Gödelian arguments against mechanism : what was wrong and how to do instead

Introduction

The question whether consciousness is a machine (mathematical structure) is one of the biggest philosophical debates of present times: lots of people are still religious, while most scientists seem to be materialists (thinking that consciousness emerges from the laws of physics)... with a few exceptions. Now if consciousness is not material, while science is one of the noblest activity of the mind, this raises the question whether the immateriality of consciousness also plays any crucial role in scientific thinking, like in the rest of life, so that scientificity itself is irreducible to any mechanical process.
In scientific fields other than pure mathematics, non purely mechanical thinking can of course be crucial to make sense of problems which are somewhat fuzzy, not completely formalizable, and discover relevant approaches; but there does not appear to be any clear way of justifying such a fuzzy claim, of the distinction between the insights of a creative mind and a bunch of para-logical computations.
Now for activities of purely mathematical research, which gains the advantage of some mathematical clarity, different candidate aspects may be considered: The possibility of the latter was unknown before Godel's discovery of his Incompleteness theorem, which gave actual sense to the concept of unprovable truth and formally established that, roughly speaking, formally unprovable truths exist, which came as a surprise to the mathematicians of that time. But once done, the question remains, how can minds be compared with machines for their skills in this field ?

The current state of the debate

Looking at existing debates, I am extremely puzzled at the terrible state of the debate. It is normal that anti-mechanists have to make their case and withstand criticism. But what came is just a couple of amateur anti-mechanists unseriously uttering a few random naive suggestions. Resultingly, the job of refuting that almost empty set of candidate "arguments", to which the crowd of mechanists rushed, was very easy, but it is ultimately rather pointless to observe continuous refutations of a defense which was never seriously attempted in the first place.
I won't report much of existing refutations here, but just a few crucial points, including some I did not see yet in the few articles I looked through.
A general mistake was, mistaking the incompleteness theorem as if it already contained something of a positive case, or even, if it already sufficed. Should we repeat : it is a theorem of arithmetic. Therefore, all what it proves, can be automatically verified. And no strictly logical deduction from that theorem, ever has a chance to teach us something about knowledge abilities of the mind and how these might differ from formal deduction either: no mind in, no mind out.
A specific confusion by some, is to forget that the actual way of looking "outside" to conclude in the truth of the unprovable Godel proposition G, is not any miraculous insight of the mind, but a mere arithmetic deduction from looking at the case when the considered theory is consistent. Indeed the truth of G obtained in the proof of the first incompleteness theorem, turns out, in the light of the second incompleteness theorem (showing G formally equivalent to the formula "theory T is consistent"), to be the mere explicitation of the assumed premise in the implication "if T is consistent then G is unprovable there".
After Godel who first had suggestions on the topic, came Lucas who, in an act of supreme laziness, tried to switch the burden of proof, requesting mechanists to make their case, and uttering suggestions on how such a case might be criticized. This is of course illegitimate : since the knowledge abilities of machines is limited, mechanists, who believe that they are machines, cannot have a problem with the event of not knowing answers to challenges.
Then came Penrose whose argument starts from the following idea of what a mind might know, which a machine couldn't : "I know that I am sound". This is quite a ridiculous proposition of difference; to show why, let us develop a few more points of pure logic, showing deep similarities between some aspects of minds of mechanists (idealized to satisfy, for the sake of the argument, some conceivable qualities pointed out by these anti-mechanist debaters), and pure mathematical theories, so that, in these aspects, no problem can be shown with the idea that they were machines.

Algorithms cannot identify themselves

Just like mechanists think that they are algorithms but do not know which algorithm they may be, algorithms themselves do not know which algorithms they are either; the same goes for mathematical theories. Indeed One may consider the possibility for a mechanist to discover which algorithm he is, by biological studies: observing and analyzing the working of brains... however such a knowledge would be empirical, not mathematical. It could not be any absolute knowledge. Indeed for a mechanist thinking he is making biological experiments on brains, no mathematical necessity, be it formal or intuitive, can ever refute the risk for himself to actually be a mere brain in a vat being experimented by someone else, receiving virtual reality sensations of "discovering things about the brain", which would actually differ from the true ways the brain works.

