This book, by a mathematician, introduces a relatively new approach in math called “category theory,” but without the math. Instead of math, it illustrates basic concepts by using recipes for food.
Obviously, it is not about making explicit mathematical analyses. But it does introduce a fair number of ideas that can be of philosophical interest.
It is often said that category theory is about “the mathematics of mathematics,” but that hardly tells us what it is really about. So I would put it this way: It is about defining numbers in terms of how they have relationships with other numbers rather than seeing the numbers as existing in an absolute sense (meaning existing on their own, regardless of anything else).
But that is not to say that it’s “relational” in the sense of relativism, the theory that one truth is as good as any other. Rather, as I understand it, it means “relative” in the sense that Newton uses an absolute reference scheme by which to measure the world, but Einstein uses a relative one. In other words, we can come to understand the word “little” in comparison to many, many things that are “big” rather than by positing that there is such a thing as the pure essence of “little-ness.” In category theory, this relational approach is now applied to mathematics itself.
Mathematician Tai-Danae Bradley was recently on a podcast with physicist Sean Carroll, and I was somewhat surprised to hear Carroll adopting premises similar to category theory himself. So category theory seems to be catching on. Bradley had been discussing the word “firetruck” and how it could be understood in relation to other words, as opposed to the word representing an object out there in the real world. Commented Carroll,
…You sneaked in what to many people would be a massive metaphysical claim…. So I think that the perspective that you’re sort of falling down on and which is actually one I’m quite sympathetic to, is probably something like pragmatism in the philosophical tradition, William James and people like that, right: the meaning of a word is its use. But I do think that there are other people who would say no, no, the meaning is the firetruck out there [in the world.]
Bradley had been discussing applying category theory to language, not just to math.
So, what is category theory? The book under review is a straightforward introduction to seeing math as about relationships, without getting into the arcane mathematics itself.
Beyond saying that the book is about absolute vs. relative—it is about how numbers can be defined by their relationships with other numbers and how proofs can be made in terms of that—I am not going to go into the specifics of category theory here, except to make a few peripheral points for context.
The notion that things are defined by their relationships, rather than being defined by having pure essences, is not a new one. I remember an old philosophy professor of mine once telling us that the four greatest philosophers of all time were Plato, Kant, Shankara, and Nagarjuna. (Note the conspicuous absence of Aristotle and of anything empirical. Was this professor a Platonist, or what?).
Anyway, Nagarjuna, from India in the second century, argued that what is ultimately real are relationships, not the physical or the concrete. And he is famous for pithy statements along the line of “it is there, but it isn’t there,” meaning that the relationships are real but not the physical objects in and of themselves. (I addressed my own take on that in my review of Ladyman’s book Every Thing Must Go, where I asked, if everything is just relationships, as Ladyman and Nagarjuna claim, then what is having the relationships? To my thinking, physical objects can have both relational and physical characteristics). Still, I agree with Ladyman that, to the extent that things have relational traits at all, that is sufficient to (in Ladyman’s words) “purge things of their intrinsic natures.” They are no longer just about their own inner essences regardless of anything else.
My own interest in Nagarjuna’s premise lies in how it can be understood in terms of energy making arrangements of itself. Instead of having “things-in-themselves” (as in Plato and Kant), there are “things-in-arrangements.” And being in an arrangement makes it so that what else is in the arrangement with an object contributes to what that object even “is” in the first place (thereby purging it of its intrinsic nature). And it makes it so that things have relationships with what else it is sharing an arrangement with. (I would have discussed Nagarjuna in the Ladyman review, but I was already running long).
In any case, the point is that in category theory we are talking about math, not physical objects, and math, as a system of logic, can much more easily be understood as being only about relationships.
Then a second ancillary consideration about understanding things in terms of their relationships is that we often hear nowadays (and I agree) that it has become generally recognized that the equations of science describe relationships and not rules. (Starting with the Enlightenment and through the 19th Century, the equations were usually thought of as rules being carried out). But why should we think of equations as describing relationships?
I am not going to go through all the arguments here, but I’ll give an example. We are most of us familiar with the equation describing the relationship between the circumference of a circle and its diameter, C = πd. Notice that that is showing how, if one variable changes, then so must the other. They change at the same time—the change together—which makes them related. It is not that there is some rule existing outside the Universe making things happen. It is just that we can’t have the circumference be one way unless we also change the diameter along with it.
