Are Kurt Gödel’s Theorems Overstretched, Overused, and Overplayed?

 In this instance, the philosopher J.R. Lucas relied heavily on Kurt Gödel's first incompleteness theorem to advance his position against “mechanism” and “machine minds".

J.R. Lucas and Kurt Gödel

“Does Godel’s incompleteness theorem prove God? The Incompleteness of the universe isn’t proof that God exists. But it is proof that in order to construct a rational, scientific model of the universe, belief in God is not just 100% logical… it’s necessary.”

— Perry Marshal [See source here.]

“Gödel’s work is not overrated in terms of its creativity. It is overrated as to its relevance.”

— Justin Rising [See source here.]

“[Gödel’s theorems have] survived over eighty years of fairly active scrutiny from the mathematical community. That’s as close to a guarantee as you’re going to get.”

— Eugene Bucamp [See source here.]

Is John Lucas’s well-known paper ‘Minds, Machines and Gödel’ a perfect example of Gödel’s incompleteness theorems being overstretched, overused, and overplayed? It’s hard to say. However, Lucas himself said (in another paper) that we can “adapt and apply [the first theorem] in innumerable different circumstances”.

So Lucas’s “Godelian argument” is part of a long and popular tradition.

Take the case of the mathematician and educator Morris Kline.

Kline made a grand claim about Gödel’s Incompleteness Theorems when (in his Mathematics: The Loss of Certainty) he said that it was a

“response to Leibniz’s 250-year-old dream of finding a system of logic powerful enough to calculate questions of law, politics, and ethics”.

Were Gödel’s theorems really a response to Leibniz’s dream?

Perhaps they were just Gödel’s way of showing mathematicians that… well, axiomatic systems (which are of sufficient complexity to express the basic arithmetic of the natural numbers) can’t be both fully consistent and complete —and that’s it!

Yes, this is a version of Gödel’s theorems without any knobs on them.

So, again, can Gödel’s theorems be applied outside mathematics?

The science writer John Horgan certainly applied them to the theories of physics. Or, more accurately, he told us that Gödel’s

“incompleteness theorem denies us the possibility of constructing a complete, consistent mathematical description of reality”.

Clearly there’s a jump here from Gödel’s incompleteness theorems to physical reality… Or at least there’s a jump from Gödel’s theorems to a “consistent mathematical description of reality”.

Is that jump justified?

Well, there’s certainly no consensus on this issue.

Indeed, Gödel himself wasn’t too keen on applying his theorems to physics — especially to quantum physics. According to the mathematician and theoretical physicist John D. Barrow:

“Godel was not minded to draw any strong conclusions for physics from his incompleteness theorems.”

In addition, the German computer scientist Jürgen Schmidhuber has argued against this fixation on Gödel’s theorems. More clearly, Schmidhuber believes that Gödel incompleteness is irrelevant when it comes to computable physics.

With more relevance to J.R. Lucas’s own Gödelian argument.

Another common supposed result of Gödel’s theorems is the argument that that they demonstrate a limit to artificial intelligence.

Perhaps this is a more feasible idea because it’s about the (meta)mathematical limitations of artificial intelligence — and thus Gödel’s theorems are relevant.

Would an indefinite advance in AI be halted by the results of Gödel’s theorems? After all, they show us (or at least they show some mathematicians and philosophers) that if a mathematical system can’t be both complete and fully consistent, then any project that relies on mathematics (i.e., AI) will never be both complete and fully consistent. Thus, there will be a limit to AI when it comes to artificial minds and/or artificial consciousness.

Returning to the general theme of the wide popularity and use of Gödel’s theorems.

When Gödel is recruited to fight various philosophical and religious causes, what often matters to the fighters is the fact that (in this case at least) the first theorem

“can also be taken as giving us a certain type or style of argument, which we can understand, and, once having got the hang of it, adapt and apply in innumerable different circumstances”.

[See source of quote here.]

Again, all this has meant that highly-technical metamathematical theorems have been extensively used to advance all sorts of philosophical, religious and moral positions.

What’s more, J.R. Lucas himself happily admitted that it’s not literally all about Gödel’s first theorem being taken in its raw form, and then being algorithmically applied to the problem of “minds and machines”.

So let’s tackle John Lucas’s philosophical positions here.

Gödel’s Theorem + Philosophy

In his paper ‘Satan Stultified’ (1968), Lucas wrote the following words:

“The application of Gödel’s theorem to the problem of minds and machines is difficult. Paul Benacerraf makes the entirely valid ‘Duhemian’ point that the argument is not, and cannot be, a purely mathematical one, but needs some philosophical premises to be able to yield any philosophical conclusions. Moreover, the philosophical premises are of very different kinds.”