Machines can simulate "creativity" as well

Also the "creativity" of possibly discovering a truth by means yet unsuspected and not logically deduced from previous thoughts, is so easily imitated by the possibility for a machine to use an axioms system, only using some axioms at first (whose consequences are anyway valid), then suddenly starting to use other available axioms not yet previously used. Any clearly expressible difference between the "creativity" of minds and machines behavior is so boringly absent in this aspect of things.

Do mathematical theories regard themselves as sound ?

Now when, inside a given theory T, an object version of the same theory is considered, what qualities may it appear to have ? Its consistency cannot be proven, says the incompleteness theorem. And even if we added to T the formula CT of its consistency, to form its Gödelized theory T', this theory T' still could not prove its own consistency. But... is this "fear" of inconsistency a very serious one ? And if we wanted to recursively add to a theory the successive formulas of consistency of the previous version, would we be making here any act of mindful creativity or not ?
In fact, no and no.
Instead of discussing consistency, it may be more relevant to discuss soundness. Here let us qualify a theory as "sound" if all its theorems expressed in first-order arithmetic, are true in the standard ℕ. This means it is true in some model whose set of integers is elementarily equivalent to the standard ℕ (we might even require that it has a model whose set of integers is standard, which isn't strictly equivalent but the difference doesn't matter).

Gödelisation is no imaginative act

People just learning the incompleteness theorem showing that from the consistency of a theory we cannot deduce the constency of its Gödelised (the next theory obtained by adding to it its formula of consistency), may have a little surprise at this : Gödelisation provably preserves soundness ! Indeed, if a theory is sound then it is consistent, so that its consistency CT is a true arithmetical property...which, added to the axioms, preserves their soundness (it cannot logically imply any false arithmetical formula F because the soundness of the theory prevents it from proving the false arithmetical formula that CT ⇒ F). Here we cannot replace the Gödelisation step by that of adding the statement of soundness, because, while "consistency" is an arithmetic formula, "soundness" isn't.
This allows for inductive reasoning: for any number n and any sound theory T, its n-th gödelised Tn is sound, thus consistent. Therefore, like any consistent theory, Tn and thus also T faces the possibility (undecidable from its own viewpoint) to see that (the image of) Tn is consistent while that of Tn+1 is inconsistent. Anyway the inconsistency of (the image of) Tn+1 logically implies that (the image of) T is unsound.

Any consistent theory "almost" sees itself sound anyway

Some theories, and even "natural" ones like arithmetic or ZF, have the following interesting property: their axioms list is infinite, and the statement of consistency (as formalized in the theory, thus relative to the current model) of any finite subset of this list, is a logical consequence of other axioms coming later in the list. In the case of ZF, even the statement of "soundness" has this property (while arithmetic cannot formulate soundness). So, when such a theory looks at itself (a version of itself produced by the same rule), it will see any of the axioms (and the resulting theorems) produced by that theory in a "reasonable" amount of time (where "reasonable" means : clearly a standard number, while standardness is inexpressible inside the theory) as "certified" to be consistent (or even sound) by the "insights" that it will "feel" when it will output its own later axioms and their logical consequences. Now if that looks like an amazing property of a theory, the rest of possible theories are not much worse anyway. Indeed, any consistent theory defined as the output of an algorithm, will anyway regard as "sound" all the logical consequences that itself (seen as object) will produce in "any describable time" (actually standard time, while standardness is not definable)... very simply because it will agree with itself about these outputs and logical consequences. Precisely, any theory strong enough (using further objects beyond arithmetic) to have a general definition of the interpretations of all arithmetic formulas (seen as objects), will see all the arithmetic theorems derived "in reasonable time" by an image of itself, as arithmetically true and thus free of arithmetical contradictions (regardless the length), but just cannot exclude the risk for the proofs of that theory using objects beyond numbers, to lead to contradictions after "unreasonable" times, as their general consistency could not be checked by interpreting these formulas on the corresponding objects of the current model.
The point of restricting the recognition of the truth of theorems to those "proven in a reasonable time" logically matters: soundness implies consistency when both concepts are equally applied to a theory seen as an object. When, in the section on the concepts of truth in mathematics, we mentioned that "The formal consistency of this set theory" do not result from "The arithmetic theorems in its framework", this was meant under the crucial restriction of only accepting those "arithmetic theorems" which can be proven "in a reasonable time", to not be confused with the statement of soundness which includes the truth all formulas whose image in some model might be "theorems" with possibly "unreasonably long" proofs.

