Tuesday, 24 June 2014

Sean Carroll: "Physicists should stop saying silly things about philosophy"

Readers probably saw this already, but I mention it anyhow. Physicist Sean Carroll has a 23 June 2014 post, "Physicists should stop saying silly things about philosophy", on his blog gently criticizing some recent anti-philosophy remarks by some well-known physicists, and trying to emphasize some of the ways physicists and philosophers of physics might interact constructively on foundational/conceptual issues. Interesting comments underneath too.

Saturday, 21 June 2014

Trends in Logic XIV, rough schedule

We now have a rough version of the conference schedule, including all the speakers and their titles. Here.


Friday, 20 June 2014

Preferential logics, supraclassicality, and human reasoning

(Cross-posted at NewAPPS)

Some time ago, I wrote a blog post defending the idea that a particular family of non-monotonic logics, called preferential logics, offered the resources to explain a number of empirical findings about human reasoning, as experimentally established. (To be clear: I am here adopting a purely descriptive perspective and leaving thorny normative questions aside. Naturally, formal models of rationality also typically include normative claims about human cognition.)  

In particular, I claimed that preferential logics could explain what is known as the modus ponens-modus tollens asymmetry, i.e. the fact that in experiments, participants will readily reason following the modus ponens principle, but tend to ‘fail’ quite miserably with modus tollens reasoning – even though these are equivalent according to classical as well as many non-classical logics. I also defended (e.g. at a number of talks, including one at the Munich Center for Mathematical Philosophy which is immortalized in video here and here) that preferential logics could be applied to another well-known, robust psychological phenomenon, namely what is known as belief bias. Belief bias is the tendency that human reasoners seem to have to let the believability of a conclusion guide both their evaluation and production of arguments, rather than the validity of the argument as such.

Well, I am now officially taking most of it back (and mostly thanks to working on these issues with my student Herman Veluwenkamp).

Already at the Q&A of my talk at the MCMP, it became obvious that preferential logics would not work, at least not in a straightforward way, to explain the modus ponens-modus tollens asymmetry (in other words: Hannes Leitgeb tore this claim to pieces at Q&A, which luckily for me is not included in the video!). As it turns out, it is not even obvious how to conceptualize modus ponens and modus tollens in preferential logics, but in any case a big red flag is the fact that preferential logics are supraclassical, i.e. they validate all inferences validated by classical logic, and a few more (i.e. there are arguments that are valid according to preferential logics but not according to classical logic, but not the other way round). And so, since classical logic sanctions modus tollens, then preferential logics will sanction at least something that looks very much like modus tollens. (But contraposition still fails.)

In fact, I later discovered that this is only the tip of the iceberg: the supraclassicality of preferential logics (and other non-monotonic systems) becomes a real obstacle when it comes to explaining a very large and significant portion of experimental results on human reasoning. In effect, we can distinguish two main tendencies in these results:
  •       Overgeneration: participants endorse or produce arguments that are not valid according to classical logic.
  •       Undergeneration: participants fail to endorse or produce arguments that are valid according to classical logic.

For example, participants tend to endorse arguments that are not valid according to classical logic, but which have a highly believable conclusion (overgeneration). But they also tend to reject arguments that are valid according to classical logic, but which have a highly unbelievable conclusion (undergeneration). (Another example of undergeneration would be the tendency to ‘fail’ modus tollens-like arguments.) And yet, overgeneration and undergeneration related to (un)believability of the conclusion are arguably two phenomena stemming from the same source, so to speak: our tendency towards what I call ‘doxastic conservativeness’, or less pedantically, our aversion to changing our minds and revising our beliefs.

Now, if we want to explain both undergeneration and overgeneration within one and the same formal system, we seem to have a real problem with the logics available in the market. Logics that are strictly subclassical, i.e. which do not sanction some classically valid arguments but also do not sanction anything classically invalid (such as intuitionistic or relevant logics), will be unable to account for overgeneration. Logics that are strictly supraclassical, i.e. which sanction everything that classical logic sanctions and some more (such as preferential logics), will be unable to account for undergeneration. (To be fair, preferential logics do work quite well to account for overgeneration.)

So it seems that something quite radically different would be required, a system which both undergenerates and undergenerates with respect to classical logic. At this point, my best bet (and here, thanks again to my student Herman) are some specific versions of belief revision theory, more specifically what is known as non-prioritized belief revision. The idea is that incoming new information does not automatically get added to one’s belief set; it may be rejected if it conflicts too much with prior beliefs (whereas the original AGM belief revision theory includes the postulate of Success, i.e. new information is always accepted). This is a powerful insight, and in my opinion precisely what goes on in the cases of belief bias-induced undergeneration: participants in fact do not really take the false premises as if they were true, which then leads them to reject the counterintuitive conclusions that do follow deductively from the premises offered. (See also this paper of mine which discusses the cognitive challenges with accepting premises ‘at face value’ for the purposes of reasoning.)

