I am strongly reminded of the final fate of the Newtonian n-body problem.
For n=3 Sundman (1913!) solved the problem in terms of a convergent power series (This was generalized to n>3 by Q. Wang around 1990).

The problem with this solution is that the power series converges so slowly that it is useless for all practical purposes.

Uwe Brauer

In fact, all of us IT “trainspotters” can see the tension between the power of digital computing and the practical plans of organizations, including when there are “bugs in the system” that might just be there to bug you (a thought I previously mentioned to a working programmer, of whom I know many and to which decently-remunerative position I aspired after my dreams collapsed and before my head expanded, was that the great social-theoretic *moralist* of the postwar era, Michel Foucault, probably knew quite a few early and *late* programmers). And the truth — as I was telling QC — is that the Australasian logical viewpoints I adverted to before, where you work with a logic that can work around contradictions or work with the possibility of plural logical consequence relations, makes sense from an EE standpoint: though it be impolitic for the Beavertonian to say it who really knows about the floating-point Pentium bug way back when, and it’s probably politic to say solid-state physics is not the exactest of sciences.

So on the practical level, which is in my view fundamentally a computational level, workarounds are real and we might never really know whether there was some catastrophic/chaotic/”quantum” phenomenon that caused more SAT solutions some one way. On the other hand, though, if there is gonna be a P/NP solution rather than arsing around with woo-woo conjectures and overshooting the moon with measure theory it’s got to be on the *theoretical* level, that is the logical level: and from the perspective of recursion theory BAs are essential and for them to behave properly the question must be closed for unrelativized TMs, negatively. On the first-order level, the fuzzword “oracle” should be precisified in terms of quantifiers if possible (I guess the lesson of Church-Turing is about how even the very best and most essential “programming” considerations can never motivate such a choice, though): I think it is, and what people want to say by giving P=NP oracles is covered by first-order completeness, that is to say a recursive procedure (proofs using a Hilbert calculus and a few rules of inference, e.g.), strengthened by recourse to a deterministic querying of the power set (models of quantification), can solve SAT lickety-split. It just can’t solve its own problems, i.e. the set of first-order truths about propositions is only r.e. and not recursive.

Beyond that, my little joke intended to evoke the enigmatic teenage saying “See you on the flip side” contains a meaningful conjecture, though: *jeden und keine* problems are NEXPTIME-hard, since if NP=DEXPTIME=problems solvable either by rote exploration of the power-set or “ingenious” guesses therein, NEXPTIME is equivalent the computational power necessary for second-order logic, which as we all know contains all of mathematics but also either contradiction or incompleteness. Maybe people should buy the antipodean story on that level, but I’m just a West-Coast guy and so I’m interested in these considerations: the complete second-order logic, Henkin semantics, isn’t really “second-order” at all; it’s a way of linking “real abstractions” to the concrete first-order relations that make them real. Furthermore, once you get to the first-order level, it’s my contention (though this may just be me being stupid) is that all first-order truths are parts of decidable subtheories of first-order logic *a la* Presburger arithmetic, although we can’t have an algorithm for determining whether any given subtheory is decidable on pain of revoking Goedel.

So don’t give up on concrete computational problems and the ‘logical’ solutions to them, just because Everything Could Be Illuminated — certainly not on the basis of the Wittgenstein quote I now “remember” was in *Philosophical Grammar*: I was once given the ‘exactly wrong’ interpretation of it by one of my CS superiors, but Monty Python told you something useful for once about that man.

I think a lot of educated people can feel Vico’s conjecture pretty deeply, although I’m not necessarily into deep feelings beyond a point, but my contention (which dates back to earlier in the decade, when I somehow assumed the P/NP problem had been solved) is that this is Satisfiability in another form: if such a practice were actual, and we can imagine this with Wittgenstein [In *Philosophical Remarks* he asked "What is the meaning of this grove of trees?" as seriously as only a gardener's assistant in a monastery could] without really ever knowing whether it was real, we would have a “logical picture” we would have to determine a sense for by assessing its compositional articulation.

This would essentially amount to testing for satisfiability, eh? We’d try out various linguistic interpretations until we hit on one that was The Right One, although I’d like to think it could still be wrong. In fact, of course — the formal considerations adduced above plus the old adage that “No man can know the hour of his own death” — which is true enough for the Standard American English semantic value of “No” — mean that physical reality can’t “tell us stuff” over and beyond the open normativity of language through mouth or hand. It just exists, and even the neural nets that would supposedly underwrite the legitimacy of such a practice are just big FSMs (see, although computer science is not such a big deal as mathematics, since it’s logic and applied logic at that, for “limited purposes” it’s useful enough).

On the other hand, there is a “truth in painting” and a truth in beauty: that is to say, both aesthetic truth and the properties of things which are “so money” have that kind of ‘irreal’ character and sometimes it’s worth your while to try and figure out in what direction physical objects and commerce involving them are tending, so to speak. [However, I will say that *pace Erdos* mathematics doesn't have to be beautiful, 'cause it often isn't to my eye and those of many others -- probably just a better idea to cultivate that impression.] However, when you choose to operate on that level you’ve vastly exceeded the resources of first-order logic, tied down by Loewenheim-Skolem and Compactness but IMO the most expressive complete logic: and though I have a personal joke about EXPTIME that’s dear to my heart, I think sometimes you just want to say “NEXPTIME, yo.”