The Gödelian Argument Against Machine Minds Relies On Skilled Metamathematicians Seeing “Gödel-Truths

 

Did the philosopher J.R. Lucas and the mathematical physicist Roger Penrose rely on the rare abilities of a very-small set of metamathematicians to establish their "Gödelian argument" against the possibility of "machine minds"?

Left-to-right: Kurt Gödel, Roger Penrose, and J.R. Lucas. (Note the words “Gödel’s brain (at least”)”.)

J.R. Lucas’s Gödelian Argument

J.R. Lucas and Kurt Gödel

The following passage is one way in which J.R. Lucas’s “Gödelian argument” is expressed:

“For any logical system L, a sufficiently skilled mathematical logician [ ] can construct some statements T(L) which are true but unprovable in L. (This follows from Gödel’s first theorem.)”

Elsewhere, we also have this:

“Appealing to Gödel’s incompleteness theorem, [J.R. Lucas] argues that for any such automaton, there would be some mathematical formula which it could not prove, but which the human mathematician could both see, and show, to be true.”

Note that in both passages “a sufficiently skilled mathematical logician” and “the human mathematician” are mentioned. So now we have this question:

How applicable is Gödel’s theorem to arguments about all human minds?

This is the kind of metamathematical knowledge human beings must have in order to see Gödel-truths.

John Lucas also approached the possibility of artificial minds from another angle. The following is one account of his position:

“‘Minds, Machines and Gödel’ is J. R. Lucas’s 1959 philosophical paper in which he argues that a human mathematician cannot be accurately represented by an algorithmic automaton.”

Here again, it must be stressed that Lucas was referring to the mind of “a human mathematician”. This raises the question as to how applicable and relevant the-seeing-of-Gödel-truths is to all human minds. And, as a final example, Lucas then went one step further when he not only referred to “a competent mathematician”, he also added that such a mathematician must be someone “who understood Gödel's proof”.

On a related theme.

Did Lucas rely on the existence of perfectly consistent (indeed perfectly rational) mathematicians in order to advance his Gödelian argument against machine minds?

Many scientists and philosophers believe that human reasoning is inconsistent. So that must include mathematicians. Indeed, it must even include those who can (to use Lucas’s own word) “see” Gödel-truths.

More particularly, doesn’t Lucas’s argument depend on the minds of only those metamathematicians who study Gödel’s theorems being fully consistent?

Lucas himself (implicitly) recognised this possible interpretation when he wrote the following words:

“A number of thinkers have chosen to impale themselves on the inconsistency horn of the dilemma. We are machines, they say, but inconsistent ones. In view of our many contradictions, changes of mind and failures of logic, we have no warrant for supposing the mind to be consistent, and therefore no ground for disqualifying a machine for inconsistency as a candidate for being a representation of the mind.”

Again, doesn’t Lucas’s argument actually depend on a tiny number of metamathematicians being able to see some (i.e., not all) Gödel-truths?

Moreover, as the philosopher Judson Webb argued in his 1968 paper ‘Metamathematics and the Philosophy of Mind’, one also needs to ask questions about whether human beings (actually, a small subset of mathematicians) can really see the truth of Gödelian sentences.

Perhaps a better questions would be: What is it to see a Gödel-truth?

To repeat.

The basic argument here is that “the mathematician” (whoever that is) can see Gödelian (unprovable) truths. Lucas took that to have the consequence that there must be at least one thing about human beings (or, actually, the minds of a small subset of mathematicians!) which can’t be predicted by machines, computers… or by anything else.

So doesn’t Lucas’s Gödelian argument — at least in a strong sense — render the minds of all non-mathematicians (i.e., not only “machines”) suspect too? After all, most human minds can’t recognise the truth of Gödel sentences! [See note.] In fact, most mathematicians aren’t metamathematicians. This means that they too may not be able to see Gödel-truths.

If this is the right way of looking at this, then it also means that only some minds can discover unprovable truths. Or, more correctly, only some minds can see the truth of some Gödel sentences…

That’s unless J.R. Lucas simply meant that all human minds have the potential to see Gödel-truths…

Do all human minds have that potential?

It depends on how that latent potentiality is cashed out.

Moreover, how could we know that all human minds do have that potential?

That question is asked because various commentators (e.g., cognitive scientists, philosophers, psychologists, etc.) have themselves questioned the idea (or belief) that human minds — or individual human minds — can gain the unequivocal truth of formally unprovable Gödel sentences.

