Gödelian Incompleteness → Creaturely Freedom

It seems we cannot be free.

To each moment of decision, the schedule of inputs is what it is, and as completely constituting the matter of our decision, so it would seem that it completely forms our act therein. We choose what we wish to do, e.g., given our understanding of our circumstances as we find them as each new moment of life arises; but it does not seem that we choose our wishes, nor does it seem that we can choose what, how much or how well we understand. Decision begins with wishes and circumstances as all alike data.

Nor do we seem to be able to choose the way that we choose. The operation of decision – which is our lever of control over our experiences – is not itself subject to our decisions. We are not in control of our means of control.

It seems to us that we choose freely from among options, to be sure. But then, the entire schedule of options really open to us at any moment, however uncountably vast their number, are just as definite ex ante as the facts already accomplished that constitute the causal basis of decision.

Thus the bases, procedure and options of our decisions, being given to each moment of decision ab initio and so unchangeably, would seem to determine us to but one such option, again ab initio and unchangeably. What seems to us to be the free choice of a moment in our lives might then be no more than what it feels like to proceed from the entire schedule of the initial matter thereof to the one option that satisfies the desires felt as an aspect of those data.

Where in this account is there room for freedom?

That room may be found in Gödelian Incompleteness. But to see how this is so, we shall have to traverse several steps.

Consider first that in order to be a cosmos in the first place – a world, properly so called – the events that constitute it as such must if they are to hang together coherently as a causal system, and without conflict or contradiction, be completely ordered to each other, and so therefore in every respect amenable to complete and consistent specification by a series – an immense series, literally, as we shall see – of propositions in a stack of logical calculi consistent in themselves and with each other. This stipulation is the implicit presupposition of all science, all planning, indeed all life: that the world is in the first place orderly throughout, and in the second, and therefore, also intelligible.

Excursus: The fact that worlds must be ordered and intelligible under the terms of consistent logical calculi accounts for their spookily mathematical character.

On Gödelian Incompleteness, no one of the stack of logical calculi under which a world is ordered, and which it expresses, can account for it completely. Nor therefore can any finite stack of logical calculi: a finite stack of incomplete calculi is itself incomplete. Yet the cosmos does in fact exist, and consistently expresses logical calculi; while as ontologically definite it is by definition also logically complete; so the stack of logical calculi that can completely and consistently specify it is itself both complete and consistent. The stack is therefore infinite, and the specification string for any moment of any world is likewise infinite: immense.

This means that no finite entity ordered by that stack of logical calculi can be competent to specify itself. It cannot then be competent to understand or even know all that there is to know about itself. Nor a fortiori can it be competent to understand or know all that there is to know about any other.

The same holds for any finite congeries of finite events: it cannot possibly account for itself. Only an infinite mind could account for it, or for any part of it; or could, therefore, specify it, or any part of it. If anything is to be ordered, an infinite mind is needed.

Such is the Lucasian Argument from Gödelian Incompleteness.

Thus for any finite mind, howsoever sapient, reality is irreducibly mysterious.

Then while the data of any creaturely decision – its circumstances, its aims, its very way of becoming and process of decision – must be complete and consistent under the terms of the infinite stack of logical calculi, they cannot even as a whole and integral system account for themselves, in whole or in any part. Even as accomplished and definite, the cosmos cannot explain itself.

Thus the schedule of inputs to any decision cannot comprehend itself; neither then can it comprehend the incipient decision, of which it is the matter. Then before any event is complete, no finite mind can completely specify its causes; nor can any congeries of finite antecedent facts. Nor then a fortiori can any or all of them specify, or thereby determine, any other act – such as those in their futures.

 Excursus: This is one reason that free creatures cannot create free agents such as themselves.

Our causal inputs cannot themselves even determine their own characters ex post, even though they are already entirely determinate. They cannot understand themselves, or their pasts. Nor a fortiori can they determine their future in our acts.

Likewise the intentions of any act cannot by the agent thereof be completely specified or therefore understood.

