This is not at all what I had in mind.— Albert Einstein, in response to David Bohm's hidden variable theory.
A no-go theorem is a formal proof that a certain kind of theory cannot work. (The term no-go theorem seems to be used mainly in physics; I find it useful in a more general context.)
A valid no-go theorem identifies a hopeless avenue of research; but in some cases, it also identifies a potentially valuable avenue for research. This is because in some cases, the no-go theorem is commonly understood more broadly than its actual technical result. Hence the no-go theorem is actually showing that some specific tactic doesn't work, but is interpreted to mean that some broad strategy doesn't work. So when you see a no-go theorem that's being given a very broad interpretation, you may do well to ask whether there is, after all, a way to get around the theorem, by achieving what the theorem is informally understood to preclude without doing what the theorem formally precludes.
In this post, I'm going to look at four no-go theorems with broad informal interpretations. Two of them are in physics; I touch on them briefly here, as examples of the pattern (having explored them in more detail in an earlier post). A third is in programming-language semantics, where I found myself with a result that bypassed a no-go theorem of Mitchell Wand. And the fourth is a no-go theorem in logic that I don't actually know quite how to bypass... or even whether it can be bypassed... yet... but I've got some ideas where to look, and it's good fun to have a go at it: Gödel's Theorem.John von Neuman's no-go theorem
In 1932, John von Neumann proved that no hidden variable theory can make all exactly the same predictions as quantum mechanics (QM): all hidden variable theories are experimentally distinguishable from QM. In 1952, David Bohm published a hidden variable theory experimentally indistinguishable from QM.
How did Bohm bypass von Neumann's no-go theorem? Simple, really. (If bypassing a no-go theorem is possible at all, it's likely to be very simple once you see how). The no-go theorem assumed that the hidden variable theory would be local; that is, that under the theory, the effect of an event in spacetime cannot propagate across spacetime faster than the speed of light. This was, indeed, a property Einstein wanted out of a hidden variable theory: no "spooky action at a distance". But Bohm's hidden variable theory involved a quantum potential field that obeys Schrödinger's Equation — trivially adopting the mathematical infrastructure of quantum mechanics, spooky-action-at-a-distance and all, yet doing it in a way that gave each particle its own unobservable precise position and momentum. Einstein remarked, "This is not at all what I had in mind."John Stewart Bell's no-go theorem
In 1964, John Stewart Bell published a proof that all local hidden variable theories are experimentally distinguishable from QM. For, of course, suitable definition of "local hidden variable theory". Bell's result can be bypassed by formulating a hidden variable theory in which signals can propagate backwards in time — an approach advocated by the so-called transactional interpretation of QM, and which, as noted in my earlier post on metaclassical physics, admits the possibility of a theory that is still "local" with respect to a fifth dimension of meta-time.Mitchell Wand's no-go theorem
In 1998, Mitchell Wand published a paper The Theory of Fexprs is Trivial.
The obvious interpretation of the title of the paper is that if you include fexprs in your programming language, the theory of the language will be trivial. When the paper first came out, I had recently hit on my key insight about how to handle fexprs, around which the Kernel programming language would grow, so naturally I scrutinized Wand's paper very closely to be sure it didn't represent a fundamental threat to what I was doing. It didn't. I might put Wand's central result this way: If a programming language has reflection that makes all computational states observable in the syntactic theory of the language, and if computational states are in one-to-one correspondence with syntactic forms, then the syntactic theory of the language is trivial. This isn't a problem for Kernel because neither of these conditions holds: not all computational states are observable, and computational states are not in one-to-one correspondence with syntactic forms. I could make a case, in fact, that in S-expression Lisp, input syntax represents only data: computational states cannot be represented using input syntax at all, which means both that the syntactic theory of the language is already trivial on conceptual grounds, and also that the theory of fexprs is not syntactic.
At the time I started writing my dissertation, the best explanation I'd devised for why my theory was nontrivial despite Wand was that Wand did not distinguish between Lisp evaluation and calculus term rewriting, whereas for me Lisp evaluation was only one of several kinds of term rewriting. Quotation, or fexprs, can prevent an operand from being evaluated; but trivialization results from a context preventing its subterm from being rewritten. It's quite possible to prevent operand evaluation without trivializing the theory, provided evaluation is a specific kind of rewriting (requiring, in technical parlance, a redex that includes some context surrounding the evaluated term).
