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.

9 thoughts on “On Some Happy Corollaries of Gödel’s Incompleteness Theorems

  1. Thank you for this interesting post, Kristor, there is a lot of meat on these bones you’ve offered up and you’ve given me some things to think about. Here’s a few:

    1) I hope you don’t mind what seems to me an intense oversimplification of your post, but your last section–

    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.

    –is not that true for any theory which is true? Truth–big-T-Truth–is necessarily internally consistent, maybe it would be reasonable to say “complete” because big-T-Truth is necessarily transcendent and so inexpressible except as God Himself. Our material manifestations of Big-T-Truth on this side of the veil are little-t-truths. So if Gödels theory is t-true, then it participates in what is T-True, and so implies all that is both T-True and t-true. Likewise the iron law that 2+2=4, or that the mitochondria is the powerhouse of the cell.

    2) The fullness of a human person, then, cannot be actual until it is complete. A living person cannot be complete because that person is still living. Neither can a persons past be considered complete because there is more past being added every moment. Yet–complete does not mean dead, because inherent to the nature of a person is a spirit, and that spirit is immortal.

    Although, now that I think about it, spirits are immortal but spirits do not change. Well, nix that, because whatever state a spirit is in in heaven will change when glorified and resurrected–and when glorified and resurrected be in it’s final state, and so unchanging, and so complete. Then and only then can the fullness of a human person be discretely understood, because then and only then is the fullness of a human person actual. I hope this is an orthodox notion!

    I can’t claim to have a full understanding of this series of posts, I’m going to have to chew on it a bit; I’m just riffing off that last bit that made me think that all Truth necessarily must point to all other Truth; it’s almost incoherent to conceive of any truth as existing in a vacuum.

    • You are welcome, Scoot. Thanks in turn for your constructive comment.

      1. Yes, all truths cohere. But not all truths contradict, say, determinism. It is true that I walked the dog yesterday. But that particular truth is compatible with determinism. Gödel’s Incompleteness Theorems, on the other hand, are not. The Incompleteness Theorems and determinism cannot all be true, in just the way that circles cannot be square. The Incompleteness Theorems are true, ergo determinism is false.

      2. It’s not quite an orthodox notion, because it is incompatible with the resurrection of the body. Theologians pretty much agree that the resurrection of the body entails its resurrection to life and to motion, and to an environing world of the same dimensional character as this one, with space, time, gravity, and so forth, to which bodies like ours are fitted; and, indeed, to a climate such as might be found on Earth, with food, animals, weather, etc. The liveliness of the resurrection body entails its continued motion, action, and change. So stuff will keep happening to us, we’ll keep doing things and having adventures.

      Where this gets complicated is with the metaphysics of the Beatific Vision, which mystics agree is not characterized by temporal passage, the way our embodied life is. In the BV, we partake life sub specie aeternitatis, precisely because in the BV we are apprehending God: because he is simple, to know him is to know as and what he knows, at least in part. It seems to be possible to do this while living embodied in a temporal world: while the manner of his ascent was mysterious to him, Paul did not have to die or leave the planet in order to be taken up to the third heaven (2 Corinthians 12:2). In my own flesh – in this very body – shall I see God (Job 19:26).

      In short, there can be no ontological conflicts between temporal life and eternal life: the Incarnation is sufficient warrant of the mutual agreement of time and eternity (the Lógos didn’t stop being eternal with the Incarnation (to stop being eternal is a contradiction in terms); the eternal nature of the Lógos did not drive out the temporal nature of Jesus). This harks back to your first point: to say that there are no conflicts between temporal life and eternal life is after all to say (among other things) that the temporal truths and the eternal truths perfectly agree.

      So, the resurrection body will be able to look at and partake the Eternal One. Heck, we already do this at Mass, albeit as veiled.

      The life of the resurrection body shall never be completed, for it is an everlasting life. It will be a life without end, and also without any evils – including the evil of boredom. It will be endlessly fun, and we shan’t be able to get tired of it.

      One last note: from a temporal perspective, the character of the everlasting life of a resurrection body cannot ever be completed; but from the perspective of eternity, all moments whatsoever, of all lives, are present, and known, in a single act; so that God does know the completed character of each everlasting life.

      This is an aspect of the fact that in God, whose omniscience is infinite, the entire infinite stack of Gödelian logical calculi is both consistent and complete. Because logical calculi can be completed only by recourse to and in terms of supersidiary logical calculi that include them as subsidiaries, completion of any one consistent logical calculus is possible only insofar as the entire infinite stack of such calculi are comprehensively expressed. By definition, all truths must be known to omniscience; so, all truths are in and by him expressed, and thus all logical calculi are completed – but only by and in virtue of omniscience.

      Anyway, it may be then that in the BV, we shall be able to glimpse our infinite lives (and everything else) the way God does: all at one fell swoop. If so, then perhaps we shall in the BV find ourselves – our entire sempiternal lives – perfectly at rest, as he is. This is Donne’s One Equal Music, that comprises and funds all musices – and, so, all matheses:

      Bring us, O Lord God, at our last awakening into the house and gate of heaven, to enter into that gate and dwell in that house, where there shall be no darkness nor dazzling, but one equal light; no noise nor silence, but one equal music; no fears nor hopes, but one equal possession; no ends nor beginnings, but one equal eternity: in the habitations of thy majesty and glory, world without end. Amen

      • But not all truths contradict, say, determinism.

        I have found that this attribute is what makes a good lie. Most people can spot a wolf, but it’s harder to spot a wolf in sheeps clothing. All the most dangerous and popular heresies begin with a drop of truth which is then diluted with toxic falsity.

