On 06/15/2024 04:45 PM, Moebius wrote:
Am 15.06.2024 um 19:30 schrieb Jim Burns:
>
It is an essential aspect of mathematics that,
when we are discussing a thing or things,
we are not discussing a different thing or things.
>
Right. The context should be a certain/unique theory.
>
Well, that's a certain aspect of the philosophy of
mathematics, in formalism, that _definition_ is
constant.
Then, a certain aspect of the philosophy of mathematics
is _abstraction_, so when we are talking about some
things, we are talking about the abstraction of things,
and also any things which so concretely represent the
abstraction.
This is a great sort of difference between model theory
and proof theory, about any representations of models
of things, and the model itself, as mathematical.
(Logical.)
When talking about mathematics the entire domain of
discourse, mathematical things, it is all the things.
When talking about mathematical things, and other
mathematical things, it is all the things, all one thing.
A domain of discourse, a "universe" or "world" of a
kind of thing, type of thing, class of thing, set of
things, with its own closed categories of axiom, and
closed categories of definition, may be considered
a particular or certain theory. Whether it's, "unique",
at all, gets into equi-interpretability with other
theories, when according to model theory that all and
only what result theorems or facts of the theory model
only and all what result theorems of facts of the other,
is what makes abstraction and "mathematical logic" and
model theory.
What you seem to decide is that a certain formal language,
is closed, and that all its terms are well-formed, and
that all its terms are meaningful, and that all or any
its axioms are independent or at least non-contradictory
is a usual thing, a theory of language.
Anyways "nominalism" and "fictionalism" aren't included
in some mathematical philosophies, except insofar as
they're models of mathematical reasoning, meta-models,
while in some mathematical philosophies, the only
true meta-model is also the model itself. This is
usually called "platonism", that the objects of mathematics
exist and we only "discover" them not "invent" them.
I.e., mathematical abstraction is objective,
while poetical abstraction is subjective.
In mathematics, some things reflect a multitude or
even a continuum of things, and not reflecting that
they have these properties these objects these things,
is missing out and called "ignorance", and when
confronted with these things, then being non-mathematical
about the mathematical logic part, is called "hypocrisy".
V
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