It Began with Babbage (10 page)

Truly spectacular water clocks were made during the early Middle Ages by Chinese, Arabs, and Europeans.
⁴⁸
But, undoubtedly, the most visible, successful, and consequential automaton was the mechanical weight-driven clock—the successor to the water clock. The most reliable evidence to date places the invention of the mechanical clock to some time between 1335 and 1344 in Milan and Padua, Italy.
⁴⁹
The Padua clock of 1344 has been associated fairly firmly with Jacopo di Dondi (1290–1359), a scholar and physician who supervised the construction of this clock and may have contributed to its design.

So mechanical automata have a venerable pedigree. In “Essays,” however, they are not what interested Torres y Quevedo, but rather automata that imitated not human movement but human thought—to the extent that such a machine could, conceivably, replace humans.
⁵⁰

This, of course, is a very different notion of automata from its original roots. Torres y Quevedo was contemplating “thinking” machines. People such as Babbage and Ludgate had designed machines that embodied their respective theories of automatic computation. Torres y Quevedo, however, abstracted still further—machines that imitated the “thoughtful action” of humans. And thought included, certainly, calculation, but went beyond to making judgment, making choices. These are thoughtful actions, and machines that do this are his second type of automata. He called the study of such machines
automatics
.
⁵¹

Such automata must have “sense organs”—“thermometers, magnetic compasses, …, etc”; they must have “limbs”—mechanisms that execute the operations demanded of them. However, the essential feature of such automata is that they be “capable of discernment”
⁵²
—that they have the capacity to use information received immediately from the environment or even information obtained at some prior time to dictate their operations. Such automata must, then, imitate organisms in controlling their behavior according to information received from outside their own physical boundaries. Remarkably, Torres y Quevedo wrote, they must
adapt
their behavior according to “changing circumstances.”
⁵³

Torres y Quevedo's theory of automata is that such a machine is an
adaptive system
that can alter its “conduct” according to “changing circumstances.” He does not stop with this general definition. His science of automatics goes deeper. His aim in “Essays” was to try and show, at least theoretically, that such an adaptive automaton can be created according to principles (“rules”) that can be built into the machine itself.
⁵⁴
To illustrate this theoretical possibility, he resorted to real devices (he was, after all, an engineer), using electrical switches, each of which could be in one of a number of positions (“states,” in
present-centered language), representing input information to the automaton, and electromagnets that could be “excited,” thus attracting their respective armatures as instances of operations.

He probed further. He referred to the French philosopher René Descartes (1596–1650), who in his most famous work,
Discourse on Method
(1637), discounted the possibility of automata being rational in the way humans are. In response, Torres y Quevedo imagined a machine in which there are “thousands or millions” of switches, each of which can be in one of as many different positions (states) as there are distinct written characters (such as letters, numbers, punctuation marks).
⁵⁵
He wrote that it is quite possible that these switches could be set to represent any phrase or sentence, short or long. A speech, then, would be represented by the setting of switches. In fact, he was anticipating the possibility of storing symbol structures (a phrase or a speech) electrically, made up of letters, numbers, punctuation marks, and so on—that is, he was imagining the possibility of an automaton that processes symbols of
any
kind, not just numbers
à la
Babbage (to whom, incidentally, he refers extensively in his essay). But, Torres y Quevedo stopped short of actually positing machines that
reason on their own
. Like Lovelace, who cautioned that the Analytical Engine could not originate computational action, so, too, did Torres y Quevedo respond to Descartes somewhat cautiously. Descartes, he wrote, had been mistaken in assuming that for an automaton to provide reasoned responses it must do the reasoning itself. Rather, it is the creator of the automaton who does the reasoning.
⁵⁶

Still, in Torres y Quevedo's “Essays,” we find the first inkling of a general theory of what a computing device might be. It is an automaton capable of symbol processing and adapting itself to variations in its input.

  
1
. A. B. Bromley. (1982). Charles Babbage's Analytical Engine, 1838.
Annals of the History of Computing, 4
, 196–217.

  
2
. See, for example, F. B. Artz. (1980).
The mind of the Middle Ages
(3rd ed., Rev.) Chicago, IL: University of Chicago Press.

