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When doing mathematics and providing models that satisfy a given theory, we differentiate between standard and non-standard models. Now, assume you are a platonist and believe that the objects described by these models do really exist.

Now would these standard models be "standard" because of some pragmatic or epistemic reason? For instance, it could be that the standard model of arithmetic is simply the one we are most interested in or most familiar with (or perhaps easiest to work with for some reason). It could just be, as the synonymous usage of "intended model" might suggest, something we intend to talk about for various pragmatic or epistemic reasons.

But it is also possible that the standard models are in some sense metaphysically privileged, perhaps because they are more fundamental than the non-standard models. This sort of an understanding is suggested by David Lewis when he sketches his solution of the Plus-Quus problem (found in "New Work for a Theory of Universals"; see here for a brief description of the problem). His solution involves invoking degrees of naturalness among properties. Some properties are perfectly natural, these are the most fundamental. Other properties can, however, be more or less natural than one another. Part of Lewis's theory of reference (sometimes called "reference magnetism") has it that naturalness makes properties "reference magnets", in the sense that--- ceteris paribus ---the most natural of a range of possible referents for a predicate is the best candidate for reference and the one most likely to be referred to. His solution, then, is to say that "plus" (our normal addition function) is a better candidate for reference than "quus" (which matches our addition function up until some number n no one has calculated then delivers the result 52 for any number >n) because it is more natural. Thus it seems that Lewis's solution metaphysically privileges the mathematical function (since naturalness is a metaphysical notion). A similar attitude might be taken towards standard models, that they (or the concepts they use) are more natural than non-standard models.

Are there philosophers who have explicitly discussed the metaphysical status of standard models? Are there philosophers whose views on mathematics seem to suggest something of the sort?

  • Let's not assume we're Platonists. It's a bad idea. We'll just get tripped up by language games involving a poorly-defined notion of what it means to exist. – Rex Kerr Mar 12 '13 at 18:04
  • @RexKerr But then my question doesn't get off the ground ;) – Dennis Mar 13 '13 at 1:18
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(This is more in the nature of a comment to your question, than an answer; but its a little too long to be one).

a. To complicate your reading of standard & non-standard models, one could consider the real line. This seems eminently like a natural object; however it has no natural notion of infinitesimals. This makes it clumsy for the purposes of calculus & analysis. One can introduce a non-standard real line that actually does have them, and this was done originally by utilising non-standard models by Robinson.

So here we have a non-standard model to introduce natural features to the real line. Certainly it seems that which model is standard depends on epistemic, pragmatic or aesthetic reasons.

b. That there can be standard & non-standard models for some theory is indisputible, but can we be more precise about this? We can, there is the property of categoricity in Model Theory. If it is satisfied it simply means that even when there are more than one model of theory, they are in fact all isomorphic - so in a sense we have only one model. For a first-order theory this is fact not possible. But when we consider this along with the size of the model more can be said. It turns out (when the language of the theory is countable that) all models of each cardinality are isomorphic.

Then we can see at least that the standard model occupies a privileged place - it is at the beginning of this transfinite count, this ladder of models.

The other interesting thing that can be done is to turn this ladder into a line by topologising it. This makes it look a lot like the real line (except of course its still very different), and then the standard model is at the beginning of a line of models.

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Badiou explores the metaphysics of models in his concept of the model. I think he sees a model as representing a mathematical concept. I'm not able to say much more than that.

  • I worry in your first section that you might be running together Lewis's technical notion of naturalness with the expression, e.g., "that's a very natural way to do something". What makes me think this is that you refer to the real line as a "natural object", but for Lewis naturalness is a feature of properties. The observation that the standard real line lacks some desirable features (hence non-standard analysis) is interesting, though. – Dennis Mar 12 '13 at 14:52
  • @Dennis: I'm using the word 'natural' as in the 'real line' is a 'natural object' there as a mathematician may use it. What exactly this means I haven't seriously thought about. I don't think that there is a direct relationship with Lewis's technical notion of naturalness. – Mozibur Ullah Mar 12 '13 at 17:28
  • @Dennis: In Category Theory 'naturality' has a mathematical definition, again I don't think this is anything to do with Lewis's notion. I'm not sure how to go about describing it without a certain setup. But to give an example properties of vector spaces that don't rely on a basis are natural, in a manifold that don't rely on coordinates are natural. – Mozibur Ullah Mar 12 '13 at 17:33
  • @Dennis: What does Lewis mean by properties? I'm thinking that one way to link up 'naturality' of the 'real line' is to think of it as embodying certain 'natural properties', and then to show that these properties in fact define the 'real line' uniquely upto isomorphism. – Mozibur Ullah Mar 12 '13 at 17:35
  • @Dennis: I've also realised that 'standard model' actually has a technical sense which I haven't fully grasped. There is also a notion of 'minimal model' too (its minimal amongst standard models). – Mozibur Ullah Mar 12 '13 at 17:36
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Is the mathematical notion of a “standard model” a metaphysical or a (purely) epistemic distinction?

The meta-theory in which a model of a formal system lives should not be confused with metaphysics. The intended structure "A" is normally given explicitly in the meta-theory. Note also that isomorphism is a well defined notion in the meta-theory. Because "Th(A)" (= the set of formulas valid for A) is often not "computable", we will take a recursively enumerable (="computable") subset of formulas $\Phi$ from "Th(A)", which captures essentially all important aspects of "A". A model "B" of $\Phi$ with "Th(B) != Th(A)" will be called non-standard model. There is also the case of a non-isomorphic model "B" with "Th(B) == Th(A)", but even so this is not a "standard model", we wouldn't necessarily call this a non-standard model.

Are there philosophers who have explicitly discussed the metaphysical status of standard models?

How about Alfred Teitelbaum or Alfred Horn? (These are not philosophers, I know.) Seriously, I think the notions of standard and non-standard model arouse out of necessity, and are not directly related to metaphysical questions. On the other hand, many theories have distinguished models:

  • A common example are "free models". This example is related to the definition of homomorphisms for models (especially then condition "if R(x_1,...,x_n) then R^h(h(x_1),...,h(x_n))"), which models relations according to the semantics of equality.

  • As another answer already pointed out, also models of minimal cardinality are often distinguished.

Are there philosophers whose views on mathematics seem to suggest something of the sort?

The question what makes certain models distinguished or less complex than others has certainly drawn some attention. However, I only know of mathematicians (like Kolmogorov) who tried to propose answers to these questions, but haven't even read their original papers.

  • I was being sloppy in my presentation to avoid too much stage-setting. Yes, the models needn't be taken metaphysically seriously. The motivation behind this question is treating the models of mathematics the same way Lewis treats the Kripke frames of modal logic when he suggests that these "possible worlds" should be understood to be concrete worlds just like our own (not merely mathematical formalism). Is the import of your last paragraph to suggest that what privileges models is not "standardness", but rather simplicity (or, relatively less complexity)? – Dennis Mar 13 '13 at 1:24
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    I had to look up Alfred Teitelbaum… TIL :) – DBK Mar 13 '13 at 11:34
  • @DBK Haha, same thing happened to me. Now I'm just waiting to find out that Tarski is superman. – Dennis Mar 13 '13 at 13:16

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