        It is true that I walked the dog yesterday. But that particular truth is compatible with determinism.

        On the other hand–there are mundane truths that are as you say, compatible with multiple models of truth. This makes logical sense too–if every truth pointed to transcendent Truth then there would be no need for education. “2+2=4 therefore God exists” would be a logically valid statement in such a world (well, it is logically valid now just in a different way). Education (science, in it’s classical definition) is the quest for the set of transcendent Truths which connects to the most mundane truths. This is why physicists want a theory of everything, because they want to create a truth which is both finite and transcendent.

        Thank you for expanding on the Resurrection–it’s a concept I am still trying to grok and it’s coming slowly.

        The liveliness of the resurrection body entails its continued motion, action, and change. So stuff will keep happening to us, we’ll keep doing things and having adventures.

        God is simultaneously perfect and active, is he not? Is “perfect” a valid synonym for “complete” in this sense?

        Another question that comes to mind is–what change matters to the question of completeness? Sin will be impossible to glorified and resurrected souls, because they will have kept their allegiance to God through their earthly life and been purified in purgatory–so not impossible the way a square circle is impossible, but impossible the way a loyal husband considers it impossible to cheat on his wife. I digress: Because our spiritual being will not change, even if our physical world (which will also be glorified) will change, is not the spiritual essence an (if not the) important element to rendering something complete?

        The life of the resurrection body shall never be completed, for it is an everlasting life.

        –although, it seems like i’ve got a fundamental misunderstanding about what you mean when discussing completeness and incompleteness, so before I dig too much more I will do a careful reading of your linked articles and some google-fu.

        By definition, all truths must be known to omniscience; so, all truths are in and by him expressed, and thus all logical calculi are completed – but only by and in virtue of omniscience.

        This makes it seem like perfect = complete, if complete is only achievable by being God. I know there is some theological concept of becoming one with God without losing the self, but even when glorified and resurrected we would not be omniscient, but perhaps perfectly sentient.

        “One Equal Music”–what an image. Thank you for that!

      • Different sorts of beings are perfected in different ways, because their natures differ. A crystal is not perfect the same way that a squirrel is perfect, e.g.

        God is perfect in being logically complete (there are other ways he is perfect, too), for that is proper to the nature of ultimacy.

        Man’s nature is perfected when he is human without defect, when his human nature is perfectly fulfilled and expressed. And it is essential to human nature that we should be embodied, live, and so move, experience, act. It is, in other words, essential to human nature that we should at every moment of our lives be logically incomplete. This is another way of saying that it is essential to our nature that we should be partiscient: that no matter how much we might learn, we would still have infinitely many more things to learn about. It is also another way of saying that it is essential to our nature that we should always have lots more that we might, and ought, do.

        Logical incompleteness entails epistemological incompleteness and ontological incompleteness.

        2 + 2 = 4 therefore God exists.

        There is indeed an Argument from Logical Truth (which includes the mathematical truths). In a nutshell, it is: Necessary Truth iff God; Necessary Truth; ergo, God. The first premise is entailed by the definition of “God,” which specifies (among other things) his perfect knowledge: his omniscience.

  2. May I attempt to summarize your argument? Freedom is a gap in causality. Whereas causality can be explained — or at any rate described — freedom remains a mystery and is therefore an instance of inexplicable grace.

    • I would say that the whole shooting match is an instance of grace, including all the causally closed bits.

      As constituting a coherent world, the past is without causal defect or gap. But the present is not at all like the past, because it is the process of becoming The present is a moment of action that has not yet resulted in a completed act. Until a quantum of becoming has finished becoming and come fully to be in act – has become fully actual – it simply does not yet exist as any definite thing, so as to have characteristics like relations to other things: causal influences and effects.

      Part of coming to be is choosing what shall be included in the causal past, and what to make of it.

      Jumping back to Gödelian terms, the past is consistent, and can be specified (at least in principle) by a logical calculus. But that logical calculus cannot be completed, and so nor can a past be completed; which is to say, that the process of becoming cannot be completed (even though a cosmos can be completed).

      Each present moment of becoming remains a mystery until it is past and wholly definite, for the simple reason that the present moment of becoming is not yet fully concrete, so as to be amenable to understanding or explanation. When we understand things, we are always understanding what is wholly in act, either as past, or as to us aeviternal, or as eternal.

      So, yes: to say that a moment of becoming is inexplicable is to say that it is free. Its causal relations to its past simply don’t exist until it, too, is past.

  3. Pingback: Kristor & Ilíon: Gödel, Creation, Evil, the Satan, &c. – The Orthosphere

  4. One thing Goedel’s Theorem proved was that some propositions, Goedelian propositions, derived from an axiomatic system could nonetheless not be proved by that system. God’s existence cannot be proved (it would mean giving into one of the Three Temptations of Christ). Goedel shows that even in mathematics some propositions can be seen to be true without being provable so our inability to prove God’s existence and yet to perceive that he is necessary is on a par with other truths.

    Further evidence that the earthly realm is not self-sufficient is indicated by the fact that symbolic logic has to include completely stupid and illogical items just to function. It is thus broken and limps. Logic requires that if the consequent of a conditional is true, then the entire proposition must be regarded as true. If p then q. p is the antecedent. q is the consequent thus “If Jimmy loves Doreess and has never eaten swiss cheese, drunk a cola, or stared at a cat then the speed of light is 299,792,458 m/s.” There is nothing logical about such absurdities.

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