Infinitesimal calculus

Gottfried Wilhelm Leibniz (left) and Isaac Newton (right), developers of infinitesimal calculus

Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s. John Wallis exploited an infinitesimal he denoted $\tfrac{1}{\infty}$ in area calculations, preparing the ground for integral calculus. They drew on the work of such mathematicians as Isaac Barrow and René Descartes. Infinitesimal calculus consists of differential calculus and integral calculus, respectively used for the techniques of differentiation and integration.

Newton sought to remove the use of infinitesimals from his fluxional calculus, preferring to talk of velocities as in "For by the ultimate velocity is meant ... the ultimate ratio of evanescent quantities". Leibniz embraced the concept fully calling differentials "...an evanescent quantity which yet retains the character of that which is disappearing", and his notation for them is the current symbolism in calculus.

In early calculus the use of infinitesimal quantities was unrigorous and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley described infinitesimals in his book The Analyst in 1734.

Several mathematicians, including Maclaurin and d'Alembert, attempted to prove the soundness of using limits, but it would be 150 years later, through the work of Augustin Louis Cauchy and Karl Weierstrass, where a means was finally found to avoid mere "notions" of infinitely small quantities, that the foundations of differential and integral calculus were made firm. Cauchy developed a versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. Weierstrass formalized the concept of limit and eliminated infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities.

This approach formalized by Weierstrass came to be known as the standard calculus. Informally, the expression "infinitesimal calculus" became commonly used to refer to Weierstrass' approach but has become something of a dead metaphor.

Modern infinitesimals

Main article: Hyperreal number

After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach, called non-standard analysis, uses technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus.

Varieties

• Differential and integral calculus: together, the original infinitesimal calculus, attributed to Newton and Leibniz.
• Standard calculus (based on the approach of Weierstrass)
• Non-standard calculus (based on Robinson's approach to infinitesimals)
• Smooth infinitesimal analysis (based on the approach by Lawvere)

References

• Baron, Margaret E.: The origins of the infinitesimal calculus. Dover Publications, Inc., New York, 1987.
• Baron, Margaret E.: The origins of the infinitesimal calculus. Pergamon Press, Oxford-Edinburgh-New York 1969. (A new edition of Baron's book appeared in 2004)
• Lavendhomme, R.: Basic concepts of synthetic differential geometry, Kluwer, Dordrecht, 1996
• O'Connor, Michael: An Introduction to Smooth Infinitesimal Analysis