Is Mathematics Physics?

Is mathematics physics?

I think of mathematics as the physics of necessity. The major difference between physical entities (like atoms, the magnetic field $\mathbf{B}$, wavefunctions, etc.) and pure mathematical entities, like $\pi$ or $\aleph_0$ or $SU(3)$, is their modal status.

[Update 28 May 2014: I clarify what I mean a bit more. The idea, which I haven't quite figured out properly yet, is this. The idea I'm defending here was mentioned before in an earlier M-Phi post, Abstracta - The Way of Modal Invariance. As we move from possible world to possible world, the intrinsic properties of a pure mathematical object, like $\pi$ or $\aleph_0$, remain invariant. In all possible worlds, $\pi > 3$. And, in all possible worlds, $\aleph_0 > n$, for any $n \in \mathbb{N}$. Pure mathematical entities are, in some important sense, modally invariant: they do not "change" their relations amongst each other temporally (in time) or modally. However, the modal status of physical / concrete entities is different: their properties and relations to each other change as we move from world to world. In some worlds, there are atoms; in some there are no atoms. In some worlds (perhaps ours), $\nabla \cdot \mathbf{B} = 0$, and in some worlds, $\nabla \cdot \mathbf{B} = 4 \pi \rho_m$ (where $\rho_m$ is a magnetic density; see this for magnetic monopoles). In some worlds (ours, say, $w$*), the set $B$ of Beatles (for definiteness, say 1 January 1966) has cardinality 4; and, in some world, say $w_1$, the set of Beatles has cardinality 3; and in some world, say $w_2$, the set of Beatles has cardinality 0. One might say that really there are three different sets, $B_{w^{\ast}}, B_{w_1}, B_{w_2}$. Or we can say that the property being a Beatle has different extensions at different worlds, because the contingent facts are varying. In these cases, the changes across time and across worlds reflect that the relevant facts are contingent, and involve concrete entities - entities which change over time and from world to world.]

There is a quite different, and opposing, idea that the role of mathematics in science is to "represent". There is a serious attempt to establish this view:

Field 1980, Science Without Numbers. 

This is a brilliant piece of work, full of extremely interesting ideas and insights. It introduces two main lines of argument:

• First, it invokes conservativeness to explain both the "insubstantiality" and utility of mathematics (that is, it explain how one might accept and use a theory $T_m$ referring to mathematical objects without thinking the theory is true - rather it is a conservative extension of a purely "nominalistic" sub-theory $T_n$, which is the theory one takes to be true); 
• second, it invokes representation theorems to explain the representational role of a mathematicized extension $T_m$ of a purely "nominalistic" theory $T_n$. 

In the first case, the central conservativeness claim is:

(Con) For any $\phi \in Sent(L_n)$, if $T_m \vdash \phi$, then $T_n \vdash \phi$.

In the second case, the central representation claim is

(Rep) Any model $\mathcal{M} \models T_n$ can be "nicely embedded" in a model of $T_m$.

People interested in these topics need to learn about the conservativeness results and the relevant representation theorems. They lie at the heart of the debate. I would call Field's book a classic of analytic metaphysics, along with other classics, such as Frege's Foundations of Arithmetic (1884), Russell's Principles of Mathematics (1903), Carnap's Der logische Aufbau der Welt (1928), Lewis's On the Plurality of Worlds (1986) and Parts of Classes (1991).

But I think Field's approach, despite its many important insights, is problematic for a number of reasons, quite complicated ones, and impossible to summarize easily. Many of the reasons are set out in detail in,

Burgess & Rosen 1997: A Subject with No Object.

I find the Burgess & Rosen response to the claim that "the role of mathematics is to represent" definitive. (However, very recently, there has been a piece of work developing a similar view to Field's: this is Artnzenius & Dorr (2012): see below.)

There has been a recent turn to "instrumentalism". To me, this seems a desperate move. The instrumentalist about mathematics claims that there is a separation of purely "nominalistic content" of a mixed assertion, e.g., a physical law like $\nabla \cdot \mathbf{B} = 0$, but without saying exactly what that content is. To me, this seems like mysticism.