In other words, what needs to be conceptualized when discussing human reasoning is not only how reasoners infer conclusions from prior belief, but also how reasoners accept new beliefs and revise (or not!) their prior beliefs. Now, the issue seems to be that logics, as they are typically understood (and not only classical logic), do not have the resources to conceptualize this crucial aspect of reasoning processes – a point already made almost 30 years ago by Gilbert Harman in Change in View. And thus (much as it pains me to say so, being a logically-trained person and all), it does look like we are better off adopting alternative general frameworks to analyze human reasoning and cognition, namely frameworks that are able to problematize what happens when new information arrives. (Belief revision is a possible candidate, as is Bayesian probabilistic theory.)

Tuesday, 17 June 2014

Diff(M) vs Sym(|M|) in General Relativity

In General Relativity, whole "physical universes" are represented by spacetime models, which have the following form,
$\mathcal{M} = (M, g, T, \phi^{(i)})$
Here $M$ is some differentiable manifold, $g$ and $T$ are $(0,2)$ symmetric tensors, and the $\phi^{(i)}$ are various scalar, spinor, tensor, etc., fields representing matter, electrons, photons, and so on. The laws of physics require that the "metric" tensor $g$ and the "energy-momentum" tensor $T$ be related by a differential equation called "Einstein's field equations". The details are not important here though. (For the metric tensor, some authors write $g_{ab}$, many older works write $g_{\mu \nu}$ and some just $g$. Nothing hinges on this; just clarity.)

Suppose we consider a fixed spacetime model $\mathcal{M} = (M, g, T, \phi^{(i)})$. This is to represent some whole physical universe, or world, let us call it $w$. Let $|M|$ be the set of points in $M$. (We can call it the "carrier set".)

It is known that one may apply certain mathematical operations/transformations to the model $\mathcal{M}$ and also it is part of our understanding of General Relativity that the result is an "equivalent representation" of the same physical universe. This is all intimately related to what has come to be called "the Hole argument".

The mathematical operations are certain bijections $\pi : |M| \to |M|$ of the set of points in $M$ to itself. If $\mathcal{M}$ is our starting model, then the result is denoted $\pi_{\ast}\mathcal{M}$.

[To define $\pi_{\ast}\mathcal{M}$, the whole model is "pushforward" under $\pi$; we really just take the obvious image of every tensorial field $g, T, \dots$ under the map $\pi$: in geometry there are "pushforwards" and "pullbacks", and one has to be careful about contravariant and covariant geometric fields; but when we are dealing with mappings that are bijections, it doesn't matter.]

Which of these maps $\pi$s are allowed? That is,
for which maps $\pi: |M| \to |M|$, do $\mathcal{M}$ and $\pi_{\ast}\mathcal{M}$ represent the same $w$?
It is sometimes claimed that the relevant group of transformations, for General Relativity, is $\mathsf{Diff}(M)$. This is the set of bijections of $|M|$ to itself which leave the differential structure of $M$ invariant. I.e., the automorphisms of $M$. Since $M$ is a differentiable manifold, they are diffeomorphisms. Let me call this,
Weak Leibniz Equivalence:
if $\pi \in \mathsf{Diff}(M)$, then $\mathcal{M}$ and $\pi_{\ast}\mathcal{M}$ represent the same world.
But I say that the relevant group of transformations is much bigger, and is $\mathsf{Sym}(|M|)$, the symmetric group on $|M|$. That is, the relevant group is the group of all bijections of $|M|$ to itself:
Leibniz Equivalence:
if $\pi \in \mathsf{Sym}(|M|)$, then $\mathcal{M}$ and $\pi_{\ast}\mathcal{M}$ represent the same world.
This is the main point made in the Leibniz equivalence paper linked here. I sometimes give this as a talk, with usually some physicists, philosophers of physics and mathematicians there. At the moment, I get 50% I'm wrong and 50% I'm right.

There's a much more general formulation, which is very simple (and is essentially the content given in R.M. Wald's classic textbook on GR, p. 438), and which implies the above, and it's this:
Leibniz Equivalence:
If $\mathcal{M}_1$ and $\mathcal{M}_2$ are isomorphic spacetime models, then they represent the same physical world.
The mistake that people keep making, I say, is that they claim that the points of the manifold must be permuted smoothly. This, I claim, is not so. The points in $|M|$ can be permuted anyway one likes, so long as one applies the operation to everything - topology and differential structure included!

