Wednesday, December 08, 2010

English Numbers

Hartley Slater in The Reasoner 4(12), 175–6, tried to show, from the fact that the number of elements in the empty set is zero, that zero is not, as a matter of English grammar, the empty set—and in general, that numbers are not sets—because we don’t say that the number of elements in the empty set is the empty set. But things aren’t quite that simple.
......To begin with, Slater’s example of zero—which is often defined to be the empty set in pure mathematics—was an unfortunate choice, because mathematicians introduced zero relatively recently. Consequently English remains rather ambivalent about its status. There being no elephants in this room, for example, it’s false that there are a number of elephants here. So from it being true that there are zero elephants here, surface grammar might seem to indicate that zero isn’t even a number (a cardinal number). But zero is of course a number (the number of elephants in this room, the number of elements in the empty set).
......For another example, the numbers one, two, three etc. correspond to the positions first, second, third and so forth. And since no sequence has an element before the first one—that’s what ‘first’ means—so, in that ordinary sense, there’s no zeroth element, and so again, surface grammar seems to indicate that zero isn’t a number (an ordinal number). Nonetheless, there’s a more mathematical sense in which whenever an element is indexed by 0, it’s a zeroth element.
......In many mathematics textbooks there’s a Chapter 0, for example, containing the set-theoretic basics. Of course, such chapters don’t amount to much evidence that mathematicians take numbers to be nothing more than sets. Mathematicians make the standard identification of numbers with sets in order to prove theorems from set-theoretic axioms. They are thereby following in the footsteps of those who did geometry by proving theorems from Euclid’s axioms. And surely few if any geometers thought that there was nothing more to space than Euclid’s axioms. Space was rather the obvious space around us, of which Euclid’s axioms were taken to be true (and obvious enough to be the premises of proofs).
......Now, even if the space that we see around us is Euclidean—having been constructed as such by our brains from our sensory input—it’s surely not unlikely that what Aristotle meant by ‘space’ is non-Euclidean. So, similarly, even if our concept of zero comes (for example) from reifying the definitive property of an absence, it doesn’t follow that it’s impossible that Euler’s ‘0’ referred to an empty set. Indeed, the standard empty set can be an urelement—can be anything that has no members (where membership is an axiomatic primitive)—because its job within standard set theory is simply to have no members, and so in that sense (at least) zero can be an empty set.
......But more to the point, Slater’s argument may beg the question. That’s because if ‘the empty set’ was a definite description of zero then we could say that the number of elements in the empty set is the empty set (for all that we wouldn’t usually). After all, we can say that the number of ones in zero is zero. In general, for natural numbers n, the number of ones in n is n. Perhaps it would be more natural for us to say that two twos are four (for example), and hence that the number of twos in four is two. But such equations all follow from the natural numbers—most obviously those greater than 1—being essentially sums of ones, which seem to be some sort of collection, perhaps not unlike sets of points in that, while their elements are in obvious ways identical, they are distinguished in ways that derive from their origins (as positions in space, in the case of points).
......Two twos are four because any two things plus another two things are four things. And in English, there being a number of things of some kind is just there being some things of that kind. So again, surface grammar indicates that numbers—most obviously those greater than 1—are some sort of collection. And we might expect mathematicians to be the experts on what exactly numbers are. So, since mathematicians prove theorems about numbers from set-theoretic axioms, we’ve some evidence that numbers are sets.
......Still, such evidence is compatible with numbers being axiomatic sets only in a rather abstract way (cf. how the integers with addition are an abelian group). Slater’s argument was based on surface grammar, so it was presumably that numbers are not sets in some more obvious sense. So note that collections in the usual, informal sense can be variable, like a stamp collection, or non-variable, like a chess set. A fundamental question in this area is therefore whether mathematicians have discovered that numbers behave like sets—at least to the extent that the natural numbers are, collectively, non-variable—or whether they’ve just tended to assume that (even though we can’t so easily assume that cardinal or ordinal numbers are non-variable, in light of the famous set-theoretic paradoxes).
......Mathematicians don’t prove the standard Axiom of Infinity—which asserts the existence of a set containing one element for each natural number (amongst other axioms giving such sets the properties one would expect of non-variable collections)—but rather prove theorems from that axiom (along with the others), or work from some other foundation. Philosophical arguments are therefore needed, to assess whether the standard axioms are giving us a scientifically adequate description of the natural numbers or not. But arguments based on surface grammar are unlikely to be of much help in this area. After all, they can’t even show zero to be a number. (For a more apposite sort of argument, see my 2003: ‘Infinite Sequences: Finitist Consequence,’ The British Journal for the Philosophy of Science 54, 591–9, and my 2010: ‘Two Envelopes, two paradoxes,’ The Reasoner 4(5), 74–5.)


Xamuel said...

Hah, I could repeat, verbatim, the very first comment I made on this blog: whether "zero" is the empty set, or an equivalence class with a couple elements, or an equivalence class with countably many elements, or an equivalence class with cardinality-of-the-continuum many elements, depends whether you mean zero the natural, zero the integer, zero the rational(or real), or zero the complex number, respectively. (Whew, sorry for the run-on sentence!)

Mathematicians *define* zero to be the empty set. We *define* numbers to be (insert definition here). We then explore the properties of these things for their own sake; these properties would not change if we called the objects "paperclips" instead. The apparent relationship to real life is a bonus.

enigMan said...

Yes, philosophical problems are never resolved to the point of actually going away... Now, you say that the relationship to real life is a bonus. I would say that it is essential.

Were there no relationship to real life, between any of these 'paperclips' and the informal natural numbers that we learn about in primary school, would such 'paperclips' be studied by mathematicians?