Occasionally, when discussing these topics, there is unclarity about co-ordinate systems. The whole idea goes back to Descartes who, putting it anachronistically,  noticed an "isomorphism" between geometric space and $\mathbb{R}^3$. This is why we talk of cartesian co-ordinates, etc. Instead of talking about points, lines, surfaces and regions in space, we can talk of polynomial equations, $f(x,y,z) = 0$ (where the values of variables are real numbers) and use algebraic methods.

Let $(U,\phi)$ be a co-ordinate system on spacetime; a map that takes each spacetime point $p$ in the spacetime region $U$ to its co-ordinates $(x^0,x^1,x^2,x^3) = \phi(p)$, where $x^0,x^1,x^2,x^3 \in \mathbb{R}$. Let the image of $U$ under $\phi$ be $\phi[U] \subseteq \mathbb{R}^4$. The map $\phi$ has to preserve the physical topology of spacetime itself: the pre-image $\phi^{-1}[V]$ of any open set $V \subseteq \phi[U]$ has to be open in the physical topology.

Suppose that $\mathbf{B}$ is a physical vector field on spacetime (e.g., the magnetic field). Then there is a co-ordinate representation $\mathbf{B}^{\phi}$ of $\mathbf{B}$ relative to $\phi$. I.e., for any $x \in \phi[U]$,

$\mathbf{B}^{\phi}(x) = \mathbf{B}(\phi^{-1}(x))$

Then the function

$\mathbf{B}^{\phi} : \mathbb{R}^4 \to \mathbb{R}^3$ 

represents the vector field $\mathbf{B}$ on spacetime. The fact that $\mathbf{B}^{\phi}$ represents the function $\mathbf{B}$ does not imply that $\mathbf{B}$ is not a function. It is simply one function representing another function.

Here I am just describing the usual practice of mathematical physics. It does not bear on "nominalism" unless one denies that there is such a function as $\mathbf{B}$, the magnetic field. If one wishes to argue, e.g., that $\mathbf{B}$ is not a function, or that we may dispense with it somehow, then one needs to present a detailed argument. Hartry Field did present an argument that, while physical fields such as $\mathbf{B}$ are indeed functions (his example involved the gravitational potential $\Phi$ and the mass density field $\rho$, but the underlying issues and reformulation method are the same), even so, an "physically equivalent" theory can be formulated using only primitive relational predicates on space-time points; since then, almost no one has tried, because it is very hard technically to carry this through. The only exception is Frank Arntzenius & Cian Dorr 2012, "Calculus as Geometry". Their idea is not to deny that $\mathbf{B}$ is a function, but rather to expand the concrete ontology, and declare the whole fibre bundle concrete! They have concretized the abstracta. A review of their work by David Baker is here.

While mixed sets of concreta and physical functions (like $\mathbf{B}$, $F_{ab}$, wavefunctions, etc.) and so on do, perhaps, "explain", this is not the issue. For example, if we wish to refer to the average height of a Beatle, we assert that

• there is a set $B$ of Beatles, 
• each Beatle $p \in B$ has a height-in-metres $h(p) \in \mathbb{R}$, 
• there is a cardinal number $N = |B| \neq 0$ of this set, 

and we define the average height(-in-metres) by adding these heights $h(p_1) + \dots$, and dividing by $N$. This is how mathematics is actually applied, in practice.

People assume the disciplines of mathematics and physics are "separate" and seem astonished that someone might dispute this. But it seems to me they are not separate. It's more accurate to say that they are the same thing: mathematics = physics. This kind of view is also advocated by Professor Max Tegmark of MIT. The difference between mathematics and physics has little to do with epistemology. It seems right that the central difference is that physics deals with contingencies and pure mathematics with necessities. For example:

it is contingent that $\nabla \cdot \mathbf{B} = 0$.
it is necessary that $(\mathbb{N}, <)$ is well ordered.