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So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily practicing mathematicians) I am curious as to what lessons mathematicians draw from the existence of non-standard models.

Within philosophical circles it seems to be the consensus (or at least a fairly popular view) that what non-standard models show us is something about the limits of formalization. For instance, Haim Gaifman in a lecture delivered to the AMS Special Session Nonstandard Models Of Arithmetic And Set Theory (January 15-16, 2003, Baltimore, Maryland) notes the following:

If set theory is about some domain that includes uncountable sets, then any countable structure that satisfies the formalized theory must count as an unintended model. From the point of view of those who subscribe to the intended interpretation, the existence of such nonstandard models counts as a failure of the formal system to capture the semantics fully.

Now among those who subscribe to this sort of view, they tend to take the failure of categoricity in a first-order theory of Peano Arithmetic to show us that it is a second-order formulation of Peano Arithmetic which is needed. I've always taken this to result from a view that it is the semantic, rather than the syntactic, side of mathematical theorizing which holds some primacy. Often this is coupled with a view taken from Hilbert that mathematical theories (at least those that seem to have an intended interpretation) implicitly define some concept or some structure which various isomorphic models satisfy. On this sort of view, it is the concept which is of primary interest and the deductive systems are a means to discover a bit about (but generally, for reasons of incompleteness, only a bit about) what, for lack of a better word, you might call the nature of this concept.

TL;DR What lessons have mathematicians drawn from the existence of non-standard models?

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    $\begingroup$ Why draw lessons from them at all? They’re interesting objects of study in their own right, and sometimes useful tools. Apart from that, I agree (vehemently, if not violently!) with @André’s comment. $\endgroup$ – Brian M. Scott Apr 8 '13 at 5:10
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    $\begingroup$ The question about non-standard models is interesting. For most mathematicians, they are of no particular interest, mathematics is very diverse. For some, they tell us more about the world, just like continuous functions with "pathological" properties do. $\endgroup$ – André Nicolas Apr 8 '13 at 5:25
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    $\begingroup$ In my humble opinion, mathematicians are better at using tools than drawing lessons. Non-standard models are useful as tool, and one might even risk a statement that tools is precisely what they are. For instance, if you want to prove a first-order statement about "the" reals, it might be useful to use a non-standard model that offers infinitesimals (and then shift back to whatever model is "standard"). Generally, I would say that existence of non-standard models tells us that there are a lot of models that approximate the standard one pretty well. $\endgroup$ – Jakub Konieczny Apr 8 '13 at 7:48
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    $\begingroup$ I don't understand the question. Can you give an example of a lesson? What are they for? $\endgroup$ – Qiaochu Yuan Apr 8 '13 at 8:06
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    $\begingroup$ @Dennis: Amusingly, I would draw the opposite lesson: when we learn two things are indistinguishable, we shouldn't be trying to distinguish them! $\endgroup$ – Hurkyl Feb 1 '14 at 1:55
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I do not know about any lessons, but the conclusions are pretty much agreed. As you said, it implies that first order formal systems are not strong enough to model a single "structure" (or model). Because of Godel incompleteness theorem you will always have additional models than the one you intend to formalize. On the other hand, almost everybody agrees that going to full second order logic (and fixing the semantics) is not a solution either, because second order systems have a lot of problems of their own (there is a trade off), for instance, Quine pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not logic, properly speaking. In the end, what this means depends on you philosophical point of view. If you are a Platonist, it means that no formal system will be capable of proving all the truths of the structure of your choice (for instance, N). If you are a nominalist, it means that, at least in practice, first order formals systems describe an infinite number of "structures" (or models) (you would need a non-recursive enumerable infinite number of axioms to pinpoint a single model).

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    $\begingroup$ Actually the GIT and plurality of models results are not the same thing. Elementary Euclidean geometry is a complete theory and yet it also has infinitely many models. Even if you take the non-recursive theory of natural numbers, it still has infinitely many models. The problem is first order logic cannot discriminate amongst the infinite. $\endgroup$ – The_Sympathizer May 19 '15 at 11:50
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As you point out, Gaifman wrote:

From the point of view of those who subscribe to the intended interpretation, the existence of such nonstandard models counts as a failure of the formal system to capture the semantics fully.

Note that this is not merely about first-order logic but a broader observation.

To answer your question, some mathematicians have drawn the lesson that the entity called the intended model does not exist and that belief in such an entity is an obstacle to progress.

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  • $\begingroup$ Thanks for your contribution. I'm sympathetic to such a claim, and find partial expression of it in some of Joel David Hamkins's work on the set-theoretic multiverse and his arguments that satisfaction is not absolute. Do you know of other examples in print? $\endgroup$ – Dennis Aug 21 '17 at 16:38
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    $\begingroup$ Yes indeed ;-) @Dennis $\endgroup$ – Mikhail Katz Aug 21 '17 at 16:43
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    $\begingroup$ Let me observe - and this is meant positively - that one need not adopt the viewpoint of the last few words of this answer completely for this answer to be useful. In particular, a weaker stance (which I personally hold) is that disbelief can be helpful to mathematical progress, without making any negative claim about belief. (Also note that genuine disbelief in standard models raises its own problems. For example, what do we mean when we say that a statement about the natural numbers is true? This isn't a stupid question - my point is just that it is forcefully raised by this answer.) $\endgroup$ – Noah Schweber Aug 21 '17 at 16:47
  • $\begingroup$ @NoahSchweber good point! I take it the crucial point is that a dogmatic belief that we have fixed on the standard model, and dismissal of non-standard models in light of this belief, is "mathematically bad" since it preempts potentially fruitful investigation of such models. $\endgroup$ – Dennis Aug 21 '17 at 16:56
  • $\begingroup$ When the Western Roman Empire began to disintegrate, Augustine of Hippo developed a doctrine disassociating the survival of the empire from the survival of civilisation. @NoahSchweber $\endgroup$ – Mikhail Katz Aug 21 '17 at 16:59

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