Tuesday, 17 July 2012

Abstracta - the way of modal invariance

Debates about how to analyse the distinction between abstract and concrete entities have normally focused on four "ways", introduced by David Lewis (1986, The Plurality of Worlds). In his very nice SEP article on "Abstract Objects", Gideon Rosen labels and describes these as follows:
(i) The Way of Negation ("abstract objects are defined as those which lack certain features possessed by paradigmatic concrete objects")
(ii) The Way of Example ("it suffices to list paradigm cases of abstract and concrete entities in the hope that the sense of the distinction will somehow emerge")
(iii) The Way of Conflation ("the abstract/concrete distinction is to be identified with one or another metaphysical distinction already familiar under another name: as it might be, the distinction between sets and individuals, or the distinction between universals and particulars")
(iv) The Way of Abstraction ("The simplest version of this strategy would be to say that an object is abstract if it is (or might be) the referent of an abstract idea, i.e., an idea formed by abstraction"; then goes on to discuss Fregean abstraction principles)
I'd suggest another approach: that concreteness/abstractness is connected to modal variation. In particular, abstracta are modally invariant.

Canonical examples of abstracta are entities like numbers (e.g., $3$, or $\sqrt 2$, or $\pi$ or $i$ or $\aleph_{57}$) or pure sets (e.g., $\varnothing$ or $\omega$ or $V_{\omega + \omega}$). Or the assorted entities of modern algebra (e.g., groups) and geometry (e.g., manifolds). Additionally, we have the abstracta of traditional philosophy - forms, universals, properties, relations, essences.

Notice that these entities do not "change", either temporally or modally.

On the other hand, concrete - ordinary physical objects for example - do "change", both temporally and modally. Concrete entities undergo temporal and modal change.

So, it seems to me that the abstract/concrete distinction can perhaps be explained modally along these lines.


  1. Perhaps there is an epistemological ''way'' as well. For example, if I S can come to know that ∆ is F "through reflection alone" or "through rational intuition" or "through a priori insight", et cetera, then ∆ is abstract. This criterion can full under "The Way of Negation" as well: If ∆ has the property of being knowable either "through reflection alone; through rational intuition; through a priori insight", or whatever, then (so the argument goes) since concrete objects lack those properties, those things that we know (those objects that are the subjects of our propositional knowledge) are not concrete objects. But one might also just move from a premise about *how* S knows that ∆ is F to a conclusion about *what* ∆ is--in this fashion, one takes their epistemic access to ∆ to be ontologically decisive.

  2. Yes - I am very sympathetic to some of view of access like that (one finds suggestions like this in Gödel; he talks somewhere of the "abstract elements" of sensory experience). Possibly that view falls somewhere in Way of Abstraction too, not sure; but, yes, it places the initial emphasis on the epistemic access (how the abstracta are known or grasped), rather than what constitutes them non-epistemologically. I'd guess that also aligns with neo-logicist views of Wright et al.

    Russell advocates a direct acquaintance theory of universals in Problems of Philosophy.

  3. Two possible counterexamples: (1) For some philosophical theologians, God is not abstract (God's not causally inert for one thing) but is nonetheless modally invariant. (2) For some philosophers (Prior (sort of), Adams, Salmon, among many others), singular propositions about contingent individuals are ontological dependent on the individuals they are about. Hence, before you existed, there was no such proposition as "Ketland is human". Rather, it came to be along with you. Hence, it isn't temporally (or, obviously, modally) invariant. The latter problem can perhaps be got around by saying that X is invariant iff, necessarily, X doesn't change at any time at which it exists. Or something like that.

  4. Thanks, Chris. I'd thought of the God one. I'm not sure what to say. Maybe we do want to say that God isn't modally invariant after all? I don't know the answer.
    On propositions, I don't know either, but breaking it down into different theories of propositions gives different answers. A Russellian theory takes propositions to be impure structures of some sort (complexes), containing what they're about, and the theories you mention are variations on that, I think. A Fregean theory takes propositions to be Gedanken, "saturated" senses. I think on the latter view, propositions are modally invariant - but ...

    Another problem I have is (modal) Cambridge changes. So, a pure abstractum can change its relations to concreta (e.g., a real number is volume-in-m^3 of c at one world, but not at another).
    I think the problem here is to make sense of "change". In the case of concreta undergoing temporal change, one usually thinks of timeslices. But with modal changes, do timeslices correspond to counterparts of each other? I don't know, and am going to think about it!

