[Hi all! This is my first foray into academic blogging. It comes not so much from a long-harbored love of bloggery but from a desire to keep my thinking clear and simple while I write. As such, this will probably resemble a chunk of an extra-accessible (modulo written-by-me) paper rather than a genuine academic blog post. If I find this useful, I might start writing things just for this blog, but until then this is what you get. Enjoy!]
This post is about the nature of quantities. The term ‘quantity’ can be used in a number of different ways, so let me clarify that I don’t mean it in the sense that is synonymous with ‘amount’ (as in “Harjit ingested large quantities of Bud Light Platinum”). By ‘quantities’ I mean those properties that are said to be “graded” or to “come in degrees”. Less roughly, quantities are properties which are more faithfully represented by mathematical tools like numbers, functions, or vectors than by the tools of first-order propositional logic (e.g. n-place predicates). What I have in mind are things like mass, charge, length, velocity, temperature, moral goodness, and spicyness. What makes a quantity, like mass or charge, different from a non-quantitative property like <being President of the United States> or <having a spherical shaped proper part> is that—using mass for example—to say that o has mass (instantiates <is massive>) is never the end of the story about o and Mass. Specifically, there is also a question of “how much” or “which” mass o has—a question we might answer by saying, e.g., “o weighs 2kg”.
This is reason to think that quantities (specifically, the property (e.g.) <is massive>) are determinable properties, like <exercising> or <being red>, which are associated with a class of determinate properties: like <jogging>, <bench pressing>, and <doing yoga>; or <being scarlet>, <being crimson>, and <being vermilion>. Whenever something is red, it must thereby be (exactly one) particular shade of red, and whenever something is a particular shade of red, it must be red. Likewise, any massive thing must also possess (exactly one) determinate mass, and anything which possesses a determinate mass must thereby be massive.
However, quantities differ from non-quantitative determinables in that their determinates exhibit what call “quantitative structure”. Determinate mass properties, for example, stand in certain relations: we say “<being 4k> is less than <being 10kg>”; “<being 5kg> is the sum of <being 2kg> and <being 3kg>”; and “the ratio between <being 6kg> and <being 2kg> is 3-to-1”. These relations between determinate mass properties run parallel to (but are distinct from) mass-relations between massive objects: we say that “a is less massive than b”, “c is as massive as a and b taken together”, and “a is three times as massive as b”. The former structure, constituted by the distribution of these relations, is mass’s quantitative structure. What’s interesting is that it seems to be grounded in little other than the nature of mass, itself. There is, intuitively, nothing inherent in <being 2kg> or <being 4kg> that explains why 2kg<4kg (except that 2kg just is less than 4kg!).
Contrast this with non-quantitative determinables. They can have determinates that exhibit structure, but it’s always grounded in the distribution of other properties shared by those determinates (for example, sprinting is more similar to doing-jumping-jacks than to bench-pressing in that the first two are cardiovascular exercises while the latter is a muscle building exercise).
Quantitative structure turns out to be a pretty big deal, for a number of reasons. Here’s one: It looks like appeal to quantitative structure is necessary if we want to extend a reductive account of resemblance in terms of property-sharing to include quantities. On a property-sharing account of resemblance or similarity, two particulars are (to put it roughly) more similar to the extent that they have more properties(*) in common. For example, if a is a red striped square, b is a blue striped square, and c is a blue unstriped circle, then we say that (relative to color, stripedness, and shape): a and b are more similar to each other than either is to c; and c is more similar to b than it is to a.
These similarity facts hold because a and b share more of the relevant properties with each other than with c, and c has more properties in common with b than with a.
However, things don’t work out so easily for quantities. Consider three particulars, all of whom are massive, where a weighs 2kg, b 3kg, and c 30kg. Intuitively, things should be analogous to the above. That is, (relative to massiveness): a and b are more similar to each other than either is to c; and c is more similar to b than it is to a. However, the relevant property-sharing facts do not bear that out. All three particulars share the property of being massive, and no two of them have their determinate mass properties in common.
This is what I like to call the problem of quantitative resemblance. It seems simple but it’s actually quite difficult (like all the best problems are). It’s simple because the answer seems obvious: a and b are more similar to each other than either is to c because 2kg and 3kg are “closer” to one another than either is to 30kg, where ‘closer’ means closer on the metric imposed by mass’s quantitative structure. What makes it difficult is that implementing this solution in a satisfying way is no mean feat.
We could just stipulate that resemblance is a matter of having properties in common or in having “quantitatively close” values of a quantity, but this is unsatisfying. Why does the metric over mass determinates covary with the degree of resemblance between massive objects? Is this so for every quantity? Why do these relations between properties matter more than others?
I think there’s a better way. I think we can explain how and why quantitative closeness matters to similarity without deviating too far from the spirit (if not the letter) of the property-sharing account of resemblance. This, pretty much, is what I’m going to do with my dissertation.
[Thanks for reading! Next week I’m going to talk about a distinction I think we need to make to get any headway in solving this problem.] Update: This post was written in August 2013, and I think it’s likely that I’m not going to follow up on it. If you’re interested in what the follow-up post was going to say, you can check out my paper “Properly Extensive Quantities”, where I introduce the distinction I was alluding to in the above comment.
(*) – This sort of account of resemblance is generally taken to be available to those who have a sparse conception of properties (or some way of metaphysically privileging a certain subset of properties)