Gödel & Lemaître

My Chief Irish Veterinary Correspondent put it most succinctly: “Didn’t Gödel drive a stake through the heart of the concept of a ‘Theory of Everything’?” (See yesterday.) This is also my understanding: that the Austrian logician demonstrated in his two “incompleteness theorems,” published in 1931, why no such thing can work. But we are dealing today with the kind of zombie that doesn’t notice when a stake has been driven through its heart.

Let me try briefly to review both ends of that sharp stick.

Gödel’s first theorem proved that any formal system of axioms subtle and complex enough to describe even so apparently straightforward a thing as a set of numbers must contain at least one “undecidable” statement, such that even if we are certain that statement is true, the system can’t prove it. It must therefore be logically “incomplete.”

And his second theorem was like unto it:

No one can prove, from inside any formal system, that it is self-consistent. Not, “some day,” not, “maybe we missed something,” not, “give us more time” — but can’t, won’t, jamais de la vie — and in the way you can’t be a man and a teapot at the same time.

Or put this another way (and there are many, many other ways to put it). Any logical account we may want to give of the totality of our wee, finite Universe (and we know darn well it is finite, today) requires a view from outside our Universe, that is indispensable to fully understand it.

Or consider: there will always be things that one knows to be true, but cannot be strictly proved, in logic; which rise, as it were, above the rational, in an ultimately demonstrable way; which present some (often beautiful) paradox.

It follows that the mathematician, the scientist, even the engineer and technologist, and everybody else, must work on blind faith, even within their own trades. And what is reasonable is not always rational: merely consistent with reason. Blind spots must necessarily remain, for us finite creatures. What we know by common sense is thus affirmed at the highest available rational level: that we cannot know everything.

True, Gödel’s “proofs” require some brains to understand. But they also take some brains to misunderstand: to defy something that comes down, in the end, to the Law of Non-Contradiction. You cannot have your cake and eat it, too; you cannot be both A and not-A. Not even God can contradict this Law of Non-Contradiction, and anyway wouldn’t try. He never contradicts Himself because (unlike other gods) He never has to.

It was this theological insight that made Christianity the guide to empirical science: that God is self-consistent, that His deeds will always finally make sense; that although God is far beyond human reason, He has from every direction left a trail of divine light.

And note this paradox: that the condition for the nurture and mastery and growth of empirical science, was blind faith. That, among other things, God is no trickster. He is immanent, and transcendent, but distinct from his own Creation — all such things as we can know, by faith.

So to explain the Universe, the set of all sets, no matter how big it happens to be — and even if it includes a bubble bath of “multiverses” — we must step outside the Whole Thing. And this would be necessary, no matter which bubble we might happen to be locked inside.

Gödel also developed, unsurprisingly, a version of Anselm’s cosmological argument for the existence of God, expressed in slam-dunk post-Euclidean logic, that was theist like Leibniz and not polytheist like Spinoza. And yes, he was extremely familiar with Kant’s naïve attempt at refutation. (Kant, who never read mediaeval philosophy, did not actually understand Anselm’s argument, let alone the improvement on it by Thomas Aquinas.)

I’m acquainted with raw, drooling ignorance in myself. I’m surprised to find it institutionalized today, and frequently enforced, though perhaps only because I am at heart a man of the thirteenth century (like Gödel, 1906–78), and thus perhaps too easily repugned of smug atheist fools.

Einstein, incidentally, once said he only worked at Princeton so he could have the opportunity to take walks with Gödel. They often went on long ones.


Now let us return over the sea, to the (once) Catholic University of Louvain, in Belgium, and to Monsignor Georges Lemaître (1894–1966). About the same time the young Gödel was formulating his incompleteness theorems as a doctoral dissertation at Vienna, or a few years before, Lemaître was playing with Einstein’s field equations of relativity, and realizing a funny little thing. The Universe is not static. It is expanding. He did everything for which Edwin Hubble is now credited by the pop science writers, except, he did it before Hubble. And then he did an even better thing: he accounted for it.

