Wolfgang Schwarz

Implicit Definitions, part 1: Mathematics

I vaguely believe that there are no implicit definitions. So I've decided to write a couple of entries to defend this belief. The defence may well lead me to give it up, though. Anyway, here is part 1.

Explicit definitions introduce a new expression by stipulating that it be in some sense synonymous or semantically equivalent to an old expression. For ordinary purposes this can be done without the use of semantic vocabulary by stipulations of the form

N = ...	(when N is a new name)
x (Nx ...x...)	(when N is a new predicate)
x (Nx = ...x...)	(when N is a new function expression)

Let's call any successful introduction of a new expression an "implicit definition" if it is not an explicit definition. Some apparently implicit definitions are uninteresting because they are logically equivalent to explicit definitions. For example, if G and H are old (interpreted) predicates, the following stipulations might be used to implicitly define F:

x y(Fx Hxy)
x(Fx Gx)
x(Gx Fx x Hxy))

Since together these stipulations are logically equivalent to x(Fx Gx y Hxy) , this definition is really just an explicit definition in disguise.

Now, are there any essentially implicit definitions? A classical example are mathematical expressions. Some say that the axioms of set theory implicitly define "set". Hilbert says his axioms of geometry implicitly define "point". And Crispin Wright and Bob Hale argue that Hume's Principle implicitly defines "the number of". The idea is that by stipulating the truth of these principles, we somehow give a meaning to the new expressions in them. I think this can't be right.

Firstly, the very idea of stipulating truths is rather odd. Truth is not the kind of thing that can be stipulated. For example, you can't just stipulate that God created the world at 4000 B.C, not even if you claim that this is meant to define the previously uninterpreted expression "God". After all, from this stipulation it follows that something created the world at 4000 B.C., which may well be false. Likewise it follows from Hilbert's axioms and the axioms of ZF and Hume's Principle that there are lots of mathematical objects, and again, the question of mathematical existence is not the kind of thing that could possibly be resolved by stipulation, I think.

Contrary to what one frequently reads, this has nothing to do with existential import in particular: You also can't just stipulate that every God is almighty and every rabbit is a God, even though this does not imply the existence of anything. But it implies that every rabbit is almighty, which is still false. The only cases where an alleged stipulation appears unproblematic to me is where it is really an ordinary, explicit definition: Hereby I stipulate that something is a God iff it is a red apple.

Secondly, let's assume for the sake of the argument that substantial facts can just be stipulated. To take a simple example, let's stipulate the axioms of group theory, thereby defining the new name "e" and the binary operator "*":

x y z(x*(y*z) = (x*y)*z)
x (x*e = x e*x = x)
x y (x*y = e y*x = e)

In what sense have we now established a meaning for "e" and "*"? Since "e" is an individual constant, we must presumably have given it a reference. So what does "e" refer to? Is its referent an ordinary thing, like this book, or a mathematical entity, perhaps a set? Or have we, by stipulating, just created this e-thing out of nothing? That would not be a further problem, given that reality can just be stipulated. The real problem is that the axioms don't tell us. They don't tell us what this e-thing is, whether it is newly created or not. For this reason I think they do not succeed in determining a meaning. It's like trying to establish a meaning for the new name 'a' and the new predicate 'F' by stipulating that Fa.

Hilbert agrees that his axioms don't really determine a meaning for "point": They don't determine which things are points and which aren't. This book, for example, is a point under some interpretations but not under others. This is even more obvious for group theory: In mathematics, all kinds of things play the role of e. What the group axioms define is not so much "e" and "*", but rather the three-place predicate "X,e,* are a group", as follows: For any set X (used as domain of quantification in the following), thing e, and operation *, X,e,* are a group iff [here follow the axioms]. This is an explicit definition, and the expressions "e" and "*" are now simply bound variables. I don't think it makes any sense at all to speak of defining bound variables.

Well, what if we just take away the quantifier binding "e" and use the remaining formula as a definition of "e": "e is the thing such that for some set X and operation *, etc."? The problem is that if the axioms have any model at all, then anything is such that for some set X and operation *, etc., and if they don't have a model, then nothing is. So in both cases the uniqueness required by the descriptor is missiing. We could at best use this suggestion to define a predicate "is an e-thing", which would either be empty (if the axioms have no model) or universal (otherwise). Similarly, we could use Hume's Principle to define predicates "is the number 0", "is the number 1", etc., which would again be either all universal or all empty, and we could use Hilbert's axioms to define universal or empty predicates "is a point" and "is a plain". But what's the point of introducing all these predicates? And why these complicated ways of defining just another universal predicate? And won't these definitions have unwanted consequences, like that every point is a plane?

A while ago, I said that eliminative structuralism is just a simplified version of supervaluationism. In the light of the foregoing, I'm not so sure about this any more. Structuralism resolves indeterminacy of several highly and precisely intertwined concepts at once. For example, a structuralist 'precisification' of "successor", determines (or is) also the precisification of "0", "+", "*" and "number". On the other hand, the different precisifications used in a supervaluationist interpretation of a sentence with several indeterminate expressions, may be largely independent of each other.

Note that in the structuralist interpretation, the genuinely mathematical expressions are again bound variables, and therefore need no definition. On this reading, the purpose of using "successor", rather than just an ordinary variable like "f", is to indicate the relevant antecedent in the structuralist interpretation: "0 is not a successor", unlike "a is not the f of anything" makes it clear that what you mean is that for any a,f, etc. that satisfy the principles of arithmetic, a is not an f of anything.

This is of course not the only possible interpretation of mathematical expressions once implicit definitions are rejected. We can also define "0" and "point" explicitly as some kinds of set; and if we follow Lewis (1991, not the structuralist Lewis of 1993), we can explicitly define sets to be some kinds of mereological sum. Or we can refuse to define "set" at all.