Model theory makes anti-realist views unsustainable. — Tarskian

I don't see how that is. Take as an example the Peano axioms for the natural numbers. Do we have a model of them? Yes, namely $\mathbb N$, the smallest inductive set guaranteed by the axiom of infinity in ZF set theory.

But the latter is just as fictional as the former, is it not? There's no empirical proof of the existence of infinite sets. They're a mathematical abstraction.

It seems to me that models are often purely abstract mathematical entities. One can take a purely formalist view of ZF for example. 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.

Model theory makes mathematics decisively correspondentist — Tarskian

Why?

Look for example at the relationship between Peano Arithmetic theory (PA) and the natural numbers (N).

PA does not create N.

N exists independently from PA. In model theory, PA is merely deemed to correctly describe N, while N is deemed to interpret PA.

So, N is a preexisting abstraction. The truth about N, called "true arithmetic", is also deemed to exist independently of any mathematical theory.

How else would you understand this approach, besides "correspondentist"?

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.

Mathematical realism is about the independent existence of such Platonic universes.

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.

Model theory truly believes that the natural numbers exist independently from mathematics or any of its theories.

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

The natural numbers have no physical instantiation as far as we know. Their existence is only abstract, fictional if you will. Or, to a formalist, purely symbolic.

It is beyond question that the axiom of infinity gives us a model of the Peano axioms, but both structures are equally fictional or equally formal.

Mathematical realism is about the independent existence of such Platonic universes. — Tarskian

Ok ... but ZF is a symbolic system. It doesn't talk about things in the real world, only sets, whose existence is entirely formal or fictional or abstract.

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

Maybe that's a good question but I'm not sure. ZF exists independently of PA, but both are symbolic axiomatic systems.

Model theory truly believes that the natural numbers exist independently from mathematics or any of its theories. — Tarskian

I'm not qualified to agree or disagree, but it sounds suspect to me. I don't think that the model theorists every say to the set theorists, "I bet you didn't know that sets are as real as cheeseburgers." Nobody ever says that or believes it. The structures studied in model theory generally live in set theory. And there's nothing real about set theory except by virtue of our imagination and symbolic processing.

ps -- Ok I have a sharper response.

The Wiki article on model theory says:

"In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold)"

https://en.wikipedia.org/wiki/Model_theory

But then when you click on structure, it says"

"In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it."

https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

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.

In short, the structures can be taken to be every bit as syntactic as the axioms that the structures are models of.

PA does not create N. — Tarskian

Odd choice of phrasing. It might be thought of as defining ℕ.

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.

Actually, I have to agree to that.

Formalism is a very consistent idea.

Model theory can indeed be viewed as the correspondence between an axiomatic fiction and a set-theoretical one.

No matter how compelling the Platonic view, the formalist view always seems to be able to match and counter it.

While category theory is indeed "general abstract nonsense", model theory is the correspondence between two general forms of abstract nonsense.

Ok, I think that I finally have learned my lesson now. I will never try to defeat formalism again. Seriously, this was my last attempt.

Ok, I think that I finally have learned my lesson now. I will never try to defeat formalism again. Seriously, this was my last attempt. — Tarskian

So glad I could provide some insight. That was an interesting question. Formalism is saved!

PA does not define N.

N is defined as a set-theoretical construct while PA is an axiomatic theory constructed as such that all its theorems turn out to be true in N but also in somewhat similar nonstandard models.

So, N is a preexisting abstraction. The truth about N, called "true arithmetic", is also deemed to exist independently of any mathematical theory. — Tarskian

I have only a terminological question as unlike the other learned contributors here I have no formal training in mathematics. However I'm interested in the ontology of number.

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? You could very well say it is real independently of mathematical theory, but in this context I think there's an intelligible distinction between 'exists' and 'is real'. I say this, because I question whether abstractions such as numbers can be said to exist in the sense that sensible phenomena exist. But they are real, as they're the same for any observer - but they're only perceivable to a rational intelligence, so they are real in a different sense to the proverbial tree or apple of metaphysical debates.

Hope it's not a daft question.

In fact, I have also come to accept the alternative formalist view.

Since N can also be described as a set-theoretical construct, it is just another symbolic abstraction.

Let's not be mistaken. The Platonic ontology is very attractive. However, there is also no denying the formalist ontology: in the end, it is also just string manipulation. You do not need to see more in it. That is not mandatory at all.

