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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 usually been considered as logically vague.
In A blend of Geometry Theorem Proving and NonstandardAnalysis, Jacques Fleuriot offers a formalization of Lemmas and Propositions from the Principia utilizing a mix of tools from geometry and nonstandard research. The mechanization of the systems, which respects a lot of Newton's unique reasoning, is constructed in the theorem prover Isabelle. the applying of this framework to the mechanization of basic genuine research utilizing nonstandard thoughts can be discussed.

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

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Q 41 set is isomorphic to the type Let us mention some of the terminology often encountered when filters and related concepts are used. A set X ~ S is sometimes said to be large [74J or quasi-big [44] if X E :F. Other terms used include residual or generic when dealing with directed sets or Baire category theory. ) or almost always, or is :F-true or almost true, if the set {x E S. P is satisfied at x} is a member of :F. 2. :F E Filters S ~ U = :F An ultrafilter is thus a maximal filter, that is a filter that cannot be enlarged.

Once a new type has been introduced successfully, Isabelle provides coercion functions - the abstraction and representation functi<)ns - that enable us to define basic operations on the new type. Thus, in this particular example, the functions Abs_prat :: (pnat * pnat) set => prat Rep_prat :: prat => (pnat * pnat) set are added to the theory such that prat is isomorphic to {x. hyprat and its inverse Abs_prat. Using these functions and other operations from Isabelle's Set and Equi v theories, we are now ready to define operations on the positive rationals.

Our approach is purely definitional to ensure that infinitesimals and other nonstandard numbers have a sound foundation in the system. To reach our goal has required constructing the various number systems leading to the reals and then going one step further to define the hyperreals by working on sequences of reals. The hyperreals have considerable intrinsic interest since they exhibit many new properties. Moreover, as a tool, they are of great value to the formalization of analysis - an aspect that will be described as we report on the mechanization of nonstandard real analysis.

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