mathematics

The discovery of quaternions

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But on the 16th day of the same month – which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal…An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,
i2 = j2 = k2 = ijk = -1

Letter dated August 5, 1865 from Sir W. R. Hamilton to Rev. Archibald H. Hamilton.

The classic account of the discover of quaternions. Hamilton also notes in this letter that the Council Books of the Academy record that he had obtained leave to read a paper on quaternions, which reading took place on the 13th November 1843.

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Ones and Zeros: the life and work of George Boole

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Mathematician and logician George Boole died 150 years ago today, on 8th December, 1864. Today also marks the start of the year-long schedule of events UCC are running to commemorate Boole, culminating in the bicentenary of his birth on 2nd November 2015 (see GeorgeBoole.com for more).

George Boole was born in Lincoln, the eldest son in a family of modest means. For details of his life as a self-taught mathematician to first professor in UCC (then Queens College Cork) in 1849, where he lived until his death see the detailed biography here.

Boole had a large impact on mathematics, providing the basis for invariant theory, and working on differential and difference equations, and probability. Developments of his work such as set theory and boolean algebra are taught to school children today.

However, of most interest philosophically are The Mathematical Analysis of Logic, and its successor An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities published in 1854. These proposed that ideas expressed in language can be expressed in algebraic form. This combination of philosophical logic and algebra, as DeMorgan said “would not have been believed until it was proved.”

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George Boole 200 (UCC)

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This video gives a brief introduction to the importance of George Boole, who will be the subject of a year of celebration in UCC next year, 2015. George Boole’s major achievement was Boolean Algebra, a major development in logic, which Frege later built on.

For Boole’s life and his contribution to the digital age, see this video Forgotten Genius: George Boole (YouTube), and for his place in philosophy Logic –The Structure of Reason Great Ideas of Philosophy (YouTube) (Boole is featured from 15:43 to 17:42). A biography of Boole is here.

Quaternion Bridge

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The quaternion plaque on Broombridge © Cliff Bilbrey on Flickr (CC BY-NC 2.0)
The quaternion plaque on Broombridge
© Cliff Bilbrey on Flickr (CC BY-NC 2.0)

This plaque commemorates the discovery of the quaternion formula (i2=j2=k2=ijk=-1) by Sir William Rowan Hamilton on 16th October 1843, as he walked along the canal from Dunsink on his way to a meeting in Dublin. Without paper to hand he scratched the formula into one of the bridge’s stones.

There is a commemorative Hamilton Walk organised by the Maynooth University Maths department every year. There is also a virtual version here.

For a Storify of Hamilton Day 2015 and the 25th Hamilton Walk, click here.