Abstract
The study of algorithms, computations, and computability offers a major contact point between mathematics, technology, and philosophy. This chapter begins with a brief history of computations and the technical means used to support them. Summary accounts are given of two scholarly developments that provided much of the intellectual background for modern computation: attempts to express all reasoning as mathematics and attempts to reduce all of mathematics to simple, rule-bound symbol manipulation. This is followed by a discussion of the Turing machine, including a detailed explanation of why it can be said to cover all systems of rule-bound symbol manipulation. The universal Turing machine and its philosophical implications are also discussed. A two-dimensional classificatory scheme is offered for proposed constructions of computing devices with stronger computing powers than a Turing machine. This categorization serves to highlight the weaknesses of current proposals for such devices. In conclusion, it is emphasized that computation has to be understood as an intentional input-output process with high demands on reliability and lucidity. The study of computations and algorithms has much to learn from other studies of intentional human action, not least in the philosophy of technology.
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Notes
- 1.
The word “calculation” is commonly used for elementary operations, and “computation” for more advanced and complex ones. There is no sharp delimitation in usage between the two terms.
- 2.
For instance, the following rule can be used to multiply two two-digit numbers that both end in 5 and whose first digits are either both odd or both even: Add the first digits. Divide by 2. Add their product. Write 25 afterwards. In this case, (3 + 7)∕2 + (3 × 7) = 26, so 35 × 75 = 2625. (I.e., (10a + 5)(10b + 5) = ((a + b)∕2 + ab) × 100 + 25.)
- 3.
The set of problems that can be solved with a method of calculation will have to be mathematically “natural”. For instance, the above instruction can also be used to make each of the additions 332666+1, 332665+2, 332664+3, etc., but such an ad hoc collection of problems does not make it a calculation.
- 4.
At the time of writing, it is not known if the procedure for finding perfect numbers sketched out here is effective or not. If there are infinitely many perfect numbers, then the procedure is effective, otherwise not.
- 5.
Gandy (1988, p. 57) showed that the functions computable with the analytical engine “are precisely those which are Turing computable.”
- 6.
Lovelace said (p. 119) that she did not know “[w]hether the inventor of this engine had any such views in his mind whilst working out the invention.”
- 7.
One of them was Ada Lovelace’s mother, Lady Byron (1792–1869), who described it as a “thinking machine”. (Quoted in Swade 2011a, p. 246.)
- 8.
- 9.
- 10.
Quo facto, quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus.” (Leibniz 1890, vol. 7, p. 200).
- 11.
Much later, infinitesimals were introduced in nonstandard analysis, but now rigorously defined. This was largely the work of Abraham Robinson (1918–1974).
- 12.
This term is usually attributed to the German mathematician Felix Klein (1849–1925) who used it in a speech in 1895 (Klein 1895).
- 13.
“L’idée vague de continuité, que nous devions à l’intuition, s’est résolue en un système compliqué d’inégalités portant sur des nombres entiers. Par là les difficultés provenant des passage à la limite, ou de la considération des infiniments petits, se sont trouvées définitivement éclaircies. Il ne reste plus aujourd’hui en Analyse que des nombres entiers ou des systèmes finis ou infinis de nombres entiers, reliés entre eux par un réseau de relations d’égalité ou d’inégalité. Les Mathématiques, comme on l’a dit, se sont arithmétisées.” (Poincaré 1902, p. 120).
- 14.
This representability of symbols as numbers was used in masterly fashion by Kurt Gödel (1931) when he assigned a unique number to each sentence that is expressible in a logical language (Gödel numbering).
- 15.
“Ich betrachte die Arithmetik, oder die Lehre von den reinen Zahlen, als eine auf rein psychologischen Thatsachen aufgebaute Methode, durch die die folgerichtige Anwendung eines Zeichensystems (nämlich der Zahlen) von unbegrenzter Ausdehnung und unbegrenzter Möglichkeit der Verfeinerung gelehrt wird. Die Arithmetik untersucht namentlich, welche verschiedene Verbindungsweisen dieser Zeichen (Rechnungsoperationen) zu demselben Endergebniss führen.”
- 16.
- 17.
“Wir denken drei verschiedene Systeme von Dingen: die Dinge des ersten Systems nennen wir Punkte und bezeichnen sie mit A, B, C, …; die Dinge des zweiten Systems nennen wir Gerade und bezeichnen sie mit a, b, c, …; die Dinge des dritten Systems nennen wir Ebenen und bezeichnen sie mit α, β, γ, ……. Wir denken die Punkte, Geraden, Ebenen in gewissen gegenseitigen Beziehungen und bezeichnen diese Beziehungen durch Worte wie “liegen”, “zwischen”, “parallel”, “kongruent”, “stetig”; die genaue und vollständige Beschreibung dieser Beziehungen erfolgt durch die Axiome der Geometrie” (Hilbert 1899, p. 2).
- 18.
“Dieser geschilderten Auffassung der Geometrie lege ich deshalb besondere Bedeutung bei, weil es mir ohne sie unmöglich gewesen wäre, die Relativitätstheorie aufzustellen” (Einstein 1921, p. 6).
