“If the whole universe has no meaning, we should never have found out that it has no meaning.”
C.S. Lewis, Mere Christianity
To summarize Part 1, despite the best logic of the best thinkers, paradox is here to stay, thanks to the transcendence and unassailability of Gödel’s incompleteness theorems.
Call It Intuition
Almost immediately after publication, Gödel’s work had a major impact on the nascent field of computing. Alan Turing, the father of modern computer science (before modern computers ever existed) was perhaps the first to fully appreciate Gödel’s work and understand its impact for computers and artificial intelligence. After all, what is a computer other than a logic-powered rule-following machine, which sounds exactly like the formal systems that Gödel proved were subject to incompleteness.
Turing was fixated on the so-called halting problem, which is the ability to determine whether a given algorithm will ultimately finish (halt) or will run forever. In the translation from Gödel to Turing, logic became algorithm-based computers (known as Turing machines), theorems became computer programs, provability became decidability, truth became haltability, and completeness became computability.
Turing was able to prove that the halting problem is undecidable, meaning there will be haltable programs that a computer cannot determine ahead of time to be haltable. So what? Why is this important?
Simply speaking, if you want to determine whether a particular problem is computational, that it, able to be performed by a computer, you can always reduce that determination down to the halting problem. Connecting the dots, Turing essentially proved that there are problems having solutions that cannot be computed by a computer.
Putting Turing and Gödel together, we understand that no algorithm can be formulated that is capable of determining the provability of all true mathematical statements.
And yet, mathematicians determine true mathematical statements all the time, often prior to being able to produce a rigorous proof. It follows that mathematicians themselves (and, by extension, human beings) must not be using algorithms (at least not only algorithms) in the function of their brains.
We humans, unlike machines, have an intuitive intelligence that can reach beyond the world that defines our physical selves. For example, we can sense certain truths in mathematics even before they can be formally proven (and even when they cannot be formally proven).
There must be a non-algorithmic, or non-computational, aspect to human thinking. Call it intuition.
To Err is Human, How Divine!
Our minds, therefore, cannot be merely computers. As stated by Roger Penrose, acclaimed mathematician, physicist, philosopher, and Nobel Laureate:
“[A] powerful case can be made that Gödel’s results established that human understanding and insight cannot be reduced to any set of rules.”
Rebecca Goldstein, Incompleteness, pp. 201-202
While human thinking allows the discernment of truths that are not reachable via pure computation, we also know by experience that many intuited conclusions reached by humans end up being erroneous. The computer, on the other hand, is always correct in its computations, but cannot ‘see’ truth beyond what is decidable and computable.
Perhaps what we have stumbled across is the notion that infallibility in computation is inconsistent and incompatible with human intelligence and intuition.
“Turing made a great point of the fact that human mathematicians are very capable of making mistakes; he argued that for a computer to be able to be genuinely intelligent, it also would have to be allowed to make mistakes.”
Roger Penrose, Shadows of the Mind p. 129
This puts a whole new spin on the phrase, ‘to err is human,’ making it not lamentable but rather laudable in this regard.
We humans apparently rely on the capacity for mistake-making so that we can find unprovable truths. Only the mind willing (if not able) to try contemplating the incomprehensible will be able to find the truths lurking out of algorithmic reach, and then apply that understanding in an attempt to make sense out of its very existence.
“[T]he inaccuracy of human mathematical thinking is essential – allowing the mind’s inaccurate action to provide a greater power than that which would be achievable by means of any completely sound algorithmic procedure.”
Roger Penrose, Shadows of the Mind p. 129
Computers accomplish their task through brute force, crunching all the numbers until there are no more numbers to be crunched. Machine learning algorithms must process massive data sets and undergo training processes before being able to categorize and quantify patterns that humans innately recognize even in unfamiliar environments.
While playing chess, a computer will calculate vast scenarios of moves and counter moves, however far down the line is desired, to determine the best next move. The human player, on the other hand, somehow knows what potential paths to ignore and what is worth considering.
“There must be a powerfully impressive selection process that allows the conscious mind to be disturbed only by ideas that ‘have a chance.’”
Roger Penrose, The Emperor’s New Mind, p. 422
Beauty is in AI of the Beholder
It is fascinating to consider what implications arise for our understanding of consciousness and whether machines will ever be able to think and create like humans. While artificial intelligence (AI) has succeeded in producing competent prose and poetry[1] along with interesting visual art (see Figure 3), no one seriously contends that the AI itself is somehow contemplating truth or beauty.