What were Lucas’s own “philosophical premises”? Or, more accurately, which of his philosophical premises were over and above Gödel’s first theorem, and its strictly metamathematical context?

When it comes to Lucas’s non-metamathematical work, there seems to have been assumptions about the “mechanical” and what it is to be “alive”. (We also need to know what philosophical work Lucas’s hyperbolic word “ossified” is doing.) More importantly, what “philosophical conclusions” do these philosophical premises lead to, and how far do they run free of Gödel’s first theorem?

So here’s Lucas himself on his own philosophical conclusions and assumptions:

“I am concerned with the reductionist thesis that we could in principle give a mechanist deterministic account of human behavior which was complete and left no room for free will, moral responsibility or individual creativity. It is that thesis that the Godelian argument is intended to refute.”

To sum up with a final question.

Did J.R. Lucas use Godel’s first theorem simply as a tool (or even a weapon) to safeguard “free will, moral responsibility [and] individual creativity” from “mechanistic attack”?

Note:

Let’s return to Morris Kline.

As already shown, much has been made of Gödel’s theorems by non-mathematicians and by many non-philosophers. Morris Kline expressed this general idea. He said that we

“might think that Gödel’s proof implies that the rational mind is limited in its ability to understand the universe”.

Again, how could a result in metamathematics do that — even in principle?

The mind must surely be limited in some way. Perhaps that means that it could never understand everything there is to know about an infinite universe. Indeed, this is bound to be the case because only an omniscient mind could know everything there is to know about the universe.

Kline also made the point that

“though the mind may have its limitations, Gödel’s result doesn’t prove that these limitations exist”.

What is limited isn’t the mind as such: it’s that “axiomatic systems are limited in how well they can be used to model other types of phenomena”. This has nothing to do with the mind of man taken generically. It’s to do with axiomatic systems and how they model other types of phenomena.

Virtual Realities Are Real Fakes: Chalmers and Baudrillard on Reality

 

Some 40 years ago, Jean Baudrillard rejected the distinction between “reality” and the “simulacrum”. (He stated: “The simulacrum is true.”) And, in his 2022 book, the philosopher David Chalmers also tackled "simulations", "virtual worlds" and "the real”. (Much of this is done within the context of the film The Matrix.)

David Chalmers and Jean Baudrillard

(i) Introduction

(ii) Jean Baudrillard on Reality and Appearance

(iii) Simulations Are Real

(iv) Laypeople and the Real

(v) Chalmers’ on the Context of Virtual Worlds

In his book Reality+: Virtual Worlds and the Problems of Philosophy, David Chalmers writes:

“Simulations are not illusions. Virtual worlds are real. Virtual objects are real.”

He continues:

“[ ] I argue that even if we’re in a simulation like the Matrix, the world around us is perfectly real. There are still tables and chairs, planets and people.”

Elsewhere, Chalmers also tells us that the “central thesis of [his] book is virtual reality is genuine reality”.

The broad upshot here is that not all experiences of simulations are also experiences of illusions.

Simulations are only illusions when those people who experience them don’t realise that they’re simulations. However, when the individual knows that he or she is “in” a virtual reality (or simply experiencing a single simulation), then the notion of an illusion isn’t apt. [See note 1 on some simulations being “deceptions”.]

All the above, then, rules out the central idea of The Matrix. In this film, people don’t realise they’re living within a virtual reality — i.e., within the Matrix.

The word “illusion” is also put up against the word “real”.

Jean Baudrillard on Reality and Appearance

Jean Baudrillard

The Matrix’s rebel leader Morpheus (played by Laurence Fishburne) tackles what the philosopher Jean Baudrillard (1929–2007) called “the Real”. The following is what Morpheus says in that film:

“How do you define ‘real’? If you’re talking about what you can feel, what you can smell, what you can taste and see, then ‘real’ is simply electrical signals interpreted by your brain.”

Despite emphasising the “simulacrum”, Baudrillard still recognised reality — or at least he recognised a non-simulated world. Take this account of Baudrillard's position:

“Baudrillard observes that the contemporary world is a simulacrum, where reality has been replaced by false images, to such an extent that one cannot distinguish between the real and the unreal.”

Baudrillard himself goes one step further than the passage above in the following:

“ The simulacrum is never that which conceals the truth — it is the truth which conceals that there is none. The simulacrum is true.”

