To me,
(1) There is at most one half of an apple in the fruit bowl
sounds fine, but
(2) There is at most one half of an object[1] in the fruit bowl
sounds odd. (I trust that others will think the same, but feel free to correct me if I’m wrong!) Surely, if an apple is an object, half an apple is half an object? But if (1) makes sense and (2) doesn’t, does that mean a half of an apple is not half an object? Perhaps the thought which drives the sense of oddness is that nothing is a half simpliciter, it is always a half of some sort of thing: One mile is half of two miles, and if we dropped the term "mile" and started calling two mile intervals “stretches”, one mile could simply be called “half a stretch”. Yet this “halfness” is nothing ontically basic; one and the same thing is half a stretch, one mile, two half-miles, and five thousand two hundred and eighty feet.
Even so, why couldn’t we have at most half of an object in the fruit bowl if “object” picks out a genuine category? If we can eat half an apple and drive half a mile, what prevents us from throwing half an object across the room? We could certainly throw half a football. The answer to these queries might be that a given entity which is a fraction of one thing is always a whole (and thus one) of something else. E.g., one slice of a pizza is also one eighth of the whole pie. We can also note that if there is at most half of an object in the fruit bowl, it cannot be the case that there is at least one, and if the bowl is not empty that is impossible. Thus we cannot say there is at most half an object in the fruit bowl; wherever we have a fraction of one sort of object there is always at least one of different sort. All this goes to show that "object" picks out a very special category if it picks out any at all.
The above reflections seem to commit us to a Fregean view in which nothing is intrinsically one or many. This view is not without its problems: Does it make good metaphysical sense to hold that how many things there are depends on what category we apply to them? Aren’t we contradicting ourselves if we say that one thing can be identical to many? After all, one deck of cards cannot be identical to two decks, why should it be any more possible for it to be identical to fifty two cards? Yet if we do not embrace the Fregean view, how else might we explain (or explain away) the strangeness of (2)? These are difficult issues, but hopefully with your input we can get a clearer view of the matter.
[1] “Object” being construed broadly as covering anything that exists.
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7 comments:
Hi Jason,
I think you are right.
One can do analogy with a distance for example. While one distance can't be half of other distance, the distance can't be "half distance" simpliciter.
BTW, the view is not defended just be Frege. Hegel for example, argued that numbers are rations (so require two sides which will be compared), and if I remember right Frege referred to some other philosophers before him that held similar view.
Oops, "while one distance can be half of other distance" it should be.
Hi Tanasije,
Thanks for your comment. I wasn't aware that Hegel thought all numbers are ratios (you did mean ratios, right?) That sounds interesting, but isn't it trivially true in virtue of the fact that any number can be written as a ratio of itself to one? E.g., 2 can also be written as 2:1, 4 as 4:1, etc. Did Hegel have something more substantive in mind?
Also, do you remember who any of the other philosophers were? I think Wittgenstein held a similar but even more radical view, calling 'object' a psuedo-concept. But he came after Frege.
Yeah, ratios. I guess I was hungry :)
Hegel's idea is I think what you were talking about. That only by comparing something by something else you get a number (and that is where ratio comes in).
So, there Hegel doesn't speak of ratio as something which holds among numbers (e.g. 2:1),and is outside incidental relation between two of them, but about something numbers are... i.e. analyzing them as concepts.
Hope this helped.
As for other people before Frege thinking of the numbers as requiring some kind of comparison, it is just some vague memory. I will try to find something in the books, and add another comment if I do.
Hi... Half an apple is an ordinary object (e.g. it can be picked up and thrown about) so in going from (1) to (2) you have clearly generalised incorrectly. But maybe (2) could make sense, e.g. if a big indivisible was half in and half out of the fruit bowl, and we took "object" to mean ordinary object? But it does seem that in a wider sense of "object" (2) is silly, since if the bowl could divide something (into a half in and a half out) then it could not be indivisible...
Anyway, I too think that amounts (e.g. whole numbers) of stuff have to be relative to units of stuff. E.g. the deck of cards: it being given that it is a deck of cards, it is clearly a many, clearly 52 things. It is one deck (one collection, one many) and 52 cards, but of course it is also lots of molecules, 104 pictures, 4 suits, etc. The cards being ordinary objects, and the deck being the usual complete set of them, that example is both paradigmatic and deceptive.
...deceptive because one deck is the same thing, the same aggregation of physical stuff as its 52 cards, but not the same collection (given the axiom of extensionality), and yet our paradigm examples of collections are just such things as decks of cards! Mill's empiricist approach to numbers might be relevent; and I dimly recall something about Helmholtz, (but I'm less sure of that).
Hi enigman, thanks for your comment. I think we agree that there is a lot of intuitive force behind the numbers-being-relative-to-units view. But it seems we can apply Leibniz's law to raise some difficult questions for it. For if the deck is really identical to the cards, Leibniz’s Law says all of their properties ought to be the same. Yet as you say in your second post, “ […] one deck is the same thing, the same aggregation of physical stuff as its 52 cards, but not the same collection (given the axiom of extensionality) […]”
My question is: Is the deck a collection? If it is, and if it is identical to the 52 cards, then a) the 52 cards are a collection and b) they are the same collection as the deck. If the cards weren’t a collection, or if they weren’t the same collection as the deck, they couldn’t literally be identical to it, because the deck is surely the same collection as itself. So if you’re right in saying the deck and the cards aren’t the same collection, they can’t be the same aggregation of physical stuff either if being the same aggregation entails numerical identity. I think we should probably shift to a weaker view in which composition is different from identity. We can say that the deck is composed of 52 cards, or that a given card is composed of so many molecules, but they are not identical to the things which compose them because they have different properties.
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