Michael McCarrin is a PhD candidate in History of Consciousness at UC Santa Cruz. Their project explores the ways foundational assumptions of Western philosophy have affected the development of mathematics and computer science as disciplines. McCarrin served as a Summer Dissertation Fellow in 2021; the Hayden V. White Summer Dissertation Fellow in 2022, and was a member of the Coha-Gunderson Prize Selection Committee in 2023. In January, we discussed the extent to which foundational mathematics has been a colonial project, the kinds of Western biases that have permeated important historical mathematical projects, and McCarrin’s experiences serving as a judge for the Coha-Gunderson Prize in Speculative Futures.
Hi Michael! Thank you for chatting with us about your ongoing work! To begin, would you provide us with a brief overview of your dissertation project and what you are currently focused on?
Sure! I’m writing about the ways in which foundational assumptions in Western philosophy have been carried over into fields like mathematics and computer science, where they continue to influence the way we think about problems and make predictions. For example, I’ve just recently been working on a chapter explaining how current conversations about artificial intelligence are related to deeply rooted ideas of what “intelligence” means and how it works. In mathematics, the context is a little different, but the approach is similar; I’m looking at the way our ideas about number and infinity have been influenced by Western metaphysics.
In your dissertation, you argue that one “major effect of the European quest for an unassailable foundation of mathematical knowledge” was to make this foundational mathematics “unmistakably European” and therefore, a colonial project. Can you share some of the impacts of “claiming mathematics for the West”?
Yes. First though, to be clear, like many projects that contribute to coloniality, the people involved at the time did not have colonization as an explicit goal, and would probably have been very surprised (and likely annoyed) by this suggestion. They believed they were establishing eternal truths that transcended nationality and politics by removing any unproven statements from mathematics. The method they settled on was to make set theory and logic the foundations of arithmetic. The reason I’m saying this is a colonial project is just that even though mathematics as a field is enormously diverse and heterogeneous and incorporates contributions from all over the world, logic and set theory are historically European, and claiming that they are the foundation of all mathematics seems to imply that they have a special status which is elevated above everything else. Also the idea of “establishing a foundation” in itself is a very European project, and continues a long tradition in Western philosophy.
The foundational work in mathematics inscribed certain almost religious beliefs about the nature of space and time into the discourse of mathematics, which was then disseminated throughout the world.
There were a tremendous number of consequences to this effort. The ones I’m focused on are not easily recognizable as expressions of colonialism, but one way to think about the impact is by analogy to missionary work. The foundational work in mathematics inscribed certain almost religious beliefs about the nature of space and time into the discourse of mathematics, which was then disseminated throughout the world. These ideas are not usually thought of as religious or even specific to a particular culture, but I’d argue that’s because the project was actually much more successful than most missionary work in persuading its disciples that its beliefs actually were universal, objective truths.
The way this worked is a little difficult to describe without going into technical details, but the most prominent example of an effect is probably the real number system, which is the system of numbers we use to do calculus and many other things. This system is so commonly accepted that most non-mathematicians probably think of it as just “numbers,” but the way it is defined gives it some very strange and surprising properties, many of which used to be considered controversial even among famous and highly regarded mathematicians like Carl Gauss. For example, almost all the numbers in this system cannot be calculated by a computer or written out because they are infinite but patternless sequences of digits. There is no way to express them—not even by a formula or an algorithm that runs forever—but according to the most commonly accepted definitions we still believe not only that they exist but that they make up almost all of the real numbers.
Your work is deeply entrenched in the disciplines of mathematics and computer science. How does your project make use of Derrida’s critique of “logocentric bias” to historicize the rise of computer science as its own discipline?
This question is a little tricky to answer because I need to summarize Derrida’s position. But here goes: what Derrida’s critique of logocentrism argues is that the history of Western thought in general has repeatedly tried to establish objective facts about the world by basing them on some unquestionable set of assumptions which forms a foundation on which everything else can be built. He calls this structure “logocentric” because it characterizes the basic strategy of the Western concept of the logos, which is the Greek word for “rationality” and also the root of the word “logic.”
