To Square the Circle

Wittgenstein’s later philosophy, the Philosophical Investigations, examined that human language was inherently flawed because it only exists within language games. Language games are the idea that words are only meaningful because of social interpretation, because we understand a word by having encountered them being used between individuals, in context, over time. 

How do we define words? For Wittgenstein, a word is a class held together by this idea of family resemblance. To define a word, we survey the objects we already call by that name, look for overlapping similarities among them, and then stress-test that provisional definition against further cases we'd also want to call by the same name. A simple example is, how do you define a game? We can look at the examples of games, such as Chess, basketball, Jenga, and Minecraft. But there is no underlying single common denominator between all these examples, apart from the fact that they are all similar enough to be called ‘games’ by us members of the human race, because of our experience in our social contexts. Marvelously, Wittgenstein was able to describe the fallibility of the human invention of words in comprehending the divinity of our God-given human thought. 

“Augustine's fascination with words and his awareness of the difficulty human beings have in communicating their meaning to one another, even when there is no linguistic barrier to cross, made him acutely conscious of the semantic problem. He affirmed the fact that we have to use words as signs to be a consequence of our fallen estate.” 

- Henry Chadwick, from Introduction to ‘Confessions’ 

In Dante’s Divine Comedy, Dante himself journeys through Hell, Purgatory, and Paradise. As narrator, he describes in grotesque detail the sins of man and their divine punishments. Whether constant mutilation, forests of suicide, or coverage in human excrement, Dante spares no effort in his vivid illustrations of his vision, of course, in the most eloquent Italian (Dante actually pioneered the transition from Latin to Italian for literature). Throughout Purgatory and Paradise, Dante writes of all those he encounters, some performing penance for their sins, others saints in heaven who lament the imperfections of the material world. Yet at the end of his epic journey, Dante comes face to face with God himself. He is completely unable to put into words his emotions and paint the artwork of his experience. Rather, he struggles to envision how perfect God is and how mortal humanity is in comparison to his great love. 

Qual è 'l geomètra che tutto s'affige 
per misurar lo cerchio, e non ritrova, 
pensando, quel principio ond' elli indige, 

tal era io a quella vista nova: 
veder voleva come si convenne 
l'imago al cerchio e come vi s'indova; 

Like a geometer who strives but cannot uncover the principle whereby the circle might be squared, just so did I grapple with this new mystery before me, as I struggled to discern how the image of our humanity could fit within that circle. 

ma non eran da ciò le proprie penne: 
se non che la mia mente fu percossa 
da un fulgore in che sua voglia venne. 

A l'alta fantasia qui mancò possa; 
ma già volgeva il mio disio e 'l velle, 
sì come rota ch'igualmente è mossa, 

l'amor che move il sole e l'altre stelle. 

In the end, my wings were not able to take me to those heights, but like a bolt of lightning that flashed within my mind, I suddenly understood and my great desire was fulfilled. At this point my towering fantasy lost its power, but my will and desire came together like a perfectly balanced wheel turned by that Love which moves the sun and all the stars.

Translation: Michael F. Meister, FSC, PhD, (Saint Mary’s College of California) Also see: https://www.youtube.com/watch?v=xua55sg3MNk 

Pure mathematics is a fascinating case study. Proof-based mathematics is one of the most challenging ideas one can investigate due to its level of abstraction. Its study involves reading the symbols that a textbook author has written, and parsing that knowledge into an interpretation function in order to understand it, now become thought and pure matter. Likewise, when we are communicating ideas with one another in mathematics, in the attempt to create scholarly discussion, we convert our thoughts (pure matter) into symbols and equations, whether written with chalk or on a LaTeX document, which is then interpreted by our fellow scholars into their thoughts (pure matter). Mathematics is perhaps the purest form of communication known to mankind so far, as, no matter the culture or artificial borders that mankind has created, any person who has studied mathematics can understand it. The language game is so constrained and rule-governed such that the gap between ‘the finite and infinite’ collapses much faster than in natural language. Yet, mathematics does not escape language games altogether. 

Yet for centuries, mankind has attempted to harness creative expression in order to tackle this challenge of understanding the immortal. Be it music, visual art, or literature, each was created in order to circumvent the problem of inexplicability. Music, with its rhythm, meter, and occasionally lyrics, attempts to activate synaesthesia in human emotion and episodic memory. Its complexity and harmonies communicates universal complex emotional states, yet is often only revealed through analysis from scholars in the field. Likewise, visual artists have created vivid, effortful depictions of reality in their works, in the attempt to understand how we can use colour and material to communicate what reality consists in. 

I have personally found literature and poetry to be the most effective medium for communicating ideas of the infinite. The human imagination is an incredibly powerful object which excites all that can be experienced within our corporeal form. Despite words existing within language games, I have always deeply appreciated the great poets and writers for their ability to build words to entertain our fascinations. 

However, what emerges is a spectrum of failures. Natural language fails widely as its words are held together only by family resemblance, and so the gap between the symbol and the thing it gestures toward is always open, always contestable. Mathematics fails narrowly, as its rules are so constrained that the gap nearly closes, but even Wittgenstein would not let it close entirely, because even a proof depends on a community agreeing to accept it as one. And the artificial intelligence we have built to speak for us fails doubly, because it does not even reach for the gap; it manipulates the symbol without ever touching the pure matter of thought that the symbol was invented to carry. 

If humans are undivine minds that don’t understand themselves, how does it create something else divine? Yet we are pretending to play God in an attempt to create an intelligent being through means of a probabilistic machine. How precious is the ability to turn thoughts from pure matter into words that could be understood by other human beings that then turn those words into pure thought. AI language models are trained entirely on family resemblance, and have access to nothing outside the language game - no contact with the "pure matter" of thought which humans convert into symbols. We can invoke Searle's thought experiment (1980): imagine a person who knows no Chinese locked in a room with a rulebook that tells them, purely by syntactic pattern-matching, which Chinese symbols to output in response to which Chinese symbols they receive. From outside, the room passes the Turing test as its answers are indistinguishable from a native Chinese speaker. Yet intrinsically, this person is manipulating symbols with zero understanding of what any of them mean. They are executing syntax without ever touching semantics. Formal symbol manipulation, all an AI model is now capable of, is never capable of deriving meaning from words, no matter how sophisticated the rulebook gets. If human meaning is already once-removed from thought (thought to symbol), an LLM's relationship to meaning is twice-removed (symbol to statistical pattern over symbols, with no originating thought at all). 

Each of our systems tries to eliminate ambiguity, to force the infinite into a finite form. Perhaps it is poetry alone that refuses this mission. Where language, mathematics, and machine all treat ambiguity as a defect to be minimised, poetry treats it as the only instrument elegant enough to gesture at what cannot be said outright. It makes the gap between the finite and infinite itself the subject, deeply embedding its own contradiction as meaning. This is why Dante's wings fail him at the very threshold of God and yet, in failing, succeed: the fulgore that strikes his mind is not a solution to the geometer's problem, rather an experience that happens instead of a solution, and only poetic language is built to hold both the failure and the flash of understanding in the same line. Catullus, Whitman, Virgil, they all understood how integral our contradictions and multitudes are to humanity, and they embraced the impossibilities. 

"Im gemeinen Leben kommen wir mit der Sprache nothdürftig fort, weil wir nur oberflächliche Verhältnisse bezeichnen. Sobald von tiefern Verhältnissen die Rede ist, tritt sogleich eine andre Sprache ein, die poetische. 

In everyday life, we barely get by with language as we only describe superficial relationships. As soon as deeper relations are at stake, another language enters: poetic language. - Johann Wolfgang von Goethe 

YZ