By Jacques Fleuriot PhD, MEng (auth.)
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) encompasses a prose-style mix of geometric and restrict reasoning that has frequently been considered as logically vague.
In A mix of Geometry Theorem Proving and NonstandardAnalysis, Jacques Fleuriot provides a formalization of Lemmas and Propositions from the Principia utilizing a mixture of equipment from geometry and nonstandard research. The mechanization of the techniques, which respects a lot of Newton's unique reasoning, is constructed in the theorem prover Isabelle. the appliance of this framework to the mechanization of ordinary genuine research utilizing nonstandard recommendations is additionally discussed.
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Extra info for A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia
A set yES such that no member of S properly contains y. The statement of Zorn's Lemma involves the idea of a partially ordered set and related concepts. We present briefly various mathematical concepts and theorems about them needed in Isabelle/HOL to express Zorn's Lemma. Paulson has already proved Zorn's Lemma in Isabelle's Zermelo-Fraenkel set theory (Isabelle/ZF)  by mechanizing a paper by Abrial and Laffitte . Reporting on the mechanization, Paulson remarks that the formal language used by Abrial and Laffitte is close to higher order logic and thus should be useful to Isabelle/HOL amongst other proof assistants.
This relationship appears (in slightly different wording) as Lemma 12 of the Principia where it is employed in the solution of the famous Propositio Kepleriana or Kepler Problem. Newton refers us to the "writers on the conics sections" for a proof of the lemma. This is demonstrated in Book 7, Proposition 31 in the Conics of Apollonius of Perga . Unlike Newton, we have to prove this result explicitly in Isabelle to make it available to other proofs. The formalization is rather involved and proceeds by a series of construction to show that the area of parallelograms cgov is equal to that of parallelogram catd and hence that areas of circumscribing parallelograms rxfYIj and ltzk are equal.
This problem is tackled in NSA by dispensing with property (4). Instead, using the axioms of classical set theory, a set JR. , JR ~ JR*, (1)-(3), (6), but not Infinitesimal ~ JR and therefore not (4). As a result, (5) now requires Infinitesimal to be an ideal in the set of finite members of JR•. This set includes the reals and the infinitesimals amongst other numbers. Though an axiomatic approach seems the easiest way to get quickly to the infinitesimals, there is always the possibility that the set of axioms might lead to an inconsistency, as we saw above.