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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 http://philosophy.stackexchange.com/questions/28253/where-did-g%c3%b6del-write-that-first-order-logic-is-the-true-logic/28255#28255Where 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 http://philosophy.stackexchange.com/questions/34074/what-is-the-philosophical-ground-for-distinguishing-logic-and-mathematics/34077#34077What is the philosophical ground for distinguishing logic and mathematics? and http://philosophy.stackexchange.com/questions/28154/was-there-a-kantian-influence-on-hilberts-formalist-programme/28157#28157Was 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:

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. http://philosophy.stackexchange.com/questions/30033/what-is-the-underdetermination-of-theories-by-evidence-and-how-does-it-square-w/30037#30037What is the underdetermination of theories by evidence, and how does it square with scientific realism?

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 http://philosophy.stackexchange.com/questions/28253/where-did-g%c3%b6del-write-that-first-order-logic-is-the-true-logic/28255#28255 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 http://philosophy.stackexchange.com/questions/34074/what-is-the-philosophical-ground-for-distinguishing-logic-and-mathematics/34077#34077 and http://philosophy.stackexchange.com/questions/28154/was-there-a-kantian-influence-on-hilberts-formalist-programme/28157#28157 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:

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. http://philosophy.stackexchange.com/questions/30033/what-is-the-underdetermination-of-theories-by-evidence-and-how-does-it-square-w/30037#30037

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:

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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Husserl gave an insightful philsophicalphilosophical analysis of the change that occuredoccurred 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:

Husserl gave an insightful philsophical analysis of the change that occured 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:

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:

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P.S. Now let us putplace 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.

P.S. Now let us put 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. 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.

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.

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