Parsons on Structuralism

Charles Parsons, "The Structuralist View of Mathematical Objects," Synthèse 84: 303–346, 1990.

Parsons takes structuralism to be the view that "reference to mathematical objects is always in the context of some background structure."

He takes Resnik's work to be a paradigm example, just noting that Resnik likes to call structures patterns.

Structuralism is recently popular and often traced to work of Benacerraf, so Parsons notes that the view dates back to Dedekind, Cantor, Riemann for special cases, and general structuralist claims can be found in Bernays in the 1950s and Quine by the late 1960s, associated with the doctrine of ontological relativity.

The definition of the view immediately poses an obvious problem: either our structuralism cannot apply to structures themselves or it is circular.

Parsons begins by exploring the version of structuralism in which structures are characterized first, independently and prior to structuralism about other things. That view, while perhaps less interesting, is already problematic, and all the problems are inherited by the circular version, so it is worth getting clear on them in the simple case.

Parsons just takes structures, in the standard mathematical way, to be certain sets, and so he assumes that sets are given somehow prior to our structuralist theory. How to characterize sets is a problem we postpone.

Dedekind gave one of the first attempts to introduce the natural numbers (instead of just taking them for granted) in "Was Sin und Was Sollen die Zahlen" (What are the numbers and what should they be) in 1884. His account is at least a precursor to the modern structuralist ones, and serves as a good example. Note that it is contemporary with the development of set theory, and so he uses some intuitive notions closely related to what we would now take to be set theory.

He defines a simply infinite set $N $ to be one such that there is a distinguished element 0 and a mapping $S:N\longrightarrow N-\{0\}$ such that induction holds:
For any set $M$ if 0 is in $M$ and for all $x$ if $x$ is in $M$ then so is $Sx$, then $N $ is a subset of $M$.

Having given this definition, he says something mysterious:
"If, in considering a simply infinite system $N $, ordered by a mapping $\phi$, one abstracts from the specific nature of the elements, maintains only their distinguishability, and takes note only of the the relations into which they are placed by the ordering mapping $\phi$, then these elements are called natural numbers ... ."

Parsons considers first, an interpretation of what Dedekind that seems to go against this final statement of abstraction, and he discusses other writings of Dedekind that, he says, raise "doubts as to whether the [reading we are about to discuss] is the best reconstruction of his [Dedekind's] intentions."

Here is the interpretation: we take some statement to be true of the natural numbers just if it is true of all $N,0,S$ that form a progression.

Parsons calls this type of view eliminative structuralism because it accounts for, for example, talk about the natural numbers while elminating any need for commitment to the existence of natural numbers.

The view of White, which Resnik discussed, is remarkably similar. The main difference is that White says true of all structures of a certain type, while the Parsons view says true of all $N,0,S$, making explicit use of set theoretic notions instead of postulating structures.

This view is, as Parsons points out, remarkably durable: people keep returning it after trying to work out positive notions of what the natural numbers are.

Tait has tried to capture what Dedekind meant by abstraction, and I will present Vaught's mathematical interpretation of Tait's view of abstraction. We propose an "abstraction function" ${\mathcal{A}}$ that takes structures to structures and has the following properties:

\[{{\mathcal{A}}({\mathfrak{A}}){\text{ is a structure isomorphic to }}{\mathfrak{A}}\]

This, Dedekindian, version of structuralism, might be called introductive structuralism. The formal notion of abstraction does not fully capture what Dedekind says, since the structure picked by the abstraction function is a particular structure that may have properties over and above what Dedekind wants to allow. Tait wants to handle this by doing something like claiming that what is true of the abstracted structure qua abstracted structure is only what is true of that structure in the language of the structure.

Eliminative structuralism has an advantage over a view that claims that any simply infinite set can be taken to be the natural numbers.

There is a standard objection to eliminative structuralism that, while common, is just wrong. Numbers are used for things like counting (applications). That is, they are placed into external relations. If we have eliminated numbers, it is said, we have no way to do that.