So as we make our decisions, both the schedule of their causal inputs and conditions on the one hand, and on the other the exact character of the targets we aim to achieve, are incorrigibly mysterious to us; and on Gödelian Incompleteness, the incorrigibility of that mystery is logically inescapable. Yet we must act in order to be, and a fortiori to survive, to prosper, to reproduce; so, we do, laboring and travailing always under great uncertainty. And this Gödelian uncertainty is not just logical, but also, and therefore, ontological; for, whatever is must be logical. If no consistent logical calculus can be completed, then certainly no system of actual things instantiating and informed by it could be completed either.

Excursus: Thus is it that it is logically impossible for there to be such things as actualities – agents that act – that are not free. Thus is it that God cannot logically create a world of actual entities that, as free, is not subject to a Fall.

These two considerations – of the irreducibly mysterious character of the causal inputs of each act, and the irreducibly mysterious character of its intentions – capture well the phenomenal character of our choices. Why do I do what I do? I have a general idea, but am never quite sure. What am I trying to do? Rather vague about that, too.

Excursus: NB: God does completely know the exact character of his intention when he acts; he knows exactly what he is trying to achieve, because he is an infinite mind, and as such able to comprehend all the truths expressed in the infinite stack of logical calculi. Such is divine Providence. His knowledge is the factual basis in the formal realm – which is to say, in the divine mind – of our prevenient Grace, in virtue of which and by knowing it he brings each free moment from mere possibility into potentiality.

So is it then that actual beings are free ab initio, even though they be sufficiently ordered – ordered by their factual circumstances, and ordered to their intentions.

Finite being cannot be completed – either logically or therefore ontologically – by finite being, either formally or therefore actually; so is it then free per se.

Excursus: Ergo, finite beings such as we, or Apollo, cannot by definition be creators such as YHWH. This is why all traditional polytheism recurs to a Most High God, who can be such a creator as YHWH, and who created all the other gods.

The analysis differs however when it comes to whether finite beings can be completely specified by an infinite being, and if so, then as to whether that specification exhaustively determines them. But those are topics for some subsequent post. I’m already working on it.

19 thoughts on “Gödelian Incompleteness → Creaturely Freedom

  1. @Kristor – You will know this better than me – but I had thought that Godel’s incompleteness theorem applied to *closed* (‘tautological’) systems (such as mathematics or formal kinds of logic); rather than to something like divine creation – which is (presumably?) ongoing/ being added-to/ expanding?

    • What a terrific question. Yes, Gödel’s proof pertains entirely to logical calculi. JR Lucas realized that if concrete reality was ordered and therefore intelligible – which it certainly seems to be – then it had to be specifiable under the terms of a logical calculus, that was at least in principle perfectly amenable to rigorous mathematical formalization: a Theory of Everything. He saw at once that scientists since Pythagoras (at least) had been working on discovering that TOE, in complete – and, evidently, massively warranted – confidence that it was out there to begin with, and then discoverable. And he saw that the Incompleteness Theorem entailed that they could not ever complete that project, because no finite set of logical calculi can be completed.

      Another corollary is that finite creation cannot ever be completed. The bits that are already created, done, and in the past are of course complete. But although they must (by the Principle of Sufficient Reason) be completely explicable by a consistent set of logical calculi, that set must be infinite, so that we can’t ever finish explicating what has happened: there are an infinite number of true statements we could make about it, so even though those statements must be all mutually consistent, and cohere, they cannot be specified by any finite procedures.

      The set of finite procedures includes the processes of becoming – which is to say, of creation, action, and motion. Finite cognitive procedures, in which bits of knowledge about reality (i.e., information) are integrated into our stores of already known bits, are a subset of the relevant finite procedures that cannot ever finish understanding themselves, thereby enacting themselves. So creation cannot ever finish evolving – out rolling – from its First Principle, who is its First Principal.

      This world shall of course eventually end. All particular actions end; so likewise for all assemblages of such actions. For, action of any sort must be quantal; as Zeno demonstrated, and Planck verified, in no other way might any act or event be completed, definite, and so factual. Zeno showed that each bit of being – each it, each act completed – must be infinitely divisible, at least in principle (this is one of the reasons that there are infinitely many true things we can say about any of them). Thus no finite procedure can complete any part of any one of them; the aid of the application of an infinite process is needful if any act is even to start becoming. Acts are infinitely divisible in principle, but if anything is to happen concretely, they must be integral in actuality. There must be atoms of becoming, minimal quanta of action. And Planck discovered just how big the minimal quanta of action are, in our cosmos anyway.

