On Some Happy Corollaries of Gödel’s Incompleteness Theorems

I shall not now reiterate arguments I here set forth to my own satisfaction in 2012, shortly after we got started – with the corrective editorial (and indeed, therefore, also substantive) help of my old friend and interlocutor (and, as with any true friend, my teacher) Ilíon, an orthospherean and shieldmate for years before there was such a thing as the Orthosphere – but shall rather recommend that any reader of the present post who finds it at all confusing should first recur thereto, and take it, and ponder it in his heart, before adding below any quibbles or queries. Consider the arguments of that post, together with the relatively brief commentary thereto, as praeparatio for this.

The arguments I proposed in 2012 are nevertheless fundamental to what I shall now suggest, so unless you understand them already, dear reader, it would do you well first to review them.

The basic notion is that any orderly system must, as orderly (and, so, qua system, properly so called; to say “orderly system” is rather like saying “rectangular square”), be amenable in principle at least to complete – i.e., to exhaustive – nomological formalization in a logical calculus. Think, e.g., of the System of Nature, which – as Baconian science, and indeed her predecessor of the more expansive Aristotelian sort both presuppose – must be capable of formalization in a system of natural laws, or at least of natural regularities (tace for the nonce on how any given regularity gets to be anything of the sort, or what any such law might be, or how it might operate). If there is truly a System of Nature, then truly her ways must be legated, and so then legible to us, in some order that can at least in principle be set forth in some formal scheme that undergirds and supports – and, somehow, regulates and so enables – her apparent and merely phenomenal orderliness, in such a way as to secure to us in the first place such a thing as phenomena.

The notion that our acts can make some sense presupposes the System of Nature. And this is so, not only of our acts taken in respect to the Cosmological Singularity or to her eschaton, forsooth, but also therefore, and by a straightforward implication, of our acts taken in respect to this morning’s coffee, or to the peregrinations of the dog. Everything we do about the dog is done about the eschaton, and vice versa. In a coherent world – which is to say, in any world, properly so called – it could not be otherwise.

So: if a human person, or a world, or an ecology, or a society, or lo even a bit of software, or anything at all, are to be the least bit intelligible, and so therefore first ontologically coherent, and so possibly actual, and second causally effectual, why then they must each of them be first within themselves logically consistent. You can’t get ontological consistence, or, a fortiori, actual being, from illogic. Only what is first logically compossible in itself can possibly find expression in actual ontological fact.

All really achieved facts, then, are relentlessly, ineluctably logical. As being all themselves throughly logical, so all facts must logically cohere with all other facts, in mere virtue of their common mutual coherence with logic, which is to them all prior. So is every world sewn together in a seamless net. No such consistent net, no world.

You can’t make three stitches out of two, or vice versa. It’s really that simple.

Knitters, knotters and splicers, all fishermen, climbers, boatmen, seamen, surgeons, seamstresses – and all accountants – will all see this immediately, and quite concretely. Economists, too. Indeed, all husbands and housewives, in the old senses of those terms, will see this (husband and housewife are both terms subsumed in the category of economist).

Not that there are too many such pragmatical sorts, these devolute days, alas.

On this notion of ontological parsimony – which is to say, order – hangs the whole cultic notion of the seamless web – i.e., the whole notion of religion, of (literally) “things tied together” – for which the Roman executioners of that palmary ligature had no option but to cast lots, if any of them were to gain anything by it.

To cast lots is to ask the real to tell us what she is. It is to pay honest attention to the real, despite our fondest most foolish wishes. On this hangs all mancy, and indeed all empiricism – which is to say, all humility. Urim & Thummim; Lux & Veritas. Lots are the first gesture in the direction of controlled experiment.

Did Saint Longinus win that contest of lots? I bet he did, whether or not he departed Golgotha with the seamless garment intact as his own possession and relic. He departed with his sight intact, no? He saw Truth. What else could there be, to found a life true and good?

