A very small note on commensurability
Again, here's Raz:
"Incommensurability is the absence of a common measure"
(Engaging Reason, p46)
There are two ways to interpret this.
On one, commensurability consists in the fact that the two scales being compared each correspond to some third scale, in virtue of which they can be compared.
On the other, the presence of such a third scale is definite proof of commensurability, even though this is not necessary.
We should definitely take the second option. For if two scales are commensurable only if there is some third scale to which they both correspond, then nothing could be comparable to nothing but itself. Everything should be equally valuable to itself, even if all values are incomparable to each other. The first option makes this claim false.
Worse, it looks like there's some kind of third man fallacy going on if we adopt the first option. To compare the series of positive numbers and the series of negative numbers, I don't need some third set of numbers to translate each into. This leads into some kind of regress, and anyway seems obviously unecessary.
The interesting upshot for me is that my well-being and your well-being might be commensurable even though there is nothing that they have in common.
Is commensurability a property of pairs of *scales* or items?
... or perhaps, we could say, is it a property of (pairs of) types or of (pairs of) tokens?
If it's a property of pairs of individual items, then I don't think your objection works - that is, a thing can be compared to itself with reference to the (single) scale it's on, without appeal to some outside scale. Likewise with numbers on the number line.
Then we can say two items are commensurable if and only if there is some scale on which both can be measured.
If it's a property of scales, then I'm not sure what sort of implication it might have for comparing individual items like your well-being and mine.
I had imagined that
Hi Paul,
I had imagined that commensurability was a property of pairs of scales: pairs of individual items are commensurable only if they sit on a scale which is itself commensurable. What sense would it make to compare two items, without any reference to what respect (i.e. what scale) you are comparing them with reference to?
So that said, I'd be very interested if you can provide an example that demonstrates why my objection doesn't work.
On the final point, I take it that well-being is a property of a person comes in degrees: a scale. Am I missing something?
Alex
I'm not sure commensurability of scales makes sense.
Hmm... considering the point in the first paragraph of your reply, I'm now inclined to say that commensurability is a property of properties of pairs of individual items. I was mistaken in the first comment to suggest it's a property of individual items as such.
Let's take a standard sort of example for comparability -- Hamlet vs. the Mona Lisa. If commensurability is a property of individual items, as per my individual suggestion, without reference to which properties of the items are at issue, then matters quickly dissolve into incoherence. There are some common scales for some properties of Hamlet and the Mona Lisa -- amount of paper used, time taken to create, etc., and some properties for which there is clearly no common scale -- there's no way to relate the difficulty of staging Hamlet to any quality in the Mona Lisa, for example. So we wouldn't be able to say whether Hamlet and the Mona Lisa are commensurable or not.
So far, I think we agree. But saying that commensurability is a property of scales isn't the only alternative. If we take property-by-property commensurability, we can say that Hamlet and the Mona Lisa are commensurable on the property of quantity of paper consumed (because the quantity of paper consumed in each can be described on a common scale) and arguably incomensurable between greatness of drama/greatness of painting.
Making commensurability a property of properties of pairs of items might, I think, still handle your objections. So let's say a pair of properties of an item or pair of items are commensurable if and only if they can be measured along a common scale.
-The first objection: The properties of any thing are commensurable with one another because each can be described with respect to its own scale. And this is so even if those properties aren't commensurable with the properties of any other object. Imagine, for example, a martian rock with the special property of flarflness. We can say that rock is as flarfl as itself -- that it's commensurable with respect to the property of flarflness -- even if there's no other object in the universe that has flarflness. The restrictive definition of commensurability still rightly says that the martian rock is equal to itself.
On the other hand, if we say commensurability is a property of scales, then we'd have to ask whether flarflness is measurable in some external scale. Ex hypothesi, it isn't, so on the restrictive definition of commensurability, something goes wrong -- we have to say that the martian rock is (measured on a scale that is) not equivalent to itself. This is why I think your objection works if commensurability is a property of scales, but fails if it's a property of properties.
- The second objection: the series of positive numbers and the series of negative numbers both are measured (only) with respect to the property of cardinality, and there's a common scale (the whole number line) with respect to which this measurement can be conducted. Ergo, they're commensurable. If there was no such scale, then I think we'd have to say they're not commensurable (but can we even conceive of numbers such that there's no common measure of cardinality between them?).
(Hastily written, mid-move, apologies for glitches.)
Paul, I think one of us is
Paul,
I think one of us is confused, though I don't know who. I think I can state my claims in the original post independently of the contrast between pairs of scales and pairs of items. I claimed that to compare any two entities, you didn't need some third thing that they had in common. Sometimes two things are just comparable between themselves.
Think of it this way: In converting Pounds Sterling to US dollars, one doesn't need to convert both into something else - the euro, say. That would be a very odd view, if the two were only comparable via some third currency that you transfer both into. Why not just think that they can be directly compared? I was making the same point with respect to comparability/commensurability more generally. Two things (items, scales, whatever) can be compared directly, without the need for some measure *in addition* to those in which those things are themselves measured in the first place. So my well-being sits on a scale, and yours does too, and they can be compared. There doesn't have to be some third independent currency we can translate both of our well-being into in order to make the comparison.
So hopefully that makes a little clearer what I was trying to say. Could you explain how your comments fit in? I currently can't see exactly how your comments are opposed to what I've written above.
Alex
Well, chances are I'm the confused one... :-)
Hmm... perhaps breaking things down a little bit will help clarify matters for both of us. Let's just focus on the first objection you raised.
