math 5400, history of mathematicspcgibson/math5400/math5400files/...2013/12/02 · mathematics in...

MATH 5400, History of MathematicsLecture 15: 19th Century Developments in Scotland and Germany

Professor: Peter Gibson

February 12, 2013

Two twentieth century developments...

P. Gibson (YorkU) Math 5400 12.2.2013 2 / 17

Question: how did these things come about?


Scotland

Recall that a period of intellectual flowering in Scotland in the 18thcentury helped to set the stage for later developments...


James Clerk Maxwell

Three great mathematical physicists came through the revitalized Scottisheducational system:

James Clerk Maxwell (1831-1879)

Peter Guthrie Tait (1831-1901)

William Thomson, Lord Kelvin (1824-1907)


Maxwell went on to become professor at Aberdeen University and King’sCollege London. He made important contributions to:

statistical mechanics

the theory of colour

electromagnetism

His masterpiece was A Treatise on Electricity and Magnetism


Maxwell’s equations

∇ · E = ρ∇ · B = 0

∇× E + ∂B∂t

= 0

∇× B− ∂E∂t

= 0


Maxwell’s equations allowed the prediction of heretofore unknownelectromagnetic phenomena, such as radio waves and microwaves.

This led not only to the development and implementation of electricpower, but ultimately to technologies such as microwave ovens thatemerged only later.


Germany

Two universities (among many others) that played a central role formathematics in the 19th century are:

the University of Götingen (est. 1737)

the University of Berlin (est. 1810)

The University of Berlin was founded by Wilhelm von Humboldt, a highlyregarded Minister of State, philosopher and linguist.


The University of Berlin emphasized “Lehrfreiheit” and “Lernfreiheit”. Inaddition, it was easy for students to move between universities.

The University of Göttingen was founded by the George II, King ofEngland and Elector of Hanover (whose father was the employer of Leibnizat the end of his career).


There was a constellation of prominent mathematicians working inGermany in the 19th century, including:

Gauss (1777-1855)

Carl Jacobi (1804-1851)

Peter Lejeune-Dirichlet (1805-1859)

Karl Weierstrass (1815-1897)

Leopold Kronecker (1823-1891)

Ferdinand Eisenstein (1823-1852)

Richard Dedekind (1831-1916)


The genealogy of modern research mathematics can be traced back to thisday.


One of the most important mathematicians of this era was Georg FriedrichBernhard Riemann (1826-1866).


Riemann acquired his doctorate under Gauss, having studied previously atboth Göttingen and Berlin. He made fundamental contributions to manybranches of mathematics, and profoundly influenced its future direction.

Riemann helped to develop:

complex analysis

number theory

differential equations

geometry

For example, Riemannian geometry is an essential ingredient to GeneralRelativity.


Some questions

What aspects of their respective institutional environments, facilitatedRiemann’s and Clerk Maxwell’s education?

What was the significance of their individual contributions?

What later developments attest to the predicitive power of Riemann’sand Clerk Maxwell’s insights?

What new horizons of inquiry, what new and important scientificquestions were raised by their work?


math 5400, history of mathematicspcgibson/math5400/math5400files/...2013/12/02 · mathematics in...

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