The paradox of increase

Monist, The, July, 2006 by Eric T. Olson

ABSTRACT

The paradox of increase in an ancient argument purporting to show that nothing can grow by acquiring new parts. If it is sound, similar reasoning leads to the more general conclusion that nothing can ever change its parts. After discussing the implications of this principle, the paper lays out the paradox in a way that reveals the premises that figure in it. It emerges that the paradox has no easy solution, and can be resisted only by taking on one of five serious metaphysical commitments.

1.

It seems evident that things sometimes get bigger by acquiring new parts. But there is an ancient argument purporting to show that this is impossible: the paradox of increase or growing argument. (1)

Here is a sketch of the paradox. Suppose we have an object, A, and we want to make it bigger by adding a part, B. That is, we want to bring it about that A first lacks and then has B as a part. Imagine, then, that we conjoin B to A in some appropriate way. Never mind what A and B are, or what this conjoining amounts to: let A be anything that can gain a part if anything can gain a part, and let B be the sort of thing that can become a part of A, and suppose we do whatever it would take to make B come to be a part of A if this is possible at all. Have we thereby made B a part of A?

It seems not. We seem only to have brought it about that B is attached to A, like this:

[FIGURE 1 OMITTED]

We have rearranged A's surroundings by giving it a new neighbor, but we haven't given it a new part. If B has come to be a part of anything, it is the thing made up of A and B after our conjoining. But that thing didn't gain any new parts either. It didn't exist at all when we started: our conjoining B to A brought it into existence. Or if it did exist at the outset, it already had B as a part then and we merely changed it from a disconnected or "scattered" object (like an archipelago) to a connected one.

So we have failed to give A a new part. And since this reasoning makes no assumptions about the nature of A or B or the manner in which we conjoined them, it seems to follow that nothing could ever increase in size by gaining a new part. The very idea of growth by addition of parts is incoherent.

Now I believe that some things can grow by acquiring parts; but it takes a good deal of controversial metaphysics to show how they can. The paradox of increase really is a paradox: its conclusion is more or less incredible, yet we cannot resist it without accepting something that looks nearly as bad. I begin by discussing some of the paradox's implications ([section][section] 2-5). I then state it in a more careful way that makes its premises explicit ([section] 6). The remainder of the paper asks what it would take to solve it ([section][section] 7-12).

2.

If the paradox of increase prevents anything from growing by gaining parts, a similar argument--running the paradox of increase in reverse--is likely to rule out a thing's shrinking by losing parts. Suppose we want to make an object X anything at all--smaller by removing a part, Y. That is, we want to bring it about that X first has and then lacks Y as a part. Imagine, then, that we detach Y from X in some appropriate way: let us do whatever would bring it about that Y ceases to be a part of X if X can ever lose a part and carry on without it. Have we thereby made it the case that X no longer has Y as a part? Have we made X smaller? It seems not. X starts out made up of Y and something else--"the rest of X," which I shall call Z--like this:

[FIGURE 2 OMITTED]

Afterwards Y is no longer attached to Z. But what has happened to X? it doesn't seem to exist any longer; or if it does still exist, it still has Y as a part, and we have merely changed it from a connected object to a scattered one. Either way, it doesn't get any smaller by losing a part. And of course Y and Z don't lose any parts either. It seems that nothing we can do would ever make anything smaller than it was before by virtue of having lost a part.

This is the amputation paradox or shrinking argument. (2) Like the growing argument, it assumes nothing about the nature of X or Y or the manner in which Y is detached. So it threatens to show that nothing could ever lose a part: the very idea of shrinking by losing parts is incoherent.

These arguments purport to role out a thing's growing by gaining a part or shrinking by losing one. If they are sound, it is hard to see how anything could exchange an old part for a new one without shrinking or growing either. So they suggest the general conclusion that nothing can have different parts at different times: it is absolutely impossible for anything to have a certain part at one time and exist without having that thing as a part at another time. Nothing can change any of its parts. If a thing has such and such parts, this must be a permanent and unchangeable feature of it. Call this the doctrine of mereological constancy:

Necessarily, if x is a part of y at some time, then x is a part of y at every time when y exists.