Dialogue on a mathematical approach to the good
This post is a dialogue and consists of two parts. In the first part, Bartłomiej Skowron and David Spivak consider what mathematics and ethics have in common. Though some will think these notions have nothing in common, these authors share the intuition that the opposite is true. In particular, they propose that sense-making is always good and that mathematics is significantly useful in modern sense-making. In the second part, David Corfield responds, pointing out the importance of the good life, argues for strengthening cognitive science research on well-being by scaffolding it with mathematics and category theory in particular.
This post is a dialogue and consists of two parts. In the first part, Bartłomiej Skowron and David Spivak consider what mathematics and ethics have in common. Though some will think these notions have nothing in common, these authors share the intuition that the opposite is true. In particular, they propose that sense-making is always good and that mathematics is significantly useful in modern sense-making. In the second part, David Corfield responds, pointing out the importance of the good life, argues for strengthening cognitive science research on well-being by scaffolding it with mathematics and category theory in particular.
1 Part I
Bartłomiej Skowron, David Spivak
According to legend, one of the ancient philosophers gave a public lecture entitled “On the Good”. This topic attracted many listeners, who probably wanted to listen to considerations about how to live a healthy life, earn money successfully, and gain honorable positions. Nevertheless, this philosopher surprised everyone: instead of talking about these essential goods, he spoke of mathematics, numbers, shapes, etc. Few understood, and many lost interest and left. Reportedly, this lecture concluded that goodness is one, further obscuring the whole matter rather than illuminating it. Why was this philosopher talking about mathematical structures when he was supposed to analyze goodness? And why was the story of this lecture recalled in the 20th century by Alfred North Whitehead and later Felix Browder & Saunders Mac Lane? Because doing mathematics can be seen as a path to goodness, and mathematical structures as structures of goodness itself. Strange, isn’t it? Whitehead wrote: “The lecturer was competent - he was Plato.” It is from Plato that the idea that goodness and mathematics have much in common originates.
Aristotle, who disagreed with Plato, pointed out in his major ethical work The Nicomachean Ethics that the good is understood in different ways. Moreover, even the same person can understand the good differently. When a person is COVID sick, good seems to be health; when she is tired, good seems to be rest; when she acts cowardly, good seems to be courage; when she does not understand how to compose morphisms in the category Poly, good then appears to be understanding the simpler cases of natural transformations. But what should we do when the answer to the question of goodness seems so case-dependent? Is it possible to say something that isn’t empty but instead meaningful and relevant about the good? Following Plato’s intuition, we claim that mathematics may help us do this.
Let us assume, laboriously and for a moment, that goodness is not only what, for example, seems good to you, your friend, or me. Let the good be an idea, a term which we don’t intend in the everyday casual sense, but instead along the lines of Plato’s form or more precisely “idea” in the sense of Roman Ingarden: an idea is something that has some content, some rich ontological structure. For example linearity is an idea—a mathematical one—captured by the ontological structure of the category of vector spaces. In the case of the good, this content is whatever structure allows us to classify certain objects (here, you can replace object with action, person, or value; the object need only be an entity with a desired form or manner of existence) as being instances of this idea. You can think of this content as a specific shape, and of instances of the idea, say goodness, as objects in which this shape is present. To put it yet another way, it is a certain pattern of patterns, and as such can be realized in many objects. We want to know what this pattern called goodness—which remains the same, regardless of how many (if any) instances it has—is.
Nevertheless, caution is needed here: ideas are characterized first of all by the fact that they are general entities (sometimes they are called universals, sometimes they are called abstracts), which means that this shape or pattern has many empty slots, which can be filled in various ways. For example, individual actual real objects do not have such gaps. The act of filling in these gaps is called concretizing qualities, but discussing this would take us into too metaphysical an area. So let’s go back to the basic idea: make a working assumption that when we want to talk about the good, we will try to speak in essence about the idea of the good, or more precisely, about the content of the good as an idea. In this view, the mathematics of goodness becomes the mathematical study of the content of goodness: understanding these “shapes,” patterns and their dynamics. For the purposes of this blog post, let’s assume you can make some sense of what’s being said here.
