Edwina, Gary R, and Jon AS,
I agree with your points and with the quotations by Peirce.
The challenge is to find a systematic terminology that is
consistent with Peirce, with modern conventions in logic,
and with the following constraints:
1. Logic allows a variable x in ∃x to refer to refer to anything
mathematical. That implies that any x that refers to anything
in pure mathematics can be said to "exist" in some sense.
2. But what sense is that? Is it some "Platonic Heaven" for all
mathematical entities -- including the infinities of integers,
real numbers, and Cantor's hierarchies of infinity?
3. Those people who deny that anything nonphysical can exist, claim
that mathematical things "depend" on physical things for their
existence. Frege, for example, identified the number 5 with the
totality of all sets of five things in the universe. But if the
universe is finite, there must be an upper bound on the integers
that can exist. And that construction fails completely for real
numbers, functions, and higher orders of infinity.
4. Some logicians (e.g., Lesniewski, Goodman, Quine...) tried to
eliminate sets because they are abstract, and they allow new
sets to be constructed from iterations of the empty set. For
example: {}; {{}}; {{},{{}}.{{{}}}}; {{{}},{{{{}}}}}; ...
But Quine relented because he realized that sets or something
similar would be necessary to define all of mathematics.
5. In his classification of the sciences, Peirce claimed that
pure mathematics is the only independent science. Every other
science, including metaphysics, depends on mathematics. That
rules out the option of claiming that mathematics has some
kind of dependency on what happens to exist in the universe.
6. For his process ontology, Whitehead considered all physical
entities to be processes and physical objects to be slowly
moving processes. He considered all processes to be
situated in a four-dimensional space time, and mathematical
entities to be "eternal objects" in the sense that they are
outside space and time.
7. Interesting option: John Wilkins (1668), the first secretary
of the British Royal Society, developed an ontology with the
help of other members of the society. See the attached
Wilkins.png. For a copy of his book, see
https://archive.org/details/AnEssayTowardsARealCharacterAndAPhilosophicalLanguage
Wilkins' top-level distinction is Transcendental/Special.
He characterized the transcendental branch as "knowing" and the
special branch as "being". Under Transcendental, he placed
language, logic, numbers, and metaphysics.
Suggestion: Suppose we name the two branches at the top of
any ontology transcendental/physical: Transcendental would
include all abstractions that are independent of space-time:
mathematical entities, sign types, and laws of nature.
Does anyone have any preferences for or against the pair
Transcendental/Physical instead of Mathematical/Physical?
John
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