Ali, (01)
Tarski wrote many papers on geometry. As his biography says, he had
taught highschool geometry for several years. (In the 1920s, there
weren't many jobs for logicians.) (02)
> I should say that I'm familiar with Tarski's work as well, though
> on a digression, I thought he committed to points, and nothing
> else, not vanishingly small spheres, i'll have to look over his
> axioms again. (03)
That's true of Tarski's more familiar axioms for Euclidean geometry. (04)
His 1929 paper is the one that shows how points can be defined
as limiting cases of nested spheres. That means you don't have
to assume that points, lines, and spheres "truly exist" in nature. (05)
You can think of tiny spheres (or blobs) as the building blocks
of physical objects. Then the points, lines, and planes of
Euclidean geometry can be considered computationally useful
(but imaginary) approximations. (06)
We use such approximations in every field. Anybody who studied
physics "knows" that relativity and quantum mechanics are more
accurate than Newtonian mechanics. But for most applications,
the Newtonian formulas are much easier to compute, and they're
accurate to more decimal places than we can measure. (07)
So when we talk about "ontological commitment", we have to
distinguish what we "really believe" from what we assume as
a computable approximation. (08)
And by the way, this illustrates another reason why a single
universal ontology would not be used for practical applications.
To be truly universal, the ontology would have to be stated at
the most fundamental level possible. But that level is likely
to be too complex for practical computation. (Just look at
quantum electrodynamics or string theory, for example.) (09)
Therefore, we inevitably get a multiplicity of approximations
for different purposes. So even if the physicists discovered
a Grand Unified Theory, the engineers would still be using
many inconsistent approximations. (010)
John (011)
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