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Humanist Discussion Group, Vol. 17, No. 680.

Centre for Computing in the Humanities, King's College London

www.kcl.ac.uk/humanities/cch/humanist/

www.princeton.edu/humanist/

Submit to: humanist@princeton.edu

Date: Wed, 03 Mar 2004 08:28:51 +0000

From: Willard McCarty <willard.mccarty@kcl.ac.uk>

Subject: mathematics and computation

My thanks to several people who made suggestions in answer to my question

about the history of 20C mathematics in its relation to computing. After

checking out most of the sources recommended and straying into others, I'd

like to draw your attention to the following, in case my quest is of more

general interest:

1. Martin Davis, The Universal Computer: The Road from Leibniz to Turing

(New York: W W Norton, 2000).

Of all of the books I examined, this is the closest to what I was hoping to

find: clearly written, non-technical and quite insightful at key points. It

emphasizes one of the histories of computing, as the title suggests, namely

the mathematical-logical, and more or less ends the story with Turing's own

end. The story it tells hangs together by a combination of biographical and

intellectual themes, but these are very well balanced. Readings from it

would suit advanced undergraduate and (post)graduate students in the

humanities quite well.

(Anecdotal aside. The basic historiographical problem with any account like

this emerged in a Freudian typo a few minutes ago. I was recommending the

book to a young, very bright and eager nephew; I typed, The Universal

Computer: The Road from Turing to Leibniz!.... I am not implying a

criticism of Davis's fine account, rather the point that it is only one of

the histories, the one you get when you take "the" computer to be in

essence defined by what Turing did.)

2. David Hilbert, "Axiomatic Thought" (1918), in William Ewald, ed., From

Kant to Hilbert: A Source Book in the Foundations of Mathematics. Volume

II. Oxford: Oxford University Press, 1996. This paper is entirely

non-technical and a very fine example of how Hilbert thought about

mathematical thinking. It provides an illuminating definition of the

meaning of the word "theory" in mathematics and best of all a very clear

statement of his axiomatic method. The editor's prefatory comments on the

mistaken notion that Hilbert was simply a "formalist" are quite helpful.

One of course needs reference to other papers by Hilbert, including the

famous 1900 address, which is online.

3. Kurt Gödel, "The Modern development of the foundations of mathematics in

the light of philosophy", denoted as Gödel 1961/?, in Unpublished essays

and lectures, vol. III of the Collected Works, ed. Solomon Feferman et al

(Oxford: Oxford University Press, 2001), with a very helpful preface by

Dagfinn Føllesdal (pp. 364-72); the essay itself in English is online at

http://www.marxists.org/reference/subject/philosophy/works/at/godel.htm.

This, it seems, was an address that he wrote but never delivered. In it he

talks about Hilbert's Program, how he thinks it failed and he charts what

he regards as the most promising "middle way" forward -- which is through

Husserlian phenomenology to deepen the abstract concepts on which

mechanical, formalist schemes are founded. The project to deepen

foundations he shares, of course, with Hilbert and many others; what is

especially valuable here is the pointer into phenomenology (on which

Føllesdal comments at length). The result of the deepening is brought out

by Davis, p. 124: "For any specific given formalism there are mathematical

questions that will transcend it. On the other hand, in principle, each

such question leads to a more powerful system which enables the resolution

of that question. One envisions hierarchies of ever more powerful systems

each making it possible to decide questions left undecidable by weaker

systems." This is immediately recognizable as the perfective cycle of

modelling. But I am left full of questions about the phenomenology!

There are a number of other, I suppose obvious, items, such as Turing's

1936 paper and von Neumann's First Draft of a Report on the EDVAC (1945),

although these are well covered in Davis' book, which does an excellent job

of connecting Turing's work to von Neumann's.

But lest this somehow seem a claim of first discovery within our small

circle, allow me to note that Tito Orlandi has been insisting on the

importance of the mathematical topics for many years.

Comments?

Yours,

WM

Dr Willard McCarty | Senior Lecturer | Centre for Computing in the

Humanities | King's College London | Strand | London WC2R 2LS || +44 (0)20

7848-2784 fax: -2980 || willard.mccarty@kcl.ac.uk

www.kcl.ac.uk/humanities/cch/wlm/

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