What’s more, such commentators are certainly suspicious of the idea that the seeing of Gödel-truths has much — or even any — relevance when it comes to characterising all human minds, as well as distinguishing human minds from machine minds.

Now let’s tackle Roger Penrose’s related position.

Roger Penrose’s Gödelian Argument

Firstly, we’ll deal with Roger Penrose’s very own Gödelian Argument when he stated the following:

“[A]s long as you believe in the rules [of ‘any system of rules’] you’re using in the first place, then you must also believe in the truth of this [Gödel] proposition whose truth lies beyond those rules.”

… And now for the relevant clincher:

“This makes it clear that mathematical understanding is something you can’t formulate in terms of rules.”

This is where Platonic seeing comes to the rescue.

Penrose believes that no machine (or computer) can have Platonic vision. That is, he states that “we have a proposition that we can see [Penrose’s italics], by the use of insight, must actually be true”. However, “the given algorithmic action is not capable of telling us this”.

Like Lucas earlier, note that Penrose uses the plural pronoun “we” (as in “we can see”). Yet, with an admittedly wild guess, only 1 person in every 10,000 can see any Gödel-truth.

Perhaps this doesn’t matter in that all human beings may have the potential to see Gödel-truths. Alternatively, even if only 1 person in 10,000 can see Gödel-truths, that’s enough to establish Penrose’s argument against machine minds and machine consciousness. (This is roughly equivalent to arguing that even if a rare disease only affects 1 person in 10,000, then that disease is still very real.)

Penrose then goes into greater detail when he asks this question:

“Why, then, can one not simply get a computer also to follow this Gödel argument and itself ‘see’ the truth of any new Gödel proposition?”

What’s more:

“The catch lies in ‘seeing’ that the Gödel argument, in any specific realization, has actually been correctly applied. The trouble is that the computer does not have a way of judging truth; it is only following rules. It does not ‘see’ the validity of the Gödel argument. It does not ‘see’ anything unless it is conscious!”

… And neither do 9,999 people out of 10,000 “‘see’ the validity of the Gödel argument”!

Do they all have the potential to do so?

Or are these large numbers irrelevant?

Roger Penrose on Consciousness

As can be partially seen in the quoted passages above, Penrose also strongly ties consciousness (Lucas focussed on “mind”) to the seeing of Gödel-truths. In his own words:

“[T]he very essence of consciousness. It must be present whenever we directly perceive mathematical truth. When we convince ourselves of the validity of Gödel’s theorem we not only ‘see’ it, but by so doing we reveal the very non-algorithmic nature of the ‘seeing’ process itself.”

To spell this out. To Penrose, “the very essence of consciousness” is to “directly perceive mathematical truth”.

Note the word “we” again in the passage above.

This passage states an explicit and essential link between consciousness and the ascertaining of mathematical truth. Basically, consciousness is required to see mathematical truth. (Penrose does repeatedly put the word see in scare quotes — i.e., he doesn’t mean seeing through the eyes.)

This also means — to Penrose at least — that mathematical truths can’t be seen by machines or computers.

As far as can be seen, the problem here is that Penrose is mainly — or even exclusively — talking about Gödel sentences (or Gödel-truths) when he refers to “mathematical truth”. And the problem with that, again, is that not many people can see them — not even some mathematicians.

What’s more, if the seeing of Gödel-truths is fundamental for consciousness itself, then where does that leave 99% of the human population?…

Unless, like Lucas earlier, Penrose simply means that all human beings merely (or only) have the potential to see Gödel-truths.”

Conclusion

This is one part of one of the many technical expressions of Gödel’s first incompleteness theorem. Roughly speaking, this is the level of metamathematical knowledge one must have in order to see Gödel-truths.

It seems odd, and perhaps even flimsy, to rely on an extreme minority of metamathematicians seeing Gödel-truths in order to establish an extremely general and deep conclusion about the difference between human minds and machine minds.

In terms of all human beings having the potential to see Gödel-truths.

It’s very likely that many human subjects would never see such truths, no matter how much mathematical, metamathematical and logical tuition and knowledge they were introduced to. Moreover, what about those people with cognitive and brain anomalies? Does this mean that such human minds are equal to, or even less than, machine minds?

Finally, isn’t it far more likely that a machine (or computer) could see (or at least recognise and acknowledge) Gödel-truths than, say, have genuine emotions, experience the sweet taste of Coca-Cola, or compose a masterpiece of music?

