Relative Identity in Heidegger

Poor neglected blog; even my images are dead now. Ah well, mourning is for the dead.

I have noted before that Heidegger had read some Frege; this isn't a huge surprise, given that he was a student of Husserl, but it's easy to forget from our current "analytic/continental" vantage point. I just stumbled across a place that reminded me of this, and where better to make a note of it than on a dead blog?

In section 44 of "Being and Time", Heidegger is concerned with picking apart the view of truth as adequatio intellectus et rei. His point in doing this isn't to deny the view so much as to complain that it obscures what is significant about truth: if we try to have a "thing" and a "mind" already in view, and then want to add "truth" (and falsity) on top of those as a certain kind of relation between the two ("agreement", "correspondence", or the lack of this), then we have gone badly awry: instead we need to first have Dasein's openness to the world in view, and then the "mind" and "thing" which are supposed to "agree" will all show up as abstractions from a more important phenomena which originally makes claiming possible at all.

One of the ways Heidegger tries to do this is by poking at the "adequatio" relation, which Heidegger translates as "Übereinstimmung": "Was meint überhaupt der Terminus »Übereinstimmung«?", what does one in general mean by the term "agreement"? It has to be some sort of relation, it has to be a bringing-together of two things, but clearly not just any relation will do: we need to get a sort of "agreement" which relates a "thought" and a "thing" just in that the one agrees with the other with regards to truth: and this is obscure. Plausibly, the only way to pick out the right sort of relation is by already having an understanding of truth: A true thought agrees with its object just in that the thought says that things are thus-and-so with the object, and the object is thus-and-so -- and cashing out this "says that" talk already will involve a notion of truth, for saying that things are thus-and-so is just to put forward "things are thus-and-so" as true. (And so adequatio intellectus et rei is empty as a definition of truth; it moves in a circle.)

But that is not what interests me -- I want instead to point to a moment in Heidegger's discussion of relations of "agreement":

Die Zahl 6 stimmt überein mit 16 - 10. Die Zahlen stimmen überein, sie sind gleich im Hinblick auf das Wieviel. Gleichheit ist eine Weise der Übereinstimmung. Zu dieser gehört strukturmäßig so etwas wie ein »Hinblick auf«. Was ist das, im Hinblick worauf das in der adaequatio Bezogene übereinstimmt?

Auf Englisch: The number '6' agrees with "16-10". The numbers agree, they are equal in regard to "how many". Equality is one way of agreeing. To this belongs structurally something like a "regards to". What is that in regards to which the terms related by "adequatio" agree? [This is my own translation; someone please let me know if I've fouled it up too much.]

Heidegger here mentions numerical equality as one form of "agreement", and says that "agreement" always takes a complement: Two things agree in some particular respect, with regards to something: For example, '6' and "16-10" agree in coming to the same number, but not in being the same arithmetical formula.

Heidegger here puts forward (as unproblematic and not in need of argument) a view of equality as distinct from identity; equality is only sameness of number, not "sameness" as such. This is Frege's early view, in Begriffschrift; I believe he changes to his more familiar view (that equality simply is identity) in "Sense and Reference", when he settles on truth-values as the referents of sentences -- that move lets him consolidate a fair bit of his notation. (I would check this if I were not too lazy to do so.)

More interestingly (if you are me), Heidegger also puts forward as unproblematic a view of the genus of which numerical equality is a species, a view of what I think we analytics usually call "identity", as needing a complement: "Wieviel", "How much?" specifies the sense in which '6' and "16-10" are identical, they are the same number. Without some such question as this, no species of "agreement" is specified: The question "Do they agree, are they identical?" does not itself have a sense, unless the context makes clear in what respect agreement is being asked about. (It may be that the context makes clear that every respect is meant, and any dissimilarity will be an absolute lack of agreement between the one and the other.)

If I am reading Heidegger correctly here, then he agrees with Geach against Frege (and against the vast majority of analytic philosophers) in holding that "identity is relative": that to say that A and B are identical is, in the primary case, to say that they are the same in some definite respect, such as being the same color or the same make of shoe. To say that they are absolutely identical, which Frege had taken as the more primordial notion, Geach claims is only to say that they are the same in every respect: they are the same color, the same make of shoe, occupy the same location in space, etc. -- Geach uses Leibniz's Law to introduce the notion of "absolute" or "simple" identity as a defined term in logic.

To remind the reader of the opposing view: Many have held that to say that A and B are the same color is, when put in a logically regimented way, to say that

There is an X and a Y such that X is the color of A, and Y is the color of B, and that X is (absolutely) identical to Y, and that there is no Z such that Z is the color of A or Z is the color of B and Z is not (absolutely) identical to X (and to Y).

This way of rewriting "A is the same color as B" carries with it an ontological commitment to colors; Quine and Davidson take this commitment along happily (Davidson a little more happily than Quine), and so see a sort of Platonism as an unexciting logical consequence of some ordinary claims: There are colors, and there are shapes, because there are true claims on the order of "X and Y are the same shape", and writing those out in a logically acceptable way involves committing oneself to the truth of "There is a W such that W is a shape".

Geach is able to avoid these commitments: he treats "A is the same color as B" as a primitive equivalence relation in the language, and so does not need to quantify over colors to write "A is the same color as B" in ordinary first-order logical notation; it just comes out as looking like "aRb". By a neat trick, Geach notes that he can (in a sense) keep his ideology conservative as well: Quine's way of writing "A is the same color as B" requires a way to say "A has a color", which he writes as "There is an X such that X is the color of A"; Geach writes "A has a color" as "A has the same color as A", as anything which lacks a color cannot be the same color as anything: he is able to use his primitive equivalence relations to do the work of coloredness-predicates, and so doesn't need the latter as primitive terms in his language. This trick works in general for turning predicates into equivalence relations. So Geach doesn't need to include more predicates in his language than Quine did, but is able to reduce his ontological commitments. And since Geach is able to introduce a sign for "absolute" identity in his language by means of Leibniz's Law, the resulting calculus is just Quine's beloved first-order predicate calculus with identity; Geach disagrees with Quine not over a matter of regimented logical notation, but of how to rewrite ordinary language claims in that regimented form.

There are a few other wrinkles to Geach's account of relative identity, but hopefully the above is clear enough to get it in view. One aspect of his view which Geach finds remarkable is that he, unlike Frege, is able to treat statements of sameness and statements of number along the same lines: where Frege insisted that "How many?" required a complement, that statements of number were assertions about concepts, he had also insisted that "Are A and B identical?" required no complement, and that this identity was a logically peculiar notion which everyone immediately grasps in a special way. Geach thinks this was quite odd of Frege; Frege had demolished the idea that "oneness" is a special property of every object, and had left "self-identity" as a special property of every object: but in English and German both we have the phrase "one and the same" "ein und dasselbe", which Geach thinks should have already suggested to Frege that sameness and oneness ought to be handled along the same lines. I think that Heidegger had (without having much affection for logical notation) done just this: he requires an im Hinblick auf before questions of sameness are answerable. It would be interesting to see if Heidegger was consistent in this; rejecting "absolute" in favor of "relative" identity has a fair number of consequences in metaphysics, as Geach was well aware -- puzzles about whether a statue is identical with its clay fall to the ground, for example -- and Heidegger is not uninterested in a number of these metaphysical puzzles.