On Totality and Incompleteness in Man and Machine

4 Then they said, “Come, let us build ourselves a city, with a tower that reaches to the heavens, so that we may make a name for ourselves; otherwise we will be scattered over the face of the whole earth.”

​

In raising a tower to the firmament it was Nimrod’s hubris that led God* to scatter the people of the Earth and confound their language. Or so it goes in Genesis 11:4: part of an allegory laid down to illustrate the folly of pride and over-glorification of human achievement. I see in this story man’s inherent drive toward totality, whether suffused by knowledge or ecstatic in love, as a story of self-realisation. We are born with a longing to identify God with form or concept, and manifest such that we might look into his eyes as equals. Humility, however, is a virtue, and an acceptance of the plurality of man – and by extension knowledge – is the love of God’s will.

​

Relativism of language began with the tower of Babel, and I think this parable captures an incompleteness at the heart of human experience from which knowledge is derived. I want to discuss incompleteness and inconsistency in mathematics, which are themselves neighbours to algorithm and computation, the results of which pervade modern life through our technology. The nature of algorithm reveals itself in the various creative projects undertaken by artificial intelligence, and as computer algorithms generate art, they enter into the realm of the aesthetic and the divine -- that is to say, the profoundly humane. There exists an incompleteness in the logic which underlies algorithmic computation as famously demonstrated almost a century ago (see below), and I find an incompleteness in my own subjective experience of art generated by artificial intelligence. Perhaps just a matter of taste, but it’s an intriguing parallel, and we might better understand how the inherent limitations of algorithms manifest in the character of digital art by considering incompleteness in mathematics, algorithm, and computation.

​

​

At the foundation of mathematics exist various statements which are known to be true but cannot be proven as such. These self-evident truths are called axioms. Taking a selection, for example, of some intuitive arithmetic and geometric axioms:

 

• If A = B, then B = A 

• Two lines drawn parallel will never meet

• Zero is a natural number

 

Once certain axioms are established, one might then derive statements of mathematical truth by performing consistent mathematical operations to build upon them. New axioms can be added to the system, or taken away, to define the scope of the system being employed. A mathematical statement was proven true if a sequence of propositions could be constructed from the axioms which terminated with the statement in question. Indeed, at the turn of the twentieth century David Hilbert and other mathematicians subscribed to a philosophy of mathematics called formalism whereby it was posited that mathematical proof could be reduced to the operation of consistent rules on mathematical formulae without the need to consider the meaning of the formulae being manipulated. Intuition was of secondary relevance to the formalists, who regarded mathematical truth as a quality of syntax within a logical system, rather than a discovery of some fundamental aspect of reality. It was Hilbert’s dream to lay down a comprehensive set of axioms from which any mathematical statement could be either proven or disproven according to a set of derivation rules, and to prove the consistency of these axioms with one another. The edifice of mathematics would be complete and unnassailable  - it was not to be.

​

8 So the Lord scattered them from there over all the earth, and they stopped building the city. 9 That is why it was called Babel—because there the Lord confused the language of the whole world. From there the Lord scattered them over the face of the whole earth.

​

In 1931, young Austrian logician Kurt Godel published an ingenious treatise which contained what came to be known as his incompleteness theorems. Godel was probing logical systems of reasoning for completeness and consistency, inspired by the logical paradoxes demonstrated by Bertrand Russell and others. A mathematical set is a collection of objects, and Russell challenged mathematicians to consider the set of all sets that do not contain themselves (labelled R). He then asked whether R contains itself, thus instantiating a paradox. A simpler way to see this is to imagine that in a small town a barber shaves only those men (or women if he’s feeling particularly adventurous) who do not shave themselves. Who then shaves the barber? If the barber shaves himself then he breaks his own rule, and if he does not, then by virtue of being a man who does not shave himself, he becomes obliged to shave himself. Drawing on these ideas, Godel’s stroke of ingenuity was to develop a scheme, using natural numbers, to encode any statement of arithmetic with a unique number - called a Godel number. The statement 1 + 1 = 2, then, has a unique Godel number, and the relationship between Godel numbers specifies whether any given statement would result from the previous statements in a mathematical proof.

​

With an encoding scheme to hand, he then was able to meticulously build a logical system and insert a paradox of the type given above which was syntactically consistent with the system. Simply put, his statement expresses in logic: “this statement cannot be proven.” If true, then it should be able to be proven (since a consistent mathematics should be able to prove what is true), whereas if false, then it must be true that it can be proven, thus entailing a contradiction either way. The only conclusion is that this problematic statement is true, but can neither be proven nor disproven, thereby shattering the formalist aspiration to completeness. Moreover, Godel’s incompleteness theorems are equivalent to the “father of modern computer science,” Alan Turing’s, solution to the halting problem, which asks whether it is always possible to know whether a mathematically idealised computer (called a Turing machine) will finish a given computation in a finite time. Turing proved that it was not. This should be no surprise: we now know that there exist some true statements which can be neither proven nor disproven within the confines of a particular logical system, and so a computer faced with the unenviable task of deriving them by algorithmic means will go on calculating forever. The formalists were essentially trying to reduce mathematical truth down to an algorithm. For our discussion, we note that modern computers are universal Turing machines and Godel's incompleteness lives within them still.

