*physical state*. We have some sort of system $S$ (say a point particle, or a rigid body, or an assembly of particles, or a region of gas, etc.), and we think that it can "be" in some range of possible physical states.

I mention this only because it seems to me that there is a persistent mistake in some recent literature about applied mathematics and indispensability, in assuming that physical states are "concrete" entities, presumably of a spectacularly peculiar sort. But they are not "concrete". "Physical" does

*not*imply "concrete". For example, a wavefunction $\Psi$ is physical, but is not a concrete entity: it's a

*function*to $\mathbb{C}$. The wavefunction

*is*the physical state.

For example, for an $N$-particle system in classical mechanics, the physical states are $N$-tuples of ordered pairs,

$((\mathbf{r}_i, \mathbf{v}_i) \mid i \in \{1,\dots, N\})$,where $\mathbf{r}_i$ is the location of particle $i$, and $\mathbf{v}_i$ is the instantaneous velocity of particle $i$. Although certain kinds of equivalences may have to be taken into account, the points in the state space

*are*the physical states. Taken together, they form a structure, which in a sense is "like" the manifold $\mathbb{R}^{6N}$, because the locations and velocities can be co-ordinatized as triples of reals (i.e., given a co-ordinate chart $\phi$ on space, and a point $\mathbf{r}$ in space, we have $\phi(\mathbf{r}) = (x, y, z) \in \mathbb{R}^3$). In classical mechanics, when one moves to the phase space, the structure is called a symplectic manifold.

Consider a 3-particle system, and consider the state

$\alpha := ((\mathbf{r}_1, \mathbf{v}_1),(\mathbf{r}_2, \mathbf{v}_2),(\mathbf{r}_3, \mathbf{v}_3))$.This state $\alpha$ is not a concrete thing. What, for example, is the

*speed*of this state $\alpha$? Its

*location*? Such questions are absurd. A

*sequence*$\alpha$ of positions and velocities is not a concrete entity (like, perhaps, the point particle itself). And despite the claims of some fictionalists, a physical state is also not like Santa Claus or Jane Eyre, because physical states are what are our scientific theories are about, quite unlike Santa and Jane Eyre.

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