Blog from Scott Aaronson: “This post is about an idea I had around 1997, when I was 16 years old and a freshman computer-science major at Cornell. Back then, I was extremely impressed by a research project called CLEVER, which one of my professors, Jon Kleinberg, had led while working at IBM Almaden. The idea was to use the link structure of the web itself to rank which web pages were most important, and therefore which ones should be returned first in a search query. Specifically, Kleinberg defined “hubs” as pages that linked to lots of “authorities,” and “authorities” as pages that were linked to by lots of “hubs.” At first glance, this definition seems hopelessly circular, but Kleinberg observed that one can break the circularity by just treating the World Wide Web as a giant directed graph, and doing some linear algebra on its adjacency matrix. Equivalently, you can imagine an iterative process where each web page starts out with the same hub/authority “starting credits,” but then in each round, the pages distribute their credits among their neighbors, so that the most popular pages get more credits, which they can then, in turn, distribute to their neighbors by linking to them.
I was also impressed by a similar research project called PageRank, which was proposed later by two guys at Stanford named Sergey Brin and Larry Page. Brin and Page dispensed with Kleinberg’s bipartite hubs-and-authorities structure in favor of a more uniform structure, and made some other changes, but otherwise their idea was very similar. At the time, of course, I didn’t know that CLEVER was going to languish at IBM, while PageRank (renamed Google) was going to expand to roughly the size of the entire world’s economy.
In any case, the question I asked myself about CLEVER/PageRank was not the one that, maybe in retrospect, I should have asked: namely, “how can I leverage the fact that I know the importance of this idea before most people do, in order to make millions of dollars?”
Instead I asked myself: “what other ‘vicious circles’ in science and philosophy could one unravel using the same linear-algebra trick that CLEVER and PageRank exploit?” After all, CLEVER and PageRank were both founded on what looked like a hopelessly circular intuition: “a web page is important if other important web pages link to it.” Yet they both managed to use math to defeat the circularity. All you had to do was find an “importance equilibrium,” in which your assignment of “importance” to each web page was stable under a certain linear map. And such an equilibrium could be shown to exist—indeed, to exist uniquely.
Searching for other circular notions to elucidate using linear algebra, I hit on morality. Philosophers from Socrates on, I was vaguely aware, had struggled to define what makes a person “moral” or “virtuous,” without tacitly presupposing the answer. Well, it seemed to me that, as a first attempt, one could do a lot worse than the following:
A moral person is someone who cooperates with other moral people, and who refuses to cooperate with immoral people.
Obviously one can quibble with this definition on numerous grounds: for example, what exactly does it mean to “cooperate,” and which other people are relevant here? If you don’t donate money to starving children in Africa, have you implicitly “refused to cooperate” with them? What’s the relative importance of cooperating with good people and withholding cooperation with bad people, of kindness and justice? Is there a duty not to cooperate with bad people, or merely the lack of a duty to cooperate with them? Should we consider intent, or only outcomes? Surely we shouldn’t hold someone accountable for sheltering a burglar, if they didn’t know about the burgling? Also, should we compute your “total morality” by simply summing over your interactions with everyone else in your community? If so, then can a career’s worth of lifesaving surgeries numerically overwhelm the badness of murdering a single child?
For now, I want you to set all of these important questions aside, and just focus on the fact that the definition doesn’t even seem to work on its own terms, because of circularity. How can we possibly know which people are moral (and hence worthy of our cooperation), and which ones immoral (and hence unworthy), without presupposing the very thing that we seek to define?
Ah, I thought—this is precisely where linear algebra can come to the rescue! Just like in CLEVER or PageRank, we can begin by giving everyone in the community an equal number of “morality starting credits.” Then we can apply an iterative update rule, where each person A can gain morality credits by cooperating with each other person B, and A gains more credits the more credits B has already. We apply the rule over and over, until the number of morality credits per person converges to an equilibrium. (Or, of course, we can shortcut the process by simply finding the principal eigenvector of the “cooperation matrix,” using whatever algorithm we like.) We then have our objective measure of morality for each individual, solving a 2400-year-old open problem in philosophy….”