      But although each particular event must end if anything is to happen, eventuation as such does not end; for no set of finite acts or motions can suffice to express all the truths implicit in any one of them, so that the finite process of that specification cannot be completed. Creation is an everlasting process. So while our cosmos will eventually end, that will not be the end of cosmic order. Something new will always happen, to supersede, evolve, and surpass what has gone before.

      NB also that finite acts must begin as well as end, if they are to be finite in the first place; if, i.e., they are to be limited *in any way whatever.* Any event constrained by any other must be finite; ergo, must have had a beginning.

  2. In the past I would have jumped on a post like this with both feet, but I find I no longer find any interest in doing that. If you want to believe that you can use math to prove metaphysical truths about god and freedom – go for it. To me it feels more like abuse than use, but I’m not the math police. Doug Hofstadter I think had a lengthy response to the Lucasian arguments against AI, but I’m not very interested in digging it up and replaying it.

    Also fundamentally you are proving something irrelevant – that we have imperfect representation of the world and our selves, thanks to Gödel. I have news for you, you don’t need any fancy math to know that. Indeed, all our knowledge is extremely incomplete and error-prone, given the limitations of our cognitive equipment. But ignorance, and the need to act with incomplete information, does not equal freedom.

    A more interesting argument would be to try to apply Gödel’s proof to God’s cognition – maybe it implies that even he can’t have perfect knowledge of his own creation, and freedom is the consequence. Maybe that solves the obvious incompatibility between god’s eternal omniscience and human freedom.

    • In the past I would have jumped on a post like this with both feet …

      .… he said, as he jumped in with both feet. No offense, a.morphous; I’m only teasing. I’m glad you are here, as your comment provokes a response, the development of which has already taught me a few things.

      If you want to believe that you can use math to prove metaphysical truths about god and freedom – go for it.

      If you want to believe that you cannot use math to prove metaphysical truths about God and freedom – go for it.

      Quite seriously, my spiritual director made the point two weeks ago that the reason people insist that there is no way in logic to demonstrate the existence of God is that people believe what they *want* to believe. If they don’t want to believe in the existence of God, then no proof is going to seem cogent to them. All such proofs are going to appear to them as no more than vain logic chopping, which is in no way dispositive – overlooking, conveniently, the fact that the conclusion of a valid syllogism on true premises is *necessarily* true, whether they like it or not.

      I saw this dynamic at work just last week. An old friend posted an account of Gödel’s Ontological Argument – a rigorous formalization of Anselm’s argument, as Gödel himself averred – and someone commented that Gödel couldn’t have been a great logician if he proved a conclusion that is not true. D’oh!

      … ignorance, and the need to act with incomplete information, does not equal freedom.

      It goes deeper than that. It’s not just about our ignorance. Inasmuch as facts of any sort – not just the facts of our cognition, but any facts whatever (of which the facts of our cognition are but one sort) – must be ordered by a consistent and complete logic if they are to cohere at all and so be factual, they are covered by the Incompleteness Theorem. This means that as finite, and so unable in their own being to account for the infinite truths of their causal antecedents, no fact other than God can take a complete, consistent, perfect account of its causal inputs. That means that subultimate facts per se are not completely caused. And that means that acts (which end in facts) are in some measure free.

      The truths of logic are necessarily true. I.e., they are true of everything whatsoever: everything must be logical, or else be nothing at all. Thus the truths of logic are the truths also of ontologic – and so of epistemologic (the epistemic being a department of the ontologic, which is in turn a department of the logic simpliciter). No finite act being competent to a perfect account in its own being (this is how all accounts are cast: by acts of agents taking – grasping, prehending – accounts of their causal objects) of its causal inputs, so no finite act can be perfectly constrained in its becoming by those inputs.

      A more interesting argument would be to try to apply Gödel’s proof to God’s cognition – maybe it implies that even he can’t have perfect knowledge of his own creation, and freedom is the consequence.