Despite myself, though, I was heavily influenced by Wand's result and by the long tradition in which it followed. Fexprs had been rejected circa 1980 as a Lisp paradigm, in favor of macros. A rejected paradigm is usually ridiculed in order to rally followers more strongly behind the new paradigm (see here). My pursuit of $vau as a dissertation topic involved a years-long process of gradually ratcheting up expectations. At first, I didn't think it would be worth formulating a vau-calculus at all, because of course it wouldn't be well-enough behaved to justify the formulation. Then I thought, well, an operational semantics for an elementary subset of Kernel would be worth writing. Then I studied Plotkin's and Felleisen's work, and realized I could provide a semantics for Kernel that would meet Plotkin's well-behavedness criteria, rather than the slightly weakened criteria Felleisen had used for his side-effectful calculus. And then came the shock. When I realized that the vau-calculus I'd come up with, besides being essentially as well-behaved as Plotkin's call-by-value lambda-calculus, was actually (up to isomorphism) a conservative extension of call-by-value lambda-calculus. In other words, my theory of fexprs consisted of the entire theory of call-by-value lambda-calculus plus additional theorems.
I was boggled. And I was naively excited. I figured, I'd better get this result published, quick, before somebody else notices it and publishes it first — because it's so incredibly obvious, it can't be long before someone else does find it. Did I say "naively"? That's an understatement. There's some advice for prospective graduate students floating around, which for some reason I associate with Richard Feynman (though I could easily be wrong about that [note: a reader points out this]), to the effect that you shouldn't be afraid people will steal your ideas when you share them, because if your ideas are any good you'll have trouble getting anyone to listen to them at all. In studying this stuff for years on end I had gone so far down untrodden paths that I was seeing things from a drastically unconventional angle, and if even so I had only just come around a corner to where I could see this thing, others were nowhere close to any vantage from which they could see it.
[note: I've since written a post elaborating on this, Explicit evaluation.]Kurt Gödel's no-go theorem
Likely the single most famous no-go theorem around is Gödel's Theorem. (Actually, it's Gödel's Theorems, plural, but the common informal understanding of the result doesn't require this distinction — and Gödel's result lends itself spectacularly to informal generalization.) This is what I'm going to spend most of this post on, because, well, it's jolly good fun (recalling the remark attributed to Abraham Lincoln: People who like this sort of thing will find it just the sort of thing they like).
The backstory to Gödel was that in the early-to-mid nineteenth century, mathematics had gotten itself a shiny new foundation in the form of a formal axiomatic approach. And through the second half of the nineteenth century mathematicians expanded on this idea. Until, as the nineteenth century gave way to the twentieth, they started to uncover paradoxes implied by their sets of axioms.
A perennial favorite (perhaps because it's easily explained) is Russell's Paradox. Let A be the set of all sets that do not contain themselves. Does A contain itself? Intuitively, one can see at once that if A contains itself, then by its definition it does not contain itself; and if it does not contain itself, then by its definition it does contain itself. The paradox mattered for mathematicians, though, for how it arose from their logical axioms, so we'll be a bit more precise here. The two key axioms involved are reductio ad absurdum and the Law of the Excluded Middle.
Reductio ad absurdum says that if you suppose a proposition P, and under this supposition you are able to derive a contradiction, then not-P. Supposing A contains itself, we derive a contradiction, therefore A does not contain itself. Supposing A does not contain itself, we derive a contradiction, therefore —careful!— A does not not contain itself. This is where the Law of the Excluded Middle comes in: A either does or does not contain itself, therefore since it does not not contain itself, it does contain itself. We have therefore an antinomy, that is, we've proved both a proposition P and its negation not-P (A both does and does not contain itself). And antinomies are really bad news, because according to these axioms we've already named, if there is some proposition P for which you can prove both P and not-P, then you can prove every proposition, no matter what it is. Like this: Take any proposition Q. Suppose not-Q; then P and not-P, which is a contradiction, therefore by reductio ad absurdum, not-not-Q, and by the Law of the Excluded Middle, Q.
