Turing's Euclidean Stroke of Genius

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Everybody knows Alan Turing was the father of the computer. What is less appreciated is that Turing stands right next to Euclid in the pantheon of creative geniuses. What is almost never mentioned is why Turing deserves so lofty a ranking.

I'm an autodidact, so I don't know how the computer science curriculum is usually organized, but I would guess that students learn about Turing machines relatively early in their education, once they have mastered a few programming languages. I've been through a half dozen or so texts on theoretical computer science that cover this ground. They all have one thing in common: they utterly ignore Turing's fundamental insight!

Such texts teach the machine; one learns that a Turing machine involves a "tape" that slides about, with a read/write head, etc. So the student memorizes, works exercises, and comes away with a nice, formal notion of computation in terms of an abstract machine. That's the way I learned it, any way, and it seems to be the way most textbooks teach it.

Now that's a fine thing, and it explains how computers work, and how some things are "computable" and some not, and so forth. It's all very clever, but it's also entirely formal. I didn't come away from it thinking that Turing was anything but a very clever mathematical engineer. In fact the lambda calculus seems much more elegant and simple than all this business of tapes, machine states, and so forth. Turing seemed to me to be a little like Henry Ford or Thomas Edison: an inventor of a useful machine. An ingenious inventor, but still just an inventor. Studying Turing machines never gave me the feeling that I was in the presence of genius; I would have ranked many mathematicians far ahead of Turing.

Then I came across Robert Soare's paper Computability and Recursion and realized that I was getting only half the story. I'm not competent to judge the paper's technical merits, but I can confidently recommend that it should be required reading for anybody studying either Turing machines or the lambda calculus, if only for its excellent account of the history of the concepts of computability and recursion. It's really more an essay in the history of ideas than a technical paper.

The key point Soare makes, although he doesn't express it in quite this way, is this: it's all about intuition, and intuition is all about lived, human experience. The formalism of the machine comes after the leap of imagination that allowed Turing to crystalize intuitions about human experience in formal terms. Soare points out that Godel was unconvinced by Church's recursive model of computation precisely because it was unconnected to intuitions derived from lived experience.

It turns out that the key section of Turing's landmark 1936 paper, On computable numbers, with an application to the entscheidungsproblem, is section 9, wherein he grounds his formalism in an attempt "to show that the 'computable' numbers include all numbers which would naturally be regarded as computable". The next sentence strikes at the heart of the matter: "All arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically." In other words, we have certain real-world experiences calculating numbers, and based on that experience, we have an intuition that some numbers are just "naturally" computable. Now the previous eight sections of his paper described his Turing machine; his observation here amounts to an admission that there is a fundamental rift between our intuitive notion of naturally computable and the numbers computable by his machine formalism. On the one hand, it's not enough to just have a warm fuzzy feeling that the machine can compute everything we think is naturally computable; on the other hand, we can never prove that it can, since intuitions based on experience are not mathematical objects.

Turing's stroke of genius was to do for the experience of calculating what Euclid did for the experience of extensionality in space: he found a way to think about that experience in terms of an ideal, formal imaginative structure whose faithfulness to intuitive experience is indisputable. Euclid came up with notions like a point with no height or width and a line with width but no height; such ideas can never be proven to represent real space, but the intuition that they do is so compelling that it must be accepted. Turing did exactly the same thing for the experience of calculating, literally with pencil and paper.

Most of section 9 of his paper is devoted to describing the work of a human "computer". As Soare points out, in the mid-thirties the term "computer" meant a human engaged in calculation. Turing analyzed the process involved in such calculating. Post-Turing, it's easy to underappreciate what an imaginative leap this was. Calculation by a human, with pencil and paper, is essentially an analog process. If it doesn't look like that to you today, it's only because of the revolution in thought brought about by Turing. (Evidence: Godel couldn't come up with it.) He looked at the process of calculating by writing stuff down on a piece of paper divided into squares "like a child's arithmetic book", and essentially realized that we can model that analog process as a finite discrete process, a sequence of "'simple operations' which are so elementary that it is not easy to imagine them further divided." He furthermore realized that we can model the human computer's "state of mind" in finite, discrete terms. With a few other simple constraints we end up with a formal imaginative model of the work of the human computer that can be directly "mathematicalized"; we can "construct a machine to do the work of this computer", and that machine will correspond naturally, intuitively, and directly to the lived experience of performing a calculation. Turing's machine is to the experience of calculation as Euclid's one dimensional line is to the edge of a straightedge.

The sad thing is that section 9, although it is the heart of Turing's argument, is very simple; but I've yet to come across a text on theoretical computer science that even mentions it. You don't really even need mathematics to understand it, and yet it is the foundation of a massive revolution in human thinking. If a number is computable by a Turing machine, and a Turing machine is a formalization of intuition, then what does "number" mean? For Euclid, number was synonymous with length (which is why the Greeks could not stomach zero as a number); post-Turing, (computable) number is synonymous with the process of computation. For Euclid, numbers say something about extension in space; post-Turing, numbers say something about thinking - Turing's machine models the process of human computation; human computation is a thought process; therefore Turing machine is synonymous with mind. Etc. etc.

In the popular mind this sort of stuff often comes up as the question of whether or not computers can think, which most people probably think of as whether or not computers can simulate human behavior. But the real question is the other way around: is thinking essentially computational? Not so different from asking whether space is essentially geometric.

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