By Jacques Fleuriot PhD, MEng (auth.)

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) includes 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 Nonstandard****Analysis**, 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 methods, 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 innovations 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**

Constructing the Hyperreals It follows from (S) that for any formula A, whether it contains the predicate standard or not, for all standard set x, there exists a unique standard y ~ x whose standard elements z are precisely those of x and such that A holds. The set y is usually denoted by B{Z E x. A}. A point worth noting is that if A involves the predicate standard or something based on the latter such as the concept of being infinitely small, then A is no longer a formula of ZFC and so the axiom of separation cannot be applied to it.

X E X) 1\ (3U. "Ix E X. x ::; U) ===> 3u. ("Ix E X. x::; u) 1\ 'Vu'. ("Ix E X. x::; u') ===> u ::; u' 2) The Archimedean property for the reals. This simple result has farreaching implications since it rules out the existence of infinitely small quantities or infinitesimals in 1R. Any such infinitesimal in 1R would mean that its reciprocal is an upper bound of IN in 1R thereby contradicting the Archimedean property: "Ix. 3n. x < n Various mechanizations of standard analysis (see for example Harrison's work in HOL [42, 43]) have developed theories of limits, derivatives, continuity of functions and so on, taking as their foundations the real numbers.

C E chain S A super S c = 0} We tried to simplify these definitions at first by removing references to the inductive set S, since it is actually used by Abrial and Laffitte to provide typing in their version of ZF. Thus, S as a parameter seems redundant when working in Isabelle's typed higher order logic. However, relying on the type made some of our proofs about ultrafilters unnecessarily complicated and prompted us to refer explicitly to the underlying set in definitions and hence in our proof of Zorn's Lemma.