雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(103)
An abstract in French for the scientific journal Comptes Rendus. Mrs Turing helped with the French and the typing.
The lambda-calculus represented an elegant and powerful symbolism for mathematical processes of abstraction and generalisation.
λ算子能夠非常簡潔有力地對數學過程進行抽象和泛化。The 'complex' number calculus exemplified the progress of mathematical abstraction.
複數是數學抽象化的又一個進展。
Originally, complex numbers had been introduced to combine 'real' numbers with the 'imaginary' square root of minus one, and mathematicians had agonised over the question of whether such things really 'existed'.
最初人們引入複數，將實數與虛數（比如-1的平方根）結合起來的時候，數學家們感到非常糾結，不知道這樣的東西是否真的存在。From the modern point of view, however, complex numbers were simply defined abstractly as pairs of numbers, and pictured as points in a plane.
從現代的觀點來看，可以簡單地把一個複數看成一個數對，它可以形象地畫在平面座標系上，A simple rule for the definition of the 'multiplication' of two such pairs was then sufficient to generate an enormous theory.
兩個數對之間有一套簡單的乘法規則，這樣就可以產生很強大的理論。Riemann's work in the nineteenth century had played a large part in its 'pure' development; but it was also found to be of great usefulness in the development of physical theory.
在19世紀以來，黎曼的工作主要是在純數學領域發揮作用，但是人們後來發現，它們在物理領域也有很多用處。Fourier analysis, treating the theory of vibrations, was an example of this.
傅利葉分析就是一個例子。The quantum theory developed since the 1920s went even further in according complex numbers a place in fundamental physical concepts.
20年代以來的量子理論，更加深入地應用了複數的概念。None of these mathematical ideas are essential to what follows, although such connections between 'pure' and 'applied' were certainly relevant to a number of aspects of Alan Turing's later work.
這些數學概念，對接下來的故事來說並不重要。不過，這種純數學和現實應用之間的關聯，倒是和艾倫·圖靈後來的工作很有關係。
