[moved here from the old blog]

I quote a description of Zeno’s “Achilles and Tortoise” paradox from Wikipedia:

“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.” (Aristotle Physics VI:9, 239b15)In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise’s starting point; during this time, the tortoise has “run” a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

Scott Aaronson’s post in his blog Shtetl-Optimized “And they say complexity has no philosophical implications” (see more about it below) reminded me that the most natural approach to the paradox is complexity-theoretic. Indeed, we have two different timescales: the one, in which the motion of Achilles and the Tortoise takes place, and another one, in which we discuss their motion, repeating again and again

“it will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther“.

Clearly, each our utterance takes time bounded from below by a non-zero constant; therefore the sum of the lengths of our utterances diverges. However, our personal time flow has no relevance to the physical time of the motion!

Well, probably this explanation of the paradox is well-known, but the reason why I am writing this post is the next, even more fascinating story mentioned inShtetl-Optimized. In a sense, it is dual to the Achilles and Tortoise paradox (and perhaps the duality could be made explicit). It is told in Harvey M. Friedman’s lectures Philosophical Problems in Logic. Friedman said:

I have seen some ultrafinitists go so far as to challenge the existence of 2

^{100}as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 2^{1}, 2^{2}, 2^{3}, … , 2^{100}do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2^{1}and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2^{2}, and he again said yes, but with a perceptible delay. Then 2^{3}, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2^{100}times as long to answer yes to 2^{100}then he would to answering 2^{1}. There is no way that I could get very far with this.

Yessenin-Volpin’s response makes it clear that the Achilles and the Tortoise paradox is not so much about the actual infinity as of a potential infinity (or just plain technical feasibility) of producing the sequence

1/2, 1/4, 1/8, 1/16, etc.

in real time. I agree with Scott Aaronson: and they say complexity has no philosophical implications!

However, there is yet another layer in this story. Anonymous said in a discussion in Shtetl-Optimized:

What a beautifully clever way to respond to such a line of questioning!

Well, one should remember that Alexander Yessenin-Volpin (listed in Wikipedia as Esenin-Volpin) was one of the founding fathers of the Soviet human rights movement and spent many years in prisons, exile and psychiatric hospitals. He knows a thing or two about interrogations; in 1968, he wrote and circulated viaSamizdat the famous “Memo for those who expects to be interrogated“, much used by fellow dissidents.

It is remarkable how the personality of a mathematician can be imprinted on his work and his philosophical views.

Indeed, Alexander Sergeevich Yessenin-Volpin was also a pote of note. One of his poems, a very clever and bitterly ironic rendition of Edgar Alan Poe’s The Raven, is quite revealing in the context of our discussion. I give here only the first two and the last three lines of the poem. (A full text of the poem (in Russian) can be found here and here.)

Как-то ночью, в час террора, я читал впервые Мора,

Чтоб Утопии незнанье мне не ставили в укор …[...]

… Но зато как просто гаркнул чёрный ворон:«Nеvеrmоrе!»

И вожу, вожу я тачку, повторяя: «Nеvеrmоrе…»

Не подняться… «Nеvеrmore!»

To make these lines more friendly to the English speaking reader, I explain that the first two lines refer to Thomas More’s Utopia: the protagonist reads Utopiato avoid an accusation that he has not familiarized himself with the utopian teachings promoted by the totalitarian system. The three exclamations “Nevermore!” which end the poem do not need translation.

The poem is written in 1948 (by a remarkable coincidence, the year when George Orwell wrote his 1984 — the title of the novel is just a permutation of digits; in 1949, when Orwel’s novel was published, Yessenin-Volpin started his first spell in prisons). As we can see, Yessenin-Volpin, who was 23 years old at the time, developed an ultrafinitist approach to utopian theories (and especially to the utopian practice) much earlier than to problems of mathematical logic.

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