The Platonist Rod Paradox
Page last updated 25 Nov 2020
The formulation of this paradox was inspired by the Platonist interpretations of the Balls in the Urn paradox. (Footnote: Unlike the case of Balls in the Urn paradox, in this paradox no piece that is added is ever subsequently removed. )
Consider the following:
At 60 seconds to noon 9 rods of length 1 are added, and then 9/10ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.
At 30 seconds to noon 99 rods of length 1 are added, and then 99/100ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.
At 15 seconds to noon 999 rods of length 1 are added, and then 999/1000ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.
And so on…
Each step takes zero time to accomplish.
Now consider the following questions:
- At noon what is the length of the composite rod?
- At noon how many pieces are there?
- At noon what is the length of each piece?
A typical response is that:
- The length of the composite rod is 1 (since its length is always 1 after each step).
- There are infinitely many pieces (since their number increases at every step and there is no limit to the number of steps).
- The length of each piece is zero (since the size decreases at every step and there is no limit to the number of steps).
But this is not the result that the Nested Intervals Theorem of set theory gives, (Footnote: For example, as described in Ethan D Bloch’s book, The real numbers and real analysis, Springer Science & Business Media, 2011, Bloch: The real numbers and real analysis: Details. ) and which states that:
- the intersection of infinitely many nested intervals is not empty, and
- if the limiting value of the left ends of the intervals and the limiting value of the right ends of the intervals are identical, then the intersection is a single point which is that limiting value.
But the limiting value of all the left ends are all zero, and the limiting value of all the right ends are also all zero, which gives the result that at noon the rod is a single point rather than a rod of length 1. This is in direct disagreement with (1) and (2) above, hence the paradox.
So how does the paradox arise?
It arises because the process never actually completes. As in the case of the Balls in the Urn paradox, the reference to time in the description is a red herring, see the section Time as a red herring on the page Balls in the Urn paradox. Since the process cannot complete, the pieces never actually become of zero length, and must always have some finite length, regardless of how small that might be. And there is no step where the left end of any piece becomes coincident with the right end of another piece.
So the correct answer to (ii) is that there is no limiting value to the quantity of pieces, noting that there is no step at which the value actually becomes infinite (note that the term “infinite” is frequently treated as though it refers to a particular quantity, whereas in fact, it refers to a situation where there is no upper limit to a quantity).
And the correct answer to (iii) is that the limiting value of the length of the pieces is zero. Since the process never terminates, the pieces must always have some length greater than zero, regardless of how small that length can be.
In this way, the apparent paradox is resolved. The Platonist notion that an infinitely recursive process can actually “complete” leads to the paradox. This goes hand-in-hand with the Platonist notion that mathematical things actually exist independently of any definition. According to Platonist beliefs, every mathematical entity (in the above definition, every interval that can be defined by that definition) already somehow “exists” prior to the creation of the definition, and there is a non-physical mathematical “reality” where every step in the process already “exists” prior to the creation of the definition.
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