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 seen as logically vague.

In **A blend of Geometry Theorem Proving and Nonstandard****Analysis**, Jacques Fleuriot provides a formalization of Lemmas and Propositions from the Principia utilizing a mix of tools from geometry and nonstandard research. The mechanization of the strategies, which respects a lot of Newton's unique reasoning, is built in the theorem prover Isabelle. the appliance of this framework to the mechanization of undemanding actual research utilizing nonstandard suggestions can be discussed.

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**Extra info for A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia**

**Example text**

We then extend this result and show that the ultrafilter can be free as well. 1 Zorn's Lemma The existence of free ultrafilters is not obvious at first sight. To show that the ultrafilter theorem holds and to carry out our construction, we need Zorn's Lemma. This is an equivalent form of the Axiom of Choice (AC) and first needs to be proved in Isabelle/HOL. Zorn's Lemma. Let S be a non-empty set of sets such that each chain c ~ S has an upper bound in S. e. a set yES such that no member of S properly contains y.

Thus, if Infinitesimal satisfies (2), (4), (5) then Infinitesimal = JR. 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.

Pratrel AA{(ad + bc, bd)}) q) p) II = prat_less_def lip < (Q::prat) = 3T. 2. 4 Constructions Leading to the Reals prat == {x. quotient--