      You’ve nailed it, although Gödelian incompleteness has nought to do with it (because, being infinite, God is adequate to the infinite stack of logical calculi necessary to the specification of anything at all (indeed, he *just is* that stack; this is one of the things that is indicated by calling his Son the Lógos), so that he is not subject to logical incompleteness, as are all other beings), and with one minor yet crucial and massively consequential correction, bolded and italicized in the aftgoing:

      God can’t have perfect knowledge of his own creation prior thereto, and freedom is the consequence; for, there is nothing prior to God, or to his act, or therefore to his act of creation. That said, God knows his creation only insofar as it is complete, thus definite, thus real, and so possible to know. He can’t know what we shall do before we do it, because there is in him no before or after; and before we do what we do, there is nothing to know about it. So he knows what we do as we do it; and, being eternal, he knows what we do before all worlds, and as a forecondition of worlds per se.

      • Jumping with both feet would mean picking apart your argument and going over all the ways in which it makes a mockery of actual thought by blending the perfectly clear and incontrovertible mathematics with metaphysical nonsense. It’s too much work, and you wouldn’t listen to me anyway.

        Gödelian incompleteness has nought to do with it (because, being infinite, God is adequate to the infinite stack of logical calculi necessary to the specification of anything at all (indeed, he *just is* that stack; this is one of the things that is indicated by calling his Son the Lógos), so that he is not subject to logical incompleteness, as are all other beings)

        There is no “infinite stack of logical calculi”, you made that up. Gödel’s proof applies to any sufficiently powerful axiomatic system, and by your own mode of reasoning binds God as well as humans. Now, it’s true you can “fix” Gödelization by infinitely adding new axioms to your arithmetic, maybe you heard about that and garbled it, but that has nothing to do “the specification of anything at all”. (ok, there I go jumping in, but I’m going to pull myself out before getting any more of this nonsense on me).

        I saw this dynamic at work just last week. An old friend posted an account of Gödel’s Ontological Argument – a rigorous formalization of Anselm’s argument, as Gödel himself averred – and someone commented that Gödel couldn’t have been a great logician if he proved a conclusion that is not true. D’oh!

        That’s dumb, obviously the failings of Gödel’s Ontological Argument are not in logic, where he was a world-class expert, but in the relationship between logic and reality, where he was most emphatically not (he was so paranoid he basically starved himself to death).

      • Clever, clever boy.

        Gödelian incompleteness has nought to do with it (because, being infinite, God is adequate to the infinite stack of logical calculi necessary to the specification of anything at all (indeed, he *just is* that stack; this is one of the things that is indicated by calling his Son the Lógos), so that he is not subject to logical incompleteness, as are all other beings)

        There is no “infinite stack of logical calculi,” you made that up. Gödel’s proof applies to any sufficiently powerful axiomatic system, and by your own mode of reasoning binds God as well as humans. Now, it’s true you can “fix” Gödelization by infinitely adding new axioms to your arithmetic, maybe you heard about that and garbled it, but that has nothing to do “the specification of anything at all” …

        Fixing Gödelization by infinitely adding new axioms *just is* invoking an infinite stack of logical calculi; is what I *mean* by “infinite stack of logical calculi.” Add an axiom, you get a new calculus.

        It is obvious that no finite computational procedure is adequate to complete an infinite computational problem (in re the Halting Problem: a program that runs forever without arriving at an output that satisfies the criteria of the programmer is inadequate thereto: it is an inapt program, that presupposes some axiom errantly – or, is just badly written). But on Gödelian incompleteness, every computation is implicitly infinite, and so incompletable. Yet we complete computational procedures all the time. We do that by an implicit invocation of the entire stack of logical calculi, to which none of our calculations adequate; basic arithmetic then presupposes and invokes – and supervenes – the entire Gödelian stack. No infinite stack → no arithmetic.

        We do the same when we get up to fetch another cup of coffee, or pick our noses. No coffee can be fetched or nose picked without an implicit invocation of infinity. Thus God is invoked and employed at each creaturely occasion. We do not usually notice that this is so, because there is for us no other way for anything to be so; so that our invocation of the Almighty at each moment of our lives is in no way remarkable to us, or therefore worthy of our notice; which means that we invoke God at every moment of our existence without noticing that we do so.

