In his brilliant, aphoristic demolition of the modern, Pure: Modernity, Philosophy and the One, philosopher Mark Anderson explains in a few short paragraphs why nature cannot explain nature:

A particle of matter

isbecause of an act of existence for which it itself is not responsible. Itis what it isbecause of its microstructure, the specific and stable organization of its constitutive elements – in a word, its form, which it itself does not produce. The same is true,mutatis mutandis, of forces and laws of nature, which neither bring themselves into being nor cause their specific and essential character.

The materialist would like to explain the world in full by means of the attributes and arrangements of material particles in conjunction with natural forces and the laws that determine their appearance and application. But any such explanation necessarily presupposes the existence and ordered constitution of the particles, the forces, and the laws themselves. Matter and its properties, natural forces and the laws that govern them, are neither self-generating nor self-explanatory; they depend utterly upon the ontologically prior acts of existence and form. Without these metaphysical principles there can be no physical reality.

Moral: physics, being derivative, will never provide the fundamental explanation of reality. [p. 37 ff.]

JR Lucas argues that this incapacity of natural history to explain itself is entailed by Gödel’s Incompleteness Theorem. Gödel showed that every consistent formal system or logical calculus that is adequate at least to simple arithmetic (that, i.e., contains the natural numbers and the operations of addition and subtraction), is capable of generating true statements that cannot be demonstrated in the terms of the calculus. Thus no formal system is capable of demonstrating all the truths that it can express; formal systems are all therefore incomplete.

[Gödel has proved] that a human being cannot produce a formal proof of the consistency of a logistic calculus inside the calculus itself: but there is no objection to going outside the logistic calculus and no objection to producing informal arguments for the consistency either of a logistic calculus or of something less formal and less systematized. Such informal arguments will not be able to be completely formalised: but then the whole tenor of Gödel’s results is that we ought not to ask, and cannot obtain, complete formalisation. [The Freedom of the Will, p. 162]

Lucas then cashes out his argument:

If we are consistent and can say that we are, it follows without more ado from Gödel’s second theorem that we cannot be completely described as physical systems instantiating some logistic calculus. [ibid]

In proposing this argument, Lucas was concerned primarily to demonstrate the freedom of the human will. If no formal system can completely describe us, then no logic – such as the logic and order concretely embodied in, and expressed by, the past – can completely determine us. But it is easy to see, as Lucas certainly did, how this argument applies likewise equally for any system of concrete actualities, whether physically implemented or purely immaterial. No consistent logical calculus can be completed. But then, any system of concrete actualities that instantiates a consistent logical calculus – which is to say, any system of things, any world, or any portion of any world, that is either causally coherent and orderly on the one hand, or intelligible on the other – can complete its own description.

To say that no world’s logic can fully account for itself is just to say that that no world can cadge together a complete causal account of itself. But note then that to say that a world cannot wholly account for itself causally is *ipso facto* just to say that it cannot wholly cause itself. Worlds as such, then, stand in need of creators.

Thus not only is it impossible for science to explain or justify science, but it is impossible to complete the scientific formalisation – that is to say, the rigorous scientific understanding – of the world, or for that matter any portion of any world. So, science begins in mystery, issues from and proceeds in mystery, and must always point to mystery.

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Kristor:“Gödel showed that every consistent formal system or logical calculus that is adequate at least to simple arithmetic (that, i.e., contains the natural numbers and the operations of addition and subtraction), is capable of generating true statements that cannot be demonstrated in the terms of the calculus.”Actually, what you’ve said here is incorrect and quite misleading. If you consider the next thing you said, perhaps you can see the point I aim at.

“

Thus no formal system is capable of demonstrating all the truths that it can express; formal systems are all therefore incomplete.”It is not true that a consistent formal axiomatic system “is capable of generating true statements that cannot be demonstrated in the terms of the calculus” — for, to generate any newly expressed statement via the rules applied to the already-espressed statements of a consistent formal axiomatic system is precisely to “demonstrate[ it] in the terms of the calculus”.

Rather, what you should have said is something more like this: “For any consistent formal [axiomatic] system or logical calculus that is adequate at least to simple arithmetic, there exist true statements which may be expressed in terms of that system but cannot be derived from the system itself; that is, the true statement cannot be generated by the application of the system’s rules to it axioms and previously expressed statements.”

