2

Logic consists of proofs, not bold assertions. Semantics means assigning truth-values, it's kind of unavoidable, syntax alone is just string concatenation. You can't do away with axioms either.

Also, is it correct to say that a model could suffer from infinite regress? I am under the impression that self-reference is common with diagonalization constructions, it results in a regress, but one that terminates. That's how Gödel's incompleteness results work, the model provides the semantics. A lot of models are incomplete or even inconsistent, but incompleteness isn't a terrible outcome. Models can be extended indefinitely.

I don't see where semantics coming from meta-semantics comes into play, and the regression, or how value-judgments are applicable.

I read the Tarski part ”The rules of logic have been given to us by Tarski, which in turn got them from Mr. Metatarski” from Girard's On the meaning of logical rules I : syntax vs. semantics, and he seems to be more criticizing Fregean Sinn or Sense, meaning natural language applications of logic, not talking about mathematical logic.

Someone told me that the point of ”we presuppose the existence of a meta-world, in which logical operations already make sense” is one related to formal logic, where I think it's referring to natural language, or formalizations of natural language and not mathematical models. He seems to be saying something similar to post-Wittgensteinian inferentialism, I guess. Inferentialist semantics. It seems to be more a theory of natural language, and not a mathematical model

Mauro ALLEGRANZA
  • 35,764
  • 3
  • 35
  • 79
user59567
  • 29
  • 1
  • 1
    I'm sorry, user59567, I don't understand the question. – Scott Rowe Jul 13 '22 at 01:55
  • 1
    Sounds many thoughts and questions here... Re your "A lot of models are incomplete or even inconsistent", it's a basic knowledge in logic that only deductively closed *theory* may be incomplete or inconsistent which doesn't apply to *models*. Re your ”we presuppose the existence of a meta-world, in which logical operations already make sense... where I think it's referring to natural language, or formalizations of natural language and not mathematical models", not necessarily as reflected in Tarsk's infinite hierarchical of formal languages as opposed to semantically close natural language... – Double Knot Jul 13 '22 at 03:22
  • In fact on the contrary Tarski criticized the meaninglessness of using informal language to *define* truth based on his famous *undefinability theorem of arithmetic truth*. And also contrary to your thinking of "not mathematical models", Wolfram's [Ruliad](https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics) could be such a (speculated) model if it counts as mathematical... – Double Knot Jul 13 '22 at 03:48
  • 1
    Re your "Inferentialist semantics. It seems to be more a theory of natural language, and not a mathematical model", Inferentialist semantics is also commonly known as proof-theoretic semantics and equally applies to any logic/math system as Tarsk's, in addition to Wittgensteinian *meaning-is-use* doctrine applied in informal natural language... – Double Knot Jul 13 '22 at 03:51
  • Re Girard's paper: Girard point of view is not easy to grasp... having said that, when he says "when does semantics give meaning to rules?" he is speaking of logical rules. – Mauro ALLEGRANZA Jul 13 '22 at 06:51
  • 1
    But the regress' ghost is there: if we do not know how to use logical rules, how can we explain them with logical semantics, that itself uses rules? But this is the way language works. – Mauro ALLEGRANZA Jul 13 '22 at 06:52
  • 1
    ”we presuppose the existence of a meta-world, in which logical operations already make sense” Not necessarily: we have the "real world" made of common sense and natural language and the "sense" of logical operators is grounded on it. – Mauro ALLEGRANZA Jul 13 '22 at 06:53
  • 2
    What is the regress exactly, that any semantic interpretation needs another interpretation to be interpreted? If so, this is known as [Wittgenstein's rule-following paradox](https://en.wikipedia.org/wiki/Wittgenstein_on_Rules_and_Private_Language#The_rule-following_paradox), as is his solution:"*there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it"*". [Ryle's regress](https://en.wikipedia.org/wiki/Ryle%27s_regress) and [epistemic regress](https://en.wikipedia.org/wiki/Regress_argument) are similar. – Conifold Jul 13 '22 at 11:22
  • 1
    One can argue that infinite regress is only a problem of certain theoretical formulations and not a problem of reality which demonstrably does not regress infinitely (eg see are self similar strange attractors real?, It can be demonstrated that self similarity practically stops in reality after a finite number of layers) – Nikos M. Jul 13 '22 at 13:56
  • 1
    "*Is chaos only a feature of our mathematical models or is it a genuine feature of actual systems in our world?[..] For instance, empirical investigations of a number of actual-world systems indicate that there is no infinitely repeating self-similar structure like that of strange attractors. At most, one finds self-similar structure repeated on two or three spatial scales in the reconstructed state space and that is it.*" [SEP: Realism and Chaos](https://plato.stanford.edu/entries/chaos/#ReaCha) – Nikos M. Jul 13 '22 at 14:41
  • SE needs a badge for "Anthill Kicker". Some people could easily get that one. – Scott Rowe Jul 13 '22 at 15:33

1 Answers1

2

I think you're asking about whether the abstract idea of a "model" is well-founded. For a given axiomatic theory of logic, a model is intended to be some set of objects that satisfies all the formal axioms.

In theory, a model allows us to assign a truth value to every logical proposition, whether or not that proposition can be proved from the axioms.