        Whether any logical calculus has anything to do with the specification of any actual entity is quite a different matter. It can’t be possible to realize an actuality that is illogical – that, i.e., is inconsistent *with itself.* So any actual entity must instantiate some consistent logical calculi.

        … obviously the failings of Gödel’s Ontological Argument are not in logic, where he was a world-class expert …

        So you accept his conclusions? Hurrah! Welcome to the fold, a.morphous! I knew you’d make it home some day. When are you planning to start Inquirer’s Classes with some church?

        … but in the relationship between logic and reality, where he was most emphatically not (he was so paranoid he basically starved himself to death).

        Gödel’s personal worries about eating anything his wife had not prepared for him *have nothing whatever to do with the merits or demerits of his Ontological Argument, or to the truth value of its conclusion.* This is just obvious. I’m shocked that a smart guy like you would stoop to adduce his personal idiosyncrasies as relevant to the truth value of Gödel’s conclusions.

        Are your arguments bad because you happen to be suffering from a hangnail right now? Obviously not. Only a fool would think it. So likewise with paranoia, or any other human defect. I’m appalled that you would be so foolish as to suggest such a thing. I thought better of you.

        From the fact that you are paranoid it does not follow that they are not really after you. In Gödel’s case – given the radical vindication of the metaphysics of traditional Christian orthodoxy that his work in logic entails, with all the sequelae for social life of that vindication, that are so adverse to the modernist agenda of our socialist globalist Establishment – of *your* socialist globalist Establishment, let’s be clear – his paranoia is not perhaps altogether inapposite. If I was Lucifer, Gödel (with Lewis, Tolkien, Orwell, Havel, Solzhenitsyn, & alii) would be at the top of my persecution list. Gödel was perhaps correct in his paranoia. The demons – and their human slaves – probably were after him.

        The bottom line: if you are speaking Truth, the demons – and their human minions – are after you. That’s all.

      • But on Gödelian incompleteness, every computation is implicitly infinite, and so incompletable.

        I have no clue what that means or what justifies it, and I think I know this stuff fairly well. Computations are either finite or they aren’t, “implicit” is meaningless, “incompletable” is also meaningless (unless it is a synonym for infinite), and Gödelian incompleteness has roughly nothing to do with whether a computation is finite or not. Turing’s proof of the undecideability of the Halting Problem shows you can’t always tell whether a computation is finite or not, and that is analogous to Gödel’s incompleteness proof and you can (more or less) derive one from the other.

        It is obvious that no finite computational procedure is adequate to complete an infinite computational problem (in re the Halting Problem: a program that runs forever without arriving at an output that satisfies the criteria of the programmer is inadequate thereto: it is an inapt program, that presupposes some axiom errantly – or, is just badly written).

        This is a perfect example of your style: mixing perfectly well-understood mathematical concepts (the Halting Problem) with complete nonsense (or to be more charitable – with nonmathematical ideas that don’t really mix). It has nothing to do with “being inadequate to the criteria of the programmer”.

        This is what I mean by abuse of math. But I also said I wasn’t the math police, and here I go handing out citations. Sorry, do go on with your very creative style of thinking, I’m going to shut up now.

        No coffee can be fetched or nose picked without an implicit invocation of infinity. Thus God is invoked and employed at each creaturely occasion. We do not usually notice that this is so, because there is for us no other way for anything to be so; so that our invocation of the Almighty at each moment of our lives is in no way remarkable to us, or therefore worthy of our notice; which means that we invoke God at every moment of our existence without noticing that we do so.

        Well that is kind of beautiful, and it has almost no math abuse in it so it’s not triggering me.

        I’m shocked that a smart guy like you would stoop to adduce his personal idiosyncrasies as relevant to the truth value of Gödel’s conclusions.

        You (or your friend) were the one who introduced the subject of Gödel’s personal qualifications, and my point was that his errors were not in logic but in the relationship between logic and reality, something you seem also to be very confused about.

        If I was Lucifer, Gödel (with Lewis, Tolkien, Orwell, Havel, Solzhenitsyn, & alii) would be at the top of my persecution list.

