I Am a Strange Loop - Douglas R. Hofstadter [105]
Not only Bertrand Russell was blindsided by this amazingly perverse and yet stunningly beautiful maneuver; virtually every mathematical thinker was, including the great German mathematician David Hilbert, one of whose major goals in life had been to rigorously ground all of mathematics in an axiomatic framework (this was called “the Hilbert Program”). Up till the Great Thunderclap of 1931, it was universally believed that this noble goal had been reached by Whitehead and Russell.
To put it another way, the mathematicians of that time universally believed in what I earlier called the “Mathematician’s Credo (Principia Mathematica version)”. Gödel’s shocking revelation that the pedestal upon which they had quite reasonably placed their faith was fundamentally and irreparably flawed followed from two things. One is our kindly assumption that the pedestal is consistent (i.e., we will never find any falsity lurking among the theorems of PM); the other is the nonprovability in PM of KG and all its infinitely many cousins, which we just showed is a consequence flowing from their self-referentiality, taking PM’s consistency into account.
To recap it just one last time, what is it about KG (or any of its cousins) that makes it not provable? In a word, it is its self-referential meaning: if KG were provable, its loopy meaning would flip around and make it unprovable, and so PM would be inconsistent, which we know it is not.
But notice that we have not made any detailed analysis of the nature of derivations that would try to make KG appear as their bottom line. In fact, we have totally ignored the Russellian meaning of KG (what I’ve been calling its primary meaning), which is the claim that the gargantuan number that I called ‘g’ possesses a rather arcane and recherché 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,