[FOM] Vagueness/Indefiniteness in math

Ellerman, David david at ellerman.org
Sat Feb 11 14:46:52 EST 2017

Given an equivalence relation ~ on a set U, there are two types of

#1: for u~u' in U, one has the equivalence class S = [u] = [u']; but
#2: one also can consider an abstract entity u_S that is definite on what
is common to the elements of the equivalence class and indefinite on how
they differ.

In general for a quotient set, quotient group, etc. the elements of the
quotient can be interpreted either way.

The #2 abstraction is used systematically in homotopy type theory. Consider
an annulus (a disc with a hole in it). Consider coordinatized paths
f:[0,1]-->X from a base point  f(0) = x_0 to f(1) = x_0. Under deformation,
there is the #1 abstraction of the equivalence class of all paths that go
around the hole just once clockwise. But there is also the #2 abstraction
as 'the path going around the hole just once clockwise' which is definite
on that property but is indefinite about any coordinatization. The element
1 in the fundamental group for that base point (which is isomorphic to Z,
the integers) can be interpreted either way as the equivalence class or the
abstract path going around just once clockwise which abstracts away from
the different coordinatizations. It is the #2 type of abstract objects that
are considered in homotopy type theory (currently the HoTT stuff).

The (naive) theory of the  #2  abstracts  u_S is developed in this paper
<https://arxiv.org/abs/1701.07936> . It is shown how to mathematically
describe both the #1 and #2 abstracts. Then it is shown that this
mathematical treatment of the #2 abstract objects is the same as the
density matrix treatment of a superposition state in quantum mechanics
which is often considered to be "objectively indefinite." (see references
in the paper). Hence the consideration of definite/indefinite abstract
objects in the philosophy of mathematics turns out to shed some light on
the superposition states of QM.
David Ellerman

Visiting Scholar
University of California at Riverside

Email: david at ellerman.org


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