The objection is mistaken, though it is difficult to phrase the reply in a fully general way, it is easy to give an example that makes it clear that the solution is general.

Here goes: We have a structure with a domain that consists of numbers $N $, things, $T$, functions from numbers to things, $F$ and nothing else. We have some symbols that apply only to numbers ($0,S$), we have some symbols that apply only to things, and for the sake of the example, those include $C$ a unary predicate symbol that applies to all and only chairs in this room. We have a three-placed relation symbol $V$ such that the relation $V$ applies to $(a,b,c)$ if and only if $a$ is a function, $b$ is a number, and $c$ is a thing. There is a sentence in the language I have developed that says "There is a function from the numbers less than 42 one to one and onto the things $t$ such that $C(t)$.

This gives us the required external relation for a particular natural number system. We need to show that it is true that "For all natural number systems ... there is a function from the numbers less than 42 one to one and onto the things $t$ such that $C(t)$.

But that is obviously the case: given a different natural number progression, we can form a structure isomorphic to the one with which we started by swapping out that progression and plugging in the new one.

There is a much more serious problem for this type of elimination of the natural numbers. It doesn't work if there aren't any natural number progressions. We therefore seem to need to prove that there are such to complete the theory. To be fair, many philosophies of mathematics (perhaps even all the nonconstructive ones) have an analogous problem.

Nothing like Dedekind's considerations yield that there is a collection with infinitely many member in a nontrivial sense. You can replace thought of with set of to get $\{\},\{\{\}\},...$, but you can't get that those are members of a single set without using some form of an axiom of infinity.

There are other cases in which Dedekind's strategy does succeed: if we assume set theory (which we were in fact assuming), then we can prove the existence of a natural numbers progression. If we assume the the existence of the natural numbers, then we can prove the existence of the integers and the rational numbers. And so forth: Given a place to stand ... .

This provides a nice way to introduce the rational numbers without running into the multiple realizations problem.

p. 311 "One lesson we might dray is that we should distinguish what is required for a structuralist account of a particular kind of mathematical object, such as the natural numbers, and what is required to give a general statement of the structuralist view."

The version of structuralism we have considered for the natural numbers presupposed sets. We can get rid of the sets by using second-order logic instead:

Here is the key line in the definition of a natural numbers progression, copied from above:

For any set $M$ if 0 is in $M$ and for all $x$ if $x$ is in $M$ then so is $Sx$, then $ N$ is a subset of $M$.

There are lots of references to sets, but we can make them go away as follows:

For any $M$, for any $S$ if $M(0)$ and for all $x$ if $M(x)$ then $M(Sx)$, then $(\forall x)(M(x)\rightarrow N(x)) $.

That says "the same thing" but without the sets. Our conditionals (For all $N,0,S$ if they are a natural numbers progression then $A(N,0,S)$) are now sentences of pure second-order logic, no objects are assumed. This could be called a version of logicism, and everyone calls it deductivism or if–thenism. Unlike traditional logicism, we have the assumption of the properties of natural numbers progressions in the antecedent of a conditional. You can also have, among many minor variants, a first-order version of if–thenism. Parson's criticisms apply to all versions, and so we won't discuss the others further.

Of course, while we are using arithmetic as our example, the same method can be applied to many familiar mathematical systems: the real numbers, $3$-dimensional geometry, to some extent set theory, ... .

This program has been criticised for introducing induction by means of a definition, instead of deriving as a fundamental consequence of the way we understand the natural numbers.

If–thenism faces the same problem the eliminative structuralist program does of nonvacuity: that is, there had better be at least one progression, or the whole thing trivializes.

For the eliminative structuralist, at least in Parsons's version, that required the existence of a certain set with a function on it. In second order logic, it is less clear what is required: one needs only that a certain sentence is not a logical truth ("There are no progressions"), and so perhaps all we need is that it is possible for there to be a progression (Putnam). It is highly disputable what, exactly, possible might mean here. After all, isn't an abstract object that is possible for that very reason actual?