Note:

J.R. Lucas argued that women and politicians are inconsistent, as can be seen from this qualified version of an earlier claim of his. (See Lucas’s paper ‘Against Equality Again’, in which he says the same thing.) However, Lucas’s general argument is more specific than that. On my reading at least, the only people who can transcend “machines” — at least in Lucas’s Gödelian sense — is a tiny subset of metamathematicians… perhaps people just like Lucas himself.

Qualia/Zombie Scenarios and Schrödinger’s Cat: Analogous Thought-Experiments?

The philosopher David Chalmers once discussed Erwin Schrödinger’s well-known thought-experiment. Chalmers did so in order to compare what he himself is doing to what Schrödinger did way back in 1935. More specifically, Chalmers compared his qualia and zombie “scenarios” to Schrödinger’s cat. He argued that they too “bring out various plausibilities and implausibilities”.

(i) Introduction

(ii) David Chalmers on Schrödinger’s Cat

(iii) Chalmers’ Thought-Experiments (or “Scenarios”)

(iv) David Chalmers on Conceivability

(v) Schrödinger’s Cat Again

Could David Chalmers’ qualia and zombie scenarios even become part of any flesh-and-blood experiments? In more basic terms, should Chalmers use the word “experiment” about his own qualia and zombie scenarios at all?…

Yes!

A thought-experiment isn’t a physical or concrete experiment — the clue is in the name. And Chalmers himself stresses the fact that his primary concern is “conceptual possibility” (or “conceivability”), not flesh-and-blood experimentation…

Yes, Chalmers is a philosopher, not a scientist.

All that said, Erwin Schrödinger’s own thought-experiment never became an actual experiment either. In other words, it has never been carried out on a cat or on any other “classical” object. That said, successful experiments involving superpositions of “relatively large objects” have been performed. (Relatively large, but not even the size of a neuron.) [See note 1.]

So what did Chalmers have to say about Schrödinger’s Cat?

David Chalmers on Schrödinger’s Cat

Chalmers wrote:

“Perhaps it is useful to see these [zombie an qualia] thought-experiments as playing a role analogous to that played by the ‘Schrodinger’s cat’ thought-experiment in the interpretation of quantum mechanics.”

Chalmers then explained this analogous role in the following way:

“Schrödinger’s thought-experiment does not deliver a decisive verdict in favour of one interpretation or another, but it brings out various plausibilities and implausibilities in the interpretations, and it is something that every interpretation must ultimately come to grips with.”

Chalmers finally concluded:

“In a similar way, any theory of consciousness must ultimately come to grips with the fading and dancing qualia scenarios, and some will handle them better than others. In this way, the virtues and drawbacks of various theories are clarified.”

In this case at least, Schrödinger’s thought-experiment about a cat was deemed by Chalmers to be “analogous” to his own qualia and zombie “scenarios”. Yet, prima facie, it’s hard to spot anything analogous here other than that all these examples are… well, thought-experiments.

Chalmers’ Thought-Experiments

David Chalmers isn’t naïve about the arguments against his own modal theorising. For example, way back in 1995, Chalmers once happily stated the following:

“The mere logical or metaphysical possibility of absent qualia is compatible with the claim that in the actual world [there is no such thing]. [ ] Mere intelligibility does not bear on this, any more than the intelligibility of a world without relativity can falsify Einstein’s theory.”

Yet, as we’ll see in a moment, “logical or metaphysical possibility” is still very important to Chalmers — despite his prefix “mere”.

What’s more, elsewhere in the same paper Chalmers wrote:

“But it is implausible that fading qualia are possible, and it is extremely implausible that dancing qualia are possible. It is therefore extremely implausible that absent qualia and inverted qualia are possible.”

On a positive reading, it can be argued that it was right and proper that these possibilities (or though-experiments) be considered — even if, in the end, they were ruled out for the “actual world”. After all, how can we rule these possibilities out (or in) unless we firstly tackle and analyse them?

So, despite Chalmers pre-empting many of the counterarguments, logical possibilities are still fundamental to almost all his philosophical work. [See note 2.] Relevantly, some of his possibilities are ruled in, rather than ruled out.

More specifically, the philosophical notion of conceivability is at the very heart of much of Chalmers’ work.

David Chalmers on Conceivability

Without (philosophical) conceivability, Chalmers’ arguments against physicalism — and in support of the possibility of philosophical zombies, inverted qualia, dancing qualia, absent qualia, etc. — wouldn’t get off the ground in the first place.

In terms of zombies scenarios (i.e., not qualia scenarios), Chalmers himself says that

“the question is not whether it is plausible that zombies could exist in our world, or even whether the idea of a zombie is a natural one; the question is whether the notion of a zombie is conceptually coherent”.

