Quoth the raven, ‘Nevermore.’ Edgar Alan Poe Of all paradoxes of infinity, Zeno’s “Achilles and Tortoise” para- dox is one of the oldest. I borrow its description from WIKIPEDIA:
“In a race, the quickest runner can never overtake the slow- est, 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 tor- toise 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 start- ing 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
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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.
In view of our discussion in the previous section, the most natu- ral approach to the paradox is complexity-theoretic. Indeed, in the description of the race between Achilles and the Tortoise, 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 the words
“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 utter- ances diverges. However, our personal time flow has no relevance to the physical time of the motion!
Since the Zeno’s paradox is not about mathematics as such, but about its relations with the real world and about our perception of time, a complexity-theoretic approach to its solution is well justi- fied. The validity of such approach is even more evident in view of a mathematical fable which is dual, in some vague sense, to the Achilles and Tortoise paradox (but perhaps this duality could be made explicit). It is told in Harvey M. Friedman’s lecturesPhilo- sophical Problems in Logic. Friedman said:
I have seen some ultrafinitists go so far as to challenge the existence of2100as 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
21,22,23, . . . ,2100
do we stop having “Platonistic reality”? Here this . . . is to- tally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objec- tion with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then pro- ceeded to start with21and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about22, and he again said yes, but with a perceptible delay. Then 23, 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 2100 times as long to answer yes to2100then he would to answering21. 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.6
However, there is yet another layer in this story. It provides an opportunity to bring into the discussion a rarely mentioned aspect of mathematical practice: the influence of the personality of a math- ematician on his or hers mathematical outlook.
The instantaneousness of Yessenin-Volpin’s response to the line of questioning is more than a quick reflex. One should remember that Alexander Yessenin-Volpin was one of the founding fathers of the Soviet human rights movement and spent many years in prisons and exile. He knows a thing or two about interrogations; in 1968, he wrote and circulated via Samizdat the famous Memo for those who expects to be interrogated, much used by fellow dissi- dents.
One advice from theMemois worth quoting:
During an interrogation, it is already too late to determine your position and develop a line of behavior. [...] If you ex- pect an interrogation, if there is just a possibility of an interrogation—get prepared in advance.
Yessenin-Volpin was also a poet of note. One of his poems, a very clever and bitterly ironic rendition of Edgar Alan Poe’sThe 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.
Kak-to noq~, v qas terrora, qital vpervye Mora, Qtob Utopii neznan~e mne ne stavili v ukor. . .
.. .
No zato kak prosto garknul qerny$i voron: “Nevermore!” I kaqu, kaqu taqku, povtor: “Nevermore. . . ”
Ne podnt~s . . . “Nevermore!”
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 readsUtopiato 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 was written in 1948 (significantly, the year when George Orwell wrote his1984—the title of the novel is just a per- mutation of digits; in 1949, when Orwell’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