One can give a formal account of possibility in terms of possible worlds, but that isn't on for present purposes, because the possible worlds themselves, when talking about mathematical possibility will be actual mathematical objects, and so they defeat the purpose of possibilism, which is all that a theory might be nontrivial whether or not it has any actual models, so long as it is possible for it to have models. Thus, "all" possibilists take some notion of possibility to be basic, fundamental, not in need of further explantion. All possibilist theories are subject to the criticism that the understanding of the properties of possibility in fact relied on a picture of possible worlds, which ends the game.

One notion of possibility that pretty clearly is independent of possible worlds is syntactic consistency.

"Definition." A theory is consistent if there is no proof on the basis of the theory of a sentence and its negation.

Now, we have to know what we mean by theory and what we mean by proof. If these are given standard set-theoretic interpretations one finds in mathematical textbooks, then consistency is just more mathematics, and so this is a reduction like showing that there is a rational-number structure using pairs of natural numbers. That one relies on the natural numbers, this one relies on some set-theoretic version of strings of symbols or the like.

If we want full-blown structuralism about essentially all of mathematics, that notion of consistency won't do.

There is another notion of consistency that avoids the trap just mentioned: interpret proofs nominalistically, that as concrete sequences of actual inscriptions. Consistency might then mean there will never be a proof of an inconsistency or there never could be (new modality) such a proof.

We have two alternatives: do if–thenism with second-order logic or with first-order logic. With second-order logic, there is no completeness theorem and so a theory can be consistent and still not have models. Nonetheless, we might view consistency as the closest thing to a guarantee we can hope for. The first-order version is weird because we have no way to talk about the natural-numbers structure, and so all we get is a series of approximations. So we have a series of eliminations of approximations to the natural numbers, such that what is true in any of them, on the account is true of the natural numbers, but not so for falsehood. Even if this is acceptable as an account of arithmetical truth, it is rather unsatisfactory as an account of what serves as the arithmetical structure.

The advantage of the first-order case is that consistency is not only necessary but sufficient for the possible existence of a structure (or at least so one can hope to show).

At any rate, the sufficiency fails for the nominalistic version of consistency: suppose the shortest proof of an inconsistency is larger than would be possible to inscribe in any physically possible universe. Then we would have nominalistic consistency, but still no model.

We would not allow our mathematics to be hostage to our physics.

So long as we're allowing physical structures to enter in, there is a more direct path: we seem to be in great shape if there is a physical progression. We then have existence on clearly non-question-begging extra-mathematical grounds. In fact, there is such a structure: take two points. Consider the set including the first one, the point half-way from that to the second, ..., ordered by left-right.

If this counts as a physical example, it still doesn't suffice, since if it turns out, as Hilbert speculated, that spacetime itself is quantized, and so there is no such sequence, while that might affect the utility of the natural numbers, it wouldn't affect their existence. The example, while sufficient in some minimal sense, gets the modalities wrong: the natural numbers definitely exist in some sense much stronger than that in which the example does.

An analogous remark holds of nominalistic consistency.

Modalism Already discussed.

Second-order logic If Quine is right that second-order logic is just set theory in sheep's clothing, then using second-order axiomatizations in an if–thenist version of eliminative structuralism allows in the back door what it kicked out the front.

There are two well-known attempts to get around this:
Field's geometry
Boolos's plural quantifiers

The conclusion is that they too smuggle in set theory.

The coup de grâce to full-blown eliminative structuralism: We would need some justification for the claim that there could be a structure answering to the principles of set theory, just as we needed some way to get the possibility of some progression to show that the elimination of the natural numbers yields a nontrivial theory.

