I hope my work will serve to bring justification to the synthetic method besides the analytical one." After long vacillations, I have decided to use a half synthetic, half analytic form. But I soon realized that, as expedient the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. I found these theories originally by synthetic considerations. “The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. The theory is also very much inspired by the thought process and work of Sophus Lie (who developed the theory of Lie algebras and Lie groups).
Lawvere's longterm goal has been to develop a more suitable mathematical language for physics, and synthetic differential geometry emerged from his categorical dynamics program. It is a theory with a very physical and geometric spirit that rigorously captures the way physicists work with infinitesimals. In case you haven't heard of it, synthetic differential geometry is a synthetic (as opposed to analytic) approach to calculus and differential geometry developed by Bill Lawvere, Anders Kock, and several other prominent category theorists which heavily relies on infinitesimals. We note several simple applications to the classical setting, such as bounds related to the flow of vector fields, analysis of small oscillations of a pendulum, and an instance of Frobenius’ Theorem regarding the complete integrability of independent vector fields.I recently found out that the synthetic differential geometry text by Anders Kock is freely available online. We study the properties of flows and Lie brackets, particularly in relation with our proposed regularity conditions. We define the Lie bracket of two prevector fields by appropriate iteration of their commutator. We define the flow of a prevector field by hyperfinite iteration of the given prevector field, in the spirit of Euler’s method. by Stroyan and Luxemburg or Lutz and Goze.
These conditions replace the more elaborate analytic regularity conditions appearing in previous similar approaches, e.g. We introduce regularity conditions for prevector fields, defined by finite differences, thus purely combinatorial conditions involving no analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from ∗M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. Keywords Algebraic basis, Hamel basis, algebraic dual, dual operator, linear partial differential operator, linear partial differential equation, solvability, existence of a solution, generalized solution, Schwartz distribution Borel-Cantelli lemma Erdős–Rényi Theorem Fourier transform Gaussian measures Kochen-Stone theorem Locally compact abelian group, Pontryagin–van Kampen duality, Fourier transform, nonstandard analysis, hyperfinite, infinitesimal, approximation MV-algebras Pontryagin-van Kampen duality Quantitative analysis R-trees, model theory, metric structures, continuous logic, real-valued structures, model completion, categoricity, stability Representation of integers, Internal set theory, Polynomial congruence Solovay reducibility, Lipschitz continuity, Hölder continuity, computable analysis Ultrafilters, Ultrapowers, Almost disjoint families, Banach spaces, epsilon-separated sets, Measurable cardinals factoring of integers, nonstandard analysis, unlimited integers, Dickson’s conjecture hyperfinite infinitesimal approximation local algebras locally compact abelian group metastability nonstandard analysis
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