Is God a Mathematician_ - Mario Livio [127]
the proofs of the first twenty-eight: Theorems proven without the fifth postulate are discussed in Trudeau 1987.
Some of those endeavors started: An excellent description of all the attempts that had eventually led to the development of non-Euclidean geometry can be found in Bonola 1955.
The first to publish an entire treatise: George Bruce Halsted’s 1891 translation of Lobachevsky’s “Geometrical Researches on the Theory of Parallels” is included in Bonola 1955.
a young Hungarian mathematician, János Bolyai: For a biography and a description of his work, see Gray 2004. The reason I have not included a picture of János Bolyai is that the picture usually used is of doubtful authenticity. Apparently his only relatively reliable portrait is a relief in the façade of the Palace of Culture in Marosvásárhely.
The manuscript was entitled The Science Absolute of Space: A facsimile of the original (in Latin) and the translation into English by George Bruce Halsted appear in Gray 2004.
There is very little doubt, however: An excellent description of the entire episode, from the perspective of Gauss’s life and work, can be found in Dunnington 1955. A concise but accurate summary of the claims of Lobachevsky and Bolyai for priority is given in Kline 1972. ome of Gauss’s correspondence on non-Euclidean geometry is presented in Ewald 1996.
In a brilliant lecture delivered in Göttingen: An English translation of the lecture, as well as other seminal papers on non-Euclidean geometries, together with illuminating notes, can be found in Pesic 2007.
Poincaré’s views were inspired: Poincaré 1891.
in the first chapter of the Ars Magna: Cardano 1545.
In another important book, Treatise of Algebra: Wallis 1685. A concise summary of Wallis’s biography and work can be found in Rouse Ball 1908.
Opinions eventually started to change: A brief summary of the history is given in Cajori 1926.
In an article entitled “Dimension”: This article appeared in Diderot’s Encyclopédie. Quoted in Archibald 1914.
stating more assertively in 1797: Lagrange 1797.
Grassmann, one of twelve children: An excellent biography and description of Grassmann’s work (in German) can be found in Petsche 2006. A good brief summary can be found in O’Connor and Robertson 2005.
It is fascinating to follow: Relatively accessible (but still technical) descriptions of his work in linear algebra can be found in Fearnley-Sander 1979 and 1982.
By the 1860s n-dimensional geometry: A good introductory text is Sommerville 1929.
by the following “declaration of independence”: The text appears in Ewald 1996.
To which algebraist Richard Dedekind: The text appears in Ewald 1996.
Here is how the French mathematician: Stieltjer’s first letter to Hermite was dated November 8, 1882. The correspondence between the two mathematicians consists of 432 letters. The full correspondence appears in Hermite 1905. I translated the text that appears here.
“Mathematicians have constructed a very large”: The lecture can be found in O’Connor and Robertson 2007.
Chapter 7. Logicians: Thinking About Reasoning
The sign outside a barber shop: The paradox of the village barber is discussed in many books. See Quine 1966, Rescher 2001, and Sorensen 2003, for example.
Here is how Russell himself described: Russell 1919. This was Russell’s more popular exposition of his ideas in logic.
For completeness, I should note: Brouwer’s intuitionist program is summarized nicely by van Stegt 1998. An excellent popular exposition is by Barrow 1992. The debate between formalism and intuitionism is popularly described in Hellman 2006.
“the meaning of a mathematical statement”: Dummett adds that “an individual cannot communicate what he cannot be observed to communicate: if an individual associated with a mathematical symbol or formula some mental content, where the association did not lie in the use he made of the symbol or formula, then he could not convey that content by means of the symbol or formula, for his audience would be unaware of the association