History of MathematicsDevelopments in Scotland and Germany

Peter Gibson

November 19, 2019

Two twentieth century developments...

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Two twentieth century developments...

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Question: how did these things come about?

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The Scottish Enlightenment

A period of intellectual flowering in Scotland in the 18th century.

1707 Act of Unification (creating Great Britain)

Scotland had suffered a financial collapse and was poor

Following union with England, there was established a system of publicschooling, and a revitalization of the Universities. As in Europe,intellectual culture in Scotland flourished.

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Figures of 18th century Scotland

David Hume (1711-1776)I A Treatise of Human NatureI Dialogues Concerning Natural Religion

Adam Smith (1723-1790)

I The Wealth of Nations

Robert Burns (1759-1796)

John Playfair (1748-1819)

James Hutton (1726-1797)

This set the stage for later developments...

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Figures of 18th century Scotland

David Hume (1711-1776)I A Treatise of Human NatureI Dialogues Concerning Natural Religion

Adam Smith (1723-1790)I The Wealth of Nations

Robert Burns (1759-1796)

John Playfair (1748-1819)

James Hutton (1726-1797)

This set the stage for later developments...

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Figures of 18th century Scotland

David Hume (1711-1776)I A Treatise of Human NatureI Dialogues Concerning Natural Religion

Adam Smith (1723-1790)I The Wealth of Nations

Robert Burns (1759-1796)

John Playfair (1748-1819)

James Hutton (1726-1797)

This set the stage for later developments...

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Figures of 18th century Scotland

David Hume (1711-1776)I A Treatise of Human NatureI Dialogues Concerning Natural Religion

Adam Smith (1723-1790)I The Wealth of Nations

Robert Burns (1759-1796)

John Playfair (1748-1819)

James Hutton (1726-1797)

This set the stage for later developments...

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Figures of 18th century Scotland

David Hume (1711-1776)I A Treatise of Human NatureI Dialogues Concerning Natural Religion

Adam Smith (1723-1790)I The Wealth of Nations

Robert Burns (1759-1796)

John Playfair (1748-1819)

James Hutton (1726-1797)

This set the stage for later developments...

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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)

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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

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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

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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.

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Germany

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

the University of Gottingen (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.

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

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

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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)

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

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

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Riemann acquired his doctorate under Gauss, having studied previously atboth Gottingen 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.

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In summary, the enlightened culture of eighteenth century Scotland andGermany (and France) led to innovations in the schooling and Universitysystems that had enormous impact.

They set the stage for later generations of 19th mathematicians (andscientists) whose work had enormous long range impact. The modernworld as we know it is partly a result.

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