## Friday, 13 June 2014

### Metaphysics as Über-theory and Metaphysics as Meta-theory, II

Though it is common for logicians to be a bit negative about metaphysics, I am very fond of metaphysics. I can trace the reason: I purchased a scruffy copy of W.V. Quine's From a Logical Point of View (2nd ed., 1961) from a second-hand shop in Hay-on-Wye, around 1987, containing Quine's essays -- "On what there is" and other essays on related themes, such as modality, reference, opacity, etc. I found "On what there is" so engrossing that I numbered each paragraph and learnt it by heart. A few years ago, I lent this copy to a close friend, but it was never returned (aleha hashalom).

In an older post, and to some extent tongue-in-cheek responding to some criticisms of analytic metaphysics, I listed a number of achievements in analytic metaphysics. Analytic metaphysics is so closely related to mathematics that one might simply confuse the two, but this is an error. There is massive overlap between analytic metaphysics and mathematics. This is why some responded to the list of achievements of metaphysics by saying "is this not just mathematics?". Well, they overlap, and when X and Y overlap, then saying (truly) something is X, one does not establish that it is not Y. Sometimes, the criticism of "analytic" metaphysics, as opposed to "naturalized" metaphysics, is ad hominem, directed not so much of analytic metaphysics, but rather of analytic metaphysicians; they are theorists and not experimentalists, and they are bad theorists, because their knowledge of (empirical) science does not go beyond "A Level Chemistry". The important criticisms I see are: an epistemological criticism (how might knowledge of the relevant kind even be possible, entirely by a priori "armchair" reasoning?); a competence criticism ("A-Level chemistry"); an irrelevance criticism ("what a waste of time"). I don't know the answer to the first, but then no one knows how mathematical knowledge is possible, and yet mathematical knowledge exists.

It's fair to say that there is more weight in the "competence cricitism" of some modern metaphysicians, as one might call it. By and large, David Lewis tends to have a very classical picture of the (actual!!) world, with "lumps of stuff" at spacetime points (and regions), and perhaps the criticisms made against this is fair. However, one must be careful about stones and glass houses. There is some physics in Every Thing Must Go but not much: for example, no detailed computation of the electronic orbitals of a hydrogen atom using separation of variables in the Schroedinger equation, or of the Schwarzschild metric in GR, or of the properties of gases, or calculations of Clebsch-Gordan coefficients, etc. And what there is there includes a mistaken formulation of Ehrenfest's Theorem, as explained here: in the book, the equation given (twice) has the quantum inner product brackets (viz., expressions of the form $\langle \psi \mid \hat{O} \mid \psi \rangle$) misplaced in the equation.

But the basic point here is still unfair to those criticized by the "competence criticism", even if there's some legitimacy to the criticism. It is extremely difficult for someone whose specialization is metaphysics - but has not studied, say, theoretical physics or mathematics to graduate level - to acquire a detailed understanding of what can, and cannot, be said fruitfully about, say, the (alleged) implications of QM or GR. As an example of this, there's an (unpublished) article on Leibniz equivalence, and I gave it as a talk perhaps six times now, with physicists and philosophers of physics; audience response is this; 50% say it's obviously wrong and 50% say it's obviously right.

Since Frege, Russell, Wittgenstein, Carnap et al. were not experimentalists, it must be that whatever progress they made, if any at all, they must have made as theorists, and yes, in their armchair (or deckchair, for Wittgenstein). It is hard to see how an experiment might help me understand, for example, the semantics of sentences about fictional objects or possible worlds or transfinite cardinals. I may analyse the semantic content of, say, "Scott is the author of Waverley" or "I buttered the toast with a knife"; or I may try to analyse the Dirac equation. Or I may analyse, "You are almost as interesting as Sherlock Holmes is", in which a real-life person is compared with a fictional character. Consequently, "analytic" refers to a method, not to any specific content. The content is unconstrained: it may be possible words, fictional objects, moral values, topological field theories, transfinite sets, the unit of selection debate, etc., etc.

In an older M-Phi post, Metaphysics as Über-theory and Metaphysics as Meta-theory I suggested one could think of metaphysics in two ways:
• Metaphysics as über-theory (think - Plato).
• Metaphysics as meta-theory (think - Aristotle).
One could probably run through any piece of work classifiable as "metaphysics" and identify which bits are über-theoretic and which bits are meta-theoretic. It is Pythagorean über-theory that "all things are numbers" and it is Aristotelian meta-theory to say that "to say of what is, that it is, is true". So, as I want to use this word, in über-theory, one attempts an overall picture of "how things ... hang together". That is,
"The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term” (Sellars, 1962, "Philosophy and the Scientific Image of Man")
Sellars says this of philosophy in general, but I think it is an overestimate. Philosophers should be able to work on small problems without feeling the intellectual burden, a somewhat pretentious one too, of trying to understand how "things hang together". For example, I don't think Russell's "On Denoting" (1905) fits this picture at all, but I do think his Principles of Mathematics (1903) or "Philosophy of Logical Atomism" (1918-19) do.