        Well I have to admit the idea that Gödel was actually persecuted by “socialist globalists” angered or endangered by his discovery – that is a new one one me. Did you come up with that one on your own? He was close friends with Einstein, a well-known socialist globalist, for what that’s worth.

        Also a bit surprised to hear you enlisting leftists like Orwell and Havel on your side, but whatever.

      • But on Gödelian incompleteness, every computation is implicitly infinite, and so incompletable.

        I have no clue what that means or what justifies it …

        It’s simple, really. I’m surprised that you don’t see it immediately. Computation per se (whether or not it be finite) presupposes the entire Gödelian stack of logical calculi – the entire *infinite* Gödelian stack. E.g., consider just the numbers. Each of them presupposes all of them; but the extent of all of the numbers is infinite. So, no infinity, then no 5. But 5, ergo etc. 5 entails infinity, and vice versa. Thus we cannot complete the finite calculation of 2 + 3 except in virtue of numbers per se, which entail the infinity of the numbers.

        I don’t see what’s so hard about that.

        Computations are either finite or they aren’t …

        Sure, but even finite computations presuppose infinities, as of the numbers. Finite computations then supervene infinity. Infinity is the forecondition of finity, and not vice versa; for, no amount of finity could equate to infinity.

        Sorry, do go on with your very creative style of thinking, I’m going to shut up now.

        OK. I would however just suggest that it might profit you – and the rest of us here – if you were instead to try to understand what I’m getting at, and respond to that, rather than to caricatures of what we here say.

        Just saying. Up to you, whether you want to learn something from those who think differently – and, perhaps, correct them here or there – or just abhor them in public. Which latter course feels great, but is nowise productive, but rather only destructive. Up to you.

        You (or your friend) were the one who introduced the subject of Gödel’s personal qualifications …

        Sorry, I missed that. Where did any of us here (other than you) introduce Gödel’s personal character? I honestly can’t see that in this thread, prior to your adduction of his paranoia in respect to his food.
        But, in any case: what have Gödel’s personal defects to do with anything? Why are you even talking about them? Wait, I bet know: it is to tear him down, and with him all his conclusions, and with them all the corollaries thereof. I get that. Too bad. It can’t be done. His achievements are *logical.* He could be a mass murderer, and his demonstrations would stand nevertheless. And so would all their corollaries.

        The defects of the agent do not ruin his act.

        [Gödel’s] errors were not in logic but in the relationship between logic and reality, something you seem also to be very confused about

        I await your demonstration that reality is not logical.

        If Gödel demonstrated that x, then x absolutely, and without question. Not because Gödel, but because of the logic of the demonstration. Gödel himself, with all his defects, has nothing to do with the truth of his demonstrations, other than as the defective creaturely medium of their transmission to us.

        As to whether Gödel – or Havel – has been singled out for particular persecution by Lucifer, I cannot of course know, even though I know well what I should do in Lucifer’s place (being like him Fallen) about such reprobate rebels against my rule as they; as must be rather obvious. But whoever speaks the truth is ipso facto the enemy of the Father of Lies, no matter what his mundane politics. So, I take Einstein and his Spinozan ilk to be at bottom friends of the Truth, come what may – as Einstein so often himself so humbly averred.

      • Mathematics is perhaps the only area of human thought where it is possible to be absolutely certain and precise, and as you say, the truths of mathematics do not depend at all on the character or opinions or ideology or religious faith of the mathematicians involved.

        What you are writing is not mathematics, it’s a mishmash of mathematics and metaphysics and stuff that appears to be pulled out of thin air. It irritates me, because it appears to be appropriating and perverting the certainty of mathematics for purposes to which it is not suited. But maybe that’s just me. Your stuff reads as nonsense mathematically, but maybe it’s great as metaphysical fantasy loosely based on math-like concepts.

        The nature of the relationship between mathematics and broader reality is an interesting open question, or set of questions. To repeat my very obvious point for the third time: Gödel’s proof is an incontrovertibly true and accepted part of mathematics, and proves some things about mathematical objects (formal systems). Whether it thus proves the existence of God or the impossibility of AI is quite a different matter – those are non-mathematical questions, and my point about Gödel was not to attack his character but to point out that expertise in mathematics is not a guarantee of expertise in matters outside of mathematics, even if they are adjacent,

      • I shall content myself with the simple observation that in your latest comment you have offered not a single counterargument, but rather only expressed your distaste for metaphysical reasoning.