Taking the words “reality has been replaced by false images” literally, they must mean that there was a reality — and even that there still is a reality! After all, in order for Baudrillard to have recognised that (among other things) most/all(?) people can’t “distinguish between the real and the unreal”, then he must himself have… distinguished between the real and the unreal. At that point, then, the real was… real. This must have been the case otherwise Baudrillard's whole philosophy-of-the-simulacrum wouldn’t have got off the ground in the first place.

More specifically, realising that “false images” are, in fact, false must have taken some epistemological and metaphysical effort on Baudrillard’s part. Indeed, this could only have been done if Baudrillard had firstly recognised what is real.

More broadly, all this means that Baudrillard effectively placed himself in a long philosophical tradition in which various philosophers (Plato and Marx are good examples) distinguished “reality” from “appearance”. In Baudrillard's own case, then, he distinguished what is real from the simulacrum.

Relevantly enough, The Matrix’s Morpheus (mentioned earlier) stated the following:

“Welcome to the desert of the Real.”

Sure — the Real here actually refers to the “virtual world” of the Matrix. Yet that virtual world is real! Or, more correctly, virtual worlds generally were taken to be real by Baudrillard himself. Moreover, in the senses mentioned earlier (i.e., in relation to Chalmers’ own take on this issue), virtual reality must be real because there’s nothing else for it to be.

Simulations Are Real

To repeat.

In a strong and simple sense, a visual simulation, or even an entire package of simulations occurring together to create a virtual reality, can’t be an illusion in and of itself. After all, what you see, hear, touch, etc. is, in some way, still real. (That, basically, is David Chalmers’ own argument.)

Indeed, even an illusion is real…

What else can it be?

This also means that it must be what people make of a simulation which may involve illusion.

For example, if someone believes that the dinosaur in front of him is real and not a simulation (or an hallucination), then he has swallowed that particular illusion. On the other hand, since people usually know that they’re witnessing a virtual reality, then the word “illusion” simply isn’t appropriate here. However, if they don’t know that dinosaurs are extinct, then the word “illusion” is apt.

Simulations are real in another sense too.

Simulations must have (or simply may have) a physical basis in that the software for the images is implemented in a physical hardware. In addition, whatever is going on in the human sensory system and brain when simulations are experienced, then that must be physical too. [Idealists will disagree with me on this. See my ‘Bernardo Kastrup: “The brain does not generate the mind.”’]

Laypeople and the Real

If we return to Morpheus and his statement that the Real is “simply electrical signals interpreted by your brain”.

It can be argued that most laypersons are aware that “what [they] feel, what [they] smell, what [they] taste and see” are not external objects and events themselves. After all, most people don’t believe that objects and events are literally inside their minds and/or brains. What could that even mean? [Some externalists don’t like this distinction. See note 2.]

Moreover, the feel of a soft blanket, the smell of shit, the taste of Marmite and the visual experience of a sunset are even less likely to be divorced from the “electrical signals interpreted by [ ] brain[s]” by laypeople. Of course, most laypeople won’t have the scientific language to go into any great — or even small — detail about this (e.g., by referring to electrical signals, neuronal activity, action potentials, etc.). However, they still know that feels, smells, tastes, and visual experiences are to do with what goes on in their brains and/or minds.

Chalmers’ on the Context of Virtual Worlds

On a broader level, the following passage expresses how David Chalmers sees things when they’re placed in a more psychological, sociological and even political context:

“These worlds needn’t be illusions, hallucinations, or fictions. Our time in them needn’t be escapism. People already lead complex and meaningful lives in virtual worlds such as Second Life, and VR will make this commonplace.”

This again shows that we needn’t conflate illusions with simulations. Indeed, perhaps some simulations can help us understand non-simulations — i.e., physical reality itself. Chalmers himself seems to be suggesting that certain types of virtual reality and “Second Life” may be educational, as well as what he calls “meaningful”.

On another, perhaps pedantic, note.

Can it really be said that there are some people around today who “already lead complex and meaningful lives in virtual worlds”? Surely Chalmers means that only parts of such people’s lives occur in virtual worlds. Indeed, if they literally lived all of their lives in virtual worlds, then there’d be an argument that such a state of affairs couldn’t be either meaningful or educational.

Moreover, don’t virtual worlds need to be placed within the context of non-virtual worlds? In other words, if virtual worlds aren’t placed within the context of non-virtual worlds, then why use the term “virtual worlds” at all? Wouldn’t these worlds simply be… worlds?

Notes

(1) This is one definition: “simulation” = “the act of pretending; deception”. Thus, on this definition at least, a simulation can be experienced as a (kind of) illusion.

(2) The following is an account of externalism at its “extreme”:

“The extreme view of externalism argues either that the mind is constituted by or identical with processes partially or totally external to the nervous system.”