The second chapter of my forthcoming book The Annotated Turing is entitled "The Irrational and the Transcendental" and innocently begins:
We begin counting 1, 2, 3, and we can go on as long as we want.
That's not true, of course. "We" simply cannot continue counting "as long as we want" because "We" (meaning "I" the author and "you" the reader) will someday die — probably in the middle of reciting a very long (but undoubtedly finite) number.
What the sentence really means is that some abstract ideal "somebody" can continue counting, but that's not true either: Counting is a temporal process, and at some point everybody will be gone in a heat-dead universe. There will be no one left to count. Even long before that time, counting will be limited by the resources of the universe, which contains only a finite number of elementary particles and a finite amount of energy to increment from one integer to the next.
We tend to accept simple statements such as the one that begins Chapter 2 of my book because our culture is hopelessly entangled in a Platonistic view of the natural numbers. We feel that the natural numbers exist somewhere "out there" independent of the people who use them to count. This assumption is not often closely examined, but it actually implies the opposite of what we think. We like to pretend that mathematics is the most "objective" and least human-bound intellectual endeavor, but our view of the natural numbers reveals mathematics to be founded on a very human metaphysical conceit. The natural numbers are not, in fact, "natural" — that is, intrinsically part of nature — but arise out of human discourse.
Is this a problem? And if so, can mathematics ever be truly liberated from metaphysics?
This is the job that the brave mathematician and philosopher Brian Rotman has taken on in his book Ad Infinitum... (Stanford University Press, 1993). I discovered this book in 1999 while searching out resources for writing the book that eventually became The Annotated Turing, and I was amused by the book's subtitle. Actually, Ad Infinitum... has three subtitles, and it's the first subtitle that made the book come up in an Amazon search. The complete title seems to be Ad Infinitum… The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back In: An Essay in Corporeal Semiotics.
The western world has mostly been comfortable with the dual concepts of infinity described by Aristotle in the Physics, Book III, Chapters 4 through 8. Aristotle differentiated between an "actual" or "completed" infinity, of which he denied the existence, and a "potential" infinity, such as that of the natural numbers, which he found to be acceptable:
Infinity turns out to be the opposite of what people say it is. It is not ‘that which has nothing beyond
itself’ that is infinite, but ‘that which always has something beyond itself ’. [Aristotle, Physics, translated by Robin Waterfield (Oxford World’s Classics, 1996), Book III, Chapter 4, page 73.]
Although Aristotle's rejection of the completed infinity has been often ignored over the centuries — most notably in the use of "infinitesimals" in the creation of the differential calculus, and more recently, in the work of Georg Cantor and his mathematics of the transfinite — it remains the gold standard for our allowance of some forms of infinity into mathematics but not others.
Brian Rotman, however, finds even Aristotle's potential infinity to be hopeless metaphysical and seeks to eliminate even that. This is not just an issue between the two philosophies of mathematics roughly divided as Platonists (a.k.a., Realists) and Constructivists, because many of the historical Constructivists have accepted the pre-existence of the natural numbers. Leopold Kronecker famously said, "God made the integers, all the rest is the work of Man" (quoted by Rotman on page 38) and to L. E. J. Brouwer, the natural numbers were a shared Kantian intuition among all mathematicians.
Rotman instead asks:
But what if no deity created the numbers, what if they are not the work of a pure angelic being outside history, but rather the production of a secular, time-bound, and empirically tainted culture? What would follow from a conception of the whole numbers as humanly constructed? Would we not at the beginning have to repudiate their closure in the past — God's creation of them in their entirety — and replace it with an openness framed in terms of the discursively manufactured present-future: numbers as things which exist or can be imagined to exist, which are humanly constructed and constructible, through some ongoing symbolic process? [ pg. 49 ]
What Rotman attempts to introduce as an alternative is a "non-Euclidean arithmetic." Geometry as formulated by Euclid sought to derive everything from self-evident postulates; yet, the fifth of these postulates was forced to bring infinity into play by including the phrase "if produced indefinitely" (in Sir Thomas Heath's translation) in describing the conditions under which lines are not parallel. Just as nineteenth-century mathematicians formulated non-Euclidean geometries by making assumptions contrary to the fifth postulate, Rotman attempts to formulate an arithmetic where the natural numbers cannot be realized indefinitely because of the limitations of resources and human cognizance.
Ad Infinitum... is not an easy book. It requires a familiarity and sympathy with constructivist mathematics and post-modern theory. (The longest entry in the bibliography belongs to Jacques Derrida.) And although I didn't follow all of Rotman's arguments entirely — and even found myself often lost in the middle of single sentences — I found Ad Infinitum... to be a fascinating and challenging assault on conventional notions of infinity.
Although Alan Turing figures in the first of the three subtitles of Ad Infinitum..., he doesn't play a big role in this book and his imaginary computing machines are only discussed in any detail on page 99. Rotman indicates that Turing's paper "was conceived entirely within classical, infinitistic terms," which is unsurprising given how much the work of Cantor and Hilbert hangs over it.
The business of carrying out the computation, of articulating what or who might, in principle, be involved or called upon to "physically" run or be able to run his ideal machine — the ideal typist as it were — formed no part of Turing's picture. Or rather, no problematic part, since the classical context in which he operated already internalized an infinity of numbers immediately accessible to a — necessarily disembodied, dephysicalized — counting agent; hence, for example, the lack of any qualms involved in Turing's assumption of an infinite memory tape as part of his machine's specification.
The ideal and reduced typist — the computing agency, unmentioned by Turing, who would run a Turing machine — is of course the transcendentally given, infinitistic agent of classical mathematics.
It is unfortunate that Turing displays no interest in the more philosophical matters of mathematics in his paper on computable numbers (although he touches on them in the correction to his paper), but it is Turing's modeling of algorithms as computing machines that reveals the strictly physical and temporal nature of mathematics. The tape is "infinite" only in the sense that whenever an additional length is required, it is available. In use, the tape is always finite. Turing doesn't address what happens when the universe (or just the paper factory) runs out of resources and the tape cannot be extended, but the reader certainly ponders the possibility, simply because the tape is a physical object and not an abstract procedure.
Turing's paper proves that the vast majority of real numbers can't even be computed, thus calling into question even the reality of the "real" numbers. Even for those real numbers that can be computed — the algebraic numbers and a tiny subset of the transcendentals — the computation is always bound to a temporal process and obviously can never be completed. It is Turing's imaginary machine that most cogently reveals these limitations as an intrinsic part of any algorithmic process.
Moreover, there is no general finite algorithm that can analyze a Turing Machine and determine whether a particular digit or pattern of digits will ever be computed, thus implying that these digits do not exist until they are actually computed by the machine. If the computation of these digits requires more time or resources than are available in the universe, they will simply never be known.
When I set out to study Turing's paper in detail, I hardly expected it to have implications for the philosophy of mathematics. Yet to me, Turing's conclusions cast real doubts on a Platonistic interpretation of mathematics and imply instead an extreme Constructivist philosophy where mathematics is limited by time, resources, and energy. I don't think this has been widely appreciated because the Turing Machine is most commonly studied in a reformulation that computes finite integral functions rather than real numbers. It's yet one more reason why it's useful to go back to Turing's original paper.
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