Now, here is the question : can we expect, as Penrose suggested, that minds may have the miraculous quality of being so insightful about knowing their own soundness, that they could be justified to certify that another version of themselves (another mind) will anyway remain sound in all its insightful truth discoveries, even if made to work in producing such non-purely-logical truth insights during an indescribable (actually non-standard) amount of time ?

What arguments can be made instead

As a formally proven theorem, the incompleteness theorem cannot suffice to inform us about the mind and eventually show any of its abilities that would transcend any machines. However, it is still good at providing good questions. But for this if we must care to read it properly. A short misleading formulation was:
"Any (algorithmic) consistent theory is unable to prove its own consistency"
Hey, to prove what ? This is not a well-formed mathematical sentence ! Anything that a theory might be able or unable to prove, needs to be a clearly expressed formula. The formula which any good theory is unable to prove, cannot be saying "I am consistent", since theories cannot know who they are. Instead, what is involved is the consistency of a specific theory, which incidentally happens to coincide with the definition of the theory where the provability is considered, but that theory has itself no idea about this coincidence which it cannot make sense of, and which is out of subject anyway. The more accurate expression of that theorem is
For any theory defined somehow, if it is consistent then it cannot prove the formula that "The theory defined as (...) is consistent", where "(...)" is a copy of the same definition of a theory.
So the question, for which we may compare the abilities of minds and machines, is about knowing the consistency of some specific theories: as for any good theory, its consistency remains formally unprovable by some machines (namely those drawing their deductions from a no stronger theory), can the mind be better at knowing that it is actually consistent (and sound) ? Now, based on the studies of : Here is the main articulation of the argument I propose:
  1. ZF is a VERY strong axiomatic system; still, with some serious effort, a good mathematician's mind can grasp proper justifications for its soundness and consistency.
  2. Thus, according to the incompleteness theorem, any candidate formal system which may account for thoughts in general, and for the abilities of a good mathematician's mind, would have to be an even stronger system.
  3. However, the principle of natural selection is itself a rather simple kind of "algorithm". So if it had to be placed somehow in the strength hierarchy of mathematical theories, its rank there would be quite low. Then the mechanist viewpoint excludes all concepts about the hierarchy of existing infinities in mathematics as out of subject in his view of the mind as a finite computational entity which just happens to have beliefs, in a way actually disconnected from any issue of real truth. This situation is similar to the case of the mechanist concept of morality, which tries to explain morality as a behavior which happened to be adopted because it was selected by evolution for its selective advantage, but that kind of explanation remains disconnected from any issue of possible real truth of morality issues which, in actual thoughts, moral subjects usually try to grasp. So in his logic, the mechanist would have to dismiss any such beliefs that would happen to be developed by minds, as mere illusions and useful speculations; he cannot account for the possible legitimacy to actually know and acknowledge such abstract mathematical insights as valid ones.
  4. Then, this mechanist framework may account for phenomenas of "useful beliefs" selected by criteria of empirical validity in finite systems. However, any empirically useful beliefs and understandings needed for all survival purposes (the kinds of beliefs and understandings which we might expect natural evolution to select), are just about understanding the properties of large but essentially finite systems. For this, some elaboration is needed but still has nothing to do with such high positions in the strength hierarchy of formal systems (high-level abstractions about very remote infinities) ; thus it can be naturally contained in systems much weaker than ZF. So, even in terms of senseless beliefs, the mechanist thesis cannot explain such a confidence in the consistency of ZF.
Therefore, the mechanist view with the concept of natural evolution cannot account for the good mathematician's thinking abilities.

References

How can we prove the presence of soul by logic? question in Quora, with my answer.
List of references collected by James R Meyer
Math Stackexchange
Quora
On the Philosophical Relevance of Godel's Incompleteness Theorems
Penrose’s Godelian argument analyzed by Solomon Feferman, summed up here
Why is the Lucas-Penrose Argument Invalid? by Manfred Kerber
The Lucas-Penrose Argument about Gödel's Theorem by Jason Megill for the Internet Encyclopedia of Philosophy
Three Objections to the Penrose–Lucas Argument by Donald King.
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