The Annotated Turing

On the forty-second page of “The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine” author Charles Petzold wrote (some emphasis added):

was not used at the time, at least not in its modern sense. The modern usage of the word only became common in the 1960s in literature about computers.¹³

In that 1900 address, Hilbert invited his audience to "lift the veil" behind which the twentieth century lay hidden. Neither he nor anyone else could have imagined quite the spectacle they would have seen. If physicists believed they were on the verge of total knowledge, those hopes were dashed in 1905, the year now known as the annus mirabilis of physicist Albert Einsten (1879–1955). In a single year, Einstein published a doctoral thesis and four other papers that established the basic principles of relativity and quantum mechanics.

No longer was there any sense that the universe was linear, Euclidean, and fully deterministic. Space and time lost their moorings in a relativistic universe. In a famous 1907 paper on relativity, Hilbert's friend Hermann Minkowski would coin the word Zaumreit or spacetime. (Minkowski had come to G&oumalt;ttigen in 1902, but died suddenly of appendicitis in 1909.) Eventually, the century's best known result of quantuum mechanics would be something known as the Uncertainty Principle (1927).

Perhaps in response to this new displacement and uncertainty, modern art and music went in startling and provocative directions. Visual forms and objects were broken apart and reassembled in cubist paintings and sculptures. As the real-word "objective" universe became less reliable, surrealists looked inward to their subconscious lives and irrational dreams.

In music, the chromaticism of late romantics like Wagner and Debussy seemed almost tame as it gave way to the harsh dissonances and jagged new rhythms of Igor Stravinsky's Rite of Spring, which incited riots at its 1913 Paris premiere. In the early 1920s, Austrian composer Arnold Schöat;nberg's development of twelve-tone music represented nothing less than a re-axiomatization of the principles of musical harmony to create a non-Euclidean music.

Twentieth-century mathematics was not immune to these upsets. The first jarring notes sounded in 1902.

Gottlob Frege, born in Wismar, Germany, in 1848, had received his Ph.D. at Göat;ttingen two decaes before Hilbert arrived there, and then began teaching at the University of Jena, where he would remain for 44 years. The first volume of his life's work, the Grundgesetze der Arithmetik, was published in 1893, and attempted a systematic development of all of mathematics beginning with mathematical logic—a program now known as logicism. This first volume sold so poorly that the publisher didn't want to hear about the second volume, so in 1902 Frege was attempting to publish it at his own expense.

¹³See Oxford English Dictionary, 2nd edition. I, 313. Also see the opening pages of Donald E. Knuth, The Art of Computer Programming, Volume I, Fundamental Algorithms, 3rd edition (Addison-Wesley, 1997). The most famous algorithm of them all is Euclid's method to find the greatest common divisor of two numbers, but the first known usage of the term "Euclid's Algorithm" seem to date only from the early twentieth century.

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(John Wiley & Sons, June 2008. Paperback, 372 pages. ISBN: 0470229055; EAN: 9780470229057.)