Douglas Hofstadter - I Am a Strange Loop

Downward Causality in Mathematics

Indeed it is. But Kurt Gödel’s bombshell, though just as fantastic, was not a fantasy. It was rigorous and precise. It revealed the stunning fact that a formula’s hidden meaning may have a peculiar kind of “downward” causal power, determining the formula’s truth or falsity (or its derivability or nonderivability inside PM or any other sufficiently rich axiomatic system). Merely from knowing the formula’s meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically “upwards” from the axioms.

This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false (or provable or unprovable).

For instance, if I tell you, “There are infinitely many perfect numbers” (numbers such as 6, 28, and 496, whose factors add up to the number itself), you will not know if my claim — call it ‘Imp’ — is true or not, and merely staring for a long time at the written-out statement of Imp (whether it’s expressed in English words or in some prickly formal notation such as that of PM ) will not help you in the least. You will have to try out various approaches to this peak. Thus you might discover that 8128 is the next perfect number after 496; you might note that none of the perfect numbers you come up with is odd, which is somewhat odd; you might observe that each one you find has the form p ( p +1)/2, where p is an odd prime (such as 3, 7, or 31) and p +1 is also a power of 2 (such as 4, 8, or 32); and so forth.

After a while, perhaps a long series of failures to prove Imp would gradually bring you around to suspecting that it is false. In that case, you might decide to switch goals and try out various approaches to the nearby rival peak — namely, Imp’s negation ∼ Imp — which is the statement “There are not infinitely many perfect numbers”, which is tantamount to asserting that there is a largest perfect number (reminiscent of our old friend P, allegedly the largest prime number in the world).

But suppose that through a stunning stroke of genius you discovered a new kind of “Gödel ray” ( i.e., some clever new Gödel numbering, including all of the standard Gödel machinery that makes prim numbers dance in perfect synchrony with provable strings) that allowed you to see through to a hidden second level of meaning belonging to Imp — a hidden meaning that proclaimed, to those lucky few who knew how to decipher it, “The integer i is not prim”, where i happened to be the Gödel number of Imp itself. Well, dear reader, I suspect it wouldn’t take you long to recognize this scenario. You would quickly realize that Imp, just like KG, asserts of itself via your new Gödel code, “Imp has no proof in PM. ”

In that most delightful though most unlikely of scenarios, you could immediately conclude, without any further search through the world of whole numbers and their factors, or through the world of rigorous proofs, that Imp was both true and unprovable. In other words, you would conclude that the statement “There are infinitely many perfect numbers” is true, and you would also conclude that it has no proof using PM’ s axioms and rules of inference, and last of all (twisting the knife of irony), you would conclude that Imp’s lack of proof in PM is a direct consequence of its truth.

You may think the scenario I’ve just painted is nonsensical, but it is exactly analogous to what Gödel did. It’s just that instead of starting with an a priori well-known and interesting statement about numbers and then fortuitously bumping into a very strange alternate meaning hidden inside it, Gödel carefully concocted a statement about numbers and revealed that, because of how he had designed it, it had a very strange alternate meaning. Other than that, though, the two scenarios are identical.

The hypothetical Imp scenario and the genuine KG scenario are, as I’m sure you can tell, radically different from how mathematics has traditionally been done. They amount to upside-down reasoning — reasoning from a would-be theorem downwards, rather than from axioms upwards, and in particular, reasoning from a hidden meaning of the would-be theorem, rather than from its surface-level claim about numbers.

Göru and the Futile Quest for a Truth Machine

Do you remember Göru, the hypothetical machine that tells prim numbers from saucy (non-prim) numbers? Back in Chapter 10, I pointed out that if we had built such a Göru, or if someone had simply given us one, then we could determine the truth or falsity of any number-theoretical conjecture at all. To do so, we would merely translate conjecture C into a PM formula, calculate its Gödel number c (a straightforward task), and then ask Göru, “Is c prim or saucy?” If Göru came back with the answer “ c is prim”, we’d proclaim, “Since c is prim, conjecture C is provable, hence it is true”, whereas if Göru came back with the answer “ c is saucy”, then we’d proclaim, “Since c is saucy, conjecture C is not provable, hence it is false.” And since Göru would always (by stipulation) eventually give us one or the other of these answers, we could just sit back and let it solve whatever math puzzle we dreamt up, of whatever level of profundity.

It’s a great scenario for solving all problems with just one little gadget, but unfortunately we can now see that it is fatally flawed. Gödel revealed to us that there is a profound gulf between truth and provability in PM (indeed, in any formal axiomatic system like PM ). That is, there are many true statements that are not provable, alas. So if a formula of PM fails to be a theorem, you can’t take that as a sure sign that it is false (although luckily, whenever a formula is a theorem, that’s a sure sign that it is true). So even if Göru works exactly as advertised, always giving us a correct ‘yes’ or ‘no’ answer to any question of the form “Is n prim?”, it won’t be able to answer all mathematical questions for us, after all.

Despite being less informative than we had hoped, Göru would still be a nice machine to own, but it turns out that even that is not in the cards. No reliable prim/saucy distinguisher can exist at all. (I won’t go into the details here, but they can be found in many texts of mathematical logic or computability.) All of a sudden, it seems as if dreams are coming crashing down all around us — and in a sense, this is what happened in the 1930’s, when the great gulf between the abstract concept of truth and mechanical ways to ascertain truth was first discovered, and the stunning size of this gulf started to dawn on people.

It was logician Alfred Tarski who put one of the last nails in the coffin of mathematicians’ dreams in this arena, when he showed that there is not even any way to express in PM notation the English statement “ n is the Gödel number of a true formula of number theory”. What Tarski’s finding means is that although there is an infinite set of numbers that stand for true statements (using some particular Gödel numbering), and a complementary infinite set of numbers that stand for false statements, there is no way to express that distinction as a number-theoretical one. In other words, the set of all wff numbers is divided into two complementary parts by the true/false dichotomy, but the boundary line is so peculiar and elusive that it is not characterizable in any mathematical fashion at all.