- What is a supertask?
- What's the paradox? The lamp must be on and off at noon!
A time line showing when the Thompson lamp is switched on/off the first few times
- Time taken for supertask = 30s + 15s + 7.5s + 3.75s + ... = 60s x
(1/2
+ 1/4 + 1/8 ...) = 60s.
- There's nothing wrong with the task, it simply doesn't specify what happens at noon!
- Except, we know the 'law' of the lamp and the complete state of the lamp, so ...
- ... the laws must be indeterministic from noon on
- ... since the laws of physics are deterministic, either the
lamp
is physically impossible or a description of a physically possible
Thomson
lamp would determine the state at noon.
2. Zeno's Arrow Paradox
- An argument against movement which assumes that time is entirely made of indivisible instants (moments) – for example, if instants of time are like points on a line.
- Note: I said instant not instance!
- Why can't an arrow (or anything else) move during an indivisible instant?
- Problem 1: if it can't move during any instant, and time is entirely made of instants, then surely it can't move ever: when would it move?
- Responses:
- The argument involves the 'fallacy of composition'.
- The at-at theory of motion: e.g., suppose the arrow's position is (x(t), y(t)) = (10t, 10t – 5t2).
A graph of distance (x) vs height (y) showing that the arrow is at a distinct position at each time (t); the positions form a continuous series over time. E.g., (a) after .5s the location is (5m, 3.75m), (b) at 1s, (10m, 5m), and (c) at 18s, (18m, 1.8m).
- Problem 2: How does it get from one position to the next if it doesn't move during any instant?
- Response: There is no next position – between any two there is a third!
- Problem 3: if it can't move during any instant, then its speed at every instant is zero – it doesn't move.
- Responses:
- No problem for zero duration instants: 0m/0s ≠ 0m/s!
- According to the at-at theory speed is the limit as t gets smaller of the average speed: e.g., suppose x(t) = 10t, and consider the speed at t=10s – it is 10m/s not zero!
| Interval |
Duration |
Distance Covered |
Average Speed |
| 9s-11s |
2s |
110m – 90m = 20m |
20m÷2s = 10m/s |
| 9.9-10.1s |
.2s |
101 – 99 = 2m |
2÷.2 = 10m/s |
| 9.99-10.01s |
.02s |
100.1 – 99.9 = .2m |
.2÷.02 = 10m/s |
| ... |
... |
... |
10 m/s ... |
Table of distances covered over progressively
shorter
intervals around 10s.
- Bergson: 'motion is composed of immobilities'.