Parts and wholes again
August 16, 2007
by Michael
The Maverick Philosopher proposes the following paradox:
"This paradox is from Peter Geach, Reference and Generality, Cornell, 1980, 215. The following formulation is mine.
1. There is exactly one furry cat, Tibbles, on a mat.
2. Tibbles minus one hair is a proper part of Tibbles.
3. If Tibbles has n hairs, then there are at least n proper parts of Tibbles. For each of Tibbles' hairs, there is a proper part of Tibbles which is Tibbles minus that hair.
4. Each such proper part of Tibbles is a cat.
Therefore
5. There are n + 1 cats on the mat.
Therefore
6. (1) is false.
Something is wrong with this reasoning since it implies that if Tibbles has 1000 hairs, then there are 1001 cats on the mat, which contradicts (1). But where is the mistake?"
His concluding paragraph suggests that the mistake is in (4). My reflections:
The argument also seems to imply that each hair simultaneously belongs to 1000 different cats, which is odd. Further, if there are n hairs and n+1 cats, it seems like there also has to be a sad little hairless cat on the mat with the others.
At first glance I would like to take issue with (2) and (3) and then make some distinctions regarding essential and accidental parts and wholes. But perhaps my discomfiture is simply with how he's defining "proper part". Because if each hair is a proper part (part [a]), and the whole composite ([a]+[b]) minus that hair is another proper part (part [b]), then each part [a] becomes a sub-part of a part [b] (ceasing to be a part [a]?) whenever a new part [a] is counted. Doesn't the aberrant multiplication of the whole come as a result of re-counting each part multiple times?
Comments
On August 17 at 12'35 AM
, Neil wrote:
What exactly is meant by “proper part”? It all sounds dreadfully quantum, “you can’t pet the same cat twice, because when you pet the cat once, hairs come off, and you’re left with a different cat”. This may be broadly true but is entirely unhelpful.
More generally, if Tibble-n=Tibble, than Tibble+n=Tibble, which means that 5 says there are n+1 cats, which is true, so (1) is true.
On August 17 at 6'30 PM
, Joseph Method wrote:
Yeah, the mistake is totally in #5. It’s a misinterpretation of the phrase “each such proper part” in 4. Each such part refers to the part of Tibbles that remains minus n*hair, that is it’s the same part no matter what you do to it. In programming that’s called a pointer variable. That works whether Tibbles - n*hair is evaluated for each hair (each separately) or cumulatively for all of the hairs, I guess Tibbles - n!*hair.
On August 17 at 6'30 PM
, Joseph Method wrote:
Yeah, the mistake is totally in #5. It’s a misinterpretation of the phrase “each such proper part” in 4. Each such part refers to the part of Tibbles that remains minus n*hair, that is it’s the same part no matter what you do to it. In programming that’s called a pointer variable. That works whether Tibbles - n*hair is evaluated for each hair (each separately) or cumulatively for all of the hairs, I guess Tibbles - n!*hair.