When Russell's Paradox was published, the shiny new axiomatic foundations of mathematics were still less than a human lifetime old. Mathematicians started trying to figure out where things had gone wrong. The axioms of classical logic were evidently inconsistent, leading to antinomies, and the Law of the Excluded Middle was identified as a problem.
One approach to the problem, proposed by David Hilbert, was to back off to a smaller set of axioms that were manifestly consistent, then use that smaller set of axioms to prove that a somewhat larger set of axioms was consistent. Although clearly the whole of classical logic was inconsistent, Hilbert hoped to salvage as much of it as he could. This plan to use a smaller set of axioms to bootstrap consistency of a larger set of axioms was called Hilbert's program, and I'm remarking it because we'll have occasion to come back to it later.
Unfortunately, in 1931 Kurt Gödel proved a no-go theorem for Hilbert's program: that for any reasonably powerful system of formal logic, if the logic is consistent, then it cannot prove the consistency of its own axioms, let alone its own axioms plus some more on the side. The proof ran something like this: For any sufficiently powerful formal logic M, one can construct a proposition A of M that amounts to "this proposition is unprovable". If A were provable, that would prove that A is false, an antinomy; if not-A were provable, that would prove that A is true, again an antinomy; so M can only be consistent if both A and not-A are unprovable. But if M were able to prove its own consistency, that would prove that A is unprovable (because A must be unprovable in order for M to be consistent), which would prove that A is true, producing an antinomy, and M would in fact be inconsistent. Run by that again: If M can prove its own consistency, then M is in fact inconsistent.
Typically, on completion of a scientific paradigm shift, the questions that caused the shift cease to be treated as viable questions by new researchers; research on those questions tapers off rapidly, pushed forward only by people who were already engaged by those questions at the time of the shift. So it was with Gödel's results. Later generations mostly treated them as the final word on the foundations of mathematics: don't even bother, we know it's impossible. That was pretty much the consensus view when I began studying this stuff in the 1980s, and it's still pretty much the consensus view today.Going there
Having been trained to think of Gödel's Theorem as a force of nature, I nevertheless found myself studying it more seriously when writing the theoretical background material for my dissertation. I found myself discoursing at length on the relationship between mathematics, logic, and computation, and a curious discrepancy caught my eye. Consider the following Lisp predicate.
($define! A ($lambda (P) (not? (P P))))Predicate A takes one argument, P, which is expected to be a predicate of one argument, and returns the negation of what P would return when passed to itself. This is a direct Lisp translation of Russell's Paradox. What happens when A is passed itself?
The answer is, when A is passed itself, (A A), nothing interesting happens — which is really very interesting. The predicate attempts to recurse forever, never terminating, and in theory it will eventually fill up all available memory with a stack of pending continuations, and terminate with an error. What it won't do is cause mathematicians to despair of finding a solid foundation for their subject. If asking whether set A contains itself is so troublesome, why is applying predicate A to itself just a practical limit on how predicate A should be used?
That question came from my dissertation. Meanwhile, another question came from the other major document I was developing, the R-1RK. I wanted to devise a uniquely Lisp-ish variant of the concept of eager type-checking. It seemed obvious to me that there should not be a fixed set of rules of type inference built into the language; that lacks generality, and is not extensible. So my idea was this: In keeping with the philosophy that everything should be first-class, let theorems about the program be an encapsulated type of first-class objects. And carefully design the constructors for this theorem type so that you can't construct the object unless it's provable. In effect, the constructors are the axioms of the logic. Modus ponens, say, is a constructor: given a theorem P and a theorem P-implies-Q, the modus-ponens constructor allows you to construct a theorem Q. As desired, there is no built-in inference engine: the programmer takes ultimate responsibility for figuring out how to prove things.
Of the many questions in how to design such a first-class theorem type, one of the early ones has to be, what system of axioms should we use? Clearly not classical logic, because we know that would give us an inconsistent system. This though was pretty discouraging, because it seemed I would find myself directly confronting in my design the very sort of problems that Gödel's Theorem says are ultimately unsolvable.