As Penrose stated,
Indeed, algorithms themselves never ascertain truth! It would be as easy to make an algorithm produce nothing but falsehoods as it would to make it produce truths. One needs external insights in order to decide the validity or otherwise of an algorithm…. It is this ability to divine (or ‘intuit’) truth from falsity (and beauty from ugliness), in appropriate circumstances that is the hallmark of consciousness.”
Roger Penrose, The Emperor’s New Mind, p. 412
Figure 3: AI-generated portrait of MC Escher
https://www.reddit.com/r/Art/comments/mjajez/m_c_escher_me_ai_art_2021/ on April 28, 2023
Turing’s Test
What constitutes the hallmark of consciousness? It seems that the non-computational ability to intuit and discern truth and beauty is a good candidate, but how can that be measured much less determined to exist?
Turing proposed a test for whether a computer exhibits the behavior of human-like consciousness, which he called the ‘imitation game’ (the name inspired the title to the 2014 movie about Turing’s role in cracking the code of encrypted German communications during World War II).
What we now call the Turing Test involves a human being submitting written questions to two unseen subjects, one of which is another human and the other is a computer. At the end of the question-and-answer process, if the questioner cannot determine correctly which test subject is the human and which is the computer, then the computer is deemed to have successfully exhibited human intelligence.
But does successful imitation mean that the computer has achieved some level of consciousness, or has it merely simulated the behavior of a conscious human?
Have an exchange with ChatGPT and you might quickly be convinced that you are in the middle of an on-line chat conversation with a real live person. However, ChatGPT is still algorithm-driven and reliant on massive data sets and trillions of computations.
If we are to probe the non-computational, non-algorithmic qualities of human consciousness, the test for consciousness should attempt to detect those qualities. We know that the nature of human consciousness allows a human to discern intuitively, so perhaps the test subjects shouldn’t be the ones answering the questions, but instead should be the ones asking the questions.
In other words, since the Turing Test relies on the consciousness of the questioner to intuitively discern human subject from computer subject, perhaps the better way to test for consciousness is to determine the discernment capabilities of the questioner.
Rather than having a human interrogator, can a computer interrogator discern a conscious being from a non-conscious one under the protocol of the Turing Test?
This posits that it is not in exhibiting the external behavior expected from a conscious being that defines consciousness, but rather the ability to discern consciousness that defines consciousness. It’s like Rene Descartes with a twist: ‘I intuit, therefore I am.’
In strange loop fashion, you turn the microscope back on the observer and try to detect consciousness in the capability to recognize consciousness. The premise being that consciousness necessitates the ability to contemplate one’s self.
Imagine the movie Blade Runner where the test for Replicants can be successfully performed only by a non-Replicant, until a Replicant exists that cannot only fool the test as the subject but can successfully conduct the test itself.
The Meta Turing Test
Let’s try to put a modicum of rigor around this. Human intelligence has flaws, commits errors, takes shortcuts, and exhibits biases. To have true AI, those traits must be preserved. Error must be tolerated.
In reference to Figure 4, say you want to determine if a computer is capable of passing the traditional Turing Test, meaning that a human interrogator is unable to discern the computer from a human subject under Q-and-A interactions.
In my proposed Meta Turing Test, the human interrogator would be replaced with an intelligent machine, with the computer subject being a machine already shown to be capable of passing the traditional Turing Test. This time the test is not being performed on the computer subject, but rather on the computer interrogator.
If in conducting the questioning, the computer interrogator is not able to determine which subject is a human and which is a computer, then the computer interrogator may be said to be exhibiting consciousness. In other words, if the computer interrogator is fooled just like the human interrogator was fooled, then the computer interrogator may be exhibiting consciousness.
Figure 4: Turing Test Configurations
It is the intuitive recognition of consciousness combined with the capacity for error, and indeed the error in discernment, that makes the computer interrogator a candidate for having consciousness!
(In my best Jeff Foxworthy voice) “If you can’t tell that you are talking to a human or a machine, you might be a redneck!”
God is (at least) a Mathematician
All this pertains to testing machines for traces of consciousness. But what about testing human minds for the presence of rationality? To this Rebecca Goldstein, a proponent of philosophical rationalism and biographer of Gödel’s, astutely observed:
“Just as no proof of the consistency of a formal system can be accomplished within the system itself, so, too, no validation of our rationality – of our very sanity – can be accomplished using our rationality itself.”