  
3
.
Natural history
was the term commonly used until the end of the 19th century to designate the study of life at the level of the individual organism, especially organisms in their natural habitat. See M. Bates. (1990).
The nature of natural history
(p. 7). Princeton, NJ: Princeton University Press (original work published 1950). The term still prevails in some contexts such as, for example, in the term
natural history museum
.

  
4
. A. E. Lovejoy. (1936).
The great chain of being
(pp. 233–236). Cambridge, MA: Harvard University Press.

  
5
. E. Mayr. (1982).
The growth of biological thought
(p. 430). Cambridge, MA: Belknap Press of Harvard University Press.

  
6
. H. H. Goldstine. (1972).
The computer from Pascal to von Neumann
(p. 67). Princeton, NJ: Princeton University Press.

  
7
. Ibid., pp. 67–68.

  
8
. H. Hollerith. (n.d.). “An electric tabulating system”. Reprinted in B. Randell. (Ed.). (1975).
The origins of the digital computer
(2nd ed., pp. 129–139). New York: Springer-Verlag (see especially p. 129). The original source of this article is not specified by Randell. However, the following article was published with the same title: H. Hollerith. (1889). An electric tabulating system.
The Quarterly, Columbia University School of Mines, X
, 238–255.

  
9
. Ibid.

10
. Goldstine, op cit., p. 8
n
.

11
. F. P. Brooks, Jr. & K. E. Iverson (1969).
Automatic data processing: System/360 edition
(p. 120). New York: Wiley.

12
. Goldstine, op cit., pp. 70–71.

13
. Hollerith, op cit., p. 131.

14
. Ibid.

15
. Ibid., pp. 132–133.

16
. D. DeLillo (1994).
White noise
(p. 203). New York: Penguin (original work published 1985).

17
. Hollerith, op cit., p. 133.

18
. R. Moreau (1984).
The computer comes of age
(p. 24). Cambridge, MA: MIT Press.

19
. P. E. Ludgate. (1909). On a proposed analytical machine.
Proceedings of the Royal Dublin Society, 12
, 77–91. Reprinted in Randell (pp. 71–87), op cit., p. 71.

20
. Randell, op cit., p. 71.

21
. B. Randell. (1971). Ludgate's Analytical Machine of 1909.
The Computer Journal, 14
, 317–326.

22
. Ibid., p. 319.

23
. Ibid.

24
. Ibid., 318.

25
. Ludgate, op cit., p. 72.

26
. Ibid.

27
. Ibid.

28
. Ibid.

29
. Ibid., pp. 72–73.

30
. Ibid.

31
. Ibid., p. 73.

32
. Ibid., p. 74. Perforated paper tape also had a place in looms, reaching back to the 18th century. More prominently, paper tapes were used in electrical telegraphy in the mid 19th century.

33
. Ibid.

34
. Ibid.

35
. Ludgate, op cit., p. 75.

36
. Ibid., p. 74.

37
. Ibid., p. 80.

38
. Ibid., p. 81.

39
. Ibid.

40
. Quoted in Randell, 1971, op cit., p. 320.

41
. Ludgate, op cit., p. 85.

42
. C. V. Boys. (1909). A new analytical machine.
Nature, 81
, 14–15.

43
. This biographical information was extracted from several websites on the Internet, most notably,
http://www.wvegter.hivemind
. Last accessed July 19, 2013.

44
. L.. Torres y Quevedo. (1975). Essays on automatics (R. Basu, trans.). Randell, 1975, op cit., pp 87–106. (Original work published in Spanish as “Essais sur l'automatique. Sa definition. Étendue théoritique de ses applications,”
Revue Générale des Sciences Pures et Appliquées
, 601–611 (15 Nov. 1915).

45
. Torres y Quevedo, op cit., p. 87.

46
. A. P. Usher. (1985).
A history of mechanical inventions
(Rev., pp. 162–163). New York: Dover Publications (original work published 1954).

47
. H. Hodges. (1971).
Technology in the ancient world
(pp. 180–181). Harmondsworth, UK: Penguin Books.

48
. D. L. Landes. (1983).
Revolution in time
(pp. 18–20). Cambridge, MA: Harvard University Press.