But then I had a whimsical thought; the sort of thing that seems at once not-impossible and yet such a long shot that one can just relax and enjoy exploring it without feeling under pressure to produce a result in any particular timeframe (and yet, I have moved my thinking forward on this over the years, which keeps it interesting). What if we could find a way to take advantage of the fact that our logic is embedded in a computational system, by somehow bleeding off the paradoxes into mere nontermination? So that they produce the anticlimax of functions that don't terminate instead of the existential angst of inconsistent mathematical foundations?
At this point, my coherent vision dissolves into fragments of tentative insight, stray puzzle pieces I'm still pushing around hoping to fit together.
One fragment: Alfred Tarski —who fits the aforementioned profile of someone already engaged by the questions when Gödel's results came out— suggested post-Gödel that the notion of consistency should be derived from common sense. Hilbert's program had actually pursued a formal definition of consistency, as the property that not all propositions are provable; this does have a certain practicality to it, in that the practical difficulty with the classical antinomies was that they allowed all propositions Q to be proven, so that "Q can be proven" ceased to be a informative statement about Q. Tarski, though, remarked that when a non-mathematician is told that both P and not-P are true, they can see that something is wrong without having to first receive a lecture on the formal consequences of antinomies in interaction with reductio ad absurdum.
So, how about we resort to some common sense, here? A common-sensical description of Russell's Paradox might go like this:
A is the set of all sets that do not contain themselves. If A contains itself, then it does not contain itself. But if it does not contain itself, then it does contain itself. But if it does contain itself, then it does not contain itself. But if it does not contain itself...And that is just what we want to happen: instead of deriving an antinomy, the reasoning just regresses infinitely. A human being can see very quickly that this is going nowhere, and doesn't bother to iterate beyond the first four sentences at most (and once they've learned the pattern, next time they'll probably stop after even fewer sentences), but they don't come out of the experience believing that A both does and does not belong to itself; they come out believing that there's no way of resolving the question.
So perhaps we should be asking how to make the conflict here do an infinite regress instead of producing a (common-sensically wrong) answer after a finite number of steps. This seems to be some sort of deep structural change to how logical reasoning would work, possibly not even a modification of the axioms at all but rather of how they are used. If it does involve tampering with an axiom, the axiom tampered with might well be reductio ad absurdum rather than the Law of the Excluded Middle.
This idea — tampering with reductio ad absurdum rather than the Law of the Excluded Middle — strikes a rather intriguing historical chord. Because, you see, one of the mathematicians notably pursuing Hilbert's program pre-Gödel did try to eliminate the classical antinomies by leaving intact the Law of the Excluded Middle and instead modifying reductio ad aburdum. His name was Alonzo Church —you may have heard of him— and the logic he produced had, in retrospect, a curiously computational flavor to it. While he was at it, he took the opportunity to simplify the treatment of variables in his logic, by having only a single structure that binds variables, which (for reasons lost in history) he chose to call λ. Universal and existential quantifiers in his logic were higher-order functions, which didn't themselves bind variables but instead operated on functions that did the binding for them. Quite a clever device, this λ.
Unfortunately, it didn't take many years after Church's publication to show that antinomies arose in his system after all. Following the natural reflex of Hilbert's program, Church tried to find a subset of his logical axioms that could be proven consistent — and succeeded. It turned out that if you left out all the operators except λ you could prove that each proposition P was only equivalent to at most one irreducible form. This result was published in 1936 by Church and one of his students, J. Barkley Rosser, and today is known as the Church–Rosser Theorem (you may have heard of that, too). In the long run, Church's logic is an obscure historical footnote, while its λ-only subset turned out to be of great interest for computation, and is well-known under the name "λ-calculus".
So evidently this idea of tampering with reductio ad absurdum and bringing computation into the mix is not exactly unprecedented. Is it possible that there is something there that Alonzo Church didn't notice? I'd have to say, "yes". Alonzo Church is one of those people who (like Albert Einstein, you'll recall he came up in relation to the first no-go theorem I discussed) in retrospect appears to have set a standard of intelligence none of us can possibly aspire to — but all such people are limited by the time they live in. Einstein died years before Bell's Theorem was published. Heck, Aristotle was clearly a smart guy too, and just think of everything he missed through the inconvenience of being born about two millennia before the scientific revolution. And Alonzo Church couldn't, by the nature of the beast, have created his logic based on a modern perspective on computation and logic since it was in part the further development of his own work over many decades that has produced that modern perspective.
