18 citations · 50 across the 10 of their papers we have counts for
10 papers
Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimals
Jacques Bair, Piotr Blaszczyk, Peter Heinig +3
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy app…
Cauchy, infinitesimals and ghosts of departed quantifiers
Jacques Bair, Piotr Blaszczyk, Robert Ely +10
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for t…
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Tiziana Bascelli, Piotr Blaszczyk, Alexandre Borovik +7
Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and…
Interpreting the infinitesimal mathematics of Leibniz and Euler
Jacques Bair, Piotr Blaszczyk, Robert Ely +10
We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations…
Nonstandard techniques and nowhere differentiable functions I: A dense family of generalized blancmange functions
Tom McGaffey
We will give an elementary nonstandard proof that the family of generalized blancmange functions are nowhere differentiable. The proof follows from the intuitive characterization o…
Is mathematical history written by the victors?
Jacques Bair, Piotr Blaszczyk, Robert Ely +9
We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leib…
Magnification Spaces: A nonstandard approach to inverse mapping theorems
Tom McGaffey
This paper develops an infinitesimal order of magnitude coupled with overflow technique that allows nonnumerical proofs of nondegenerate and degenerate inverse mapping theorems for…
Regularity and nearness theorems for families of local Lie groups
Tom McGaffey
In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate d…
Topologizing Rings of Map Germs: An Order Theoretic Analysis of Germs via Nonstandard Methods
Tom McGaffey
Using nonstandard analysis we define a topology on the ring of germs of functions: . We prove that this topology is absolutely convex, Hausdo…
Generalized solutions of nonlinear differential equations A nonstandard jets approach
Tom McGaffey
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential…