What is the historical context and what are the philosophical implications of model theory?

Question

I tried to place Abraham's Robinson non-standard analysis in the thread of history and philosophy of logic, but got confused. I think I miss some background knowledge concerning perhaps the development of model theory and its philosophical implications. Hence my question: What is the historical context and what are the philosophical implications of model theory?

Answer 1

Husserl gave an insightful philosophical analysis of the change that occurred in mathematics over the course of 19th and early 20th century, the change some aspects of which he himself foresaw and promoted as a new way for mathematics (and science) in Logical Investigations (1900). Here is a surmise from Formal and Transcendental Logic (1929) §§29-30:

"Naturally this is done by that peculiarly logical universalization called "formalization", as a result of which all the materially determinate What-contents of the concepts — in the case of geometry, all the specifically spatial contents — are converted into indeterminates, modes of the empty "anything-whatever"... It is not just any multiplicity whatever (that would be the same as any set whatever); nor it is the form, "any infinite set whatever". On the contrary, it is a set whose peculiarity consists only in the circumstance that it is thought of with empty-formal universality, as "a" province determined by the complete set of Euclidean postulate-forms... The great advance of modern mathematics, particularly as developed by Riemann and his successors, consists... in its having also gone on to view such system-forms themselves as mathematical Objects, to alter them freely, universalize them mathematically, and particularize the universalities..." [boldface mine]

From viewing Euclidean figures as idealizations of material objects, as mathematicians did from Euclid to Kant, they went to viewing them as empty indeterminates subject to formal axioms. And then non-Euclidean geometries became possible as modifications of the axioms, where they were impossible on Kantian view. Furthermore, they were able to "particularize the universalities", find models of non-Euclidean geometries, such as Klein's and Poincare's, refill empty formalities with "matter". The same happened with complex numbers and their geometric interpretation. It became clear that different "material" theories (models) may be "equiform", formal theorems then apply to all of them uniformly, but have different "meaning".

This allowed more efficient development and application of mathematics, with seemingly disparate areas enriching each other, it also became a tool for investigating meta-logical properties of formal theories themselves. E.g. Hilbert showed independence of his geometric axioms by exhibiting models where all but one of them held; Gödel showed consistency of the controversial axiom of choice by building a model with it within the set theory without it; Skolem showed that all first-order theories, including real analysis and even Cantor's transfinite arithmetic, had countable models, etc. Skolem's and Gödel's results underscored the meaning-neutrality of the formal, and ushered in the current dominance of the first order logic, see Where did Gödel write that first-order logic is the "true" logic? But they also highlighted the limits to its expressive power, and later undermined the logicist program of Frege and Russell, and the formalist one of Hilbert, see respectively What is the philosophical ground for distinguishing logic and mathematics? and Was there a Kantian influence on Hilbert's formalist programme? Tarski showed how models allow to separate the theretofore conflated (even by Husserl and Carnap) notions of semantic and syntactic inference, which had a profound influence on analytic philosophy of science and language. He also gave a model-theoretic proof of Gödel's celebrated incompleteness theorem. Russell quipped about the import of the nascent model theory in Mysticism and Logic (1917) Ch.4 as follows:

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." [boldface mine]

One need only replace "mathematics" with "theoretical science" to recognize the drive behind some of the major trends in the 20th century epistemology of science, see e.g. What is the underdetermination of theories by evidence, and how does it square with scientific realism?

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P.S. Now let us place Robinson's non-standard analysis into this context. His stated purpose was to produce a rigorous framework that would better fit intuitions about and manipulations with infinitesimals of Fermat, Leibniz, Euler, etc. Better that is, than the Weierstrassian analysis, which effectively eliminated them. To accomodate intuitions this framework could not just be a formal theory, it had to come with a model interpreting it. The model had to contain the ordinary reals, and in a way that is equivalent to Weierstrassian analysis, i.e. it had to be an extension of them. But it had to be such an extension that all the new numbers obeyed the same "first-order" relations as the reals, so that all elementary functions would seemlessly extend to them, etc. This is Leibniz's "generality of algebra", which became the "transfer principle". So Robinson needed a model equiform to first-order analysis, but extending to infinitesimals when viewed "externally". This is what the hyper-reals accomplished. Ideologically, it was a continuation of "model-engineering" to specifications that Hilbert and Gödel earlier employed for other purposes.