From:
Philosophy and Literature

Volume 19, Number 2, October 1995

pp. 390-391 | 10.1353/phl.1995.0083

*Philosophy and Literature* 19.2 (1995) 390-391

## Book Review

*Ad Infinitum: The Ghost in Turing's Machine: Taking God Out
of Mathematics and Putting the Body Back In*

*Ad Infinitum: The Ghost in Turing's Machine: Taking God
Out of Mathematics and Putting the Body Back In,***by
Brian Rotman;** xii & 203 pp. Stanford: Stanford University
Press, 1993, $39.50 cloth, $12.95 paper.

Brian Rotman's book attempts to pull mathematics -- the last, most solid home of metaphysical thought -- off its absolutist foundations and into the roiling current of postmodernism. His argument works in what has by now become a rather standard manner. He shows how the grounding ideas of mathematics rest upon the insupportable assumption of a realm of absolute truth, "truth perfect, truth outside time, outside space, outside the material world of matter, energy, and entropy, truth somehow able to be discovered or apprehended but never invented" (p. 19). Rotman questions this Platonic grounding in a number of ways.

First, with the help of linguistics (de Saussure, Benveniste,
Wittgenstein, and Peirce) he provides a semiotic analysis of
mathematical arguments as real-world linguistic events. Proof he
looks at "not as a form of inference engaged in preserving eternal
truths, but as an historically conditioned species of rhetoric" (p.
35). Secondly, Rotman argues that one idea has most secured the
fortress-like solidity of mathematical Platonism: the idea of *ad
infinitum* or infinite iteration. Mathematics takes off from the
givenness of the natural numbers, our everyday 1, 2, 3 and so on to
infinity. It is the "so on to infinity" that frames up the house of
arithmetic as it has stood since Euclid. (Euclidean in this context
is equivalent to Platonic in the context of philosophy.) But what
is the referent of these numbers? Ultimately they are said to be
the symbolic representation of the action of counting, which is the
"mathematical ur-cognition" (p. 51). And yet we cannot imagine any
material human situation in which counting could actually go on to
infinity. Therefore, some impossible, supernatural agent (the ghost
of the title) must be projected in order for this conception to
exist. Similarly, Rotman argues against the assumption of "perfect
repeatability" because this notion depends upon some purely
imaginary, ideal identity of object or action that could possibly
be repeated. *Ad infinitum* requires and assumes both a
disembodied, metaphysical counter and a disembodied, metaphysical
counted.

So Rotman sets about putting the body back into this symbolic
system by rejecting the idea of *ad infinitum*. Instead of
assuming the existence of counting activities that run, in
principle, out to infinity without changing their nature, Rotman
posits what he calls "realizable iteration," that is, counting that
admits the necessary dissolution or change of whatever repetitive
function you begin with. Put simply, he includes the fact that
quantitative accumulation or repetition always eventually brings
about a qualitative change: at some point in everyday life a
difference of degree of the "same" thing or action inevitably
becomes a difference of kind. The natural numbers and the *ad
infinitum* remain blissfully sheltered from this inescapable
real world truth, but Rotman reformulates mathematics so as to
include this truth in the very idea of numbering.

In fact (and he acknowledges this) he theorizes counting after the model of chaos or complexity theory in the physical sciences. Realizable iteration, he says, will bring in the "qualitative differences" forgotten or repressed (the language of Nietzsche, Heidegger, and Freud runs throughout) in the metaphysical version of number, and "display [the qualitative differences] as the emergent effects of repetition" (p. 131). Even numerical addition, for instance, which had been the most linearly continuous of activities, now will be linear beginning from some initial point, but will become increasingly nonlinear as quantitative increase tends towards some point of qualitative discontinuity with what had come before. The point of discontinuity, though certain to appear, is by its nature not specifiable beyond a horizon of probability, and in this, Rotman aligns his arithmetic not only with chaos theory, but also with quantum mechanics. In an appendix, Rotman elaborates at least to a degree the "pre-elements" of his non-Euclidean arithmetic.

As chaos theory does not destroy...

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