Saturday, 20 April 2013

Theoretical Terms in Mathematical Physics

Semantics is the theory of meanings of expressions, normally with a particular, fixed language in mind. In semantics, one might be interested in:
  • the meaning of "the" (in English)
  • the meaning of "and" (in English)
  • the meaning of adverbs (in English)
  • etc.
For example, the meaning of "the" in English for expressions of the form "the $F$" (definite descriptions) was given a famous analysis by Bertrand Russell in his article "On Denoting" (1905). This analysis is contextual (i.e., "the $F$" is not explicitly defined). Following Russell, the statement
The current Prime Minister of the UK studied PPE
is analysed as,
There is exactly one current Prime Minister of the UK and this person studied PPE. 
Metasemantics is the metatheory of semantics. In metasemantics one is interested in questions like:
  • what are languages, in general?
  • what is status of claims about the semantic properties of languages?
  • how are languages acquired, spoken, implemented, cognized, grasped, etc., by minds?
One particularly pressing part of metasemantics concerns the semantics of theoretical expressions in science. A reasonable metasemantics for science aims to explain how meanings of theoretical expressions in, e.g., the language(s) of mathematical physics are grasped and assigned to linguistic strings. For example, how do we grasp the meanings of expressions in a passage like this:
Passage 1.
Consider a massive uncharged scalar field $\phi$ propagating on flat Minkowski spacetime $M$ with a potential $V(\phi)$. The field $\phi$ satisfies the Klein-Gordon equation,
$(\square + m^2) \phi + \frac{\partial V}{\partial \phi} = 0$.
Next let us consider the behaviour of this field when we consider a small graviton field $h_{\mu \nu}$ coupled to the energy tensor $T_{\mu \nu}$ of $\phi$.
(I've made this passage up, but it's the sort of thing one reads in a mathematical physics textbook or a paper.)

There are two main problems with the language of modern mathematical physics. The first is to understand how mathematical expressions obtain meaning. And the second is to understand how theoretical expressions like "massive uncharged scalar field", "potential", etc., obtain physical meanings.

We can sort of "undo" the physical content of Passage 1 as follows.
Passage 2.
Consider a scalar function $\phi$ on a differentiable manifold $M$ diffeomorphic to $\mathbb{R}^4$, with metric $g_{\mu \nu} = diag\{1,-1,-1,-1\}$. Let $\phi$ satisfy the equation,
$(\square + m^2) \phi + \frac{\partial V}{\partial \phi} = 0$,
for some $m \in \mathbb{R}^+$ and some function $V(\phi)$. Next let us consider the behaviour of this field when we consider a small symmetric (0,2) tensor $h_{\mu \nu}$ on $M$ coupled to the tensor $T_{\mu \nu}$ defined as follows ....
In Passage 2, we use notions like "manifold", "diffeomorphic", "$\mathbb{R}^4$", "scalar function", "metric", "$\mathbb{R}^+$", "symmetric (0,2) tensor". All of these can be defined in pure mathematics (and, in fact, reduced to the language of $ZF$ set theory, although this would be a bit nutty). For example,
A manifold $M$ is a topological space such that ... 
In Passage 1, however, we are imagining a possible physical world, whose underlying physical spacetime is rather like our actual spacetime would be if it were flat (i.e., Minkowski), along with certain physical fields with certain properties (i.e., a spin zero scalar field with mass $m$).

It strikes me as highly implausible to suppose that the notions from mathematical physics in Passage 1 can be somehow reduced to "logical constructions from sense data" as Bertrand Russell and Rudolf Carnap had hoped. But even so, it remains very unclear how human cognition can mentally represent how a hypothetical massive scalar field would behave under these circumstances. We can. It's just not clear how we can.

A metasemantic theory which accounts for the semantics of Passage 1 almost certainly will involve "heavy-duty" notions from Lewisian metaphysics: in particular, modality and "natural properties", etc.


  1. Why think that if we can't reduce notions of mathematical physics to sense data then we're going to have to use something like Lewisian naturalness in our metasemantics?

  2. It's a long story, and I've another drafted post on this, called "Lewisian Definitions". But here goes: roughly, once phenomenalistic reductionism via explicit definitions (i.e., the first dogma of "Two Dogmas") is abandoned, a plausibly empiricist metasemantics will instead invoke implicit definition (via conceptual roles and/or inferential roles), with no external (non-epistemically accessible) constraints on interpretations. The problem now is that this faces,

    - indeterminacy problems of the Quine-Putnam sort,
    - trivialization problems of the Newman sort (in connection with theoretical content turning into mere cardinality content)
    - trivialization problems of the Miller sort (in connection with approximate truth becoming highly language-dependent).

    To avoid these problems, one has to go Lewisian, and to invoke naturalness and natural similarities, which needn't themselves be grasped or understood at all. (One might go even further along the realist route, and invoke Platonic grasping of abstract properties and relations.)

    In metasemantics, I'm arguing that the main choices are between:

    (a) denying naturalness of properties and relations any role in metasemantics and accounting for grasp of meaning in broadly empiricist terms (roughly, experience plus implicit definition via conceptual/inferential role);

    (b) accepting within metasemantics some further assumptions about external (i.e., nothing to do with our minds) naturalness structure, which then cuts down admissible interpretations, and reduces the indeterminacies.

    At the end, I endorse (b), because the cost of (a) seems too much: it leaves too much semantic indeterminacy; it trivializes part of the content of scientific theories, as well as certain notions, like "approximate truth", that needed in scientific theory comparison.