As simply as I can manage, here’s how “sympathy” works in Hume. Let’s start a few steps back.
According to Hume, a mind is not a “thing” but an aggregate collection of ideas, impressions, and relations, organized by principles. In essence, the “mind” consists of two parallel streams of impressions and ideas which continually influence each other, as shown here. Each of the three “sets” on the diagram above (labelled “You” and “Them”) represents one mind.
For Hume a “passion” (what Spinoza calls an “affection,” what we might call and “affect” or “feeling”) is the combined effect of a particular impression (generally a positive or negative sensation) joined with the idea of a particular object (that thing which we feel positively or negatively about). Hume’s system is, IMO, unnecessarily complex compared with Spinoza’s, but we can schematize them in similar ways for our present purpose.
Without getting too complicated, the green lines marked “cause & effect” on the diagram represents the particular chain of thought or structure of association that produces in you a particular idea of a particular object. To give a very simplified example: You open the fridge and take out a carton of milk. You sniff it before you pour. It smells bad. You conclude the milk has expired. For our purposes, it doesn’t matter what the exact order and connection of ideas is; suffice to say that a particular constellation of ideas and impressions (e.g., “fridge,” “cold,” “milk,” “carton,” “smell,” “gross,” whatever) together produce a single passion, say “disgust,” which consists of the idea of spoiled milk plus the sensation of unpleasure you experience when you smell it.
This is the structure of a given passion in a single individual. So what is sympathy? Well, Hume defines sympathy as “that propensity we have to…receive by communication [other people’s] inclinations and sentiments, however different from, or even contrary to our own.” What does this mean? Basically, it means that feelings are contagious, and, as Hume explains, they are contagious in direct proportion to how much we like somebody.
Hume takes it for granted that we like ourselves. Hume also takes it for granted that the more we like someone (that is, the more intense the positive sensation produced by that person), the more inclined we are to see them as being similar to us. Now, as Hume continually insists, the essence of human understanding is the relation of cause & effect. We know - or think we know - that a particular entity acting in a particular way on another particular entity will generally have the same effect every time. (This is basically the criterion of repeatability by which empirical evidence is measured even today in the sciences).
What does this have to do with sympathy? Basically, it’s a question of what we would call “cognitive dissonance” (Hume doesn’t use this term, but I find it useful). On the one hand, I consider myself similar to this other person whom I like. Rationally, based on the principles of cause & effect, that similarity should mean that a particular thing will affect me more or less the way it affects that person. In practice, however, a particular thing often affects us differently than it does people we like (one person might find a particular smell disgusting, their best friend might not).
We are thus confronted with two opposite ideas: the notion that we should feel a particular way, and the fact that we actually don’t. This is where the question of intensity comes into play. In Hume, when two ideas are contradictory, the stronger one “wins” by that margin by which it exceeds the other. Let’s say my belief in “idea A” has an intensity of 3, and my belief in “idea not-A” has an intensity of 2. As a result of this contradiction, I will be left with the idea of A, which will now have an intensity of 1. (In this Hume differs greatly from Spinoza, who, like Freud, insists that opposing ideas can be sustained together in the mind).
This, basically, is also how “sympathy” works. If the intensity with which I like someone exceeds the intensity of my current feeling about a particular object, the other person’s idea of that object will supersede mine, but in weaker form, because of the intensity that idea must “sacrifice” in order to overcome my own idea. This sounds a lot more complex that it actually is. Let’s say you’ve just read a book and you really, really liked it. If a doofus you can’t stand tells you the book sucked, not only will you not agree with them, your fondness for the book might even increase; in this case, your positive opinion about the book (or your negative opinion about the doofus) is much stronger than their negative opinion about the book; not only that, but their criticism of the book will be one more item on the list of negative experiences you associate with that doofus, intensifying your negative impression of them and making it even less likely that you’ll sympathize with their opinion. On the other hand, if a literature professor you really respect and admire tells you the book sucked, you’re probably going to take their opinion more seriously…and maybe like the book just a little bit less. Maybe their opinion isn’t “strong” enough to make you totally dislike the book, but it will likely impact your opinion to some extent.
This is the essence of “sympathy” in Hume: emotions (“passions”) are contagious, and the chance of being infected depends on the relative intensity of your feelings about the other person and your feelings about the object in question. Or to put it much more succinctly: according to Hume, the human mind has a tendency to imitate the emotions of people we care about (in extreme cases, where the imitation isn’t checked by our own sense of self, it can become pathological, what psychiatry calls “projective identification,” a common symptom of borderline personality disorders).
Just a couple of points left to make. First, the role of contiguity and resemblance in producing sympathy. The function of resemblance is fairly obvious: first, we assume that the person we are sympathetic to is fundamentally similar to us, and second, we assume that the person we are sympathetic to is affected in a fundamentally similar way by the same objects. The function of contiguity is pretty straightforward, too: it is the way we judge whether the person we sympathize with is capable of being affected in the same way. Basically, if I hear a strange noise and I find it creepy, I’m not going to call a friend in another city to ask if the noise is creepy, even if that person is exactly like me - because that person can’t hear the noise. That is, they are not in a position to be affected by the object in the same way as I am - our positions are not contiguous.
Second, as I noted repeatedly in the previous Hume diagrams, it’s crucial, in order to understand Hume properly, to avoid making reductive or essential distinctions. For example, it would be wrong to break down sympathy as a structure into an opposition between the internal element of cause & effect and the external elements of resemblance and continuity; it would be equally wrong to consider the former temporal and the latter spatial; or the former personal or individual and the latter social or collective. Rather, the point Hume makes again and again in the Treatiseis that mental activity is an aggregate of ideas, sensations, and relations organized by a few basic principles. This definition is just as valid and just as useful when the aggregation of thoughts in question are “divided” between more than one body. In other words, your internal process of forming a judgment about a particular object (what I have noted here in green as “cause and effect”) is not a different kind of activity or process than the interpersonal process of sympathy. Rather, there is only a difference of degree: the number and variety of thoughts may be greatly different, but at the end of the day, both the internal structure of cause & effect and the interpersonal structure of sympathy are organized by exactly the same principles. This is both the central claim and the elegant simplicity of Hume’s system.