Ad Infinitum... The Ghost in Turing's Machine: Taking God Out of Mathematics and Putting the Body Back In. An Essay in Corporeal Semiotics

by Brian Rotman

This ambitious work puts forward a new account of mathematics-as-language that challenges the coherence of the accepted idea of infinity and suggests a startlingly new conception of counting. The author questions the familiar, classical, interpretation of whole numbers held by mathematicians and scientists, and replaces it with an original and radical alternative―what the author calls non-Euclidean arithmetic. The author's entry point is an attack on the notion of the mathematical infinite in both its potential and actual forms, an attack organized around his claim that any interpretation of "endless" or "unlimited" iteration is ineradicably theological. Going further than critique of the overt metaphysics enshrined in the prevailing Platonist description of mathematics, he uncovers a covert theism, an appeal to a disembodied ghost, deep inside the mathematical community's understanding of counting.

Genres: Mathematics, Philosophy, Science

224 pages, Paperback

First published September 1, 1993

Reviews

[Alexander · Rating 4 out of 5]

To be surprised not by answers, but by questions: this, in some sense, is the delight of every genuine book of philosophy, and to read Brian Rotman’s Ad Infinitum is to sip from just that cup of joy. The problem, after all, is well known: what kind of things are numbers? Or mathematical ‘objects’ more generally? Do they exist in some Platonic sky-realm, awaiting their discovery by us finite mortals? Or are they inventions of the human mind, contrivances invented for the sake of our tinkering and curiosity? How would one even go about approaching, let alone answering, these kinds of questions? Well - what if one were to approach math as a language? A fully blown system of signs, able to be analysed as signs, amenable to rich diversity of tools so far developed by linguists, philosophers and semioticians, and answering to the issues and complexes involved in all those areas of study?

Well, one answer might be - this book. Proceeding from the basis of a ‘semiotic approach’ to mathematics, Rotman brings to bear a whole suite of new questions and lines of philosophical attack that, to my mind, cannot be but indispensable for any future take on mathematical reasoning. For instance, consider the question - placed front and centre here - of what kind of agency is involved in the act of mathematical practice: Who counts? Who calculates? And specifically: who counts to infinity? Because it sure isn’t you, finite, fleshy, time-bound human-being that you are. Instead, for Rotman, all calculation involves the idealisation of agency, the projection, as it were, of an ideal calculator, an ideal subjectivity, one who counts in our place, unbound from the constraints of space and time.

And speaking of time, it’s the temporality of mathematics that is also here placed into question (an instance of the surprise I spoke of: who thinks of time and mathematics together?): how are we to make sense of this idea of ‘counting to infinity’, the three little dots (…) that indicate the effort to count on forever - ad infinitum? Tracing the genesis of both the ideal subject and the idealised, infinite time of mathematics back to the workings of the sign (that is, the workings of language), Rotman thus ties together an unexpected complex of ideas with fascinating results: mathematics, language, time, and subjectivity, each of which are called upon to think through a ‘corporeal mathematics’, one in which the body is once again admitted into the mathematical heaven from which it has long been ejected.

Hence the true import of Rotman’s project: an effort to forge a new kind of mathematical approach freed from the mystifying idealisations engendered by the inattention to the specifically semiotic nature of infinity. Referred to as an ‘non-Euclidian arithmetic’, Rotman effectively aims to do for mathematical time what non-Euclidian geometry did for mathematical space: where non-Euclidian geometry allows for parallel lines (read: infinite lines) to actually meet (and thus not continue on to infinity), so too does Rotman make the case for the temporality of 'infinite' counting to itself reach an end. An effort, in other words, to bring counting - and with it mathematics - back into the fold of the material world from which it is born. A fascinating book.

[Fraser Kinnear · Rating 4 out of 5]

Rotman’s goal is to justify mathematical intuitionism through criticizing the Platonic assumptions around infinity that most mathematicians take for granted in their formalist approach to doing math.

Rotman’s position is that we shouldn’t accept any mathematics that we can’t expect to be completed in real time. He draws an analogy of an “Agent” that calculates according to instructions from a “Subject”, with the calculation after some time being interpreted by a “Person”. Formalist mathematicians are comfortable accepting a non-time-bound “ghost” Agent (thus the book’s subtitle), which can’t possibly exist in physical reality. This book could have been much shorter, and more deeply integrated with other academic ideas, if Rotman just mapped his analogy to computers, as this discussion is such a well understood concern in computer science that we have even developed a form of notation to classify problems that face this time constraint.

In the rarefied world of mathematics, this seems like a mostly aesthetic argument, with formalists being more excited over the expanded breadth and depth of mathematics they have access to once accepting the axiom of infinity in ZFC.

However, while Rotman doesn’t address it, taking a non-Platonic position on infinity does have very interesting repercussions in physics and the ontology of physics. I had the lucky coincidence of hearing Joscha Bach discuss exactly this on Lex Fridman’s podcast, relevant quote above. In the podcast, Bach refers to a burgeoning corner of physics called digital philosophy that explores these ideas, with Stephen Wolfram perhaps as its most popular evangelist.

Physicists usually have a notion of space that is continuous. I tend to agree with people like Steven Wolfram, who are very skeptical of [such] geometric notions. I think that geometry is the dynamics of too many parts to count… there are no infinities. If there were true infinities, you would be running into contradictions, which is in some sense what Gödel and Turing discovered, in response to Hilbert’s call… There is unboundedness, but if you have a language that talks about infinity, at some point that language is going to contradict itself, which means it’s no longer valid. In order to deal with infinities in mathematics, you have to postulate their existence [axiomatically], you cannot construct the infinity. – Joscha Bach on Lex Fridman’s podcast