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. I have no other choice than to accept that viewpoint as equally legitimate.

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

What about applied mathematics? The 'unreasonable effectiveness of mathematics in the natural sciences'? The fact that mathematical predictions enable discovery of natural facts otherwise unknowable? Don't they indicate that mathematics has some real traction? Get your sums wrong, and your lunar lander crashes. That ain't abstract.

I didn't say it did. My point was more that you might be accused of accept realism in your premise, in supposing that ℕ exists. But the point is now moot. Cheers.

Applied mathematics is actually not mathematics.

As soon as it is about correspondence with the physical universe, it is about the use of mathematical language and other notions in physics, chemistry, engineering, or something else, in order to maintain consistency in the ideas being expressed.

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.

And whether there is or is not a ‘Platonic universe’ depends on who you ask, right?

Correct. Kurt Godel said yes. David Hilbert said no. They both have arguments that are equally convincing. I consider the problem to be undecidable.

I think I’m with Gödel (but it’s only a hunch.) I believe that there are real abstracts.

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

How would you classify model-dependent realism? Clearly this is not "correspondentist". You can argue that it is a form of "realism" as the title suggests, but where does the correspondence lie. I suggest that you consider that correspondence inheres within the formal system itself. When the math is applied correctly there is correspondence between the symbols, and what you call the "Platonic universe".

But "correspondence" in the common sense, means to correspond with the empirical world of observable sense objects. Your thread does not seem to make a distinction between these two very different senses of "correspondence". The common sense is correspondence with an assumed observable, sensible world, and the sense you mention here is correspondence with an assumed Platonic universe.


In the op. you do not address the nature of axioms. In reality, the axioms dictate the applicability of mathematics, and it is how mathematics is applied which determines whether it is formalist or realist. In other words, the meaning of the symbols is dependent on the context of the application, and the applicability, therefore context, is dependent on the axioms. And, I would argue that in general, the axioms are intentionally extremely vague and ambiguous in this respect, for the very purpose of allowing the mathematics the widest possible context of applicability. You will see however, axioms like those of set theory, which are explicitly realist. Such restrictions limit the applicability of the mathematics, (which is evident from the recent paradox threads of @keystone), by doing things like limiting "infinite" to fit it into the confines of "an object".

Applied mathematics is actually not mathematics. — Tarskian

If you think clearly about this idea, you will see that the inverse is actually the case. To analyze, let's separate form from content, and assume that the formalist's goal is to remove all vestiges of content from the formal system. How can this be accomplished? If the axioms have no designated relationship with anything outside the logical system, then the system my be infinitely useful, but at the same time infinitely useless, because it is robbed of all meaning. So the formalist allows a little bit of meaning, content, to inhere within the form of the structure.

For example,

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

In this example, you have an assumed "Platonic universe". This assumed universe provides the content. So consider the following two possible sets of rules for the processes of counting. The first set of rules would be to produce a bijection between the symbols and the physical objects to be counted. You want to count chairs, you biject "1" and a chair, "2" and a chair, etc.. In that case, the formal structure, and the set of rules for application are completely distinct from the content, the content being the physical objects counted, which is dependent on the application The second possible set of rules for the process of counting would be to produce a bijection between the symbols and a "Platonic universe" of "numbers". In this case, the content, being the numbers as objects, is built right into the formal structure. The limits of applicability are built into the structure, instead of being defined by a further set of rules.

If you categorize the first as "formalist", then you have a separation between the formal structure and the content (physical things) which the structure is applied to. 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. If you categorize the second as "formalist", then the content inheres within the formal structure, and there is no proper separation, as required for a true formalism.

Either way, mathematics cannot escape the need for, or its dependence on, application. There is always some form of application built into the formal structure, as axioms. Either the rules for application are a distinct part of the structure, as in the first case, or the application itself is already built into the structure, as in the second case. 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. And in reality mathematics is nothing other than application, pure means without any defined end.

There are many different, incompatible, mathematical systems. Assuming Platonism, which mathematical system is "correct", and how is this determined?

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

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. The truly mathematical version of geometry, is algebraic geometry. It revolves exclusively around dealing the roots of multivariate polynomials, which is entirely about string manipulation and does not require any visual input. 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.

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

That is exactly the essence of the formalist view:

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.

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.