I hope you have grasped the vital distinction I have tried to draw, and thus grasp how incorrect, how upside-down, what you initially said is.

And, by the by, this also explains/demonstrates that minds are not ‘formal axiomatic systems’, that minds are not computers … and that computers are not (and cannot be) minds.

Ilion, great to hear from you. You are absolutely right about this, of course. Mea culpa! I had exactly the idea in mind that you have expressed, and managed by too nice a concern for style to commit a huge error of diction. Having written the passage using “expressing,” I then grew worried about using “express” too frequently in that section of the post, so I went back and substituted “generating.” Which, of course, greatly *weakens* my argument. D’oh!

I am grateful to you for having pointed out my error.

Doggone it; I used “express” three times in the first paragraph of this comment!

Yeah, it was pretty clear to me that you probably did already correctly understand the

meaningof Gödel’s work — it’s just that it’s so easy to state it backwards, which then just reinforces the incorrect understanding that most persons who try to speak on it have.Ilion, you write:

I am in sympathy with this view, given our traditional post-Cartesian conception of the matter constituting a physical computer. But does it not depend upon a treatment of matter (or whatever the substrate of the computer might be) as an actualization of a consistent, complete formal system – which, per Gödel, no (finite) actuality can possibly be? In fact – I never thought of this before – doesn’t the Incompleteness Theorem effectually show that the Cartesian notion of

res extensa,simpliciter, is just incoherent – as in, illogical?This opens the door to something like panpsychism, at least in the form of Wheeler’s “it from bit.” If matter is indeed inherently spooky in that way, then there might conceivably be a way to implement a mind on a physical substrate. I suppose the real question is whether such a mind would really be a Turing machine. I suppose it wouldn’t. Hm.

But then there’s this: if no sort of actual substrate of a computer can be an implementation of a complete formal system, is a (finite) Turing machine even possible in the first place? It would seem that the answer is “no.” In which case, our computers are

approximationsof Turing machines.Hi ilion7. I think it is good you took the time to explain the important difference between what Kristor said and what Godel states.

You say: “this also explains/demonstrates that minds are not ‘formal axiomatic systems’, that minds are not computers … and that computers are not (and cannot be) minds.”

That is so only if our minds are taken to be formal systems which are both (i) consistent and (ii) adequate for at least simple arithmetic. It seems (ii) is obviously satisfied. But what reason do we have to think that our minds are consistent in the sense that we cannot derive from our beliefs both some formula P and its negation? In fact, prima facie, that seems patently false.

I am open to the deduction of these philosophical conclusions from mathematical results and I am quite sympathetic to the conclusions. However, some of these supposed implications from Godel I am not so sure of.

Also, my question may just be arising from a misunderstanding of the Godelian argument against mechanism! If so, please correct me. I’m taking a course dedicated entirely to the two incompleteness theorems but we cover solely the mathematical/logical aspects of them, apart from the philosophical consequences which might follow. So this is an area I am not as familiar with. Thanks.

awatkins, it seems to me that the question of whether our minds are consistent is not at all trivial. I agree that it *seems* obviously true that they are not. But it seems to me that when we derive contradictory conclusions from the same data and using the same algorithms, what is happening – what *has* to be happening – is that there has been an inadvertent error somewhere. Indeed, deriving a contradiction is a certain sign either that one has erred logically or that one’s data are noisy, and – in healthy minds – prompts deliberation that aims to discover the error and correct it.

If our minds were not anyhow consistent, how could we either detect such errors, or know when we had corrected them? So it seems to me that our minds must be consistent at some level.

Aquinas was especially good at that sort of deliberation: at repair of errors and resolution of conflicts in the philosophical discourse. He’s always showing that x = y from one point of view, but not from another.

Another way to state your conclusion is that only that which is both consistent and ‘infinite’ can explain itself.

Just so. That’s exactly the conclusion I was driving at. By extension, no finite actuality is achievable in the absence of a prior infinite actuality. Existence of any particular being presupposes being as such.

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Absolutely correct.

Ulrich Mohrhoff in his Pondicherry Interpretation of Quantum Mechanics (PIQM) shows quite convincingly that *matter itself* is a contingent reality and only can be understood as an emanation of a divine mind, ie: God.