I take your point to be the following: Some claim that it is necessary, for two objects of comparison (leaving aside for a minute what those objects are) to be commensurable, for there to be some measure external to the objects themselves by which they can be compared (the necessity claim). Every item ought to be commensurable to itself (the reflexivity claim). But the necessity claim and the reflexivity claim are inconsistent, as is shown by the example of a unique item, which, ex hypothesi, is incommensurable with everything else. Such an item (the martian rock, and its property of fleeminess) can't be commensurable with itself, because there's no external measure by which it can be compared to itself. Given an inconsistency between necessity and reflexivity, we should reject necessity.
My claim, then, was that this argument only works if we assume that the objects of comparison are scales themselves. Necessity and reflexivity are inconsistent on your argument because there may not be some kind of external scale against which a scale itself (like level of fleeminess) can be measured. However, they're not inconsistent if we're not trying to compare scales, but rather trying to compare points on scales (that is, values of properties, held by individual items). So on my account, the martian rock is commensurable with itself even if we accept necessity, because we're not asking "can fleeminess be compared on some external scale," we're asking "is there a scale on which the fleeminess of the martian rock can be compared to the fleeminess of the martian rock," and the answer is unequivocally yes, to wit, the scale of fleeminess itself. Ergo, reflexivity and necessity aren't inconsistent, and there's no reason to reject necessity.
Is that clearer?
Re: the dollars/pounds example, I think part of the problem is this notion of an "external" ("some third") scale. The Raz claim you started with is consistent with the idea that any scale covering all objects of comparison will do, but that some common scale is necessary. It might be too strong to say that the scale must be external. So a given quantity of dollars and a given quantity of pounds might be commensurable in terms of the measure/scale "dollars," and that could still satisfy the necessity claim, that there be some scale. By contrast, no similar covering scale exists to convert, say, dollars into the love of a devoted family dog, or your welfare and mine.
It seems I was the confused
It seems I was the confused one! Yes, that makes much more sense: phrasing the problem in terms of necessity conflicting with reflexity is a nice way of phrasing the problem.
I guess we probably both agree on the following claims:
1) To compare two scales, you don't need some third scale.
2) To compare two objects, you do need some scale.
I originally made the first claim, and you've now rightly pointed out that I shouldn't state it so strongly that it rules out the second.
We might then have an interesting debate whether normal, interesting comparisons are between scales or objects. I need to think about what I might say about that: perhaps you are correct, though I'm still concerned that assignment of properties (well-being, fleeminess) is already implictly on a scale, so that comparing two objects is just to compare two scales.
Alex
Third Way?
I always took it that the paradigmatic case here was the geometrical one: the legs of a right triangle are incommensurable with the hypotenuse. There is no third scale here -- rather, the point is that any scale that adequately measures the legs will be in some way defective, even if only very slightly, for measuring the hypotenuse, and vice versa. You can't identify any unit that measures them both without anything left out. So the question would not be how many scales you have (you only need one to show incommensurability in the geometrical case), but whether a scale calibrated to measure one thing adequately can adequately measure another.
You seem at one point, unless I misunderstood you, to slip from 'commensurability' to 'comparability'. This is a common slide, but I think the two are not the same; two things that are commensurable are comparable, but always in such a way that putting one in the same terms that captured the other will result in some sort of defect in the description. That is, a way of describing that is well-suited to one will always be a bit off in describing the other.
Hi Brandon, I'm not sure I
Hi Brandon,
I'm not sure I follow what's going on here.
I've never heard of the phrase "legs of a right triangle", though googling tells me it means the sides of a right-angled triangle that aren't the hypotenuse. Is that an American phrase? So that said, if I've misunderstood this, that would explain why I can't follow the rest of what you say.
So when you say that the hypotenuse and "the legs" are incommensurable, do you mean with respect to length? If so, why aren't they commensurable? As I say, I must be really misunderstanding something here.
I do indeed take commensurable to mean the same as comparable. (Raz does the same.) I again don't follow what you take the difference to be.
Alex
Triangles
It's possible 'legs' is an Americanism, although if so, I wasn't aware of it.
Take a right triangle with one of its non-hypotenuse sides vertical and the other horizontal; and pick a unit of measurement that allows the measurement of the sides to be whole number (I'll be unimaginative and use 1 unit). Now, suppose the vertical side to be 1 unit and the horizontal side to be 1 unit. By the Pythagorean theorem, the hypotenuse will be the square root of 2. This is a number that cannot be expressed in whole-number increments of our chosen unit of measure. This will be true whatever units we pick: units that allow the vertical and horizontal sides to be measured by whole numbers will never allow the hypotenuse to be measured by whole numbers, and units that allow the hypotenuse to be measured by whole numbers will never allow the vertical and horizontal sides to be measured by whole numbers. The magnitudes of the hypotenuse and the 'legs' of the triangle are incommensurable: they have no "common measure". This website has a nice discussion if you're interested in a slightly more technical presentation.
It was this that Kuhn had in mind when he proposed his incommensurability thesis for scientific theories; his point wasn't that they can't be compared but that they can't be compared according to any 'common measure' -- since we can't be infinitely precise in the relevant ways, comparison will always be approximate -- you can compare them but only at some cost to strict accuracy with regard to one. At least some people have the same idea when talking incommensurability of values and the like, or at least that's my impression; although this may well be a case where people are walking around with an equivocal term and not really realizing it. It's interesting that Raz takes it as synonymous to non-comparable.