In fact, sense-making is central to our story of how goodness and mathematics are related. To see this, let’s return to everyday life on Earth. When I need to shop, I need to find the right store where I can find the products in question. I can get to that store guided by Google Maps or just glance at the map and use my sense of direction, as long as I have that sense developed. When I stroll through the park, I can distinguish the balmy, woody notes of lavender or the earthy notes of iris, as long as I have my sense of smell sharpened. When I’m feeling lonely, I can use social sense to make some friendships that seem to respond to that loneliness. If I’m doing philosophy, and I don’t want to be naïve, I can use my sense of irony, like Socrates, and thus for an exercise, distance myself from my own philosophical beliefs. When I make a decision, dividing some goods among people, I can do it fairly if I have a sense of justice. In general, when I make a decision of a moral nature, something attracts me approvingly to good choices, and something disapprovingly outrages me when I imagine bad decisions—the moral sense is responsible for this attraction or repulsion. This is probably the case with every sense: good compositions attract us, and bad ones repel us. Fitting things together is, in fact, sense-making. In general, it seems that the art of living is the art of making sense and avoiding nonsense.
Sense-making is multi-sensory, dynamic, and complex. Our senses of things and, above all, the corresponding qualities must fit together in order to for us to act proficiently. Otherwise, there will remain a lack of meaning and an inability to navigate. In order to divide goods equitably, I must be able to both divide and also (sometimes) use my sense of sight to see the objects I am dividing, as well as my moral sense to see the merits of those subjects, which would be the basis for equitable distribution. When experienced florists arrange a bouquet, they compose the qualities of the different senses: both the color of the flowers and their scent must be well composed. These qualities must fit together so that the bouquet is good and so that we can say that this florist is good at being a florist. A good florist is a sense-composer. Certain fragrance notes may not fit together, and such combinations repel us because they don’t make sense together, whereas certain combinations are almost perfect and attract many sense-makers. Sometimes these combinations even intoxicate and please, such as that of orange, grapefruit, nutmeg, and vanilla. One might ask: on what is this sense-making founded? Behind each sense are the qualities of that sense. For example, to the sense of sight, there are qualities such as redness, linearity, and blurriness; to the sense of smell, there are qualities such as the notes of nutmeg and vanilla. The sense of irony includes qualities such as perversity. To the metaphysical sense, there are qualities such as sublimity. And to the moral sense there are moral qualities, such as the quality of justice.
How does all this relate to mathematics? Simple. Certain sense qualities fit together, making sense, and others don’t; they don’t make sense because they don’t fit together well. Sense-makers do the work of finding fitting compositions daily: we compose senses, and as long as we do it well, we make sense of the world. Then, in interaction with other sense-makers, we exchange our notions of sense, influencing and changing each other: sometimes we all match each other perfectly, and sometimes the group can’t make sense of each others’ points of view at all. Sometimes we all find the same qualities, and in this case we hook up and compose our senses together, making a synergistic group sense that allows the group to navigate the situation better than any member alone could have. This is how consciousness—not necessarily experience, but coherent sense—is created with many multi-level interactions, both individual and collective. Individual senses can come together to create a new higher-level sense, and this process is formed through an active and creative process. This way senses combine may be governed by multi-level laws of composition or fitting together. It seems that no one currently knows the general laws of sense composition: they still need to be discovered. But perhaps these laws have something to do with proper accounting, the culmination of which is mathematical and which we can try to approach with category theory.
Plato had the intuition that goodness is oneness and order. To separate good from evil is to separate order, the cosmos, from disorder. We understand order in this sense as meaning structure: that which can be mathematically grasped. This means that our notion of the good could in fact be mathematized by articulating multiscale compositional dynamics that occur within the network of sense-makers immersed in a world of sensory qualities. This way of mathematizating goodness—or at least some nearby notion—means formulating the dynamic process by which sense is made.
Plato assumed that ideas are static—timeless—and that they exist separately from material things. The considerations here indicate that all sense-making is in fact dynamic. Nevertheless, this dynamism still has a mathematical pattern, the stability of which does not consist in being static, as Plato and Ingarden think, but consists in the dynamic structure of how senses are sensibly composed. Thus, although we started from Plato’s intuition, we conclude that Plato’s theory of ideas is like Wittgenstein’s ladder that, having climbed, must be discarded.
You might assume that the good cannot be analyzed because it is simple and thus inaccessible to cognition. Or you might claim that it is a heterogeneous conglomerate without any internal uniformity or structure and that one can only randomly hit parts of it. We do not claim either of these. The shape of the good is cognitively graspable: it can be learned, and it is essentially compositional in nature. To know the good is to understand the laws of composition. All this, of course, does not mean that it is easy to see the good. Nevertheless, it does mean that it can be done in some way, an expression of epistemological, axiological optimism. However, to reiterate, it is not easy and probably will never be complete. The way leads through categorifying the content of goodness, a project which we propose to begin. Does it make sense?
2 Part II
David Corfield
Thank you both for your reflections on how we might use mathematics to address goodness. There are some helpful ideas contained in your post to get us started. In particular I agree that something like sense-making plays an important role in what we take to be good.
