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Notes

1. Gillies starts with a brief account of three political revolutions which are important models for subsequent discussion of the possibility of revolutions in mathematics:-
   • The English 17th-century revolution (deposition of Charles I followed by a dictatorship and subsequently, after the restoration of Charles II, by the “Glorious Revolution”, the replacement of James II by William and Mary)
   • The French revolution – similar deposition and execution of the King, followed by a dictatorship and subsequent restoration of the monarchy with attenuated powers (and with a similar epilogue, where a less suitable monarch – Charles X – is replaced by a more suitable one – Louis Philippe)
   • The Russian revolution where, while there is deposition and execution of the Czar, there is no subsequent restoration of the monarch.
2. Kuhn’s The Structure of Scientific Revolutions has been substantially accepted, with modifications since 1962. Kuhn’s theory is that science continues in stasis in between revolutions where the dominant paradigm is replaced by a new one. Again, three examples are given :-
   1. Copernican: convoluted path from Aristotle-Ptolemy to Newton
   2. Chemical: explanation of combustion as loss of phlogiston replaced by an explanation involving addition of oxygen
   3. Einsteinian: replacement of Newtonian mechanics by the theory of relativity