What i and -i could not be
According to realist structuralism, mathematics is the study of structures. Structures are understood to be special kinds of complex properties that can be instantiated by particulars together with relations between these particulars. For example, the field of complex numbers is assumed to be instantiated by any suitably large collection of particulars in combination with four operations that satisfy certain logical constraints. (The four operations correspond to addition, subtraction, multiplication, and division.)
An allegedly attractive feature of realist structuralism is that it is faithful to mathematical practice. Unlike various forms of eliminativism or fictionalism, we can accept mathematical theorems as literally true statements about an objective, mind-independent part of reality. Unlike classical Platonism, we don't have to assume that there is a special realm of abstract particulars. According to realist structuralism, the number 2 is not a special particular, but a "place in a structure". In fact, the number 2 figures in different structures, and thus has different properties depending on whether we do arithmetic, real analysis, or complex analysis.
OK, but what exactly is a place in a structure? If we're realist structuralists, we don't want to say that the number 2 is an abstract particular. So presumably it is itself a structural property. What else could it be?
On this view, the number 2 is a property that can be instantiated by different particulars within an instantiation of a more complex structure. More specifically, let C(x) be the conjunction of all predicates true of the number 2 in the complex field, expressed in terms of the structural relations of addition, subtraction, etc., and logical expressions. Then the number 2 is the property of being an x such that C(x).
On reflection, though, this doesn't actually work. For one thing, if we want to take mathematics at face value, we now have to say that '2+0=2' states that the result of applying the addition operation to the aforementioned property C(x) and a similarly defined complex property C'(x) is the property C(x). That is, the addition operation must be defined to operate on structural properties. But note that the addition operation '+' also figures in C(x). And here it arguably can't be interpreted as an operation on structural properties. After all, we want to say that in any instantiation of the complex field by some particulars P, addition relates the elements of P, not abstract properties that remain constant from instantiation to instantiation.
I haven't seen this problem discussed anywhere. But another problem has recently received some attention. The problem is that the numbers i and -i have the exact same structural properties (in the complex field). That is, the complete structural description of i also completely describes -i, an vice versa. So if numbers are "places in a structure", and a place in a structure is a structural property, then it seems to follow that the number i is the very same number as the number -i. But we don't want to say that i = -i.
So as realist structuralists, we shouldn't say that numbers are structural properties.
But then what are the numbers, if we don't have anything else in our ontology than properties and concrete particulars? It's not helpful, I think, to keep talking about "places in a structure", or about "parts of complex properties". These are metaphors, and as far as I can tell there is no good explanation of what they could mean.
It seems to me that what a realist structuralist should say instead is that numbers – even numbers in the context of complex analysis – are not determinate things at all: numerals like '2' and 'i' are not straightforwardly referring terms.
Rather, we have the complex structure C, in Plato's heaven. This structure has two "places" for i and -i, in the sense that any instantiation of the structure will identify two individuals as i and -i. That's what we mean when we say that i is not equal to -i. The statement (i ≠ -i) is not a statement about two specific things – two individuals, or two properties. Translated into ontologese, it is a universally quantified statement about all (possible) instantiations of the the structure C.
But now we're two thirds of the way to eliminative structuralism. According to eliminative structuralism, mathematics is not the study of a special domain at all. Not of special, abstract particulars. Nor of special, abstract structures. Rather, mathematical statements are interpreted as universally quantified statements about all (possible) instances of relevant axioms.
The upshot, I think, is that it's harder to "take mathematics at face value" than many structuralists claim. A statement like 'i ≠ -i' seems to express the non-identity of two definite things. But it's hard to see how it could do that.
Indeed, it is hard to see how on any view the terms 'i' and '-i' could have determinate reference. Even if you're a classical Platonist and believe in a special domain of complex numbers, how does our word 'i' manage to latch onto a specific element of that domain?
There might be a nice application here for Kit Fine's "semantic relationism".
Hi wo,
Interesting read! I'm certainly no expert and not necessarily a friend of structures myself, but here are some loose thoughts:
1. One thing that perplexes me a bit is why do you presume that a stucturalist would want (or need) to identify the number 2 with some property. Isn't it quite customary for a structuralist to say instead that numbers are not properties but objects of a special kind - namely, positions within the structure? For this to be convincing, of course, you'd have to take seriously the structuralist's talk of positions and it looks like you dismiss it as being too contrived and metaphorical. What would convince you otherwise? (In particular, I wonder what's your take on an axiomatic approach of the sort provided by Nodelman and Zalta.)
2. The argument leading to $latex i = -i$ seems to hinge on an indiscernibility principle, and one strong enough to guarantee that two elements (or places/positions) of a given structure are identical whenever they are related by an automorphism of the structure in question (cf. Keranen's paper). But why think that a structuralist must commit herself to such principle in the first place? Why can't she, say, treat the identity of positions as primitive and proceed from there, thus avoiding the need for an explicit identity criterion? I suppose that some epistemological considerations might be at work here, but as I said, I'm not overly familiar with this.
P.S. Oh, this could be a red herring, but what you say about $latex i \neq -i$ being a universally quantified statement bears a prima facie resemblance to Tim Button's hybrid solution and his distinction between basic and constructed structures.