For example, when Prof. Max Tegmark suggests that physics is ultimately mathematics, then that claim is an example of über-theory. I think this somehow neglects the modal contingency of concrete entities, that "how the concreta are" changes from world to word; and that rather it is mathematics that is ultimately physics -- the physics of modally invariant objects -- then that's über-theory too. If you like, mathematics is the physics of modality. The idea is that purely abstract entities, like $\pi$, $\aleph_0$ and $SU(3)$, don't modally change their relationships to one another as we let the worlds vary. There is no world "in" which $3 < 2$, $(\omega, <)$ is not wellordered, or $e^{i \pi} + 1 \neq 0$. Purely abstract objects are not even "in" possible worlds at all. The distinction between physics and mathematics is not, I think, connected to how knowledge of their objects is acquired, but is connected to the fact that relations amongst purely abstract entities (e.g., how $\omega$ is related to any $n \in \omega$) are fixed and invariant ("Being"in Plato's terminology), whereas the relations of concreta, such as e.g., Blackpool Tower and the Eiffel Tower, are a matter of change ("Becoming", in Plato's terminology). For example, at the moment, the Eiffel Tower (a concretum) is higher than the Blackpool Tower (a concretum), but this temporary French advantage over the British could, of course, be remedied by an "accident" (Team America; World Police, 4:15).

Russell's Principles of Mathematics contains a great deal of über-theory and meta-theory, but his "On Denoting" is a classic of meta-theory. Meta-theory was the central focus of Rudolf Carnap's Der logische Aufbau (1928). The logical apparatus for doing meta-theory had blossomed with the publication of Frege's Begriffsschrift (1879) and then was amplified in his later writings on semantics and applied to the case of the foundations of arithmetic in Die Grundlagen der Arithmetik (1884). Bertrand Russell joined this revolution against German idealism in 1899, after attending a conference at which Peano was present. Russell's friend, G.E. Moore, was part of this rebellion too, although not a logician. A decade later, on the advice of Frege, a young Austrian, Ludwig Wittgenstein, spent a year and half visiting Russell at Trinity College. In general, and in practice, all published work in metaphysics does both. Meta-theoretic work in metaphysics has no serious objection to it, aside perhaps from mild "competence" accusations of insufficient expertise in difficult parts of, let's say, mathematical logic or theoretical physics. This work appears alongside the work of other logicians, mathematics and computer scientists (and sometimes cognitive scientists), often in the same journals. Aside from the usual internecine waffle and squabbles -- e.g., about one's favourite "logic", etc. -- there is no deep disagreement as to methods and also as to the genuine progress that is made. For example, Michael Clark once showed me, on a blackboard in 2000, a paradox, involving an infinite list of sentences, and I was struck by how one might make it precise. When I went home and did that, I discovered that the infinite set (simplifying notation quite bit),
$\{Y(n) \leftrightarrow \forall x>n \neg T(Y(x)) \mid n \in \mathbb{N} \} \cup \{T(\phi) \leftrightarrow \phi \mid \phi \in L\}$
actually had a model: a non-standard model. That's progress and I did no "experiment".

With über-theory, it is different. For how can a priori reflection, from the armchair, tell us "how everything hangs together". Surely, that task is for the empirical scientist, and, in the end, for the physicist. In other words, it seems utterly pretentious for a metaphysician to even insinuate some ability to discover "how things hang together". I agree. Well, sort of. There are two main lines of response. The first is that while it is true that the armchair metaphysician is not performing experiments on neutrinos or gravitational waves (and neither of course is the mathematician or theoretical scientist), the armchair metaphysician is going to have, and should be expected to have, some degree of knowledge and acquaintance with science - with mathematics, with formal parts of linguistics and computer science, with parts of physics, chemistry, biology and psychology (cognitive science, more broadly). But this is material for analysis. It is not therefore a direct attempt to find out how "everything hangs together", but an attempt to see how our best scientific theories (or even how our discourse in general) depict "how things hang together". A second response focuses on what these "things" might be in "how things hang together"? The scientist may be interested in galaxies or ganglia; for the metaphysician, there also is a more or less canonical list of the kinds of things one are interested in: properties, relations, quantities, abstract entities and structures, formal systems, moral values, propositionspieces of discourse, possible worlds and fictional entities.