Slicing it Thin

Abstract

Could there be a physical object think of something brick-shaped, say two inches high with infinitely many parts think of the bottom half, the half-inch-thick layer above it, the quarter-inch layer above it, the eighthinch layer above it, and so on ad literally infinitum with these infinitely many parts all being non-abstract, physical objects, entering into the causal history of the world in distinguishable ways? A metaphysical question, surely, both in having been discussed by metaphysicians (recent example: Milov Arsenijevid [1]) and also in the unfortunate characteristic that the hardest part of answering the question is understanding it. Suppose we understand the question in the sense a physicist would find natural. Then the answer seems to be straightforwardly negative, for reasons having to do with the atomic composition of matter and the vagaries of quantum mechanics. The conceptual question that metaphysicians ask involves a more abstract notion of possibility. Well, then, is it logically possible, suspending the laws of physics and making arbitrary hypotheses about the nature of matter? Hypothesize that matter is homogeneous, exactly filling regions of Euclidean space, and that the laws of nature allow differential interaction with any solid, understanding a physical solid as the material content of a geometrical solid in the sense of Tarski [4]. Then the positive answer follows immediately: that a brick, or any other regular open point-set, divides into infinitely many disjoint regular open point-sets is a mathematical triviality. But that is too easy. Are there hypotheses, about the composition of matter and the laws of physics, which aren't too far-fetched, and allow us to make sense of the notion of a brick with infinitely many physically distinguishable layers? But how are we to understand far-fetchedness? Allowable hypotheses should be recognizable as not-too-extreme generalizations and modifications of actual physics: cf. David Lewis's discussion of Quine's construction of ersatz possible worlds in the 'Foundations' chapter of [3]. But what is too extreme? What we want is a hypothesis such that a world satisfying it would permit observations analogous to and only observations analogous to the sorts of observations on which our actual knowledge of the physical world is based, hypotheses that, so to speak, change only the empirical details of the world, but leave the phenomenology of our physical knowledge as it is. But this still sounds fairly vague. Perhaps, rather than attempting at the outset to enunciate criteria of permissibility for science fictions, we should ask what the point of these