Antoni Gaudi and learning algorithms from Nature

Happy holidays.

A few days ago, I was exploring Barcelona. This means that I saw a lot of architecture by Antoni Gaudi. His works have a very distinct style; their fluid lines, bright colours, myriad materials, and interface of design and function make for very naturesque buildings. They are unique and stand in sharp contrast to the other — often Gothic revival and Catalan Modernisme — architecture around them. The contrast is conscious; when starting out, Gaudi learned the patterns of the neo-Gothic architecture then in vogue and later commented on it:

Gothic art is imperfect, only half resolved; it is a style created by the compasses, a formulaic industrial repetition. Its stability depends on constant propping up by the buttresses: it is a defective body held up on crutches. … The proof that Gothic works are of deficient plasticity is that they produce their greatest emotional effect when they are mutilated, covered in ivy and lit by the moon.

His buildings, however, do not need to be overgrown by ivy, for Gaudi already incorporates nature in their design. I felt this connection most viscerally when touring the attic of Casa Mila. The building was commissioned as an apartment for local bourgeois to live comfortably on the ground floor off the rents they collected from the upper floors. And although some of the building is still inhabited by businesses and private residence, large parts of it have been converted into a museum. The most famous part among tourists is probably the uneven organic roof with its intricate smoke stacks, ventilation shafts, and archways for framing other prominent parts of Barcelona.

This uneven roof is supported by an attic that houses an exhibit on Gaudi’s method. Here, I could see Gaudi’s inspiration. On display was a snake’s skeleton and around me were the uneven arches of the attic — the similarity was palpable (see below). The questions for me were: was Gaudi inspired by nature or did he learn from it? Is there even much of a difference between ‘inspired’ and ‘learned’? And can this inform thought on the correspondence between nature and algorithms more generally?

I spend a lot of time writing about how we can use algorithmic thinking to understand aspects of biology. It is much less common for me to write about how we can use biology or nature to understand and inspire algorithms. In fact, I feel surprisingly strong skepticism towards the whole field of natural algorithms, even when I do write about it. I suspect that this stems from my belief that we cannot learn algorithms from nature. A belief that was shaken, but not overturned, when I saw the snake’s skeleton in Gaudi’s attic. In this post, I will try to substantiate the statement that we cannot learn algorithms from nature. My hope is that someone, or maybe just the act of writing, will convince me otherwise. I’ll sketch my own position on algorithms & nature, and strip the opposing we-learn-algorithms-from-nature position of some of its authority by pulling on a historic thread that traces this belief from Plato through Galileo to now. I’ll close with a discussion of some practical consequences of this metaphysical disagreement and try to make sense of Gaudi’s work from my perspective.

We do not learn algorithms from nature; instead, we project our algorithms onto nature in order to make meaning of it. If we do this — carefully and critically — then Nature, on occasion, can point out where we are mistaken and thus motivate us to try anew, or, more often, to refine and slightly modify our existing repertoire of algorithms. As such, algorithms are primarily in our minds, a way for us to share — and thus create — meaning and understanding. They are only read into the world by our attempts to assign Nature sense and meaning. It is not the case that there are algorithms underlying nature for us to simply discover. Much like there are not true names underlying objects.^[1]

I think there are two particularly salient features of algorithms that we should focus on: they are self-contained and communicable. The communicability suggests an analogy to language. In fact, the mention of ‘true names’ is meant to hint that our question can be informed by an old philosophical problem. Now it is merely dressed up in the modern garb of computation. As such, in this article, I will first chase a few historic threads before returning to my algorithmic philosophy and Gaudi.

The oldest precedent of this discussion for me is Plato’s Cratylus. In this dialogue, Socrates, Hermogenes, and Cratylus debate the correctness of names. Hermogenes advocates for conventionalism — that word meanings are assigned arbitrarily and conventionally by their users — while Cratylus and, to a lesser extent, Socrates advocate for naturalism: names for objects cannot be chosen arbitrarily because objects have true names and a good language reflects this. Cratylus goes as far as saying that speaking of anything by a name other than the true one is senseless; in such cases, one is referring to nothing at all. With an atypical air of confidence and good self-deprecating humour, Socrates leads us towards this true language — probably some variant of Greek for Plato — by tracing the etymology of a few philosophically significant words. By tracing this etymology, Socrates is learning the word’s true names and thus the Platonic Forms that they denote.^[2]

This ancient caricature seems easy to deny and parody, and its ethnocentric bend is evident. In fact, a lot of the modern readers of Cratylus take Socrates’ humor as a deflation of naturalism, although it is clear that contemporary commentators — Aristotle, for example — did not read these passages ironically. Of course, you can reject Plato’s method of etymology, while still disagreeing with my assessment of the relationship between nature, mind, and algorithm. After all, one might say that we use the more sophisticated processes of science rather than tracing etymology to learn algorithms from Nature. This tension should motivate us to track the idea closer to today and try to find the moment when it found more pleasant robes.