The equation is describing the world in terms of its relationships.
And that is true of a great many of the equations of science, such as Newton’s. It might not seem immediately apparent that F = ma is about the relationship between force and acceleration until we realize that the equation is a proportionality. That is explicitly stated in the verbal form of the equation, that “the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.” But a proportionality is a relationship.
One of the few philosophers who I have enjoyed reading over the years is Richard Boyd. And of course, like most philosophers I encounter, I find him to be mostly wrong, but he is at least mostly wrong in a way that I find interesting. He challenges me to put into words exactly how he is wrong. (And obviously, many people find me to be mostly wrong, as well; but that is the fun of talking about things). Boyd, for instance, has thought up “cluster kinds,” discussed in my earlier posts, which are interesting.
Well, Boyd was not happy with having to admit that the equations are describing relationships. He liked the old way of looking at the world, where the equations were about putting things into motion. So he insisted that, if the equations were describing relationships, then at least they were describing causal relationships (because that way he could keep his sense of causality even if the world was looking relational).
But poor Boyd. Of course that doesn’t work. That is because the equations describe one variable changing “as” the other is changing, simultaneously, not with one changing before the other as in causality. (In causality, one change must come prior to another in order to bring about the change in the other).
We can see that exemplified in the equation C = πd. It is not that the circumference changes “and then” the diameter does, or vice versa. It’s that they must change at the same time. That’s because they change in a way that maintains being proportional.
With F = ma, that might not be intuitive. We might think that force comes first to cause the acceleration. But that is not what the equation is saying. It is saying that force and acceleration are proportional so that they change at the same time, just like diameter and circumference in a circle.
We might argue that, yes, they change simultaneously, but there is still some outside factor causing them both to change together. And that might well be. But it is not what the equation is saying. The equation is not describing such an outside factor. Instead, it is saying that, if force or acceleration is changing, then they must change proportionally as per their relationship.
Of course, there are other types of equations rather than proportionalities—there are other types of relationships—such as statistical equations. But comparable arguments can be made about them.
And there are many other arguments against Boyd’s contention. I am not going to list them all here. But another famous one is Bertrand Russell’s, that when looking at an equation, there is no way to tell what is supposed to be the beginning (the cause) and what is supposed to be the end (the effect).
That doesn’t totally nix causality. It only says that causality does not exist in the form that the old Enlightenment philosophers said that it does, regarding being in the equations of science.
Nagarjuna, by the way, also saw the relationships he talked about as being causal relationships. He did so sort of by fiat. He just put it into his initial premise: The world, he argued, is made of causal relations. He didn’t have to worry about that fitting in with modern equations.
Artificial intelligence guru Judea Pearl has recognized how equations are noncausal and has responded by advocating that there should be arrows used instead of equal signs in the equations, to denote causality. The results are his “causal graphs.” But in actually using the causal graphs, he fudges at the very last minute and converts the arrows back into equal signs for purposes of calculation. (But that is a whole other story).
I mention it because category theory also employs arrows, but it is not in the sense of Pearl’s causal graphs. Cheng refers to the arrows as “manifestations” (as about the same thing appearing in different guises), and other authors invoke Eastern mysticism (yes, regarding math), but there are also other interpretations.
It is fascinating stuff. And Cheng’s book is a straightforward introduction, with the emphasis on concepts as illustrated by food recipes.
I confess that I have not tried Cheng’s food recipes myself. But I credit her for her wisdom in including them. She was writing back when it was a craze in novels to include some food recipes out of nowhere. It was said to really help the sales. So she is a clever woman, because back in the day this book was indeed a popular success. And there are sequels.
Also, it is full of interesting ideas that I haven’t gotten into, about the mathematical relationships themselves. They are accessible because, as I said in the opening, she is describing them just conceptually.
doing the math
For an introduction to the actual math, and to see some examples of the kind of proofs that are possible using this approach, an excellent tutorial is our own WordPress blogger Gazing by Lamplight. It is an optimal source for anyone looking to make an erstwhile effort at a fuller understanding.
Happy Holidays, everyone.