Rebecca Goldstein, Incompleteness, p. 204
In human consciousness, we find the coexistence of linked dualities: reason and imagination; logic and intuition; computation and inspiration. To divorce one from the other, or to allow one to effectively dominate the other, has dire consequences.
We often consider insanity resulting when consciousness becomes untethered from reality. Chesterton sees it a little differently, asserting:
“The madman is not the man who has lost his reason. The madman is the man who has lost everything except his reason.”
G.K. Chesterton, Orthodoxy
The point is that our consciousness exists beyond the physical realm (at least to some extent) or otherwise it could not contemplate itself, intuit higher truths, and grapple with paradox. The very fact that our reasoning mind can come into contact with concepts as abstract as ‘eternity’ suggests there is something eternal in us.
Indeed, given the abstractness of pure mathematics, we get the inkling that the part of ourselves that can contemplate mathematics is also the part that will survive our bodily death. To that end, of all that God is, we know that He is at least a mathematician.
C.S. Lewis talked about the part of our consciousness that exists beyond the physical realm when he observed that human desire appears to be oriented toward something that this world cannot satisfy. Since the natural world contains all that is sufficient to deal with our physical needs, it doesn’t make sense for our supernatural desires to be incapable of being fulfilled. As such, the very fact that we yearn for God is at least highly suggestive that God exists.
Truth and Beauty
We know from Gödel’s theorems that paradox is an unavoidable necessity. But how does the necessary existence of paradox help in exploring what it means to search for meaning?
The objectivists of Gödel’s time, represented by the Vienna Circle, believed that we can only know about the natural world, and only then through disciplined and rigorous observation. To them, discussing any meaning beyond the natural world is useless because anything beyond it is incomprehensible, and what cannot be comprehended might as well not exist.
Where does that leave discussions about what is meaningful in the observable natural world? What meaning can be derived when it is limited to only what can be proven?
The goal of proving what can be proven is to actually prove it, and once something is proven, it is certain. When a thing becomes certain, it is a mere truism, and can add nothing more other than to be a step in a proof of yet another truism.
Proving the provable is a process that results in reducing the world to an exercise of knowing what can be known. As things become known, they are relegated to the status of mere truisms, collectively trivial and tautological. Ultimately, it is a process of elimination–by eliminating the unknowns we are in essence eliminating everything that is not meaningful.
What the objectivists created for themselves was a self-contained world of exhilarating discovery and progress, at least while it lasted. Each discovery reveals a new part of the picture of the universe, and once the whole picture is uncovered, there is simply nothing left to discover.
Once all is certain, there are no more intellectual peaks to climb and humankind can finally bask in the light of its own brilliance. As Wittgenstein said at the end of the Tractatus:
“My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out though them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up to it.)”
Ludwig Wittgenstein, Tractatus Logico Philosophicus, 6.54
That final sentence might as well be the epitaph written on objectivism’s headstone. It is an end worthy of Macbeth to have reached the goal and find that, as predicted, the effort amounted to ‘sound and fury, signifying nothing.’
For the rest of us, we know that there is more than what we can observe. Logic and reason can be applied not just to studying the natural world, but also to contemplating abstract ideas and their meaning–to physics as well as metaphysics.
We know that science is not meant to provide final answers, but instead is a process of ever-improving approximation.
We know that certainty is a delusion, and that, paradoxically, truth somehow lies beyond certainty. To deny that is to deny what makes us human, to deny our ability to see truth in beauty and to recognize that the greater the beauty the more profound the truth.
Perhaps that is the root of why some intellectual elites so eagerly dismiss the reality of truth in proportion to beauty–simply because such a reality is so accessible and so egalitarian. In contrast, science is rigorous, hard work in which, “…novelty emerges only with difficulty, manifested by resistance, against a background provided by expectation.”[2]
But truth and beauty cannot be hoarded and lorded like some specialized knowledge, credentialed expertise, or peer-reviewed publication. The elites deride ‘common sense’ and the profound truths seen therein, not because they are false or lack profundity, but simply because they are common.
[1] On April 28, 2023, I entered the prompt, ‘Compose a haiku about artificial intelligence,’ into ChatGPT, and received back the following:
Silicon minds hum,
Data streams through circuits and wires,
AI dreams in code.
[2] Rebecca Goldstein, Incompleteness, p. 161, quoting Thomas Kuhn’s The Structure of Scientific Revolutions.