In 2019, the CEO of Civil Science, a consumer data and analytics company, posted a survey that showed 56% of polled respondents disagreeing with a claim that Arabic numerals should be taught in American schools. The post was both funny and sad, illustrating, at best, just how much of a disconnect with the history of mathematics the average American has, and at worst, an upsetting xenophobic reaction to the word “Arabic.” Of course, the joke here was that “Arabic numerals” is just another term for the common graphic depictions of the numbers 0-9. But perhaps more worrisome, as any high school math teacher could probably attest, is that a large percentage of the general public is blissfully unaware of the history of mathematics, including its Arabic heritage and influence from the graphic depictions of numerals to the root of the word algebra (from the Arabic Al-jabr). In fact, if we look at its history we’ll find many curious and almost awkward episodes concerning conflicts, debates, experimentation and wonder that have been largely under explored in popular culture and exist mostly as B-sides of scientific history. Episodes such as the Newton-Leibniz controversy over the claim to have invented calculus or Sir William Rowan Hamilton etching his formula for quaternion multiplication like graffiti into the wall of a bridge in Dublin, for example, illustrate just how rich and socially dynamic this history is. For such a supposedly objective discipline, math’s history is full of personal and diverse subjective narratives.
Andrew Witt’s new book, Formulations: Architecture, Mathematics, Culture, appears to be a kind of travelogue of some of these curious episodes. Witt narrates an extensive history of mathematics’ entanglement with architecture and culture at large, covering inventions and ideas like descriptive geometry, drawing machines, hyperdimensional math, stereographic drawing, chronophotography, topological surfaces, voxelization, and hyperbolic architecture. Formulations takes readers on a voyage through the conception of these topics in a somewhat chronological order (Witt tends to fast-forward and reverse through different eras over the course of a chapter), revealing the times when mathematicians thought like, and were influenced by, architects and architects thought like, and were influenced by, mathematicians. We encounter familiar names such as Gaspar Monge and Buckminster Fuller as well as less familiar yet nonetheless key figures in the math-architecture alliance such as Ivan Sutherland and J. D. Bernal. The 400-page volume concludes with a chapter titled “Of Dabblers and Virtuosos” where the author reflects on the contemporary role of these disciplinary topics in design and the cultural sphere with a crucial question: “what, then, is the place of mathematical expertise in an architectural discipline surfeited with ready-made technique distilled into consumable software components?”
To find Witt’s answer to this, you will just have to read the book. But the structure of ending with a contemporary provocation related to historical episodes pervades the entire volume. Witt’s arguments are simple and well-structured, often consisting of a situating historical event, followed by a rigorous analysis of the instrument/technique/concept in question, and ending with a relevant connection to the contemporary moment. These conclusions are typically short polemic statements on each mathematical instrument/technique/concept’s relevance in our software and computation-based world. In a way, this is perhaps the main role of the book: to provide an allegorical history of math and architecture in order to reflect on the systems that undergird the current production of design media. Math, after all, is a crucial component of software instruments that today allow us to shape the built environment. It would therefore make sense to publish a book on the lineage of geometric math and its influence on design pre-computation. But this reading would also drastically undermine the richness of the dialogues Witt engages throughout his tome. There is a bigger mission related to scientific truth and objectivity afoot.
An important question that Witt poses, albeit indirectly, is: was math invented or discovered? Formulations addresses this through various anecdotes, concluding that techniques are inventions, but math itself is a kind of designed discovery, a language that helps explain approximately how the universe works. The figures that facilitated this design/discovery have historically been hyper-curious, hyper-focused individuals, “virtuosos” in a specific subject seeking out truths. For me, this reveals a better argument for the entire project: that math is not a truth in itself, but rather an instrument derived from best guesses and speculation. Moreover, and much like architecture, it often engages approximation and tolerances. This perspective shifts the perception of math from objective gospel to subjective conjecture.
At various points throughout the book, Witt uses specific language to illustrate the tension between truth and untruth. He describes the technique of geometric triangulation as the “rational control of an irrational world” and elsewhere explains that “new geometries shook confidence in old systems of both drawing and measurement.” This rhetoric follows a Duchampian line of thinking and recalls the artist’s own quote that states: “I believe that the laws of physics such as they are, such as they have been taught to us, are not the inevitable truth... After all, every century or two a new scientist comes along who changes the laws of physics, isn’t that so?” It is telling that Witt also invokes Duchamp when discussing four-dimensional geometry. Formulations could be considered a faithful enactment of Duchamp’s revisionist observation. Acutely aware of artistic dialogues on hyperdimensionality, Witt alludes to the mystical draw of multiple dimensions for other artists like Max Ernst and Theo van Doesburg.
These links to surrealist art and experimental design are perhaps some of the most enlightening throughout the volume. Witt’s dive into Max Ernst’s La Fable de la souris de Milo explores the connection between surrealism’s fascination with scientific and mathematical imagery. But unfortunately the discussion on surrealism stops at Ernst, and while there are a few mentions of Duchamp, there is no reference to his famous quest to undermine mathematics with works such as The Bride Stripped Bare by Her Bachelors, Even (The Large Glass) and Three Standard Stoppages, both canonical works that play off the very mathematical measuring and drawing instruments Witt is so carefully analyzing. There are, however, some fascinating quotes summarizing Amédée Ozenfant and Le Corbusier’s skepticism of the fourth dimension. It appears that, while they appreciated the mathematical acrobatics it afforded, architecture was, for them, primarily a three-dimensional affair; a rather conservative view for such innovation-driven individuals. (They may be surprised to hear it, but 4-dimensional multiplication is a core component of modern 3D-modeling software.)
Formulations does feel a bit unfinished. Despite being quite lengthy readers might be left wondering about the contexts of the episodes described therein. There is no mention, for example, of the role geometric surveying played in the seizure of indigenous land in the Americas (only its military role in 17th and 18th Century France is discussed). Witt also largely avoids discussing the colonial tradition associated with mass scientific initiatives in the nineteenth century where new “natural” discoveries in the sciences led to theories and systems of classification that, in turn, facilitated racial subjugation and justified colonization. Mathematics in the era discussed participated in the same othering and exoticising practices that backed up many claims to Western hegemony. Witt does briefly acknowledge this phenomenon in certain areas. In one section on stereoscopy, he makes wide use of the term “tourist” and “tourism” to recall the enlightenment phenomenon of traveling and bringing back artifacts, animals, and people from places outside Europe. This tourism extended, Witt argues, to mathematical forms and novel geometries, yet the “stereographic tours of specific cities and regions across Europe, Africa, Asia, and North America” are described dryly as part of the touristic tradition with little mention of the colonial impulses that facilitated them.
This is not to say that every history of science must rigorously examine its attendant problems in hindsight. This would make it difficult for non-experts to discuss issues of othering, dominance, and colonization. But to eschew the clear contexts of Western hegemony in discussions of science and its cultural impact is to avoid the elephant in the room. Which is that not only have math and geometry been instruments of cultural production, they have also always been instruments of control. It is difficult to acknowledge this history, particularly when there is so much to say about the instruments themselves. Formulations hints at a critical position on control, but tends to prioritize a positivistic lens on the subject at large. The goal is always to expand the reader’s historical knowledge of mathematics. Witt’s five page conclusion is by far the strongest component of the book, and may stand by itself as an effective polemic essay. It addresses criticality as well as sets up potential arguments for others to follow, expand on, or debate. More importantly, it offers a sharply witty and relevant comment about architecture, provocatively calling it “a kind of invented truth.” If there is a better descriptor for our wacky discipline, I haven’t heard it yet.