Model theory makes anti-realist views unsustainable. — Tarskian
I certainly do not believe that mathematics revolves around the correspondence with the physical universe. By "correspondentist", I actually mean: correspondence with a particular designated preexisting abstract Platonic world, such as the natural numbers. — Tarskian
Mathematical realism is about the independent existence of such Platonic universes. — Tarskian
If these Platonic universes do not even exist, why try to investigate the correspondence with a particular theory? It only makes sense if they do exist, independent of mathematics or any other theory. — Tarskian
Model theory truly believes that the natural numbers exist independently from mathematics or any of its theories. — Tarskian
So, N is a preexisting abstraction. The truth about N, called "true arithmetic", is also deemed to exist independently of any mathematical theory. — Tarskian
So, if you want to accept N as a mathematically realist Platonic abstraction, it works. However, I have no counterargument to the idea that N can also be viewed as just abstract nonsense. — Tarskian
Model theory makes anti-realist views unsustainable. Model theory makes mathematics decisively correspondentist. Because of model theory, mathematical realism and more specifically, Platonism, are unavoidable. Mathematics is about abstract Platonic worlds and is not just string manipulation. — Tarskian
Applied mathematics is actually not mathematics. — Tarskian
Mathematics proper seeks to establish the correspondence between an abstraction and a Platonic universe -- when interpreted according to realism -- or between an abstraction and another abstraction -- when interpreted according to anti-realism. Mathematics proper is never about the physical universe. — Tarskian
However, the structure is useless without rules of application, so we proceed toward axioms of geometry, and rules of categorizing, to provide rules of application. The rules of application are still a part of the formal system, and there is no proper "formalist" separation. — Metaphysician Undercover
In no way can mathematics completely escape application, without it becoming something other (a useless bunch of symbols) than mathematics. So the inverse of your statement is actually the truth. With absolutely no application, mathematics would be absolutely nothing. — Metaphysician Undercover
https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality.
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
Model theory makes mathematics decisively correspondentist — Tarskian
Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics. The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. This isn’t necessarily due to mathematics itself, but is a consequence of the language that we use to express mathematical ideas. — Fundamentals of Model Theory, William Weiss and Cherie D’Mello
So model theory studies the structures that satisfy some axioms; buy the structures themselves are nothing more than formal systems. A set along with a collection of operations and relations.
I think that resolves your concern. One can study a set along with some operations and relations defined on it, without believing such a set is real or has concrete existence or whatever way you are expressing your concern. — fishfry
There are no sets in the real world in the sense of set theory. Show me the set containing the empty set and the set containing the empty set, which is better known by its more familiar name, 2. — fishfry
I have only a terminological question. . . You say that the 'truth about N is deemed to exist independently of any mathematical systems'. My terminological question is, is 'exist' a correct choice of words in this context? — Wayfarer
N exists independently from PA.
— Tarskian
I don't have a clear idea of what you mean by "exists" here. Same for "preexisting" in the next paragraph. — Banno
Classical Euclidean geometry is arguable not "real" mathematics. As Kant pointed out, it is incredibly married to sensory input, to the point that it is not pure reason. — Tarskian
The fact that Euclidean geometry has too much meaning and does not fit the formalist narrative, points out a problem with Euclidean geometry and not with the formalist ontology. If it is not possible to interpret it as meaningless string manipulation, then it is not real mathematics. — Tarskian
In its anti-realist take, mathematics is indeed "about nothing". In its realist take, mathematics is about an abstract, Platonic universe that is completely divorced from the physical universe. In both cases, any downstream application of mathematics is completely irrelevant to mathematics itself. That is a feature and not a bug. — Tarskian
It's (potentially) a choice between grammars, between languages. — Banno
Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences. Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.
Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.
the idealist, rather than being anti-realist, is in fact … a realist concerning elements more usually dismissed from reality.
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