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.

The Kalam Ontological Argument

The Kalam Cosmological Argument is well known: if the cosmos had no beginning, it would not require a creator. Yay, for the atheist! But then, the cosmos would be infinitely old; and, so, it would be impossible for finite events (such as all those that constitute reality insofar as we can apprehend it) to complete the infinite traversal from the infinitely distant past to any moment whatever of the cosmogonic timeline. Zeno would be pleased. There could then be no present moment, for no such present moment could ever yet have happened. Nothing whatsoever could then ever happen. But, tace Zeno, there is always a present moment, events do transpire, ergo etc. The infinity of the past is refuted by the reality of any present event (or any past event, for that matter). The cosmos is therefore temporally finite, had a beginning, so stands in need of an extracosmic cause, and so forth: God, QED.

But there is also an analogous Kalam Ontological Argument. Ontological arguments proceed from a priori premises, that do not at all depend upon a posteriori observation, such as your indisputable observation of this present moment of your experience. They work whether or not there is anything out there to be observed, or anyone to observe it.

Philosophical Skeleton Keys: The Stack of Worlds & the Literal Fall; &c.

The stack of worlds implicit in Gödel’s Incompleteness Theorems furnishes a way of understanding the Fall as having happened literally, and in (so far as I can tell) complete congruity with the latter day scientific model of our own world’s history – and, indeed, with that of any other – and with the account in Genesis.

This post supervenes two others in a series respecting divers Philosophical Skeleton Keys: first, The Stack of Worlds, and then, The Play: Its Wright, Players, & Characters. It will I think be easier to understand this post if you review them, before essaying this one.

Philosophical Skeleton Keys: The Play: Its Wright, Players, & Characters

This post is a sequel to my post on the stack of worlds. It tries to understand a few things about how a stack of worlds might work – or, perhaps, *must* work – and how those workings might help us untangle a few perplexities that have bedeviled thinkers for millennia. It is absurdly long, and for that I beg forgiveness. But I find there is little I can do about that, at present: when the inspiration comes, it comes as a unit, and the overwhelming necessity is just to get it all down before it vanishes.

Philosophical Skeleton Keys: The Stack of Worlds

This post supervenes my recent post On Some Happy Corollaries of Gödel’s Incompleteness Theorems (so you might want to review that post, and the earlier posts it cites in turn, in order to find yourself quite oriented in what follows (sorry, dear reader: not everything is TLDR)).

There is much talk in traditional cosmology of a stack of heavens above our own, and likewise of hells below. The hierarchy of angelic choirs echoes that stack. Most pagan pantheons feature such hierarchies of gods, with a Most High God above all gods, whom they worship, and who lives in the Highest Heaven which is above all the heavens. There is talk too of other worlds parallel to our own (such, e.g., as Jotunheim in the mythic scheme of the Vikings), that might communicate with each other (as at Ragnarok, when the giants of Jotunheim make war upon the men of Middle Earth and the gods of Asgard), so as to form a world of worlds.

That sort of talk struck me at first as fantastic, and so relatively irreal – despite its irresistible odor of concrete factuality, and its ubiquity in the traditions of Earth, and thus its uncanny tinct of credibility. There is also the difficulty that there is a certain beauty in the notion, that cannot be found in the flat idea that our world (however generously conceived (as with the various sorts of branching cosmoi proposed by this or that metacosmology)) is all there is. Then at last there is the ancient conviction of the Great Chain of Being, no link of which might be concretely missing if any part of the chain were to find concrete instantiation.

Kristor & Ilíon: Gödel, Creation, Evil, the Satan, &c.

In my last post on some happy corollaries of Gödel’s Incompleteness Theorems, I tipped my hat to our long time commenter (and an orthospherean from before there was an Orthosphere) Ilíon, who had corrected some early posts of mine on the topic. Ilíon and I then took up a conversation by email, which he has reproduced at his own blog. I intend to continue my conversation with him there in the comments thread.

Ilíon is – how can I say this? Ilíon is like Auster and Zippy, with whom he tangled, both. He’s one of those sharp edged minds that always manage to teach me, if only by forcing me to get really clear on what the heck I mean, and, so, think. Our converse has always been challenging, and – without fail – charitable and friendly. And edifying.

He is also willing to entertain radical hypotheses, which makes him interesting. But he is just as ready to slice them to bits with an entirely orthodox Christian razor.

I therefore recommend to you my conversation with my longtime friend, Ilíon.

 

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.

Mystery and Order; the right and left hemispheres

In The Master and His Emissary, Iain McGilchrist writes that a creature like a bird needs two types of consciousness simultaneously. It needs to be able to focus on something specific, such as pecking at food, while it also needs to keep an eye out for predators which requires a more general awareness of environment.

These are quite different activities. The Left Hemisphere (LH) is adapted for a narrow focus. The Right Hemisphere (RH) for the broad. The brains of human beings have the same division of function.

The LH governs the right side of the body, the RH, the left side. With birds, the left eye (RH) looks for predators, the right eye (LH) focuses on food and specifics. Since danger can take many forms and is unpredictable, the RH has to be very open-minded.

Goedel’s Theorem

Kurt Gödel^[1] was a Platonist,^[2] logician and mathematician who developed the intention of making a profound and lasting impact on philosophical mathematics. His next task was to think of something! Amazingly, at the age of twenty five, he achieved his goal, publishing his incompleteness theorem.

Kurt Gödel and Einstein

A good friend of Albert Einstein’s, Einstein once said that late in life when his own work was not amounting to much, the only reason he bothered going to his office at the Institute for Advanced Study at Princeton was for the pleasure of walking home with Gödel.

John von Neumann wrote: “Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.”^[3]

While at university, Gödel attended a seminar run by David Hilbert who posed the problem of completeness: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?

The Halting Problem – there is, definitively, more to thinking than computation

Alan Turing

Kurt Gödel’s Incompleteness Theorem[1] was inspired by David Hilbert’s question “Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?” Hilbert played the same role regarding Alan Turing’s proof of the halting problem. Hilbert had asked: “Is there some mechanical procedure [an algorithm] for answering all mathematical problems, belonging to some broad, but well-defined class?”[2] In German this is called Entscheidungsproblem – the decision problem.[3]

Turing found that he could answer this question by framing it in terms of a Turing machine[4] – could there be a program that could determine whether any other arbitrary computer program and input would eventually stop or just loop forever? This was called the halting problem.

“Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.”[5]