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Before the nineteenth century none of the mathematical concepts as we know them today were formally defined, but many of these concepts were already there. The founders of calculus, Leibniz, Newton, Euler, Lagrange, the Bernoullis and many others, used infinitesimals in the way shown below and achieved essentially correct results even though no formal definition was available (similarly, there was no formal definition of real numbers at the time). HISTORY OF THE INFINITESIMAL The first Mathematician to Make Use Of Infinitesimals was Archimedes (c. 250 BC )Archimedes, ''The Method of Mechanical Theorems'', see the Archimedes Palimpsest , although he did not believe in the existence of physical 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 Calculus , they made use of infinitesimals. A typical argument might go: ::To find the Derivative ''f′''(''x'') of the Function ''f''(''x'') = ''x''2, let d''x'' be an infinitesimal. Then, ::since d''x'' is infinitely 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 ''.George Berkeley, ''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. When we consider numbers, the naive definition is clearly flawed: an infinitesimal is a number whose modulus is less than any non-zero positive number. Considering positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. If ''h'' is such a number, then what is ''h''/2? Or if ''h'' is indivisible, is it still a number? Also, intuitively, one would require that the reciprocal of an infinitesimal is infinitely large (in modulus) or unlimited, but this would make it the greatest number when clearly, there is no "last" biggest number. 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 wrong, and both give the same results if used correctly. MODERN USES OF INFINITESIMALS See Also: Non-standard analysis Smooth infinitesimal analysis Infinitesimal is necessarily a relative concept. If epsilon is infinitesimal with respect to a class of numbers it means that epsilon cannot belong to that class. This is the crucial point: infinitesimal must necessarily mean infinitesimal ''with respect'' to some other type of numbers. The path to formalisation Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the Model and which collection of Axiom s are used. We consider here systems where infinitesimals can be shown to exist. In 1936 (a number system) in which this will be true. The question is: what is this model? What are its properties? Is there only one such model? There are in fact many ways to construct such a One-dimensional Linearly Ordered set of numbers, but fundamentally, there are two different approaches: : 1) Extend the number system so that it contains more numbers than the real numbers. : 2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers. In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the Hyperreal s and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called Nonstandard . In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number which is less, in absolute value, than any positive standard real number. In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels i.e, in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level and there are also infinitesimals with respect to this new level and so on. All of these approaches are mathematically rigorous. This allows for a definition of infinitesimals which refers to these approaches: A definition :An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number. What standard and nonstandard refer to depends on the chosen context. 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 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 need not be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above. REFERENCES
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