To Chalmers, then, “the notion of a zombie” is “conceptually coherent”.

However, let the English philosopher Philip Goff put the case-for-conceivability in more technical terms.

Firstly, Goff expresses his view about the importance of conceivability (leading to possibility) in this way:

“If P is conceivably true, then P is possibly true.”

So is a cat being both alive and dead at one and the same time “conceivably true”? It depends on what, exactly, is conceived. More clearly, what is it to conceive of the same cat being both alive and dead at one and the same time?

Goff then expresses his position in possible-worlds jargon:

“If P is conceivably true (upon ideal reflection), then there is a possible world W, such that P is true at W considered as actual.”

Or, less technically, Goff actually refers to Chalmers when he says that

“Chalmers holds that every conceivably true proposition corresponds in this way to some genuine possibility”.

Thus, if it’s “conceivably true” that the same cat be both alive and dead at one and the same time, then there is a possible world W in which that cat is both alive and dead.

It’s not clear, here, how shifting this issue to a “possible world” negates the problem we have with the psychological notion of conceivability.

Oddly enough, according to many-worlds theorists, the alive cat and dead cat are indeed in two different worlds — if not in two possible worlds. That is, this superposition would actually occur in a multiworld - it’s not a (mere) possibility. (Such physicists believe that the alive-and-dead-cat superposition would be actual if the experiment were ever successfully carried out.)

Schrödinger’s Cat Again

Erwin Schrödinger himself never believed that a cat could be both alive and dead at one and the same time…

In basic terms, Schrödinger intended his thought-experiment to illustrate the bizarre nature of the Copenhagen interpretation of quantum mechanics. Indeed, Schrödinger took it to be a reductio ad absurdum.

Chalmers, too, sees himself as considering various logical possibilities, and (as it’s often put) seeing where they lead.

Oddly enough, Schrödinger can be seen as actually arguing against taking outlandish possibilities (in this case, superpositions) seriously. After all, aren’t there literally innumerable possibilities that can be — or even must be — considered? Of course, it can now be argued that some possibilities are (well) more possible than others. They may also may be more interesting, relevant, useful, scientific, etc. than others. And all that may apply to Schrödinger’s cat, as well as to Chalmers’ zombie and qualia scenarios.

Ironically enough, Schrödinger concocted a thought-experiment (which is about the logical possibility of a cat being both alive and dead) to advance his case against the possibility of a particle (i.e., not a cat) being in various superpositions within quantum experiments.

All that said, it can now be acknowledged that both Schrödinger’s and Chalmers’ thought-experiments do indeed (as Chalmers put it earlier) “bring out various plausibilities and implausibilities”. And, surely, that’s a good thing for both physics and philosophy.

Notes:

(1) Here are two examples:

“Two distinct types of experiments — both of them carried out by several groups independently — have shown that vast numbers of atoms can be placed in collective quantum states, where we can’t definitely say that the system has one set of properties or another. In one set of experiments, this meant ‘entangling’ two regions of a cloud of cold atoms to make their properties interdependent and correlated in a way that seems heedless of their spatial separation. In the other, microscopic vibrating objects were maneuvered into so-called superpositions of vibrational states. Both results are loosely analogous to the way Schrödinger’s infamous cat, while hidden away in its box, was said to be in a superposition of live and dead states.”

(2) For example, in David Chalmers’ brilliant book, The Consciousness Mind (1996), there are comments and arguments about logical possibility and conceivability on almost every page. Indeed, this was the book which jumpstarted Chalmers’ career.

Dennett Quines Qualia: A Bitter Taste Is Not a Quale

 What is it for a lemon to be “objectively bitter”? According to the philosopher Daniel Dennett (who died in April), it’s “to produce a certain effect in the members of the class of normal observers”. To Dennett, then, bitterness is a relational property. That basically means that a specific instance of a bitter taste is not a quale.

Daniel Dennett demanded that bitterness be objective (or intersubjective). And it can only be objective if it’s somehow behaviourally expressed by the class of normal observers.

Yet even if there are examples of uniform overt behaviour (such as verbal reports), do we learn anything about bitterness itself via such behavioural responses? Don’t we, instead, just learn about how people react to bitterness?

The qualiaphile argument will be that uniform reactions to bitterness aren’t themselves bitterness. So this is almost equivalent to arguing that the throwing of a ball isn’t the same as the smashing of a window. In other words, the throwing of a ball resulted in the smashing of a window, just as the tasting of something bitter leads to a certain effect in the members of the class of normal observers.

