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 usually been seen 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 tools from geometry and nonstandard research. The mechanization of the tactics, 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 hassle-free actual research utilizing nonstandard ideas can also be discussed.
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Additional resources for A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia
Below, we briefly mention how our formalization in Isabelle/HOL compares with the one in Isabelle/ZF. The definitions used by Abrial and Laffitte require the choice operator since starting from AC, they prove Hausdorff's Maximal Principle and then derive Zorn's Lemma. Unlike its ZF counterpart, Isabelle/HOL provides such an operator, the so-called Hilbert description operator, c. Thus, the formulation of the various theorems in Isabelle/HOL is somewhat simpler than that given by Paulson for ZF.
1. 4 Other Geometric Properties The other main geometric notions include those of similar and congruent triangles. We look at the second notion more closely. Intuitively, two triangles are congruent if one figure can be moved without changing its size or shape, so as to coincide with the other. There are two possible approaches for the mathematical treatment of congruence. 4 Formalizing Geometry in Isabelle 29 to describe its essential properties, and then prove the theorems that follow. However, congruence can also be treated definitionally in terms of lengths (distance) and angles.
Thus, UFT is strictly weaker than AC, and WUF is weaker still. To prove WUF, we show that the set of naturals is not finite by an inductive proof and then discharge the premise of the UFT corollary. This ultrafilter need not be explicitly defined: it does not matter which ultrafilter on IN is used. The set of all free ultrafilters on IN determines a set of isomorphic fields from which we can choose any member to be the set of hyperreal numbers. Thus, in our formalization, we use Hilbert's €-operator to define UIN: UIN == (cU.
A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia by Jacques Fleuriot PhD, MEng (auth.)