      • Well, what I said is that your theory doesn’t make sense, so I can’t make a counterargument, it is “not even wrong” as the saying goes. I did say it was possibly “great as metaphysical fantasy”, but I guess that is faint praise.

        I don’t think it’s quite right to say I have a distaste of metaphysical reasoning. In fact I quite admire your skill at it, even when it does not convince me. But I like it better when you do it without misusing mathematics.

        But yeah I don’t in general trust metaphysical reasoning. It requires more faith in the power of language and reason than I possess.

      • If what I have written is incoherent, so that it is strictly inconceivable, thus meaningless, and so as you say cannot therefore be either wrong or right, but is rather simply misthought – as “square circle” is a misthought – then it should be easy for you to show where that incoherence lies. You don’t. Instead, you just complain that it reads *to you* as nonsense mathematically, and leave it at that.

        Well, OK, we get it: a.morphous can’t make head or tail of what Kristor has written. So … what? You tell us about your feelings of confusion, but not about your thought. So what you say tells us nothing about what I wrote, or what might be wrong with it. It tells us rather only some stuff about a.morphous.

        I am not surprised or perturbed that what I wrote reads to you as mathematical nonsense, inasmuch as what I wrote is *not math,* but metaphysics. It is not a mathematical extension of Gödel, but an application to ontology. If you are reading it as math, you are engaging in a category error, in rather the way that a mathematician might err if he were to take the physical constant c as nothing but a variable.

        That said, metaphysics – like physics – is formalizable. Gödel formalized his Ontological Argument. The tricky part is to figure out exactly what the terms of the formalization truly denote, so that we might then tell whether the valid conclusions of the formalization really pertain to the actual world. Math suffers that difficulty less than other formalizable discourses – economics, physics, and so forth – because the entities it considers are purely abstract, whereas those of other disciplines have concrete denotations.

      • So what you say tells us nothing about what I wrote, or what might be wrong with it. It tells us rather only some stuff about a.morphous.

        Well you are probably right, I find that a more interesting topic. At any rate, I should have chosen my words better and said that your theory does not make sense to me. It obviously makes sense to you and maybe everybody on earth except me.

        But you don’t seem to appreciate what I’ve written; I have said repeatedly that while it doesn’t make sense as math it might make sense as something else. Which means that I am at least trying to understand it, even if I am not ultimately succeeding.

        That said, metaphysics – like physics – is formalizable…The tricky part is to figure out exactly what the terms of the formalization truly denote, so that we might then tell whether the valid conclusions of the formalization really pertain to the actual world.

        Uh, yah think? That’s not just “the tricky part”, that is the whole enchilada of a whole host of philosophical questions.

        Math suffers that difficulty less than other formalizable discourses – economics, physics, and so forth – because the entities it considers are purely abstract, whereas those of other disciplines have concrete denotations.

        FWIW this echoes and agrees what I have been saying above, that absolute precision is the norm in mathematics but not in other areas of thought.

        The question of whether metaphysics is formalizable is interesting. My first answer would be no: the ultimate aspects of reality are always going to elude representation and reason. That intuition, I realize, is grounded out in my own personal abuse of Gödelishness. it’s related to that kind of metarecursive/strange loop sort of thinking. But I would not claim that Gödelian reasoning proves me correct; that would just be stupid.

        I would think you would agree given you say “reality is irreducibly mysterious”, but apparently the mystery is no match for your logic.

      • A.morphous, I thank you with a full heart for your continued engagement with this stuff, and for your work in trying to understand what I write. I take it as a compliment – and a complement.

        The question of whether metaphysics is formalizable is interesting. My first answer would be no: the ultimate aspects of reality are always going to elude representation and reason. That intuition, I realize, is grounded out in my own personal abuse of Gödelishness. … But I would not claim that Gödelian reasoning proves me correct; that would just be stupid.