Sometimes this is called "gauge equivalence". Personally I don't care one way or the other about the terminology. However, note that Leibniz equivalence is analogous to the standard case of gauge equivalence - the U(1)-gauge symmetry that characterizes electromagnetism. Let $\mathbb{M}^4$ be Minkowski space, and let $A$ be the 1-form electromagnetic potential. Let $\Lambda$ be a smooth scalar field on $\mathbb{M}^4$. Let $d\Lambda$ be its derivative. Then the gauge equivalence principle for electromagnetism is that $A$ and $A + d \Lambda$ are "physically equivalent". I.e.,
$(\mathbb{M}^4, A)$ and $(\mathbb{M}^4, A + d \Lambda)$ represent the same physical world.
[I'm not really very knowledgeable of the philosophy of physics, and the various revisions and so on proposed, for example, against standard quantum theory, etc.: things like Bohmian mechanics, the GRW theory and so on. Here I'm just writing about classical General Relativity.]

[UPDATE (19 June 2014): I changed the text a teeny bit and added some links to the background maths.]

Saturday, 14 June 2014

Relativization of quantifiers and relativizing existence

One can relativize a claim by inserting a qualifying predicate for each quantifier. For example,
(1) For any number $n$, there is a number $p$ larger than $n$.
(2) For any prime number $n$, there is a prime number $p$ larger than $n$.
This is called relativization of quantifiers. Whereas (1) is kind of obvious, (2) is not. Formally, (1) is called a $\Pi_2$-sentence, as it has the form, roughly,
(3) $\forall x \exists y \phi(x,y)$
Suppose (1) is true. When we relativize to the subdomain of prime numbers, it expresses a different proposition, and we can consider whether it remains true in the subdomain which is the extension of the relativizing predicate. I.e.,
(4) $\forall x (P(x) \to \exists y (P(y) \wedge \phi(x,y))$.
ln fact (4) is true, but it expresses something stronger than (3) does. We might write the relativized (4) more perspicuously as,
$(\forall x \in P)(\exists y \in P) \phi(x,y)$.
$(\forall x : P)(\exists y : P) \phi(x,y)$.
$(\forall x)_P(\exists y)_P \phi(x,y)$.
Nothing hinges much on this: it is pretty clear what is meant either way.

Suppose we relativize to a finite set. Let $D(x)$ mean "$x$ is either 0, 1, 2 or 3". Then
(5) $(\forall x \in D)(\exists y \in D) \phi(x,y)$
is now false.

If $\Theta$ is the original claim, then we sometime denote the claim relativized to $P$ as $\Theta^P$. The fact that $\Theta$ is true does not in general imply that $\Theta^P$ is true. In general, if $\Theta$ is a true $\Pi_1$-sentence, then its relativization $\Theta^P$ is true as well. (In model-theoretic lingo, we say that "$\Pi_1$-sentences are preserved in substructures".) On the other hand, if $\Theta$ is a true $\Pi_2$-sentence, then its relativization need not be true, as we saw above.

Here is a vivid example. Imagine a society which contains Yoko, who happens not to be married to herself, and in which the following $\Pi_2$-sentence is true:
(6) Everyone is married to someone.
Now restrict this claim to the unit set, $\{Yoko\}$. Clearly,
(7) Everyone who is Yoko is married to someone who is Yoko,
is false.

This tells us a bit about how to relativize the quantifiers in a sentence to a predicate.

It may be annoying to keep relativizing univocal quantifiers, and one might prefer a many-sorted notation, in which distinct styles of variables are used to range over separate "sorts". So, for example, in textbooks and articles, we generally know that
the letter "$n$" (and probably "$m$") is going to denote a natural number.
the letter "$r$" (and probably "$s$") is going to denote a real number.
the letter "$z$" is likely to denote a complex number.
the letter "$t$" is likely to denote a time instant.
the letter "$f$" is likely to denote a function.
the Greek letter "$\phi$" is likely to denote either a mapping or a formula.
the Greek letter "$\omega$" is likely to denote either the set of finite ordinals or an angular frequency.
the upper-case Latin letter "$G$" is likely to denote either a graph or a group, and "$g$" will denote an element of the graph or group.
With capital Latin letters, "$A$", "$B$", "$C$", $\dots$, all bets are off! But "$X$" or "$Y$" are likely to denote sets. So, if you see, e.g., the equation,
(8) $f(t) = r$
then intuitively, the intention is that the value of the function $f$ at time $t$ is some real $r$.

