Re: The reality of sets, on a scale of 1 to 10 (theory of theories)

On 3/25/2025 1:00 PM, Ross Finlayson wrote:

On 03/25/2025 09:40 AM, Jim Burns wrote:

>

>
Platonists need not eschew formalism,

Given the usual meanings of those terms,
not.eschewing looks like a very good trick.
Too bad about there being essentially zero chance
of you (RF) explaining how that'd be done.
https://plato.stanford.edu/archives/spr2016/entries/platonism/
⎛ Platonism is the view that
⎜ there exist such things as abstract objects —
⎜ where an abstract object is
⎜ an object that does not exist in space or time and
⎝ which is therefore entirely non-physical and non-mental.
https://plato.stanford.edu/archives/spr2016/entries/formalism-mathematics/
⎛ The guiding idea behind formalism is that
⎜ mathematics is not a body of propositions representing
⎜ an abstract sector of reality
⎜ but is much more akin to a game, bringing with it
⎜ no more commitment to an ontology of objects or properties
⎝ than ludo or chess.

Platonists need not eschew formalism,
nor for that matter positivism and logicism,
indeed the objects of the theory are sort of the same.

The theories are the same.
Their consequences are the same.
However,
Platonist.objects and formalist.objects
are not the same. Thus:
⎛ Most writers on the subject seem to agree
⎜ that the typical working mathematician is
⎜ a Platonist on weekdays and
⎜ a formalist on Sundays.
⎝
   — Philip J. Davis.
...because which they are doesn't really matter.

Long ago I adopted the style
to not cut the context because
it's too easy for it to be mutilated.

I have been admonished, sometimes,
for my posts being too long.
I don't consider myself ruled by those admonishers,
but I am posting in order to communicate.
I consider it a good idea to listen to
what they find unhelpful.
Sometimes, I even try to explain further
something which missed the bull's.eye
on my first attempt!
----

the inconstancy of either platonism or formalism
is merely self-deceiving vacillation,

>
If a typical working mathematician,
on Sundays, affirms the Intermediate Value Theorem
as a formalist, and,
on weekdays, affirms the Intermdiate Value Theorem
as a Platonist,
what is it that vacillates?

>
For example,
when a formalist platonist arrives at
axiomless natural deduction,
and then a formalist non-platonist
forgets or ignores that,
that's vacillation.

https://en.wiktionary.org/wiki/vacillation
⎛
⎜ 1. Indecision in speech or action.
⎝ 2. Changing location by moving back and forth.
⎛ 'There's glory for you!'
⎜
⎜ 'I don't know what you mean by "glory,"' Alice said.
⎜
⎜ Humpty Dumpty smiled contemptuously.
⎜ 'Of course you don't—
⎜ till I tell you.
⎜ I meant
⎜ "there's a nice knock-down argument for you!"'
⎜
⎜ 'But "glory" doesn't mean
⎜ "a nice knock-down argument,"'
⎜ Alice objected.
⎜
⎜ 'When I use a word,'
⎜ Humpty Dumpty said in rather a scornful tone,
⎜ 'it means just what I choose it to mean—
⎜ neither more nor less.'
⎝
-- Lewis Carroll, _Through the Looking-Glass_