I've got one more puzzle piece I'm pushing around, that seems like it ought to fit in somewhere. Remember I said Church's logic was shown to have antinomies? Well, at the time the antinomy derivation was rather baroque. It involved a form of the Richard Paradox, which concerns the use of an expression in some class to designate an object that by definition cannot be designated by expressions of that class. (A version due to G.G. Berry concerns the twenty-one syllable English expression "the least natural number not nameable in fewer than twenty-two syllables".) The Richard Paradox is naturally facilitated by granting first-class status to functions, as λ-calculus and Lisp do. But, it turns out, there is a much simpler paradox contained in Church's logic, involving less logical machinery and therefore better suited for understanding what goes wrong when λ-calculus is embedded in a logic. This is Curry's Paradox.
I'll assume, for this last bit, that you're at least hazily familiar with λ-calculus, so it'll come back to you when you see it.
For Curry's Paradox, we need one logical operator, three logical axioms, and the machinery of λ-calculus itself. Our one logical operator is the binary implication operator, ⇒. The syntax of the augmented λ-calculus is then
T ::= x | c | (λx.T) | (TT) | (T⇒T)That is, a term is either a variable, or a constant, or a lambda-expression, or an application, or an implication. We don't need a negation operator, because we're sticking with the generalized notion of inconsistency as the property that all propositions are provable. Our axioms assert that certain terms are provable:
- For all terms P and Q, if provably P and provably (P⇒Q), then provably Q. (modus ponens)
- For all terms P, provably P⇒P.
- For all terms P and Q, provably ((P⇒(P⇒Q))⇒(P⇒Q)).
(λx.F)G → F[x ← G]That is, to apply function (λx.F) to argument G, substitute G for all free occurrences of x in F.
To prove inconsistency, first we need a simple result that comes entirely from λ-calculus itself, called the Fixpoint Theorem. This result says that for every term F, there exists a term G such that FG = G (that is, every term F has a fixpoint). The proof works like this:
Suppose F is any term, and let G = (λx.(F(xx)))(λx.(F(xx))), where x doesn't occur in F. Then G = (λx.(F(xx)))(λx.(F(xx))) → (F(xx))[x ← (λx.(F(xx)))] = F((λx.(F(xx)))(λx.(F(xx)))) = FG.Notice that although the Fixpoint Theorem apparently says that every F has a fixpoint G, it does not actually require F to be a function at all: instead of providing a G to which F can be applied, it provides a G from which FG can be derived. And —moreover— for most F, derivation from G is a divergent computation (G → FG → F(FG) → F(F(FG)) → ...).
Now we're ready for our proof of inconsistency: that for every term P, provably P.
So, why did I go through all this in detail? Besides, of course, enjoying a good paradox. Well, mostly, this: The entire derivation turns on the essential premise that derivation in the calculus, as occurs (oddly backwards) in the proof of the Fixpoint Theorem, is a relation between logical terms — which is to say that all terms in the calculus have logical meaning.
Suppose P is any term. Let Q = λx.(x⇒P). By the Fixpoint Theorem, let R be a term such that QR = R. By writing out the definition of Q and then applying the β-rule, we have QR = (λx.(x⇒P))R → (R⇒P), therefore R = (R⇒P).
By the second axiom, provably (R⇒R); but R = R⇒P, so, by replacing the right hand R in (R⇒R) with (R⇒P), provably (R⇒(R⇒P)).
By the third axiom, provably ((R⇒(R⇒P))⇒(R⇒P)); and we already have provably (R⇒(R⇒P)), so, by modus ponens, provably (R⇒P). But R = (R⇒P), so provably R.
Since provably R and provably (R⇒P), by modus ponens, provably P.Note: I've had to fix errors in this proof twice since publication; there's some sort of lesson there about either formal proofs or paradoxes.
And we've seen something like that before, in my early explanation of Mitchell Wand's no-go theorem: trivialization of theory resulted from assuming that all calculus derivation was evaluation. So, if we got around Wand's no-go theorem by recognizing that some derivation is not evaluation, what can we do by recognizing that some derivation is not deduction?