number that I called ‘g’ possesses a rather arcane and recherche number-theoretical property that I called “sauciness” or “non-primness”. You’ll note that in the last couple of pages, not one word has appeared about prim numbers or non-prim numbers and their number-theoretical properties, nor has the number g been mentioned at all. We finessed all such numerical issues by looking only at KG’s secondary meaning, the meaning that Bertrand Russell never quite got. A few lines of purely non-numerical reasoning (the second section of this chapter) convinced us that this statement (which is about numbers) could not conceivably be a theorem of PM.
Consistency Condemns a Towering Peak to Unscalability
Imagine that a team of satellite-borne explorers has just discovered an unsuspected Himalayan mountain peak (let’s call it “KJ”) and imagine that they proclaim, both instantly and with total confidence, that thanks to a special, most unusual property of the summit alone, there is no conceivable route leading up to it. Merely from looking at a single photo shot vertically downwards from 250 miles up, the team declares KJ an unclimbable peak, and they reach this dramatic conclusion without giving any thought to the peak’s properties as seen from a conventional mountaineering perspective, let alone getting their hands dirty and actually trying out any of the countless potential approaches leading up the steep slopes towards it. “Nope, none of them will work!”, they cheerfully assert. “No need to bother trying any of them out — you’ll fail every time!”
Were such an odd event to transpire, it would be remarkably different from how all previous conclusions about the scalability of mountains had been reached. Heretofore, climbers always had to attempt many routes — indeed, to attempt them many times, with many types of equipment and in diverse weather conditions — and even thousands of failures in a row would not constitute an ironclad proof that the given peak was forever unscalable; all one could conclude would be that it had so far resisted scaling. Indeed, the very idea of a “proof of unscalability” would be most alien to the activity of mountaineering.
By contrast, our team of explorers has concluded from some novel property of KJ, without once thinking about (let alone actually trying out) a single one of the infinitely many conceivable routes leading up to its summit, that by its very nature it is unscalable. And yet their conclusion, they claim, is not merely probable or extremely likely, but dead certain.
This amounts to an unprecedented, upside-down, top-down kind of alpinistic causality. What kind of property might account for the peculiar peak’s unscalability? Traditional climbing experts would be bewildered at a blanket claim that for every conceivable route, climbers will inevitably encounter some fatal obstacle along the way. They might more modestly conclude that the distant peak would be extremely difficult to scale by looking upwards at it and trying to take into account all the imaginable routes that one might take in order to reach it. But our intrepid team, by contrast, has looked solely at KJ’s tippy-top and concluded downwards that there simply could be no route that would ever reach it from below.
When pressed very hard, the team of explorers finally explains how they reached their shattering conclusions. It turns out that the photograph taken of KJ from above was made not with ordinary light, which would reveal nothing special at all, but with the newly discovered “Godel rays”. When KJ is perceived through this novel medium, a deeply hidden set of fatal structures is revealed.
The problem stems from the consistency of the rock base underlying the glaciers at the very top; it is so delicate that, were any climber to come within striking distance of the peak, the act of setting the slightest weight on it (even a grain of salt; even a baby bumblebee’s eyelash!) would instantly trigger a thunderous earthquake, and the whole mountain would come tumbling down in rubble. So the peak’s inaccessibility turns out to have nothing to do with how anyone might try to get up to it; it has to do with an inherent instability belonging to the summit itself, and moreover, a type of instability that only Godel rays can reveal. Quite a silly fantasy, is it not?
Downward Causality in Mathematics
Indeed it is. But Kurt Godel’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 “Godel ray” (i.e., some clever new Godel numbering, including all of the standard Godel 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 Godel 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 Godel 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 Godel 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, Godel 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