        Figuring out the proper and correct denotation of the terms of a formalization of anything other than math will ever be incorrigibly tricky – i.e., incompletable (in practice, I mean) – because reality is indeed irreducibly mysterious at bottom (the formal analogue of that basic mystery is to be found in axioms), if only because there are of any concrete an infinite number of true statements we could make. On Gödel, even purely abstract formal systems cannot be consistently completed; how then might we ever complete any consistent formalization of concretes? Gödel’s argument does not *demonstrate* that we cannot do so, to be sure. But it is hard to see how we could, if the thing cannot be managed even with geometry.

        From that it does not of course follow that we cannot learn any truths at all by our formalizations of concretes. We see now but darkly, and through a glass; but we do see. If we did not, why then we’d have no notion that we don’t yet see very well. So formalizations of physics, or economics, or biology, or, yes, metaphysics, can still help us discern truths. We can at least learn from them what meanings of terms just will not work in a consistent formal system; we can, that is, at least approach perfect consistency – although not completeness – by ruling out inconsistent notions. So we can see that, e.g., nominalism can’t be true, for it is autophagous.

  3. Unlike the Formless One, I know you’re on to something, Kristor.

    Godel’s Theorem, in brief, says that any logical system big enough to contain arithmetic is formally incomplete: the system contains propositions concerning valid concepts within the system which are logically neither provable nor disprovable. More axioms are needed to tie them down. But then you have a larger system which must contain additional formally undecidable propositions, etc.

    So the Theorem has a floor (must contain arithmetic) but no ceiling. If reality is governed by logic (which basically means that its parts have natures that can be identified with precision), then any system describing reality, as long as it includes arithmetic, comes under the jurisdiction of the Theorem

    Some people find “mystery” unacceptable. e.g., the mystery of the Trinity (one God but three different Persons.) But mystery in the sense of not having a plausible mechanism / proof will always exist. It’s a theorem.

    • Mystery is a term denoting the incorrigible perplexity of finity in its grapple with the infinite. It is Jacob’s wrestle in the night; to which, he erected an altar, and so a civilization.

      Theology that makes plain common sense to common finite minds – that, i.e., is easy to understand – cannot be true. Any theology that is adequate to reality must be impossible for us to understand as we understand breakfast and the bus schedule. Indeed, when we examine them closely, lo, we find that we *don’t* understand either our morning toast and eggs or the bus schedule (let alone the bus or, a fortiori, the egg) – and, indeed, that we *just can’t* understand them. All careful philosophical effort tells that truth, and indeed hits us over the head with it: reality is fundamentally mysterious. That’s all.

      NB: this does not at all mean that reality is anywise unintelligible, or chaotic.

  4. Though many believe that connecting metaphysics and Godel’s theorems is inherently suspect, it is not. Godel himself went to meetings of the Vienna Circle, who rejected all metaphysical statements as meaningless. He familiar with positivism, but rejected it. In fact, he said somewhere that his motivation for seeking the incompleteness theorem was *because of* his beliefs about the reality of mathematics.

    Many logicians at the time believed that math was reducible to syntactic rules, so they didn’t try to look for incompleteness. A notable exception was Emil Post, who famously claimed that he “would have proven Godel’s theorem in 1922 had he been Godel” (in other words, he had the idea, but not the techniques).

    Ironically, in the course of studying for his citizenship hearing, Godel believed he had found an inconsistency in the US Constitution, which would allow the country to be turned into a dictatorship legally (here is a description by Oscar Morgenstern, who was there: https://robert.accettura.com/wp-content/uploads/2010/10/Morgenstern_onGoedelcitizenship.pdf). Maybe it sounded crazy in 1947, but in 2022 …

    After Godel’s death, it was found that he had many notebooks (I think written in shorthand), marked with the subject they dealt with, among which were politics, theology, and philosophy.

    Godel’s biographer John Dawson actually looked among these notebooks for a description of the inconsistency in the Consitution but wasn’t able to find anything, so it’s unknown what it was.

    • Godel: “I have discovered a truly marvelous proof that the Constitution can lead to dictatorship, which this margin is too narrow to contain.”

      (With apologies to Fermat.)

  5. Freedom is observed reality to the non-npcs. Any theorem that implies there can be no freedom can only apply to the demon possessed npc trash.

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