Suppose you could make the following observations:

- Ling (零) was a pianist
- Every pianist has a favourite student
- Different pianists have different favourite students
- Ling was not the favourite student of any pianist*
- Anything that Ling knew and that every pianist teaches to his favourite student ends up known by everyone in the Ling School of Piano (it’s like martial arts)

If the first three observations continue to be true, the Ling School of Piano will obviously go on forever. From all five, you can deduce a lot of mathematical results about it, starting with most of arithmetic . Piano arithmetic turns out to be just the same as ordinary arithmetic†.

On the other hand, if you make these observations (per Dennett)

I am a mammal

The mother of every mammal is a mammal#

you can deduce that there have been infinitely many mammals. Which turns out not to be the case.

The problem was identified by the Greeks, if not earlier, as the logical paradox of the heap (σωρίτης, “sorites”). Suppose:

I have a heap of sand

If I remove a single grain of sand from a heap, I still have a heap of sand

From this, we can deduce that even when I’ve removed every grain of sand I will still have a heap of sand. Which seems silly. Most philosophers‡ who have considered the question say that the argument is wrong somehow.

There is a randomised-trial version of the sorites: “Should we randomise the last patient?”

Note that:

It is unethical to choose a treatment at random when there is reliable evidence as to the best treatment

Therefore, it is unethical to conduct a randomised trial that is larger than needed to produce reliable evidence

Stopping the trial one person earlier won’t affect whether it produces reliable evidence

From these principles we could deduce that randomised trials of non-zero size are never ethical.

While this is a coherent viewpoint, it’s an extreme one. The output of the argument seems much stronger than the inputs. The paradox of the heap is reassuring: not that it implies randomising the last patient is actually ethical, but that it supports thinking about the question in real settings rather than as an exercise in piano arithmetic.

**he was a brilliant pianist, but he liked thrash metal and dyed his hair*

*† at least, as applied to the ‘standard’ members of the Ling School — there could be nonstandard members who aren’t the favourite student of any standard member*

*# that’s kind of the point of mammals*

*‡ but not all; these are philosophers we’re talking about*