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200
Archimedes
For a set to fulfill the Archimedean property, it must contain no infinitely large or small quantities. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs. -
435
Zeno: Achilles and Arrow Paradoxes
Achilles
Zeno's Paradoxes point to the need for a continuum, as treatments of numbers at the time led to counter-intuitive mathematics. -
Leibniz
"We have to make an effort in order to keep pure mathematics chaste from metaphysical controversies... we use infinites and infinitely smalls as an appropriate expression for abbreviating reasonings." (Guicciardini, 1738).
"You are right in saying that all magnitudes may be infinitely subdivided... I conceive no physical indivisibles short of a miracle, and I believe nature can reduce bodies to the smallness Geometry can consider" (Reyes, 173). -
Newton
Contradiction: Newton found fluxions to make intuitive sense for physics and astronomy, but also recognized that there is no way to define and defend a rigorous definition of infinitesimals.
In 1680, he published Geometria curvilinea, in which infinitesimals were not used. 'Method of first ratios of nascent quantities and last ratios of vanishing quantities.' "Placed a layer of mathematics [and ambiguity of language] between Newton's Calculus and his use of infinitesimals" (Reyes, 168). -
Naysayers
"How can one talk of dividing what were referred to (by Newton at least) as indivisibles outside the realm of paradox and absurdity?" (Reyes 166). "Infinitesimals belonged to the realm of theology and faith and not that of mathematics, which of necessity occurs within the secular boundaries of finite human reason" (Reyes, 166). -
Cauchy and Gauss
Saw the infinitesimal as something whose limit was zero. Brought rigor to calculus, helping to abandon the infinitesimal. Though they had no ontological objection to the idea, they saw limits as the future of the Calculus. -
Cantor
The Continuum Hypothesis, yet no committment to infinitesimals. Perhaps supporting the CH, a daunting task, was enough that Cantor did not feel that putting his weight behind infinitesimals, another unproven idea, would be in his best interest. -
Charles Dodgson (Lewis Carroll)
"Dodgson's basic idea of the number system was an extension of the notion of "denseness" to the existence of an extended real number system that would include infinitesimal and infinite numbers obeying the same laws as ordinary real numbers." (Abeles, 14)
"What the Tortoise said to Achilles" reasons that Cantor's reasoning that there can be multiple levels of infinity, but no infinitesimals, is a flawed logic.
Links necessities of non-Euclidean Geometry with a non-Archimedean number system. -
Non Standard Analysis (NSA)
The Difference between Reals and HyperrealsRobinson Abraham Robinson connected the original formulation of the infinitessimal, as posited best by Leibniz, with a system for computing and working with infinitesimals.