49
. Usher, op cit., p. 196.

50
. Torres y Quevedo, op cit., p. 87.

51
. Ibid., p. 88.

52
. Ibid.

53
. Ibid.

54
. Ibid., p. 89.

55
. Ibid., p. 90.

56
. Ibid., p. 91.

IN 1900, THE
celebrated German mathematician David Hilbert (1862–1943), professor of mathematics in the University of Göttingen, delivered a lecture at the International Mathematics Congress in Paris in which he listed 23 significant “open” (mathematicians' jargon for “unsolved”) problems in mathematics.
¹

Hilbert's second problem was: Can it be proved that the axioms of arithmetic are consistent? That is, that theorems in arithmetic, derived from these axioms, can never lead to contradictory results?

To appreciate what Hilbert was asking, we must understand that in the
fin de siècle
world of mathematics, the “axiomatic approach” held sway over mathematical thinking. This is the idea that any branch of mathematics must begin with a small set of assumptions, propositions, or
axioms
that are accepted as true without proof. Armed with these axioms and using certain
rules of deduction
, all the propositions concerning that branch of mathematics can be derived as theorems. The sequence of logically derived steps leading from axioms to theorems is, of course, a
proof
of that theorem. The axioms form the foundation of that mathematical system.

The axiomatic development of plane geometry, going back to Euclid of Alexandria (fl. 300
BCE
) is the oldest and most impressive instance of the axiomatic method, and it became a model of not only how mathematics should be done, but also of science itself.
²

Hilbert himself, in 1898 to 1899, wrote a small volume titled
Grundlagen der Geometrie (Foundations of Geometry
) that would exert a major influence on 20th-century mathematics. Euclid's great work on plane geometry,
Elements
, was axiomatic no doubt, but was not axiomatic enough. There were hidden assumptions, logical problems, meaningless definitions, and so on. Hilbert's treatment of geometry began with three undefined
objects—point, line, and plane—and six undefined relations, such as being parallel and being between. In place of Euclid's five axioms, Hilbert postulated a set of 21 axioms.
³

In fact, by Hilbert's time, mathematicians were applying the axiomatic approach to entire branches of mathematics. For example, the axiomatization of the arithmetic of cardinal (whole) numbers formulated by the Italian Giuseppe Peano (1858–1932), professor of mathematics in the University of Turin, begins with three terms—“number”, “zero”, and “immediate successor”—and are assumed to be understood. The axioms themselves are just five in number:

1. Zero is a number.

2. The immediate successor to a number is a number.

3. Zero is not the immediate successor of a number.

4. No two numbers have the same immediate successor.

5. The principle of mathematical induction: Any property belonging to zero, and also to the immediate successor of every number that has the property, belongs to all numbers.

Exactly a decade after Hilbert's Paris lecture, British logician and philosopher Bertrand Russell (1872–1970), in collaboration with his Cambridge teacher Alfred North Whitehead (1861–1947), published the first of the three-volume
Principia Mathematica
(1910–1913)—not to be confused with Newton's
Principia
—which attempted to develop the notions of arithmetic from a precise set of logical axioms, and which was intended to demonstrate that mathematical knowledge can be reduced to (or, equivalently, derived from) a small set of logical principles. However, Russell and Whitehead did not address Hilbert's second problem.

Hilbert returned to the foundations of mathematics repeatedly throughout the course of the first three decades of the 20th century, establishing what came to be known as “Hilbert's program”.
⁴
In 1928, in an address delivered at the International Congress of Mathematicians in Bologna, Italy (the home, incidentally, of the world's first university), and in a monograph titled “Die Grundlagen der Mathematik” (“The Foundations of Mathematics), he asked: (a) Is mathematics
complete
, in the sense that every mathematical statement could be either proved or disproved? (b) Is mathematics
consistent
, in the sense that a statement such as 2 + 2 = 5 could never be arrived at by a valid proof, or in the sense that two contradictory propositions
a
=
b
and
a
≠
b
could both be derived? (c) Is mathematics
decidable
, in the sense that there exists a definite method that can be followed to demonstrate that a mathematical statement is true or not?