An introduction to the history of mathematics
Calculus The history of the calculus and its conceptual development , Carl Boyer. Boyer spends a great deal of time discussing the earlier development of calculus, up to Newton and Leibniz, and only the very last section dealing with developments since then. However, you might want to read: The higher calculus: a history of real and complex analysis from Euler to Weierstrass , Umberto Bottazzini.
Just what it says. To continue the study, read: Lebesgue's theory of integration: its origins and development , Thomas Hawkins.
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A continuation of the study of the development of analysis. Modern Mathematics is a term by no means well defined. Algebra cannot be called modern, and yet the theory of equations has received some of its most important additions during the nineteenth century, while the theory of forms is a recent creation. Similarly with elementary geometry; the labors of Lobachevsky and Bolyai during the second quarter of the century threw a new light upon the whole sub ject, and more recently the study of the triangle has added another chapter to the theory. Thus the history of modern mathematics must also be the modern history of ancient branches, while sub jects which seem the product of late generations have root in other centuries than the present.
The seventeenth and eighteenth centuries laid the foundations of much of the sub ject as known to-day. The discovery of the analytic geometry by Descartes, the contributions to the theory of numbers by Fermat, to algebra by Harriot, to geometry and to mathematical physics by Pascal, and the discovery of the differential calculus by Newton and Leibniz, all contributed to make the seventeenth century memorable.
The eighteenth century was naturally one of great activity. Accompanying this activity, however, was a too implicit faith in the calculus and in the inherited principles of mathematics, which left the foundations insecure and necessitated their strengthening by the succeeding generation. The nineteenth century has been a period of intense study of first principles, of the recognition of necessary limitations of various branches, of a great spread of mathematical knowledge, and of the opening of extensive fields for applied mathematics.
Especially influential has been the establishment of scientific schools and journals and university chairs.
To the influence of the schools and the journals must be added that of the various learned societies3 whose published proceedings are widely known, together with the increasing liberality of such societies in the preparation of complete works of a monumental character.
The study of first principles, already mentioned, was a natural consequence of the reckless application of the new calculus and the Cartesian geometry during the eighteenth century. This development is seen in theorems relating to infinite series, in the fundamental principles of number, rational, irrational, and complex, and in the concepts of limit, contiunity, function, the infinite, and the infinitesimal. But the nineteenth century has done more than this. It has created new and extensive branches of an importance which promises much for pure and applied mathematics. A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject.
An Introduction to the History of Mathematics by Howard Eves
An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers.
A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject.
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Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers. Pythagoreans I. Parmenides and Zeno's Paradoxes I.