Set theory is so far from anything physical, temporal, or mental, for example just in its sheer size, that anything like a realization of it, wherever you find it, is going to consist, in effect, of mathematical objects, and so the elimination fails. Worse, you can't easily just give up on set theory, since any theory of structures adequate to a structuralist story will have the problematic features of set theory.

Parsons thinks (in his usual cautious and qualified way) that this shows that any successful global structuralist philosophy of mathematics would have to be "non-eliminative."

So now, we are up to discussing a form of structuralism like that of Resnik: he thinks that there are positions in patterns.

Here the point seems not to be to show how to live without mathematical objects, but to say what kinds of things mathematical objects are, which turns out to be something like, positions in patterns or structures. Notice that it is not the structures themselves that are the objects, but the positions in them.

The obervation that leads to non-eliminative structuralism is that the objects of mathematics have nothing to them except their relationships to other objects. They do not seem to have internal or intrinsic or ... properties. That is a chief feature of what noneliminative structuralism is trying to capture.

There is some reason to doubt this in the case of sets. Consider the number of chairs in this room (40) and the set of chairs in this room. If one of the chairs ceased to exist, that wouldn't do anything to the number 40. The same does not seem to be true of the set of chairs.

Perhaps mathematics should not be about sets of chairs, but only pure sets or at least only sets of mathematical objects, but the intuition remains in such cases that a set of things really intrinsically has the things in it, and that is not a structural property.

To make what is for our purposes the same point in a different way, one can take any progression that doesn't include Julius Caesar, any position in the progression, and replace what is there by Julius Caesar, and you still have a perfectly good progression in which Julius Caesar is, for example, now the successor of 39. Now take set theory, and replace $\{\varnothing \}$ by Julius Caesar. We no longer have a membership relation.

This point is worth making because most structuralists have gone straight from considerations about cases like the natural numbers to global structuralism, without stopping to see whether the point is general.

When structuralists say that objects in different structures have no identity conditions between them, that is too fast, since the structure we are considering must vary by context: it we are considering numbers and sets, it is normally appropriate to consider a single structure with both numbers and sets in its domain. Thus, which questions will make sense will depend heavily upon the context in which the questions are raised in a way that is not well understood.

Resnik has made similar remarks, that we already discussed, and so I'll say no more here.

It is usual to think that the key point about mathematical objects according to the structuralist is that they are incomplete (2 is less than 3, but it is neither a set nor not a set). Mathematical objects share that quality with fictional objects (which is one of the attractions of fictionalism as a philosophy of mathematics).

Parsons points out mathematical objects are incomplete in a sense that is much more radical than are fictional objects. Sherlock Holmes is, in the story, a detective, in the ordinary meaning of the word. He lives on Baker Street, a street that actually exists, and so forth. He is a cocaine user, and it is real cocaine that he is supposed be using, though there is no fact of the matter about the brand of his works.

Mathematical objects are not just incomplete, they have no "real" properties independent of their theory: no mathematical detectives or streets. When a number is less than another, the notion of "less than" is not some independently given prior notion, it is just as incomplete as the objects to which it applies.

Finally, what is a structure? Structuralism is about objects in structures not about the structures themselves, and so, to get started, we need a nonstructural characterization of structures. There is nothing impermissible about climbing up that ladder and throwing it away, ending up with structurally described structures, but we can't start with them.

Parson's favorite account is linguistic: the domain of a structure is given by a predicate, and the relations and so forth by relations and so forth suitable related to the predicate:

"$x$ is a natural number"

"$x$ is a natural number, $y$ is a natural number, and $x$ is less than $y$"

That will give the background without postulating any entities to serve as structures, and the things of which the domain predicate holds will be the structuralists numbers.

We can talk about a particular progression, for example, as follows:

"$x$ is a string of strokes"

"$x$ and $y$ are strings of strokes and a string of the type of $x$ occurs in $y$"

This breaks the circle, but it gives an account of structuralist mathematical objects in terms of quasiconcrete objects, namely, words and phrases. -- ProfessorShaughanLavine? - 18 Apr 2005