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HISTORY OF THE INFINITESIMAL The first Mathematician to make use of infinitesimals was Archimedes , although he did not believe in their existence. See the article on How Archimedes Used Infinitesimals . The Archimedean Property is the property of an ordered Algebraic Structure of having no nonzero infinitesimals. In India from the 12th Century until the 16th Century , infinitesimals were discovered for use with Differential Calculus by Indian Mathematician Bhaskara and various Keralese Mathematicians . When Newton and Leibniz developed the Calculus , they made use of infinitesimals. A typical argument might go: ::To find the Derivative ''f'' ::since d''x'' is infinitesimally small. This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work '' The Analyst : or a discourse addressed to an infidel mathematician''. The fundamental problem is that d''x'' is first treated as non-zero (because we divide by it), but later discarded as if it were zero. It was not until the second half of the Nineteenth Century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a Limit . In the 20th century, it was found that infinitesimals could after all be treated rigorously. Neither formulation is right or wrong, and both give the same results if used correctly. MODERN USES OF INFINITESIMALS Infinitesimals are legitimate quantities in the , and the standard part of ''x'' + d''x'' is ''x''. Alternatively, we can have Synthetic Differential Geometry or Smooth Infinitesimal Analysis with its roots in Category Theory . This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., ''not'' (''a'' ≠ ''b'') does not have to mean ''a'' = ''b''. A ''nilsquare'' or '' Nilpotent '' infinitesimal can then be defined. This is a number ''x'' where ''x'' &2 = 0 is true, but ''x'' ≠ 0 can also be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above. SEE ALSO |
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