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Negri and collaborators call "geometric" theories that can be axiomatized by FOL formulas of a special sort, namely universal closures of $A\to B$, where $A$ and $B$ do not contain $\to$ or $∀$, see e.g. Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. The terminology goes back to, at least, ...

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Poincare mentions that inductive reasoning, in his Geometry and the Imagination, that is, generalising from the special to the general case as being the essence of mathematics, and that of science. I don't know if Pierce was familiar with this work, but I assume it is likely that he was ... (It's worth adding that this is not special to science, it's a ...

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The geometry in geometric theories, as in topoi, comes from topology and hence also the name topos. They, that is the topoi, are seen as generalisation of a topology. This generalisation was by Grothendieck and generalises that of a topological covering which is, in his language, a sieve. And in this way we get elementary topoi, which was lawvere's ...

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This answer is CW, so I don't gain reputation for someone else's work. To quite Kevin Arlin's answer to the same question on math.stackexchange: Geometric logic constitutes the logic, models of whose theories are preserved by geometric morphisms between topoi. Geometric morphisms are those appropriate to toposes viewed as generalized spaces, for instance, ...

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And even before units were introduced in physics, people knew, for example, not to add oranges to say, apples. At least this is what I was taught in primary school, and I imagine that this goes back a long way. Unless some evidence turns up showing that Russell or Whitehead were directly influenced by the notion of dimension in physics, I'd say this is just ...

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