Although the ‘true names’ or ‘true language’ perspective has lived on most recognizably among some philosophers and mathematicians,^[3] these rationalist versions would probably be passed over in silence by physicist and many other scientifically, especially empirically, minded readers. However, that doesn’t mean that we won’t find something akin to ‘true names’ if we undress the empiricist’s zeitgeist.

Let’s trace the thread to Galileo. Physicists often paraphrase his Assayer aphorism that

this grand book — I mean the universe — which stands continually open to our gaze … is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it

For Galileo, this passage would have been much more literal than for the modern reader. God had revealed his will to us through two great works: the Bible and the world around us. The astronomer’s friend, Pope Urban VIII, would help us understand the great Latin book, while Galileo was happy to teach us how to read the mathematical book.^[4] But with the first and second industrial revolutions, we killed God and forgot the theological origin of the book of nature. The modern reading became metaphorical. There is now no author for the text but there is still the (usually) implicit assumption that the writing is intelligible. Although actually reading nature to the laity might still require the inspired genius of scientific saints like Einstein or modern lab-coat wearings priests.

As an aspiring member of the modern day ecclesiastic class of experts, it is in my interest to believe that I am learning to read eternal truths rather than writing ones that spring to mind. It is the convenience of this belief that feeds my skepticism. But that doesn’t mean that I want to replace the scientific reading of nature by some other mystic method for learning eternal truths. Since no such method exists. Nor does it mean that all beliefs about the world are equal and grounded only in convention. I am not willing to swing as far towards conventionalism as Hermogenes. Rather, I think that we have to take a Kantian stance in between. The world in itself — the noumena — is not directly accessible nor necessarily intelligible. Instead, we rely on the phenomenal world that structured the noumenal through our methods of making sense of it. The most salient features of the structuring come from our minds and the culture that helps shape them, but the structure also involves our bodies and technologies. Thus, when we arrive at some truth, we must recognize that part of it reflects the world and part of it reflects our mind and tools.

What does this mean in practice? Although the distinction is largely metaphysical, it does change which generalizations we consider first. Suppose that I keep seeing the multiplicative weight updating rule or power-laws crop up during my investigations. If I believe that I am learning algorithms from nature then it feels natural to generalize that there is some deeper Truth about the universe that my observations are hinting at. However, if I believe that I am projecting algorithms onto nature then this just highlights my limited repertoire of mental models. From this perspective, when different fields keep finding similar things — especially foundational things — then my first instinct shouldn’t be to celebrate the unity of nature but to mourn the lack of diverse perspectives and approaches. And since my tools constrain or shape the sort of truths I can arrive at, I must be extra mindful of the fact that I cannot simply choose independently problems to work on or questions to answer; I have to look at problem-tool pairs.

Of course, we don’t need to look for important examples buried in the philosophy of science. There are also direct concerns on how we implement social algorithms and how we make claims of ‘objectivity’ or ‘correctness’ for inferences made about people from big data or other models. But that discussion is better saved for another post.

The perspective I sketched above makes Gaudi’s opening quote on the imperfection of Gothic revival easier to unravel. His references to tools (compass, formulaic industrial repetition, buttresses, crutches) highlight the interplay between the method and the final product. An interplay that most architects and designers have to be attentive to. It also pushes us to look into Gaudi’s method for representing, conceiving, and communicating his vision. Whereas most of his contemporaries relied on architectural drawings, Gaudi preferred to build three-dimensional models and improvise during construction.^[5] The few drawn plans that he did produce were mostly to appease the authorities that demanded them. His physical models allowed him to externalize computation to nature, instead of trying to learn its algorithms and disrupt them by the projections of an internal representation. To design his arches, for example, he would sketch the floorplan on a board and then hang strings with weights on them from where he wanted the columns to start, adding more strings as he saw fit. This created an upside down force model. A photo of the resulting suspension when turned right-side up provided the arches and columns that Gaudi wanted. In this way, his method was not self-contained and thus less algorithmic.^[6]

Gaudi escaped the rationalist rigidity of Gothic revival by projecting fewer of his own algorithms onto nature. He recognized that he couldn’t learn algorithms from nature to then simply apply this knowledge to construction. He had to draw inspiration from and work with nature for a final product. If he had tried to extract and codify the essence of a natural style then I suspect that he would have instead learnt a pattern of his own projection and recreated it with formulaic industrial repetition. Instead, we got the Case Mila, Sagrada Familia, and other great architecture in and around Barcelona.