As for my own starting point to approach the question, I’d like to consider first what might appear to be the hardest part of the problem. When Ancient Greek philosophy considers the good, it is always as part of the particular question of what it is for a human to lead a good life. This is a central concern of each of Socrates, Plato and Aristotle. Now of course the good life has you enjoy good things. It can have you appreciating via the senses, say, a florist’s bouquet, the florist having done a good job in composing the scents and colors of the bunch. It can have you appreciating a good coffee at a convenient store. It can have you communing with good friends for an evening. But the good life involves so much more than these. Rather than a series of satisfactory experiences, it encompasses a certain way of living. Naturally it involves a life that makes sense of features of its environment, that allows for a good fit between an individual and their niches. But the good life is a life rich with meaning, and a life full of meaning-making.
‘Making sense of’ is a phrase generally applied to an existing phenomenon, making sense of the scene before you, making sense of what someone is saying to you, making sense of a mathematical argument. Making meaning, on the other hand, is not making meaning of anything, it’s a creative process. It may be the setting up of a project, like we’re doing here. It may be planning to lead a life together with someone and enacting this plan. Leading a good life is living a life full of meaning, following a vocation, sustaining a flourishing community, creating meaning for yourself and others.
Well that only seems to have complicated the problem. Perhaps we might have hoped to be able to say something about how goodness applies to entities of particular types - a good apple, a good book, a good sheepdog, a good soccer match. Now we have to deal with the thorny issue of characterizing the good life, as though there could be something mathematically tractable about this.
Do we see existing work out there to learn from? Well, of course plenty has been said about the good life. Indeed, the wisdom traditions of the world, from the East to the West, have a great deal to say. ‘Know thyself’ advises Socrates. ‘Inculcate the virtues’, says Aristotle. And Eastern traditions of Buddhism and Daoism have their own say: Follow the Noble Eightfold Path, Follow the Way.
So plenty of verbal descriptions of what the good life looks like, and discussion of the means to transform oneself to pursue it. Can we really hope that category theory can have something to say on the subject? Well, there’s plenty of work at the moment from the interface of cognitive science and wisdom studies, which looks to see whether the former can make sense of the Western and Eastern traditions. Here we see a strong convergence at the moment with the following three approaches:
- Predictive processing/active inference on well-being – Mark Miller, Brett Andersen;
- Relevance realization and the meaning crisis – John Vervaeke;
- Hemispherical balance, the left serving the right – Iain McGilchrist.
Notably these people are talking to each other (e.g., here and here). They’re even publishing together, see, e.g., the joint work of Vervaeke with Miller and Andersen. The unanimous conclusion is that we face a problem today and that we need to change the way we live in the world, in particular we need to develop our meta-cognitive capacities to equip us with the ability to apportion wisely appropriate care to our meaning-making projects, and the ability to seek the right challenges to push our boundaries.
These approaches also allow a great deal to be said on what goes wrong in a life. From the predictive processing side, the optimisers that we are get stuck in feedback loops, the loop of addiction, of depression, and so on. Or we rely too heavily on left-hemispherical thinking and become too instrumental in our dealings with the world. Departures from the right balance are involved in mental disorders.
Now perhaps here we have a window of opportunity, since in particular the predictive processing approach commences from an explicitly mathematical point of view in its characterization of the “Bayesian Brain”. To do this there needs to be mathematics that deals with recursive systems. One where layers of the hierarchy adapt according to bottom-up and top-down processing. We need to have our higher-level cognitive systems engage in self-monitoring, modeling the way we model the world to predict if that modeling is working or needs correction. We need the cognitive apparatus to apply itself to itself.
Take Vervaeke’s description of Jung’s idea of the Self:
“What’s the self? Well, it’s kind of the archetype of the archetypes. It’s like Plato’s notion of the good, which is the form for how to be a form. The eidos of the eidos. It is the virtual engine regulating the self organization of the psyche as a whole. It is the principle—the self is the principle of autopoiesis itself. It’s the ultimate virtual engine that constellates all the other virtual engines so that the psyche can continue its process of autopoietic self-organization. Remember when a system is self-organizing its function and its development are completely merged. It develops by functions and it functions by developing.”
Readers of this blog will know that we have at least one claim for a fruitful place where ACT meets such a complex system, and that is in the work of one of us, David Spivak. Another hint that we could take this further is that another Topos Institute member, Toby St Clere Smithe, is looking to use ACT precisely to capture the workings of the Bayesian Brain. Further relevant work on category-theoretic cybernetics may also help us address the question of the well-being of a complex system. My proposal then is that we take up such work and see if ACT can help us articulate what the cognitive scientists are telling us about well-being.