Abstract
The foundational work of Alan Turing and contemporaries on computability marked a turning point in the development of mathematical sciences: It clarified in a rather absolute sense what is computable in the setting of symbolic computation, and it also opened the way to computer science where the use of algorithms and the discussion on their nature was enriched by many new facets. The present essay is an attempt to address both aspects: We review the historical development of the concept of algorithm up to Turing, emphasizing the essential role that logic played in this context, and we discuss the subsequent widening of understanding of “algorithm” and related “machines”, much in the spirit of Turing whose visions we see realized today.
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Notes
- 1.
On the occasion of the Alan Turing Year 2012, a new presentation of Turing’s work, including hitherto unpublished papers, is available in the volume [5].
- 2.
Among the more comprehensive sources we mention [6].
- 3.
For an interesting account on the “House of Wisdom”, we recommend [11].
- 4.
For a more detailed summary of Al Khwarizmi’s life see, e.g., [27].
- 5.
- 6.
Leibniz wrote this letter [13] of 1671 to a duke and not to a colleague; hence he used German rather than Latin, with some Latin words inserted: “In Philosophia habe ich ein Mittel funden, dasjenige was Cartesius und andere per Algebram et Analysin in Arithmetica et Geometria gethan, in allen scientien zuwege zu bringen per Artem Combinatoriam […]. Dadurch alle Notiones compositae der ganzen welt in wenig simplices als deren Alphabet reduciret, und aus solches alphabets combination wiederumb alle dinge, samt ihren theorematibus, und was nur von ihnen zu inventiren müglich, ordinata methodo, mit der zeit zu finden, ein weg gebahnet wird.”
- 7.
“Verbi gratia quia Homo est Animal rationale (et quia Aurum est metallum ponderosissimum) hinc si sit Animalis (metalii) numerus a ut 2 (m ut 3) Rationalis (ponderosissimi) vero numerus r ut 3 (p ut 5) erit numerus hominis seu h idem quot ar id est in hoc exemplo 2,3 seu 6 (et numerus auri solis s idem quot mp id est in hoc exemplo 3,5 seu 15.”
- 8.
From [14]: “Nam ubi semel a Missionariis haec lingua introduce poterit, religio vera quae maxime rationi consentanea est, stabilia erit et non magis in posterum metuanda erit Apostasia, quam ne hominess Arithmeticam et Geometriam, quam semel dedicere, mox damnent.”
- 9.
From [14]: “Aliquot selectos homines rem intra quinquennium absolvere posse puto; intra biennium autem doctrinas, magis in vita frequentalas, id est Moralem et Metaphysicam, irrefragabile calculo exhibebunt.”
- 10.
“Eine Lösung dieser Aufgabe im allgemeinsten Falle dürfte vielleicht mit unüberwindlichen Schwierigkeiten verbunden sein.”
- 11.
Cited from [10, p. 117].
- 12.
F. Rieger, Das Gesicht unserer Gegner von morgen, Frankfurter Allgemeine Zeitung, 20th Sept. 2012.
- 13.
G. Baum, Wacht auf, es geht um die Menschenwürde, Frankfurter Allgemeine Zeitung, 16th June 2013.
- 14.
“Freiheit gegen Algorithmen”, Spiegel Online, 21st June 2013.
- 15.
This aspect, with a focus on the role of algorithmic game theory, is developed at length by F. Schirrmacher, a leading German journalist, in [20], a bestseller on the German book market.
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Thomas, W. (2015). Algorithms: From Al-Khwarizmi to Turing and Beyond. In: Sommaruga, G., Strahm, T. (eds) Turing’s Revolution. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22156-4_2
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