Notes

1. I’ve touched on these ideas of the connections between algorithms, the world and mind before in my posts on Gandy’s physicalist and Post’s Kantian variants of the Church-Turing thesis. As might be obvious from those articles, my position is closer to Post’s than to Gandy’s.
2. If we must learn about the world around us and not just the Forms then Plato allows this, too. This is because the world around is just a shadow of the Forms, so knowing the forms is strictly better. Of course, for Plato, the Forms and not the imperfect world was the target of Knowledge, so he didn’t bother to explicitly worry too much about the latter.
3. To stay with the steps of language, it is tempting to look to the analytic philosophy of the linguistic turn. Here, we can see Plato dressed up in the styling of the Logical Atomism of Russell and, to a lesser extent, early Wittgenstein. They maintain the idea of words denoting of the world and the dream of an ideal language. However, the logical atomist knows that her common language is limited, and since she might have learnt to read the ancient Greeks in their original, is aware that their language fares no better. Overall, Plato’s (pseudo)historic perspective is distinctly non-modern. The modern scholar expects to progress to the ideal language, not find it in the past. She aims to build the perfect language for herself through reductive analysis. In logic, she hopes to eliminate all ambiguity in finding a one-to-one correspondence between words and objects, propositions and facts. For the logical atomists, the world and the true language become the same and all hard questions are reduced to the epistemology of learning the logic from our experience with the world and the structure of our current faulty language of descriptions.
4. Of course, claiming the authority to interpret the book of God is not the best idea during the Thirty Years’ War and can land one before the Inquisition. The popsci accounts of Galileo like to imagine him as a champion of scientific truth and dispeller of Church mythology, a radical martyr of a new scientific world view. But it is not clear to me that the dispute that occurred was really as far outside the Church mainstream as we like to tell it. It can be viewed as a part of the ongoing theological debate at that time over who can interpret the holy books and which books should have primacy. The emerging Protestant sects were encouraging the laity to read and interpret the Bible for themselves, while the Catholic church saw the priest as a necessary mediator. Galileo was claiming to interpret God’s work and suggested that his reading of the book of Nature was authoritative. The claim of authoritative reading was the point at issue, and the fact that the reading resembled something that would eventually become science was purely incidental.
5. His magnum opus – – Le Sagrada Familia — is a testament to improvisation: when I visited it over 90 years after Gaudi’s death, it was still under construction and projected to remain under construction for another decade. One of the things that draws me to medieval Gothic cathedrals – – the ones that were built between the 12th and 16th centuries and inspired the Gothic revival or neo-Gothic in the 18/19th century – – is that a single cathedral would have been constructed over several lifetimes and worked on by many architects and master craftsmen. This makes the construction procedure not self-contained, with the mental models of many agents shaping the final product. It allows a break of the “formulaic industrial repetition” and lets the cathedral adjust to how it has been worn and ‘mutilated’ by time. When it is being finished, the new architects can already see how nature has taken its course with the older parts of the cathedral and accommodate this in their development. By contrast, the buildings of the Gothic revival and neo-Gothic were constructed much faster and from the vision of a single designer; making the projections of their creator much more evident. Although many other pieces by Gaudi, like the Casa Mila, were conceived and built solely by him — in collaboration with many artists, architects, and craftsmen; to whom he gave great autonomy — the distributed construction of Le Sagrada Familia points to another departure from neo-Gothic and a return to a more traditional style.
6. The lack of clarity on what I mean by self-contained is a hole in my views on this topic, but hopefully, doesn’t leave my position in tatters. To see this, consider the example of hanging up clothes to dry. I can specify a procedure like: draw a rope between the two posts in the backyard to make a clothesline, start with your largest bedsheets and drape it in the middle of the clothesline, if only one of the edges touches the ground then pull on the other edge until it hangs at the same height as the first, if both edges touch the ground the tighten the clothesline, etc. I would consider this to be an algorithm. However, like Gaudi’s method, it relies on gravity and physical geometry for solving subproblems like sag of the clothesline or the midpoint of the sheets. Yet it feels that these aren’t the ‘central’ parts of the clothes hanging algorithm but are just well used physical constraints. It feels like they aren’t essential to an abstract representation of what’s happening. For Gaudi, on the other hand, gravity does the ‘real work’ of forming the representation.