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[Jon Zingale]

EricS,

You write:







*I bring up this debate in mathematics because it seems significant to
mehow long and how intensely it has been going on, with both sides wanting
anotion of “truth”, and neither being able to claim to have achieved it in
termssatisfied by the other.  If the intuitionists had never been able to
build areal system around their position, the formalists could just declare
victoryand go home.  But the debate seems still live, even within math and
not only inphilosophy, with clear trade-offs that there are proofs that
each side willaccept that the other rejects.*

It would surprise me to meet a mathematician who feels intensely one way
or an other about a particular choice of topos. For mathematical-logicians,
what seems more interesting are the geometric morphisms between toposes.
I would argue that the formalists to some extent *did* *just declare
victory*
many times over and that their are still pockets of scientific/mathematical
culture that believe everything can be *reduced to bits*. Still, and not
just as
with the intuitionists, richer toposes are there to be found and explored.

My two favorite examples come from algebraic geometry and from
quantum cosmology. In the former case, Grothendieck arrives at the
idea of a non-boolean topos while writing the foundations of algebraic
geometry. In the latter, Fontini Markopoulou-Kalamara
<https://en.wikipedia.org/wiki/Fotini_Markopoulou-Kalamara> develops her
non-boolean topos in the context of quantum gravity†.

Jon

†) Tangentially related to other parts of the overall discussion, Fotini
is also a design engineer working on embodied cognition technologies.
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