Stab the heart of God, and bathe in his flux, letting be what he would have be. What else could be at all worth doing, by comparison?

OK then, back to the topic at hand: what is, must be logically consistent. But – and here’s the rub – on Gödel, what is logically consistent – and finite – cannot be complete.

So, sure, on a first order take, what that means is that nature cannot explain itself, and so therefore nature cannot explain anything; for as consistent it cannot recur to itself (for, circularity is lethal to explanation). Thus is theism proved, as trivially (albeit not as obviously) as any truth of arithmetic.

Consider then the corollaries, and so the meat of this post.

First, the fully specified character of a human person at time t cannot account for the character of that same person at any subsequent time. For, while the person at t is in principle amenable to a consistent formal specification, nevertheless any such formalization must (per Gödel) be incomplete, and so must for its next iteration be waiting for further specifications, none of which might ever possibly suffice to a complete specification of that person. The human person, qua concrete instantiation of a logical calculus – of, i.e., an essential form, as that of Zippy, or of Kristor, or of Lawrence – cannot be completed.

Because no one instance of a human life can complete it, and so bring it to an absolute end, so there is no way to get one instance of Zippy without getting an endless series of such instantiations.

This in a nutshell is why the character of the human person must evolve without ceasing: none of its specifications of itself at any time t can possibly be complete. Gödel has shown that human persons are immortal, as a matter of mere logic.

On Gödel, then, we are immortal.

Hey, Zip! Love you, brother! See you soon. I hope you can poke some holes in what I have so far said.

OK: because on Gödelian Incompleteness the fully specified character of a human person at time t cannot completely account for the character of that same person at any subsequent time, therefore is the freedom of our will also demonstrated.

Take my past as wholly defined, which of course it is (otherwise, it would not be past). On Gödel, my past is a logically consistent system; in no other way might it be logically coherent, ontologically possible, or then real. But also on Gödel, my life is not complete, despite its coherence: it can develop and elaborate in ways that are not explicit in its past, and cannot from them be demonstrated as ineluctable. So, we are not bound and determined by our pasts, but rather execute real choices. The reality of choice at time t is logically entailed by Gödelian Incompleteness.

Following on the same arguments, the human person is not exhaustively specified by any number of her past physical aspects. As being determinable, any such past aspects must be first wholly determinate, which is to say, that they must be aspects of the actual past of that person, and so (as consistent – this being the only way they might cohere ontologically, so as to be factual) formally incomplete – and, then, actually incomplete. So the human person has always room to evolve.

To say that a thing has room to evolve is to say that its evolution is not entirely determined by its past. A thing that is entirely determined by its past just is that past, and has no being apart from that past. But this now is not that then, ergo etc.

Thus it is that on Gödel, physicalism is false. Persons are not just material – which is to say, precisely, that they are not functions entirely of their pasts. I.e., the human spirit is not material.

Consider now in summary what we have so far mined from Gödel: the human person is not merely material; the human person is free. Materialism and physicalism are false. Traditional psychology is buttressed. So is common sense.

Pretty good, right?

There is more.

Gödelian Incompleteness renders all determinism absurd. There is on Gödel no way to obtain a complete and consistent world system: no logical calculus encoding the regularities of a world can ever completely account for – can ever demonstrate – all the truths it is capable of expressing. Rather, on Gödel, worlds are in and of themselves sempiternal, and cannot be completed by themselves (albeit that they may be superseded by something else – by some other world, as with traditional Christian eschatology). What this means in practice for us, here and now, is that we are not determined ex ante. Our future is our own, and does not derive ineluctably from our past.

For what it is worth. I welcome comments and questions. Ilíon, I hope you are out there and paying attention, to keep me on the strait and narrow.

The bottom line is that Gödelian Incompleteness demolishes determinism, atheism, physicalism, and materialism; and it supports immortality, theism, and all the other appurtenances of traditional Christianity.

As Gödel himself well knew.