There are many different, incompatible, mathematical systems. — Michael

Could you give me an example of two incompatible mathematical systems? There is a lot of research in inconsistent mathematics, but I am not sure that this is what you mean?

Could you give me an example of two incompatible mathematical systems? — Tarskian

There is a universal set in New Foundations but not in ZFC.

The surreal number line, unlike the real number line, includes infinity and infinitesimals.

Model theory makes mathematics decisively correspondentist — Tarskian

Does it?

Are you maybe taking the wiki definition "and their models (those structures in which the statements of the theory hold)" to mean real world structures? From that view, it would make sense why you think it is correspondist.

Because University of Toronto professors define such that it seems perfectly agreeable with formalism:

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

Even if we do have mathematics that are supposed to match real world phenomenons — correspondist —, like ancient mathematics was, this doesn't seem to show anything other than mathematics being possibly an empirical endeavor. Or rather, that the application of mathematics is necessarily empirical — by the very meaning of the word "application" of course. We know that pure mathematics is "conclusion follows from the premises", without any need for correspondence with real world phenomenons.

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

That would be a defeater for immanent realism and psychologism, but not for platonism. But I agree with your overall post.

I am not a platonist, but I imagine they might say all those structures exist independently as abstract entities. If my description is right, it becomes clear how that is just formalism with extra steps (not parsimonious) and an additional ontological commitment.

You might be interested in my thread. https://thephilosophyforum.com/discussion/15080/grundlagenkrise-and-metaphysics-of-mathematics

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

Only a terminological question? By no means -- it's the question. @Banno is getting at the same thing:

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

Absent an agreed-upon use of the existential quantifier, you can read the "ontology of numbers" question pretty much any way you want. I started a thread here a while back that might be of interest.

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

That's the point, mathematics is always "married" to something, be it the world of sensory input, or the alternative, Platonic universe.

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

You're jumping to a conclusion. How does the imaginary Platonic universe provide a less problematic grounding for meaning than the sensed universe?

You yourself described how the formalist approach does not get rid of correspondence, it only replaces the objects of the sensible universe, with the objects of the Platonic universe, as that which the mathematics corresponds with. But there is a huge problem here, the objects of the Platonic universe are simply whatever strikes the fancy of the mathematician. So for example, the mathematician might think, "I wish I could count to infinity". Then, one could create an axiom which states that the natural numbers are countable, and stipulate a 'set of the natural numbers', and voila, the mathematician has counted to infinity, within that imaginary universe of Platonic objects.

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

What you describe is what formalism would be like if it could achieve its goal in an absolute sense. It would be "about nothing" but as I indicated, at the same time it would be "about everything", providing infinite applicability, and at the same time, as you say, no applicability. But of course, no mathematician would seek this, because it would be incomprehensible nonsense.

So formalist mathematics is always tainted, and rules of application always inhere within the axioms, whether the influences are the sensible universe, the Platonic universe, or both. Your claim, "any downstream application of mathematics is completely irrelevant to mathematics itself" is simply false, because that's what the axioms of mathematics do, lay the grounds for application.

Oh, that thread dropped of my list. I didn't see your last reply. Still the most anoying question on the forums.

Yes, that's the issue on Tarskian's thread.

It brings out the conflict in my own arguments, between Midgley and Davidson, and provides something of a logical frame for that discussion. No small topic. So I'm not surprised to see it here as well.

But for here, it seems that has come around to an antirealist position, after 's "the structures themselves are nothing more than formal systems". A more direct rout than I was taking. Nice work.

's is not just a "terminological question". It's (potentially) a choice between grammars, between languages. Which implies quantifier variance. Which I think we (you and I) are inclined here to deny.

It's (potentially) a choice between grammars, between languages. — Banno

I prefer to think of it more as an ontological question. As the SEP article on Platonism in Philosophy of Maths says:

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.

Or as Rebecca Goldstein says of Gödel:

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.

I think the resistance to this idea is because it suggests that the non-physical reality of number, hence as the SEP excerpt says challenges the prevailing naturalist dogma. But then, in the SEP entry on Idealism, which you have previously quoted, it is said that:

the idealist, rather than being anti-realist, is in fact … a realist concerning elements more usually dismissed from reality.

All of this is a case in point.

I prefer to think of it more as an ontological question. — Wayfarer

Indeed, a distinction that I can't make sense of. Ontology is choosing between languages. It consist in no more than stipulating the domain, the nouns of the language.