting and no big deal. The sky is not falling. And neither are bridges.
To summarize, the short polemical but still technical paper says:
- something about mathematics has a component that is close to spiritual
- actual infinity is misguided
- the consistency of ZFC is overwhelmingly believed (since there is no proof (yet?)) but there are some doubters.
- there is a straightforward and simple exposition as to why one might doubt consistency
The last however...
Technically this short essay is an extended less technical explanation of one chapter "Is exponentiation total?" of his book "Predicative Arithmetic" Princeton U. Press, 1986. Some background is necessary here. First, philosophically, logic is in a bizarre sense he playground of the wildly perverse skeptic. When doing logic, important questions are things like "How do you know 'A and B'? ", "If you didn't have the rule for eliminating double negation, what sort of things can you prove?". And technically, this can lead to an ordering of more and more powerful (but also therefore more questionable) logics. One philosophical problem is that of 'predicativity'; a set is impredicative if it has to be defined by, in a technical sense, already being defined. Another is totality of the exponential function; certain theorems are provable only if you -assume- that the exponential function is total (defined for every natural number).
Now impredicativity might seem a little questionable (we're all use to recursive definitions, but that 'necessarily self defining' is kinda strange). But how can one -possibly- doubt that exponentiation is total? It just -is- total of course...right? Well, that's the point of the chapter and this longer article, that for purely technical reasons, successor ('plus one') is total by an axiom (you really have to start somewhere), but the recursive definition of addition is total (using Peano arithmetic), also multiplication is total (again by PA) and ...well...no...you can't similarly prove in this way that exponentiation is total (the technical block being that exponentiation is not so conveniently associative like addition and multiplication).
Frankly I'm not sure if Nelson shows that it is -impossible- to prove that it is total, but I'll take his word for it (not a particularly good mathematical attitude (see...not particularly skeptical..therefore my statement that logic or even mathematics is the most skeptical science of all. What I haven't made this statement before? OK...but that's for another place and time )).
From this kind of provability statement (of both 'it is possible to prove...' and 'it is impossible to prove...' (and many other technical arguments) one gets an ordering of logical systems that are equivalent to mathematical statements (this is the product of work on reverse mathematics). First there's PA (Peano Arithmetic), EFA, PRA, RCA_0, WKL_0, ACA_0, ATR_0, Pi_1^1-CA_0, Z_2. This ordering of logics (and equivlanet mathematical systems, grows in strength (can prove more and more).
Anyway, that's it though. That is the entire warning sign that Nelson is giving: that the totality of exponentiation must be assumed as an axiom because it can't be proven from PA, this fact is a sign of the impending doom, of the total collapse of contemporary mathematics. Why? because it calls into question the consistency of mathematics.
How could mathematics not be consistent, you wonder. Well, that's mathematical skepticism for you. How do you know that a+b = b+a? It's obvious right. The whole point of proof (the modern mathematical/Greek defined idea) is that you can look and intelligently guess but in some instances be wrong. And since being wrong is Bad, you always want to be right, and you want to make sure that you are never ever wrong. And the way to do that (circularly of course!) is proof. Someone makes a statement...how do you know they're telling the truth? You doubt them? You are skeptical? They must -prove- it, and that proof goes for math. And the power of math (= symbolic reasoning) is that at some level it is terribly obvious (of course addition is commutative!) but sometimes not (the sum of reciprocals of squares = pi^2/6), so if you can doubt the harder one, you really should doubt the easy one too because you may have missed some extremely small detail. And note that most of these math 'things' are learned, i.e. remembered, even if derived at one point. That's what skepticism does for you.
But back to consistency. Is math...sorry let's pick a specific 'system'...is arithmetic consistent? That is, given the rules of construction numbers, the operation of addition,.,multiplication, an induction scheme, and some minimal logic, how do we know if, using that logic, we aren't able to prove a statement A and also it's negation -A (that's what consistency is, the inability to claim both A and its negation at the same time). And why is that important, other than being terribly confusing (so which is true, A or -A?) ? With a minimal logic (I mean that not in a technical sense but informally) and the two separate statements A, and -A, it is normal expected practice that one can then deduce -anything- (that's just an obtuse way of saying ex falsum quodlibet, from falsity one can infer anything (and 'A and -A' is pretty much no matter what considered always to be false (again a weird way to say something pretty obvious)).
So there's arithmetic (Peano Arithmetic is the mathematical system I described), and since a little bit of logic goes with it, some things can be proven (e.g. a+b=b+a). But consistency (by Goedel's theorem) cannot be proven within that proof system. So assuming only PA (Peano Arithmetic) we don't know if it is consistent or not. We have lots of human data to corroborate that it is (no one has yet found a mistake in arithmetic) but no one has yet proven, no one has yet -proven- that arithmetic is consistent. (everything has to be qualified, not only is skepticism -the- skeptical science, it is also the defining science...you better say exactly what you mean and mean exactly what you say...so there are proofs of -relative- consistency (one system is as consistent as another).
OK, but what is the point to this concept, consistency? It's all about skepticism and strength of provability. Since (by Nelson) the question 'Is exponential function total?' is not provable in PA, if you ever need to use the exponential on arbitrary naturals, you have to -assume- that it is total. Pretty obvious right, but still by following the rules, you need to assume it. And really, anything you assume is questionable, not just as to proof (that is established already that we cannot prove it (can't prove the negation either)) but questionable as to proof (they're different proof and truth). and something that is so obviously true (Nelson doesn't say it this way but he's leading us this way) as exponential being true, but it is not provable in a restricted system, well, that might mean for the much stronger, richer system of all math (which includes this PA as a small but integral part) where there are going to be -lots- of extra assumptions, much less 'obvious' than 'exponential is total', then with more and more axioms, there''s going to be more and more doubt if they are consistent. The higher up in that list the more questionable things become (in one very formal sense, but not in another where all these facts and constructions in the stronger mathematical systems are human proven (i.e. Brouwer's Fixed Point Theorem is equi-...pollent(?) to WKL_0, but that's a -theorem, right? ).
So I think what Nelson is getting at is that if you can doubt 'exponential is total', then there's a possibility that arithmetic might have some weird bizarro situation which is inconsistent. And if one can derive an inconsistency both A and -A, then you can derive anything, and then any statement you make could be provable.
The point is, that that hasn't happened yet. And would be pretty messed up if it had. And really, in the finite world of PA that we've explored no inconsistencies have been found, then, even if we go beyond a horizon and find some inconsistencies, it hasn't affected our current playpen of mathematical toys. Outside of that warm dependable cocoon maybe things fall apart, but inside, everything is all right. Nelson may be saying things are scary outside, and bad things may happen there (we don't know and every time we shine a flashlight there, nothing bad is found), so far, no contradictions.
Which is to say, whatever collapse Nelson is thinking of, it won't have an affect on anybody except i.e. real mathematicians and those who use mathematics won't be affected in the least. It will certainly have an effect (if it happens) on science writers trying to make a cool story for all the non-technical people, and for the vast majority of technical people who will need years of non-useful irrelevant study to grasp the subtlety presented here.
So, if you're concerned that mathematics might imlode because of some internal inconsistency, the only evidence so far for it, from a person (Nelson) who is actively seeking such inconsistency, is because exponentiation is not associative, well, I don't think that's very concerning.
Oh yeah, and even if something else shows up that truly shows that arithmetic is inconsistent, it won't afffect any mathematician's day to day work. A number of popularizations will come out of it. Some NYT headlines. But that's the extent of it.
