Harrison White and Mathematical Sociology

Harrison White’s sociology was always, I believe, a mathematical sociology, although it became more impressionistic later, as he worked with the strengths that come with age and not those granted to youth. It is important to remember that mathematics is not about numbers—real mathematics abhors numbers—but about relations. In all things, sociology loped behind Harrison White, and it says a great deal about academia that this figure who should probably have been the center of a determined effort to raise our discipline, while having a fine “career,” and certainly being intellectually influential, really didn’t bring the field with him. Now, when I am working on a piece, I find myself going back to his work from the 1960s, when he was an Assistant Professor, to see if I can catch up to where he was.

There are a number of things I don’t quite understand about Harrison White. One of them is why, when I contacted him as an advanced graduate student from Berkeley, he was so intensely generous. I had met with him when I considered Columbia for graduate school, but I was sure I didn’t want to do network analysis, which I associated with Ron Burt and confused with path analysis, because it seemed conservative, and so, like all good Marxists-in-training I went to Berkeley. (And how funny that then I would end up a gigantic fan of Burt’s ... but that’s a different story; also, however, the story of a great visionary of network phenomenology.) But proving Marx right about the connection between ideology and social structure required modeling structure, and for small groups, that seemed to require thinking in terms of networks, and I was told to talk to Harrison.

I sent him a number of long rambling things I had written, which he dutifully read, and he invited me to meet him in his apartment for coffee. “You have something, but you’re like a goddam babe in the woods,” he told me. “Go to Irvine, learn some math, and you have a chance.” When I said I couldn’t do that, as I had a baby, he threw up his hands. But he showed me the places to start (the work of John Boyd and Douglas White in particular).

But he didn’t just tell me to learn math. He saw that I was reaching toward a phenomenological solution, and he told me to read Heidegger (especially Being and Time). And when I said, hoping to be a bit shocking, “Oh, I never thought I’d read that Nazi,” White didn’t blink. He said levelly, “John, read the Nazi.” Grappling with combinatorics and with Heidegger was instrumental in getting me out of the rut I think I, like most of us, was in. And when I slowly worked out ideas that were probably simplified versions of things he had already thought of, he was astoundingly supportive, encouraging, and generous. I was shocked to find that when my first book came out, he had contributed a very kind blurb on the back, and it didn’t subtract from the sentiment that the book he praised didn’t seem to me to be precisely the same one as what I thought I had written. But who knows, maybe he was right?

Just as supportive were a number of his students or collaborators: Arabie, Bearman, Boorman, Breiger, DiMaggio, Pattison, and Winship were all profoundly kind and generous in giving me a second education in formal sociology ... and Ed Laumann was his gruff, direct, and hilarious self. In fact, their influence was easier to see because White scattered so many seeds so widely that it could be immobilizing to really follow what he was saying. His 100 suggested topics at the end of Identity and Control might be read as his frustration that there were so many obvious paths forward that not enough people were taking. (I think it was from one of those, certainly somewhere in his work, that I took the idea of reading materials science to learn about the range of order in a solid [Martin et al., 2016].)

All this happened during the spell of Identity and Control (White, 1992), which I read a few times. (The first time I met Harrison, he asked me if I had read it, and I admitted that I gave up at page 100. “Dammit!” he swore, “that’s where everyone stops! And that’s where it gets good!”) There are crib notes, Tilly’s most notably, if you want to memorize the lingo, but that doesn’t seem to me an important part of the book. In fact, I am no longer sure how important it is compared with White’s other work; just as the most beautiful pebble that you see in a stream can appear dull and interchangeable with others when you take it out and it dries, Identity and Control lost a great deal of luster when removed from the insanely brilliant energy that White plastered all around everything having to do with it.

With White the mathematical sociologist, you could generally follow where he was going once he made clear where he was starting out. But with the White of Identity and Control (I&C), I think most of us experienced what an information theory perspective might tactfully call “high surprisal.” But as Weber knew, this increases charisma. Without White right there, chances were, you couldn’t quite reproduce his logic. So there could be no replacing him.

Now, my interpretation of this work, and of its strengths, is wholly mediated by Boorman’s analysis, and, just as with Boorman’s (2024) treatment of Sun Tzu (Boorman likes cryptic sources to serve as an aegis), sometimes I think he is swapping in what White should have said. And here, there are a number of profound ideas that can indeed be of use to the field. In particular, we may have a way of thinking about the active, indeed, strategic, side of social life without digging ourselves into the hole regarding subjective orientation, where action theory is still spinning its wheels, throwing up mud, but not moving forward.

Action theory a la Weber and Parsons was largely an attempt to generalize an orientation taken from neo-classical economics, one based on a particular model of the actor. This has strengths for many questions, but leads to many unproductive investigations of the emergence of order on the part of actors with particular orientations. The part of I&C that I understand is an alternate generalization, one that can be thought of as answering the question, “what similarities do we find between how 19th century French painters, and 20th century American cement manufacturers, organize themselves?” So maybe a lot of the details don’t quite make sense to me anymore, but still, if you want to understand social order and not start from the individual/social bifurcation, but the mesh, and to see structure and strategy as two sides of the same coin, this is still the place to get your orientation. And maybe we can even do that sort of complex, topological, and wholly abstract mathematical sociology implied.

But I don’t think we’re there yet. Maybe most of us need to really pursue White’s work in temporal order of its appearance. Sure, people write about networks and they think they’ve generalized blockmodels. Maybe. But we haven’t yet grappled with and assimilated the 1960s work (as Boorman would say, I draw your particular attention to 1963a,b), let alone the core ideas from the 1970s, especially Boorman and White (1976). Harrison says somewhere that training a new generation of rational choice or causal modelers is like finding a bunch of brilliant engineering school students and convincing them to set up shop in depleted and abandoned coal mines. Why not see what we can do with the many starts he left us in terms of combinatorics, in terms of algebras of permutation matrices, in terms of role algebras? We basically turned our back on this, making only a little use of the simplest clustering into blocks for a single network. White was a thinker of the caliber of Herbert Simon, and while Simon basically had a whole field form in the wake he left behind him, outside of White’s students, and a few younger colleagues, we haven’t devoted the person-hours to this task that it deserves. ⁽¹⁾