While these issues seem fairly clear, can sense be made of relativizing existence itself? That is, can we make sense of a claim like:
(9) $x$ and $y$ "exist in different senses"

For example,
(10) The Eiffel Tower and $\aleph_0$ exist in different senses.
(11) Dame Kelly Holmes and Sherlock Holmes exist in different senses.
We usually think such claims are meaningful -- surely they are. But what exactly do they mean? Probably, something like this,
(12) $x$ and $y$ are (from or members of) different kinds of things.
And this seems to mean,
(13) there are kinds (types, ontological categories, ...) $A,B$ such that $\square[A \cap B = \varnothing]$, and $x \in A$ and $y \in B$.
There are two necessarily disjoint categories and $x$ is in one, and $y$ is in the other.

Quine wrote a famous paper, "On what there is" (1948). Normally, following Quine, we treat "what there is" and "what exists" as synonyms. But it is not very interesting to inquire as to what "exists", if one insists that "exists" be a predicate. If one insists that "exists" be a predicate, then what then becomes interesting is what this predicate "$x$ exists" means. Everyone agrees that ordinary usage counts as grammatical both:
(14) There exists a lion in the zoo.
(15) Sherlock does not exist.
The first is normally, and uncontroversially, formalized using the quantifier "$\exists$" and the second seems, on its surface, to involve a predicate.

[I have a mini-theory of what "$a$ exists" means. I think a claim of the form "$a$ exists" means "$\exists x H_a(x)$", where $H_a$ is, loosely speaking, the property of being $a$.]

Quine stressed that the meaning of the symbol "$\exists$" is explained as follows:
(16) $\exists x \phi$ is true if and only if there is some $o$ such that $\phi$ is true of $o$.
In other words, we explain the meaning of "$\exists$" using "there is". I can't quite see how it might work otherwise, except: by a proof-theoretic "implicit definition", via introduction and elimination rules.

Consider the following idea: the idea that the following two claims
(17) $\exists x \phi$ is true
(18) there is nothing that is $\phi$ 
are compatible.

One finds something like this being advocated as a solution to some problems in the foundations of mathematics. I think - but I am not sure - that Jody Azzouni's view is that (17) is compatible with (18). This would imply that there being no numbers (say) is compatible with the truth of mathematics. I cannot make good sense of this, mainly because the technical symbol "$\exists$'' is introduced precisely so that (17) and (18) are incompatible. Similarly, claim like,
(19) The sentence "There are numbers" is ontologically committed to there being numbers
is simply analytic, since it is part of the definition of the phrase "ontological commitment".

Suppose someone says there are things that don't exist (e.g., fictional objects or perhaps mathematical ones). I assume that, in their idiolect, "exists" means "has some property", but what this is has been left unspecified. If so, it means
(20) There are things which lack property $\dots$.
And what this $\dots$ is, is somehow left unspecified. A crucial ambiguity can arise. For example, the claim
(21) Numbers don't exist.
can be taken to mean,
(22) If there are numbers, they don't "exist"
(23) There are no numbers.
With a charitable interpretation, the first claim (22) is true, but not very interesting, because "exists" probably just means (in the speaker's idiolect) "is a concrete thing". No one in the world asserts that numbers are concrete things! The second claim, (23), is exciting: it denies that there are numbers.

Returning to relativized existence claims, like a claim of the form
(10) The Eiffel Tower and $\aleph_0$ exist in different senses,
I don't really see how making sense of such a claim requires anything other than working with many-sorted logic, where the sorts are thought of as having some deep metaphysical significance. For example, the assumed significance might involve a Platonic theory of Being vs. Becoming, and then we might take (10) to be based on an assumption like
(24) The Eiffel Tower belongs to the world of Becoming, while $\aleph_0$ belongs to the world of Being.
One would need to be careful about trying to make this kind of approach work with a 1-sorted logic, for example using a pair of quantifiers $\exists_1$ and $\exists_2$, as a famous argument shows that an assertion of existence-in-sense 1 is logically equivalent to an assertion of existence-in-sense 2:
$\vdash \exists_1 x \phi(x) \leftrightarrow \exists_2 x \phi(x)$.
Proof. Suppose $\exists_1 x \phi(x)$. Skolemize, to give $\phi(t)$, where $t$ is a skolem constant. By Existential generalization, $\exists_2 x \phi(x)$. So, $\exists_1 x \phi(x) \to \exists_2 x \phi(x)$. Similarly in the other direction.

I believe that Kurt Gödel says somewhere that no sense can be made of relativizing existence itself, and Quine also makes a similar point in various writings.

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.