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Mathematics and the Scientific Revolution

After the decline of Greek mathematics, and a period in which religious authority stultified intellectual creativity, the renaissance opened a new era which lead to the revitalisation of science and mathematics.

Descartes' philosophical methodology, involving scepticism (rejection of authority) and rationalism (an assertion of self confidence) shows the shift of mood. Though Descartes' method is deductive, he rejects (Aristotelian) logic as sterile.

Gallileo provided a new view of science, emphasising descriptive and quantitative explanation. Science for Gallileo involves the derivation of a body of knowledge from a small number of mathematically formulated physical principles.

This new science, most clearly illustrated in Newton's mechanics, provided the stimulus for major developments in mathematics in the 17th century and the rapid development of analysis through the 18th.

Co-ordinate Geometry
Co-ordinate geometry was introduced by Fermat and Descartes, ignoring rather than solving the foundational problems which had prevented the Greeks from taking this step (viz: the lack of any well understood number system which could account for incommensurable ratios).

This development is important to science because it makes geometry quantitative and permits the use of algebraic methods. Geometry must be quantitative for it to be useful in science and engineering, and algebraic methods permit more rapid development of mathematics than the less systematic (if more rigorous) methods required by the Greek axiomatic approach to geometry.

The Calculus
Developed independently by Newton and Leibniz, the calculus is the foundation on which a large part of the mathematics required for science is built. The development of the calculus depended upon a number system which not only includes irrational (numbers necessitated by incomensurable ratios), but even infinitesimal numbers, which sometimes behave like zero and sometimes don't.

Objections, particularly to the latter, failed to halt the onward sweep of mathematics in the service of science.

Analysis
During the 18th century the calculus was further developed and other branches of analysis were rapidly opened up. These developments lacked logical rigour and were guided by intuitive and physical insights.

Though the lack of rigour was noted, for example by Bishop Berkeley, and many attempts were made to put things right, this did not inhibit the development of the subject. It was not until well into the 19th century that rigorous foundations for this plethora of new mathematics could be discovered.

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the particular to the general, and it is extremely peculiar that such a procedure does not lead to more of the so-called paradoxes.
Abel, 1826
The Formalisation of Mathematics

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