- 19.
Arguably, this was the realization of Ada Lovelace’s above-mentioned “science of operations” that could apply to “letters or any other general symbols” as well as numbers (Lovelace [1843] 1989, pp. 117 and 144).
- 20.
“naturgemäß und konsequent…den Zahlzeichen und den Buchstaben in der Algebra gleichstellen und ebenfalls als Zeichen auffassen, die an sich nichts bedeuten, sondern nur Bausteine für die idealen Aussagen sind”, “nur Objekte für die Anwendung unserer Regeln” (Hilbert 1928, p. 8).
- 21.
More precisely: a deductive system, i.e. the combination of a set of axioms and a set of rules for making derivations based on them.
- 22.
“ein Verfahren”. Hilbert and Ackermann (1938), p. 91.
- 23.
“das Hauptproblem der mathematischen Logik”, Hilbert and Ackermann (1938), p. 90.
- 24.
In many accounts of Turing’s work, this analytical work is not adequately described. Robin Gandy (1919–1995), who was Turing’s graduate student, rightly called it a “paradigm of philosophical analysis” (Gandy 1988, p. 86).
- 25.
See for instance Turing (1937a, p. 231, [1948] 2004, p. 9), Cleland (2002, p. 166), Israel (2002, p. 196), and Sieg (1997, pp. 171–172, 2002, pp. 399–400). Misunderstandings on this are not uncommon, for instance Arkoudas (2008, p. 463) claims that “the term ‘algorithm’ has no connotations involving idealized human computists” and that Turing just “referred to human computers as a means of analogy when he first introduced Turing machines (e.g., comparing the state of the machine to a human’s ‘state of mind,’ etc.)”. A careful reading of Turing’s 1936–7 article will show that Arkoudas’s interpretation cannot be borne out by the textual evidence.
- 26.
Turing still used the word “computer” in this sense a decade later, see Turing ([1947] 1986, p. 116) Gandy (1988, p. 81) proposed that we use “computer” for a computing machine and “computor” for a computing human. Some authors have adopted this practice, e.g. Sieg (1994). However, the difference between the two spellings is easily overlooked. To make the difference more easily noticeable, I propose that we revive the word “computist” for a human performing calculations.
- 27.
In this example, there are in fact three symbols, since the empty space and 0 are not treated in the same way, as can be seen from rules 1 and 18 in Textbox 9.2. It is perfectly feasible to use only two symbols; we can for instance replace each symbol space with two adjacent symbol spaces such that 00 represents an empty space, 01 represents 0 and 11 represents 1. See Sect. 9.5.2.
- 28.
Halting is dealt with in different ways in different versions of Turing machines. A common construction is to let there be a combination of a state and an accessed symbol for which there is no instruction. When the computist arrives at that combination (s)he is assumed to halt since she has no instruction on how to proceed.
- 29.
- 30.
This argument presupposes that there is only a finite number of different positions that a symbol can have within a symbol space. This is a reasonable assumption, given the function of symbol spaces, as explained above.
- 31.
One aspect of this priority for simplicity is that each operation is assumed to affect only a minimal part of the tape. This feature can be described as a locality condition or, better, a set of locality conditions for reading, writing, and moving (Sieg 2009, pp. 584–587).
- 32.
Based on this omission in Turing’s text, Copeland (1998) claims that Turing machines can compute Turing-incomputable functions, namely if they perform infinitely many operations in finite time.
- 33.
The option of erasing a symbol to replace it by a blank square was not included in Turing’s account, and it does not either seem to have had any role in later versions of the Turing machine.
- 34.
As we saw above, Turing argued that the process could be so constructed that only one symbol at a time is observed. Consequently, “symbols” can be replaced by “symbol” in this statement. Cf. Turing (1937a), pp. 231–232, 251 and 253–254.
- 35.
Cf. Hofstadter (1979), p. 562.
- 36.
- 37.
Several other equivalent characterizations have been added to the list, first of them Emil Post’s (1936) proposal, which had some elements in common with Turing’s but was conceived independently. As clarified by Soare (1996, p. 300), Post’s ideas were much less developed than those presented by Turing.
- 38.
He wrote later that he was “completely convinced only by Turing’s paper”. (Letter from Gödel to Georg Kreisel, May 1, 1968, quoted by Sieg (1994, p. 88)).
- 39.
The reference about the necessity of proving preliminary theorems refers to a technicality clarified by Sieg (1994, p. 112).
- 40.
See also Gödel (1958).
- 41.
According to one estimate, the universe can register up to 1090 bits (Lloyd 2002). Obviously, the practical limitations for a computing device ever to be built are much stricter.
- 42.
Charles Babbage put much effort into making his computing machines operate by switching reliably between discrete states (Swade 2011b, pp. 67–70).
- 43.
Cf. Smith (2006).
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Hansson, S.O. (2018). Mathematical and Technological Computability. In: Hansson, S. (eds) Technology and Mathematics. Philosophy of Engineering and Technology, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-93779-3_9
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