When you say "model-theoretic regress" I believe you are talking about the regress from talking about propositions and axioms as purely textual things, to talking about objects in the model, to which the textual propositions supposedly refer.

I agree with you that this is problematic, because of the Pragmatic Maxim.

Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

The notion of a model is unjustified, because it is disconnected from any physical object that the mathematician works with. The objects a human mathematician directly interacts with includes the text of the proof, diagrams and images on paper or in their imagination, and other activity of their own neurons. The mathematician performs calculations involving these physical objects. The mathematician is probably limited to what a Turing machine can do, if we suppose that the mathematician's brain can be simulated on a Turing machine, which seems likely, or at least has not been ruled out.

The notion of a model - that yields a true or false answer to every question we ask in the logical theory, whether or not the question can be answered according to the formal axioms - goes beyond anything a Turing machine could do, and goes beyond anything the mathematician could do. The model "decides" everything. What practical connection does the mathematician have to the model? If the model says a certain proposition is true - though this proposition can be neither proved nor disproved by any axiom (or meta-axiom, or informal rule of thought) the mathematician knows - then in what practical way can this possibly matter to the mathematician?

To take a specific example, consider the continuum hypothesis. The continuum hypothesis is independent of ZFC; we may take the axioms of ZFC and assume additionally the continuum hypothesis, or we may take the axioms of ZFC and assume additionally the negation of the continuum hypothesis. Usually mathematicians do not declare whether they are taking the continuum hypothesis to be true or to be false. It does not matter to their work.

So, if they haven't even considered the continuum hypothesis, what model are they working in? No model of ZFC + (continuum hypothesis) is also a model of ZFC + (negation of continuum hypothesis). By declining to take a position on the continuum hypothesis, a mathematician cannot be working in any specific model, and thus their propositions cannot "refer" to any specific model.

Even if the mathematician does declare their stance on the continuum hypothesis for the purposes of a particular paper, there are an infinite set of other propositions, that, like the continuum hypothesis, are also independent of ZFC, and which the mathematician has not declared a stance on. So, again, the propositions they derive cannot refer to any specific model.

causative
  • 10,452
  • 1
  • 13
  • 45
  • 3
    "The notion of a model is unjustified, because..." Then you have to discard almost all of mathematics from ancient Mesopotamia to the invention of mathematical logic, which was almost all done on the basis of a model. – David Gudeman Jul 13 '22 at 06:05
  • 1
    "So, if they haven't even considered the continuum hypothesis, what model are they working in?" It doesn't matter. The proofs apply to any model that satisfies the axioms that they use. – David Gudeman Jul 13 '22 at 06:07
  • 1
    @DavidGudeman Yes, the proofs apply to any model that satisfies the axioms - which means that nothing in the proof refers to any *specific* object in any *specific* model! I don't think the Mesopotamians were using anything like the modern notion of a model of a logical system; the closest thing they had were physical pictures of a few circles and other geometric objects, or similar pictures in their mind's eye. They needed no notion of the set of all possible arrangements of geometric objects. – causative Jul 13 '22 at 06:10
  • 1
    OK, you're right that they didn't deal with models, but they did deal with model objects, things like natural numbers and ratios and geometric lines which are not physical objects. Up to the nineteenth century, all mathematics was done by thinking about these sorts of objects, not by thinking about textual axioms. Models are just a formalization of the connection between notation and traditional mathematical thought. – David Gudeman Jul 13 '22 at 08:13
  • @DavidGudeman Pragmatism has no issue with mathematical propositions and symbols referring to pictures and physical objects. The issue is only when we start talking about infinite models that we can't specify, axiomatize, or even truly think about. – causative Jul 13 '22 at 11:56
  • 1
    It certainly is a problem when we start talking about things we can't think about! Tattoo that on your arms, everyone! – Scott Rowe Jul 13 '22 at 15:36
  • You seem to be skipping over a category: things that are neither pictures nor physical objects, nor infinite models--things like numbers. Also there are plenty of physical systems we can deal with mathematically that we can't fully specify or axiomatize or fully grasp (if that's what you mean by "truly think about"). The only thing left of your objection seems to be infinity, and infinity is nothing but an abstraction that does not include certain limits such as a maximum number. If you can't use such abstractions, your math is always limited in scale. – David Gudeman Jul 13 '22 at 16:21
  • @ScottRowe We can talk or think about models of ZFC only as long as they remain indefinite and unspecified. We can't talk or think about any *specific* model of ZFC, because to distinguish a specific model from all the others would require giving a position on the continuum hypothesis and on an infinite number of other hypotheses. – causative Jul 13 '22 at 18:17
  • @DavidGudeman I did some looking and it seems that Peano arithmetic has a standard model, unique up to isomorphism, and the unique theory that has the truths of this model is called True Arithmetic (TA). So in the case of the natural numbers we can at least identify which model we want to refer to. So arguably you could say that theorems about the natural numbers are referring to this model, unlike when we talk about ZFC. (However, no Turing machine can calculate the theorems of TA, not even in the limit.) – causative Jul 13 '22 at 21:10
  • I guess I am more familiar with KFC than ZFC. There aren't an infinite number of choices there and you don't have all the time in the world to order. Perhaps the person in question should choose and then they wouldn't have an infinite number of models to deal with? – Scott Rowe Jul 14 '22 at 10:02