[ went here from the old blog ]
I cite a description of Zeno 's `` Achilles and Tortoise
`` paradox from Wikipedia:
- `` In a race, the speediest contrabandist can ne'er catch the slowest, since the chaser must foremost hit the point whence the prosecuted got, so that the slower must e'er maintain a lead. '' ( Aristotle Physics Six: 9, 239b15 )
In the paradox of Achilles
and the Tortoise
, we ideate the Grecian hero Achilles in a run with the footslog reptilian. Because he is so fast a contrabandist, Achilles graciously permits the tortoise a start of a hundred pes. If we say that each racer begins running at some unvarying velocity ( one in no time and one rattlingly slow ), so after some finite time
, Achilles will hold run a hundred pes, taking him to the tortoise 's getting point; during this clip, the tortoise holds `` run '' a ( much shorter ) distance, state one pes. It will so take Achilles some farther time period to run that distance, during which the tortoise will progress further; then another time period to make this 3rd point, while the tortoise travels before. So, whenever Achilles hits someplace the tortoise holds been, he still holds further to move. Hence, Zeno tells, swift Achilles can ne'er catch the tortoise. Therefore, while mother wit and common experience would maintain that one smuggler can catch another, according to the above statement, he can not; this is the paradox.
Scott Aaronson 's station in his blog Shtetl-Optimized
`` And they state complexness holds no philosophic entailments ''
( see more about it below ) reminded me that the most natural attack to the paradox is complexity-theoretic. So, we hold two different timescales: the one, in which the gesture of Achilles and the Tortoise haps, and another one, in which we discourse their gesture, reiterating time and time again
it will so take Achilles some farther time period to run that distance, during which the tortoise will progress further
Clearly, each our vocalization takes clip sprung from below by a non-zero invariable; hence the amount of the lengths of our vocalizations diverges. Withal, our personal clip flowing holds no relevancy to the physical clip of the gesture!
Goodly, likelily this account of the paradox is well-known, but the ground why I am composing this station is the following, even more absorbing tale named in Shtetl-Optimized
In a way, it is duple to the Achilles and Tortoise paradox ( and perchance the dichotomy could be done explicit ). It is stated in Harvey Metre Friedman 's lectures
Philosophic Jobs in Logic
Friedman stated:
I hold seen some ultrafinitists travel so far as to dispute the being of 2 100
as a natural figure, in the sense of there being a series of `` points '' of that length. There is the obvious `` draw a line '' remonstrance, enquire where in 2 1
, 2 2
, 2 3
, 2 100
make we halt holding `` Platonistic world ''? Here this... is whole guiltless, therein it can be easily be replaced by 100 points ( names ) divided by commas. I raised merely this expostulation with the ( extreme ) ultrafinitist Yessenin-Volpin during a talk of his. He enquire me to be more specific. I so continued to get with 2 1
and enquire him whether this is `` existent '' or something to it event. He virtually forthwith stated yes. So I inquired about 2 2
, and he again stated yes, but with a perceptible postponement. So 2
3
, and yes, but with more postponement. This kept for a few more times, till it was obvious how he was caring this remonstration. Surely, he was readied to e'er reply yes, but he was attending take 2 100
times equally long to respond yes to 2 100
so he would to answering 2 1
There is no manner that I could get rattlingly far with this.
Yessenin-Volpin 's response does it clear that the Achilles and the Tortoise paradox is not suchly about the existent eternity as of a possible eternity ( or only apparent proficient feasibleness ) of producing the sequence
1/2, 1/4, 1/8, 1/16, etc.
in existent clip
I hold with Scott Aaronson: and they state complexness holds no philosophic entailments!
Nonetheless, there is yet another bed in that narration. Anonymous told in a discourse in Shtetl-Optimized
What a beautifully cagy manner to react to such a line of oppugning!
Goodly, one should think that Alexander Yessenin-Volpin ( named in Wikipedia as Esenin-Volpin
) was one of the founding begetters of the Soviet human rights move and passed many geezerhood in prisons, exile and asylum. He cognizes a thing or two about questions; in 1968, he pent and spreaded via Samizdat
the famed `` Memo for those who expects to be interrogated
``, much utilise by fellow objectors.
It is singular how the personality of a mathematician can be formed on his work and his philosophic positions.
So, Alexander Sergeevich Yessenin-Volpin was likewise a pote of note. One of his poems, a rattlingly cagy and bitterly wry rendering of Edgar Alan Poe
's The Raven
, is rather revealing in the context of our discourse. I give here but the first two and the last three lines of the poem. ( A full text of the poem ( in Russian ) can be bumped here
and here
Nitrogen
vrmr!
N
vrmr...
`` N
vrmore!
To do these lines more friendly to the English speaking reader, I explicate that the first two lines name to Thomas More
's Utopia
: the booster reads Utopia
to avoid an accusal that he holds not familiarise himself with the Utopian pedagogies advanced by the totalitarian system. The three exclaimings `` Nevermore! '' which cease the poem make not postulate rendering.
The poem is indited in 1948 ( by a singular happenstance, the twelvemonth when Eric arthur blair pent his 1984
the rubric of the novel is only a substitution of figures; in 1949, when Orwel 's novel was printed, Yessenin-Volpin begined his first trance in prisons ). As we can see, Yessenin-Volpin, who was 23 ages old at the clip, developed an ultrafinitist attack to Utopian theories ( and especially to the Utopian pattern ) much earlier than to jobs of symbolic logic.
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