The foundations are “unquestionable” in the literal sense of the word: the insight of the critique is that if you start to look into these assumptions you’ll find they are not actually the foundations, but rather the consequences of some deeper set of foundations. So they can only be considered the foundations as long as they remain unquestioned. What Derrida points out is that we can’t seem to stop ourselves from asking these foundational questions, so we have to keep adding new foundations to the structure to support it. In other words, the repeated establishment and collapse of new foundations is endemic to the structure of Western thought.
David Hilbert in 1907
My argument is that this pattern is exactly what European mathematicians were unwittingly participating in when they tried to establish a foundations of mathematics in the 19th century. The effort culminated in “Hilbert’s Program,” a project led by influential mathematician David Hilbert which proposed to “finish” mathematics by creating a formal axiomatic system that was complete, consistent and decidable. Complete means that every true statement could be proven from the axioms; consistent means that it was impossible to prove two conflicting statements; and decidable meant that given any correctly written statement you could always decide whether or not it was true.
The program collapsed in 1931 when Kurt Gödel proved no system powerful enough to express arithmetic could be complete. Five years later Alan Turing showed no such system could be decidable. To do so, he developed a model of computation that became a early landmark in the budding field of computer science.
So the basic idea that connects this historical event to Derrida’s critique is that it demonstrates an instance of the much broader pattern Derrida is trying to describe. Further, it can be seen as part of a much larger reconfiguration, occurring over centuries, of everything that was based on this foundation-oriented strategy.
What is one “metaphysical bias” you perceive in the field of contemporary mathematics that warrants further study?
The use of the set as the basis of number. A set is a mathematical object that represents a collection of things. So you can have a set of teacups, or a set of integers between 0 and 10. And sets can also be infinite–for example, the set of natural numbers contains all the numbers you could ever count to.
One of the questions that was debated in the 20th century as part of the project of establishing a more secure foundations of mathematics was how to define numbers themselves. For example, does the number 3 represent a place in a sequence (the 3rd apple on the table) or does it represent a quantity, or something else? (For some reason everyone seemed to agree that it had to be one thing or the other.)
Since the set is fundamentally about a collection of elements, using it as the basis for number indicates that the people arguing for the definition of number as a quantity won out over those arguing it had something to do with sequence. This resolution has consequences for mathematics because sets make it extremely easy to represent quantities but relatively cumbersome to represent sequences. Furthermore, sets can’t grow or shrink; they have no primitive notion of process. We can talk informally about “constructing” a set but technically this is only a way of specifying which already-complete set we’re referring to.
A direct result of this property is that in order to have infinite sets (which can only be established by axiom), we have to imagine them as being both infinite and already complete at the same time—a situation that was considered controversial for many centuries and which leads to some very counterintuitive properties, such as the ability to divide a set into two equal parts that are both the same size as the original.
To be clear: there are lots of people thinking about all kinds of alternative ways of thinking of things in mathematics, and there have been for a long time. So I’m not the first or only person to suggest we consider different approaches–far from it. What I’m advocating for, as an application of my work, is just one more reason to take those suggestions seriously.
You were also part of the selection committee for the 2023 Coha-Gunderson Prize. Can you share a bit about your experience? What were you looking for in the submissions? What surprised or excited you about the work you saw?
Absolutely. This was a great experience—it was really fun to see the different ways people interpreted the call for submissions, and I got to see a ton of really fascinating, thoughtful art. I try to keep an open mind about what I’m looking for because the prize is framed in a way where the only thing that’s really required is that the submission be creative work that has something to do with imagining the future. Beyond that, I think I tended to favor the pieces that were a little further along in their process, where it seemed like the creator had already spent a fair amount of time with the piece, as opposed to the many amazing ideas for projects people submitted that were just starting or in earlier stages. I think as a group we also tried to recognize a variety of different kinds of work. I probably have other preferences and biases I’m not aware of but I think those are the major factors.
Finally, can you share your favorite spot on UCSC’s campus?
The whole campus is my favorite spot. If I have to pick something smaller I’d say the fire trail that goes from the 3 barrels down to the entrance of Pogonip. I think on the running app Strava someone titled it “Descent into Endor” and that’s just what it looks like. It’s really fun to run down through the redwoods in the morning.
Banner Image: Redwoods on UCSC campus.