![]() Archimedes discovered infinitesimals and used them to obtain many results in geometry, but his published proofs did not mention infinitesimals, and his letter to a friend, in which he explained infinitesimals, was lost until recently and so infinitesimals had to be rediscovered in Europe (much later!) and were then used to calculate areas and volumes. Much before: Eudoxus created the method of exhaustion which can be used to prove some things that today are proved via the theory of limits. Note: It's very interesting that Newton developed many "versions" of calculus: one time he even sought to divorce it from algebra and create a "geometric calculus" more powerful than that of Barrow, by using things like similar triangles and "ultimate ratios", but he also had one more "algebraic" with relied more on polynomials and series, and he experimented many ways of presenting his calculus from "the start". In this sense, Newton discovered/created calculus. ![]() Newton, being very comfortable with algebra and analytic geometry, after having learned the "geometric calculus" of Barrow soon turned it into a machine of solving problems. Barrow discovered the fundamental theorem of calculus, but he did not know much of algebra and analytic geometry. Barrow was very interested in chronology and because of that he studied kinematics, optics and mathematics. Newton was tutored by Barrow, who was a very competent classical geometer (i.e. Then Leibniz discovered/created calculus. Huygens accepted to tutor him and taught him analytic geometry and presented texts of Pascal and Fermat to him. Leibniz easily solved the problem proposed. Huygens tested him with a problem of finding the sum of a given sequence. He found Huygens and told him about his discoveries and asked to be tutored. Leibniz started with sequences of numbers, and after discovering some results about them he sought a tutor in mathematics. So, please make corrections in commentaries/edits when necessary.įrom what I've read, it was more or less like this:Ĭalculus was discovered/created independently by Newton and Leibniz. ![]() I'm not a historian nor have studied these things deeply. For this reason it is more meaningful to acknowledge the syntactic contribution of the pioneers of the calculus than to apply the essentially anachronistic straitjacket of "mathematical rigor" in analyzing their work. This observation is sometimes expressed by claiming that the pioneers of the calculus were not "rigorous" but this I believe is a misleading statement: Gauss and Dirichlet similarly did not have access to modern sematic theories and in this sense would also not be "rigorous", but in fact there are almost no errors in their work. The semantic side of the story did not emerge until much later. They developed the calculus in the sense that they understood its procedures and syntax, which remain today largely as they have been developed by Leibniz, Euler, Lagrange, and others. To answer your question, I would say that the contributions of Leibniz and others would today be viewed as centering on the syntactic side of the mathematical endeavor. These semantic issues only began to emerge starting around 1870, and resulted in the modern foundations for analysis in particular and much of modern mathematics in general. (2) On the other hand, we have the issue of the ontology of mathematical objects, i.e., what are the basic objects mathematics uses, how one axiomatizes their behavior and justifies their "existence" relative to appropriate foundational theories. These procedures can be characterized as syntactic material because at this stage in the development of mathematics, its practitioners were not overly concerned with semantic issues (see below). (1) On the one hand, we have the procedures employed by Leibniz and other pioneers of the calculus, and their procedural moves in deriving various new results such as Leibniz's derivation of the product formula for differentiation, and Euler's various solutions of the Basel problem, to give some elementary examples. The distinction is between the following two items. ![]() To understand the relation between Leibniz's theory, on the one hand, and modern ϵ-δ definitions and constructions of the continuum, on the other, it is helpful to make the following distinction.
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