Lemaître is the true and only author of the “Big Bang” hypothesis, which in wake of yesterday’s “gravity wave” announcement is once more confirmed to be at the heart of all astrophysics. The priest himself called it the hypothesis of the Cosmic Egg: that our universe began as a “primaeval atom”: an extremely small fraction of the radius of a proton which, oddly enough, blew out to its present, rather larger size.

For this he was ignored, or mocked. The expression “Big Bang” was itself coined by the atheist Fred Hoyle to make fun of it, and has stuck because it still appeals to the craving of materialists for a static Universe, infinite in scale. They can’t handle something that began; there must be something before that “just happened,” to no good end, for no good reason, in the infinite regression of a hall of mirrors. They must absolutely insist on the meaninglessness of it all; a succession of nothings. For otherwise they must face down the very God that they have been avoiding.

But that Cosmic Egg was quite a something; quite a nuanced, profound something; and rather consequential, as we have come to see. For it carried the possibility of our own biological existence.

The primaeval atom; the egg; the Seed, as I think of it myself, implanted in the soil of the Holy Spirit. Which burst forth in a million stars; in a million million million of them. I can understand this in a way consistent with both reason and faith; I cannot understand it in a way consistent with a long yawn. As Einstein said (to much subsequent ridicule), “God does not play dice with the universe.” The Maker of that Seed knew what He was doing; this certainly is what Georges Lemaître understood.

I’ve mentioned Lemaître in Idleposts before (there’s a search function in this website, y’know); he is perhaps my biggest modern scientific hero, after Pierre Duhem. As other truly penetrating intellects, he cannot be properly appreciated by the post-modern mind, which accepts only Prometheans as heroes — i.e. men who seem to stand against God in rebellion; tricksters angling to steal His fire, and repeat the sin of Adam. Whereas, Lemaître merely served God, with real distinction.

The hypothesis of the Cosmic Egg found the light of day in the same year of grace 1931 (as Gödel’s key publication). And like Gödel, Lemaître stood modestly, yet also bravely athwart the crass metaphysical assumptions of “modernity.” It wasn’t his egg that was so provocative, in itself. Rather, it was what the egg said.

It said that our Universe is finite. Generations of the cleverest scientific minds had taken material infinitude for granted. It was necessary to all their thinking: an infinite amount of space and time in which anything we see could have gradually “evolved,” like Darwin’s beasts, by pure happenstance at their infinite leisure.

Cut the time-line short and we are dealing with “miracles” instead — with things that happen not slowly but suddenly, while casually ignoring all our human expectations. And these not small things, either.

Well, sometimes you just have to tough it out. There is currently no way home to an infinite Universe, and no way foreseeable. The “multiverse” conjecture does not get us there, it only displaces the question — kicks it a little farther down the road. Human reason can run, but it cannot hide: sooner or later it is staring once again at the inescapable, ineluctable, Fact of God.

Einstein himself was at first scandalized, by Lemaître’s hypothesis, when Eddington (who’d been among Lemaître’s teachers, and thought him the brightest student he’d ever had) brought it to his attention. “Your math is correct, but your physics is abominable,” Einstein told the priest.

But within a couple of years he had come around, and realized that his own unspoken assumption of a static, infinite Universe was unsustainable. Indeed he came to call that Cosmic Egg “the most beautiful thing” he’d ever seen. Other fine minds likewise came around, though as I recall, by the 1960s, some were still fighting, still defending the body that was dead in the cosmic water.

Since the 1990s, we have known that the Universe is not only expanding, but accelerating outward. It is icing on old Lemaître’s cake. We have also come to realize there are irregularities in the rate at which the stars recede; that there are mysterious Great Attractors scattered here and there through intergalactic space. Indeed, yesterday’s formal announcement of the demonstration of “gravity waves” lets us hope for insights into these irregularities.

Beyond those we continue to find, Horatio, that there are “more things in Heaven and Earth.” Our choice is to take this with awe; or with the deathly grin of those whose faith is not in God, but in their glib, sorry, mechanistic contraptions — in a scientism that real science continues to kick away.