Moreover, it’s possible that the same piece of food could have a uniform “certain effect” even if it tastes differently to each person tested. Or, alternatively, something that tastes the same can have different certain effects in the class of normal observers…

So does this mean that Dennett was fusing (or conflating) cause and effect here?…

Possibly not.

How could investigators (or laypersons) know that the same piece of food tastes differently to different people, or that it tastes the same?

Dennett argued that we could only know this with the help of overt behaviour, as well as by noting physiological changes…

These standard philosophical possibilities (e.g., a piece of food having a uniform “certain effect” even if it tastes differently to each person tested) will have been rejected by Dennett simply because they’re all unscientific in nature. He might have told us that we have no way of knowing if these possibilities are ever realities. So, if that’s the case, then such possibilities are merely idle from a scientific — if not a philosophical — point of view…

But hang on a minute here.

Haven’t qualiaphiles — and many others — simply assumed that uniform reactions to bitterness aren’t themselves constitutive of what bitterness is?

What is the remainder which qualiaphiles believe is being left out?

The quale that is bitterness?

What is that?

More importantly, what is this bitter quale that is something more than the verbal reports, reactions, physiological changes, etc. of human persons?

Isn’t such a quale an example of Ludwig Wittgenstein’s own beetle-in-a-box?

Well, if the quale of bitterness is private, directly apprehensible, atomic, intrinsic (i.e., non-relational), exhaustive, ineffable, incomparable, and incorrigible, then it pretty much seems to be a beetle in a box.

[All this partly depends on whether Dennett’s characterisation of qualia — or his account of what qualiaphiles take qualia to be — is the only one available or possible. In fact, there are alternative approaches. See my ‘Did Daniel Dennett Believe that Qualia Are Real?’ in which Owen Flanagan’s alternative take on “the way things seem to us” is tackled.]

Daniel Dennett’s Wittgensteinian Position

The American philosopher Alvin I. Goldman expressed Dennett’s position on these possibilities when he says that Dennett’s

“claim is that there is no way to distinguish between these competing stories either ‘from the inside’ (by the observer himself) or ‘from the outside’, and he appears to conclude that there are no genuine facts concerning the putative phenomena experience [or differences] at all”.

This reiterates Dennett’s Wittgensteinian point that there are no facts, data… or anything available about the intrinsically private. In other words, we can know nothing of other people’s qualia — as least as qualia are seen by qualiaphiles. And, on an another Wittgensteinian theme, if we can’t know anything about other people’s qualia, then we can’t know anything about our own qualia either.

To be clear.

Qualiaphiles deem qualia to be private… things. Public language, on the other hand, is intersubjective. Yet we use a public language to communicate supposedly private qualia.

If a quale were genuinely private, then how would we know that other people were referring to the same thing which we refer to (say, the bitterness of a particular lemon)? To us (if not to them), then, other people’s qualia would be like beetles in a box. That’s partly why Ludwig Wittgenstein concluded that our language for sensations (if not qualia) isn’t only public: it must also refer to public items.

To sum up.

A particular quale (e.g., the bitterness of a lemon) can’t be a beetle in a permanently-closed box. If it’s in any kind of box, then it’s in an open and publicly-accessible one. Thus, the named “object” (i.e., the quale) must “drop out of consideration as irrelevant”.

Yes, but qualia are real!

Despite all the above, there must still be a sense that qualia are indeed private. This seems to be (literally) self-evident.

For example, if I don’t communicate my experiences of qualia with either linguistic or physical behaviour, then no one else would know that I’m experiencing the bitter taste of a lemon. However, I’m still experiencing this case of bitterness. Thus, only I know that I’m tasting a lemon which is bitter. Therefore, this bitter quale is private.

Moreover, surely there’s more to qualia than our knowledge of them. They must have both an ontological and an experiential status…

The problem is that (as Wittgenstein himself argued about sensations) as soon as we say anything about qualia (especially when using the jargon of philosophy — and the term “qualia” is itself jargon!), then they become (for other persons, and even for oneself) public things…

What’s more, this means that such things aren’t qualia at all!

Moreover, many philosophers will deem the very idea of a quale-in-itself suspect.

Does that idea even make sense?

Does it serve any purpose?

Conclusion

Yet, again, isn’t it possible that one can accept that qualia have no factual or scientific status at the very same time as accepting that they are… well, real?

It’s here that Dennett might have asked the following question:

If qualia have neither scientific nor factual status, then exactly what kind of status do they have?

What is the answer to that question?