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 taste of getting interested in some topics (theories, definitions...) rather than others
- The ingenuity of guessing truth, and then possible proofs before actually establishing them; discovering proofs more efficiently;
- Understanding what "truth" means
- Knowing a truth even while no proof exists

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

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.

- For an algorithm: just add in its code a conditional instruction with a special condition that was never fulfilled yet : it remains unnoticed until the condition suddenly comes true.
- No reasonable axiomatic set theory can, without contradiction, prove any theorem stating which axiomatic set theory it exactly is. Indeed if it could, since any consistent theory is incomplete, it remains possible without contradiction, to add more axioms to it... which would thus, without contradiction, make false the initially proven theorem stating which axiomatic theory it was. Contradiction.

In fact, no and no.

Instead of discussing consistency, it may be more relevant to discuss

This allows for inductive reasoning: for any number

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 ?

"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 isAny (algorithmic) consistent theory is unable to prove its own consistency"

So the question, for which we may compare the abilities of minds and machines, is aboutFor 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.

- The mind has a natural sense of the concept of real truth in arithmetic seen as a realistic theory; this still isn't remarkable, but then, can a mind actually know "in the absolute", as particular cases of arithmetic truths, the fact that some specific theories are sound and consistent ?
- Generally speaking, automated proof checkers can verify when a formula is deducible in reasonable time from a given axiomatic theory. They can "know" that one theory is consistent just if they were programmed, as their source of "knowledge", to assume for this a stronger theory. But where could this choice of framework come from ? Which framework can it specifically happen to be, and how can such a "knowledge" be qualified, in comparison to that of the mind ?

- The philosophical aspects of the foundations of mathematics, continued up to the issue of the concepts of truth in mathematics, which explains the whole issue of the strengths of axiomatic set theories,
- My argument for the soundness and consistency of ZF

- 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.
- 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.
- 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. - 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.

List of references collected by James R Meyer

Math Stackexchange

Quora

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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