Raymond Flood - Shaping Modern Mathematics: The 19th Century

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source: GreshamCollege           2012年10月1日
The 19th Century saw the development of a mathematics profession with people earning their living from teaching, examining and researching and with the mathematical centre of gravity moving from France to Germany. A lot of the mathematics taught at university today was initiated at that time. Whereas in the 18th Century one would use the term mathematician, by the end of the 19th Century one had specialists in analysis, algebra, geometry, number theory, probability and statistics, and applied mathematics. This series of free public lectures looks at the shaping of each of these mathematical areas and at the people who were involved.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and...
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1. 56:31 Ghosts of Departed Quantities: Calculus and its Limits
In 1734 Bishop Berkeley published a witty and effective attack on the foundations of the calculus as developed by Newton and Leibniz.  But it took nearly 90 years for the calculus to be given a rigorous foundation through the work of the prolific mathematician, Augustin-Louis Cauchy, who formalised the concept of a limit and created the specialism now called analysis.
2. 54:14 Polynomials and their Roots
We are familiar with the formula for solving a quadratic equation where the highest power of the unknown is a square.  The quest for a similar formula for equations where the highest power is three, four five or more led to dramatic changes in how this question was regarded. Powerful techniques in algebra were developed following work by Abel and Galois in the 19th century to show that there is no such formula when there are powers higher than four.
3. 1:02:49 From One to Many Geometries
For 100 years up to the end of the 19th century the study of geometry was completely changed with the development of non-Euclidean geometries and the use of techniques to think of geometries in higher dimensions - a development essential to Einstein in his development of the theory of General Relativity.
4. 1:00:20 The Queen of Mathematics
Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." The properties of primes play a crucial part in number theory.  An intriguing question is how they are distributed among the other integers.  The 19th century saw progress in answering this question with the proof of the Prime Number Theorem although it also saw Bernhard Riemann posing what many think to be the greatest unsolved problem in mathematics - the Rieman Hypothesis.
5. 59:44 Are Averages Typical?
 Not necessarily, for example the average person has fewer than two legs!  This is because some people have fewer than two legs but nobody has more than two, so dividing the total number of legs by the total number of people to get the average gives a number less than two.  Average does not mean typical!  The lecture will examine how the work in the 19th century of such mathematicians as Florence Nightingale, Adolphe Queteller and Karl Pearson on describing and quantifying variation and uncertainty laid the foundations for the theory of statistics as a mathematical discipline.
6. 57:03 Modelling the World
An area of which 19th century British mathematics could be uniformly proud was applied mathematics where new techniques were used on a wide range of problems.  Figures such as William Thompson (later Lord Kelvin), Peter Guthrie Tait, George Stokes and James Clerk Maxwell succeeded in applying mathematics to understanding the physical world.  They worked on many topics including mechanics, thermodynamics, electricity and magnetism, hydrodynamics and the theory of gases.  This lecture will introduce and discuss some of their influential achievements.

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