OP's specific phrasing is that they "map symbols to symbols". Analog computers don't do that. Some can, but that's not their definition.
Turing machines et al. are a model of computation in mathematics. Humans do math by operating on symbols, so that's why that model operates on symbols. It's not an inherent part of the definition.
> OP's specific phrasing is that they "map symbols to symbols". Analog computers don't do that. Some can, but that's not their definition.
How is that not symbolic? Fundamentally that kind of computer maps the positions of some rods or gears or what have you to the positions of some other rods or gears or what have you, and the first rods or gears are symbolising motion or elevation or what have you and the final one is symbolising barrel angle or what have you. (And sure, you might physically connect the final gear directly to the actual gun barrel, but that's not the part that's computation; the computation is the part happening with the little gears and rods in the middle, and they have symbolic meanings).
Computers are functional mappings from inputs to outputs, sure.
Analog fire computers are continuous mappings from a continuum, a line segment (curved about a cam), to another continuum, a dial perhaps.
Symbolic operations, mapping from patterns of 0s and 1s (say) to other patterns are discrete, countable mappings.
With a real valued electrical current, discrete symbols are forced by threshold levels.
> Symbolic operations, mapping from patterns of 0s and 1s (say) to other patterns are discrete, countable mappings.
What definition of "symbolic" are you using that draws a distinction between these two cases? If it means merely something that symbolises something else (as I would usually use it), then both a position on a line segment and a pattern of voltage levels qualify. If you mean it in the narrow sense of a textual mark, that pattern of voltage levels is just as much not a "symbol" as the position on the line segment.
Your comment is only true if you take an excessively reductive view of "symbol."
You keep referring to what we are interested in, but that's not a relevant quantity here.
A symbol is a discrete sign that has some sort of symbol table (explicit or not) describing the mapping of the sign to the intended interpretation. An analog computer often directly solves the physical problem (e.g. an ODE) by building a device whose behavior is governed by that ODE. That is, it solves the ODE by just applying the laws of physics directly to the world.
If your claim is that analog computers are symbolic but the same physical process is not merely because we are "interested in" the result then I don't agree. And you'd also be committed to saying proteins are symbolic if we build an analog computer that runs on DNA and proteins. In which case it seems like they become always symbolic if we're always interested in life as computation.
A symbol is a discrete sign that has some sort of symbol table (explicit or not) describing the mapping of the sign to the intended interpretation