​

The utility of Godel’s encoding scheme was that it was more general than the system of logic it described. It therefore allowed for meta-mathematical commentary, and for Godel to construct his paradoxical statement which he knew to be (syntactically) true but had not proved within the logical system. His theorems demonstrate that the generality of a given language allows one to construct and verify true statements which lie beyond other, more specified, languages. I think this point is a profound one, and speaks to the way the mind handles information and abstraction. In his book The Emperor’s New Mind, Roger Penrose extends this idea of knowledge without formal proof to argue that our capacity for insight-based approaches to knowledge suggests the mind is not algorithmic in nature. Whilst this is a controversial and complex argument, I think it fair to say that insight – that is to say knowledge without logical proof – is of critical importance to creative endeavour. The artist is guided by an intuition, or by the magnetism of ancient symbols which strike to the depths of the heart. The way in which emotive art communicates directly with its audience testifies to the lack of a need to prove the knowledge it embodies. Conceived in the mind of the artist, great art is self-evident.

​

Consider now the Library of Babel: an audacious online project which aims to generate every possible one-page sequence of 3200 characters and store them in a catalogue. Every single unique poem, play, thesis, lyric, script, confession of love and story of death exists within these bounds. Every single truth expressible by language has been iterated into life. If you gave enough monkeys with typewriters enough time they would eventually write Shakespeare: the old thought experiment writ large! This was an astonishing find to me, and I began to type in an original scrap of poetry I had conceived of recently, watching nonplussed as it was displayed on screen with cool alacrity. Initially, I was spooked by the library's immensity. I felt suddenly unoriginal given that an algorithm had arranged every combination of letters up to 3200 characters (and what’s to say it won’t go further) I might conceive of in my lifetime. In a very real way I think that’s justified, but on further reflection, I thought that this gargantuan flex of computational muscle is particularly telling in the relationship between humans, computers, and art. Although the algorithm had generated every possible sequence of characters, it had done so indiscriminately, such that for every passage of interpretable genius, there exist countless permutations of the same phrases which are pure gibberish. Although impressive, there is little intelligence in the action of the computer here, and in AI-generated art, I’ve encountered a sense of artificiality, a hollowness that speaks to a lack of humanity. An incompleteness in the creative expression of machines, which might just result from an incompleteness at their mathematical foundation.

​

This only seems reasonable; for one, the human mind, and the behaviours it produces, are orders of magnitude more complex than artificial attempts at intelligence. This means we are difficult to predict: creativity displays highly chaotic dynamics. Again, we might appeal to Godel by suggesting that in order to conceptualise human creativity in a formal system of logic, we would need a more general language of logical description. For our brains to use their own cognitive products to describe themselves at greater generality seems like an impossibility. Moreover, human self expression is the integrated product of every single second of life experience, and as the artist grows and matures her work becomes the integrated expression of her being. Art, and insight, is as broad as the spectrum of experience and human beings evolved to be generalists across a variety of habitats. It is, in fact, through the lens of inherited evolutionary priors that we are able to perceive and make sense of art at all. In a sense, art is as broad as the entire history of human consciousness. Whilst artificial intelligence is able to faithfully replicate Beethoven’s style to complete his tenth symphony within local constraints of style and form, I get the sense that no programme extant today could lay the groundwork for an eleventh symphony. Logical incompleteness, and hence the limits of computation, arise from the magnitude and complexity of our unique evolutionary and personal histories which shape the architecture of our brains.

​

We might conceive of human creativity as the connection and/or synthesis of two ideas in a network in a novel or powerful way. This connection must span a certain affective “space” between memories, emotional states, or artefacts of life, say. In designing artificial intelligence programmes, we are opting for algorithms with explicit programmatic and implicit mathematical boundaries. Creativity draws on a certain breadth, then, which is antithetical to the nature of algorithm**. Just as in the natural sciences, computation is foremost a tool, rather than means of  creative insight in of itself.

​

Recently, an artificial intelligence programme created a digital piece which won first prize in the Colorado State Fair’s digital art category. Theatre D’opera Spatial, the piece submitted, is an ethereal composition with a certain deftness of touch and delicacy of colour. Three figures stand draped in sumptuous robes with backs turned, gazing into a gleaming world beyond, immersed in the dreamscape of a Venetian space opera. I find this piece more convincing than others I have come across, though still characteristic of the technologist’s proclivity for logic as expressed in science fiction, rather than the deep humanity of artistic genius. Critics may fail to credit the technique required to produce a piece of art aided by AI, whereby aside from expertise in programming, a painstaking process of fine tuning and experimentation is often required. This particular composition was the result of over eighty hours of experimentation with text prompts to feed the algorithm, and I think there's a decent case to be made that the technique of the artist aided by AI deserves the respect given to a master craftsman who knows his tools intimately. Ultimately, the question of whether the idea supersedes the technique in artistic importance has been one of great debate harkening back to the Renaissance. I think that this particular medium deserves respect, but is no substitute for the character and humanity of the great artists. Theatre D’opera Spatial is an impressive alliance between man and machine, but is certainly no masterpiece.

​

​Knowledge lives with incompleteness and ambiguity in the garden of the mind while art and spirituality gesture toward the mystery of being. Instances of incompleteness in the logic of formal systems of computation signal the mind’s capacity to think beyond the machine, if not up to its speed, and my experience of artificially generated art likewise comprises a feeling of incompleteness. I hope this semantic and aesthetic parallel has been of interest, as the complement between man and machine in scientific and artistic endeavours is a fascinating story for the future. Whilst projects such as the Library of Babel employ machines to drive towards totality, the story of the tower of Babel suggests that incompleteness is the loving will of God, and is best placed to preserve and ground us in our humanity.