  5. The proposal will pinch when it comes to lots of marginal cases. Is the game of chess an abstract object? Dummett says so with great confidence; but it 's changed over time and it might have had slightly different rules. Are impure sets abstract objects? Nominalists in the Quine/Goodman tradition have always thought so. But they are plausibly contingent — the set {you, me} would not have existed if either of us had not existed — and temporary. On a slightly wilder note, I'm tempted to think that even the pure abstract objects (e.g., the numbers) are contingent entities, since we can consistently describe a nominalistic world, and since the desciption would be consistent with the essential truths. I may be wrong about that; but I'm not *obviously* wrong. So I don't think it's a conceptual truth that abstract objects are modally invariant. A contingently existing non-spatiotemporal, acausal item would not count as *concrete* just because it was contingent.

  6. Many thanks, Gideon. Good points.
    I think the hard cases are the ones you mention -- mixed or impure abstracta. I'm very unsure what to say.
    Maybe modal & temporal individuation for games might be similar to modal & temporal individuation for languages. I tend to think the languages don't really change; rather they are different, and very finely individuated temporally and modally (though they may be be very similar). So, "chess" might be similar to "English", in this respect, a non-uniqueley denoting singular term (unless one very explicitly disambiguates).
    Chris's example of God might cause trouble as well.
    The problem of mixed or impure abstracta more generally is the main difficulty, I think. Are the impure abstracta changing as we go from world to world ... this might imply that universals are undergoing modal change. What I want to say is that the concreta are changing mutually, and in relation to the abstracta, which aren't genuinely changing.
    For the pure abstracta, I'm a bit more confident, since I don't think pure abstract are "in" the possible worlds to begin with; they're more like a fixed background outside modal change; but I can only make this more exact with a model-theoretic representation of possible world, by having a two-sorted setup, with a concrete domain and an abstract one (like your two-sorted setup in your 97 book with John Burgess). But the models can't *be* possible worlds, as they are structured sets and so violate Leibniz equivalence, since one can have isomorphic but distinct structured sets (i.e., merely haecceistically different possible worlds), which have to be "quotiented out" somehow. (Maybe possible worlds are propositions expressed by Scott sentences.)

    Still, at the moment, the modal invariance idea is a vague suggestion, that I occasionally mention to people in the pub, and I don't know if it will fly, if I can come up with answers to these hard problems.

  7. I'm quite interested in the perennial tendency of philosophers to insist that what appears to be 'change' in the abstract realm — e.g., change in the English language — is really something else: a non-rigid term taking on different referents at different times. Why not take appearances at face value and say that English — the language itself — comes into being, changes, and maybe disappears? The lurking suspicion is that philosophers are seduced by a syllogism: Languages are abstract objects; abstract objects don't change; ergo ... But in this context, the second premise begs the question. Common sense tells us that languages change, and a bit of reflection tells us that they're abstract. So why aren't languages (and the like) just counterexamples to the second premise?

  8. I think common sense may be wrong here.
    A physical system S has temperature 300K at time $t_1$ and temperature 290K at later time $t_2$. We do say "the temperature of S has changed from $t_1$ to $t_2$", but we mean that $Temp_K(S, t_1) \neq Temp_K(S,t_2)$. The number 300 didn't turn into the number 290 (the relations amongst the abstracta are in some sense fixed). Rather, the physical system changed.
    Language change may be analogous. Suppose L and L' are languages, and we might ask for their individuation criteria, when are they identical? A proposal is:
    (*) L = L' iff L and L' have identical phonology, syntax, semantics and pragmatics.
    So, if L and L' differ on even a single lexical element, for example, they're distinct.
    Given some empirical data, this will imply that if L and L' are the idiolects spoken by, say, you and me, then L and L' are distinct, even though common sense suggests we both speak "the same language" - English. But then, what is English if it is neither of our idiolects? Presumably, one might idealise somehow, or aggregate somehow, or something like that, and explicitly define some language L*, with an explicit listing of lexicon, description of the phonemes and so on. But then no one speaks this language L* (e.g., no one has its whole lexicon).
    And this, i.e., (*), gives us that individuals in a speech community speak differing idiolects, though usually very similar. This is somewhat counter-intuitive, but I think it's ok for linguistics.
    (I wrote something on this a couple of months ago
    For games, maybe something similar.
    But then also for novels, symphonies, and Beatles songs ... so Paperback Writer was discovered, rather than created: but I go with the counter-intuitiveness.