A Critical Introduction to Mathematical Structuralism

Mathematical structuralism is a theory in the philosophy of mathematics which argues that mathematical objects are defined solely by their place in mathematical structures.

Mathematical structuralism adopts a position that's common to most other philosophical structuralisms in that it denies that there are any “intrinsic properties” of objects. It even denies the very existence of objects apart from structures.

This means that we don't have objects (or things): we only have structures and relations. In terms of mathematical structuralism only: we don't have numbers until we also have structures and relations. In other words, numbers are born of their structures and relations.

Mathematical structuralism has been dated back to David Hilbert’s The Foundations of Geometry of 1899. In that work Hilbert states:

“We think of . . . points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as 'are situated', 'between', 'parallel', 'congruent', 'continuous', etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.”

Here it can be seem that mathematical objects gain their identity relative to their “relations” to other things. Thus “points, straight lines, and planes” are defined by relational terms such as “between”, “parallel,” “congruent,” and “continuous”. This entire package then takes the form of a structure (or a system). Thus the points, straight lines and planes have relations of betweenness, congruence and being parallel to other things by virtue of being part of a whole structure (or system) in which these relations can occur.

Hilbert was even more explicit about his own (proto)structuralism in the following correspondence with Gottlob Frege (as quoted by Stewart Shapiro):

“Every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points, I think of some system of things, e.g., the system love, law, chimney-sweep . . . and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras’ theorem, are also valid for these things . . . [A]ny theory can always be applied to infinitely many systems of basic elements.”

In terms of Paul Benacerraf's initial reasons for formulating mathematical structuralism.

Benacerraf firstly noted that algebraic theorists had no position on the ontology of mathematical objects. Such theorists were only concerned with their “structure”. Thus Benacerraf asked himself whether or not what is true of algebraic theories is also true of other mathematical theories.

As for mathematical structuralism itself, Hartry Field puts the mathematical structuralist position very clearly in the following:

“The core idea – which I'll call the structuralist insight – is that it makes no difference what the objects of a given mathematical theory are, as long as they stand in the right relations to one another.”

Clearly it's the case that in the passage above objects are played down and structures are played up. We can gain some purchase on what a structure is by talking about the “relations” (or the “right relations”) which “objects” need to have “to one another”. However, two obvious points need to be stated here:

i) It is objects which have these relations to one another.

ii) It is objects which are part of a structure.

These points will be tackled later. For now, let Benacerraf give a more detailed account of mathematical structuralism. He does so in the following:

“For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that.”

In simple terms, we can say that the number 1 is (partly) defined by being the successor of 0 in the structure determined by the/a theory of natural numbers. In turn, all other numbers are defined by their respective places in the number line. (See more detail on this later.)

It's worth noting here that “objective truth” (or at least truth) isn't rejected or denied by mathematical structuralists: it's just the account of how that truth comes about which is different to other accounts. Put simply, mathematical objects don't bring about objective truth: abstract structures do. Another way of putting this is to say that nothing is said about any mathematical object other than its place in a structure. Thus it seems to follow that there is no ontology of mathematical objects offered by mathematical structuralists.

Do Structures Give Birth to Numbers?

According to Benacerraf, the “structure” of a “particular sequence” provides the meat (as it were) of a number. Or, in an alternative phraseology, the structure (or the set of all similar structures) actually gives birth to the number. Firstly we have the structure, and only then do we have the number. We don't, in other words, firstly place a number within a structure because that would mean that the number already exists. Instead, the structure brings forth the number. What is important is the abstract structure, not the abstract object.

The idea that numbers are given birth to by structures is shown in the following passage:

“Only when we are considering a particular sequence as being, not the numbers, but of the structure of the numbers, does the question of which element is, or rather corresponds to, begin to make any sense.”

So firstly we have the structures. And then we have numbers as “elements” of these structures. But what can we make of these abstract structures before they have their elements or numbers? What do they “look” like in their naked form? To use a term and question from Quine (though he was referring to abstract propositions): what are the “identity conditions” of these abstract structures?

Again, how can we make sense of an abstract structure (or even of a “particular sequence”) without numbers (or at least without something) other than the abstract structure itself? In this case, Benacerraf's abstract structures appear to be like the bare substratums of certain ontologists or even like Kant's well-known noumena.

Of course others (including Benacerraf himself) have noted the problem with accepting abstract structures though not accepting abstract mathematical objects. The obvious problem is this:

If we can't gain epistemological access (whether causal or otherwise) to abstract numbers, then how can we gain access to abstract structures?

Thus, to repeat, Benacerraf's position is that he wants to get rid of abstract mathematical objects; though he's happy with abstract structures. Thus that means that abstractions (in and of themselves) aren't the problem. Take this passage:

“Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions.”

Thus if Benacerraf was attempting to get rid of numbers as abstract entities, then why didn't he also have a problem with “abstract structure”? Some or all of the questions asked about abstract (or Platonist) objects can now be asked about abstract structures. Indeed “progressions” themselves are also an abstraction.

In addition, if you take away the numbers from these abstract structures, what is left? Just a pure or naked abstract structure? But what is that? It's surely numbers that gives some kind of shape or reality to abstract structures or progressions. Indeed it's hard to even conceive what these structures could be without numbers or other mathematical objects (such as lines and planes in geometry, elements and operations in abstract algebra, etc.).

Types of Mathematical Structuralism

Prima facie, one may ask whether or not mathematical structuralists deny the existence of mathematical objects entirely or simply have a unique position on them.

One thing structuralists do share is that mathematical objects are “incomplete” in that only the structures they belong to make them complete. (Objects fill in the dots provided by structures.) However, an incomplete object is still an object of a sort.

Such objects are also said to lack “intrinsic properties”. So now we ask the following question:

Can an object with no intrinsic properties be an object at all?

(The denial of intrinsic properties in ontic structural realism, for example, does lead to the denial of objects/things themselves.)

Having said all that, certain brands of structuralism do endorse abstract mathematical objects of various kinds - not only structures. (Indeed it's hard to even imagine maths without abstract objects of some kind.) Some mathematical structuralist positions can even be deemed to be examples of Platonism.

Platonist Structuralism

Mathematical realists (or Platonists) believe that abstract mathematical objects exist independently of the mind. They also deem them to be eternal and incapable of change. (Of course this isn't to say that there's only an either/or situation when it comes to the philosophy of mathematics: i.e., either mathematical realism or mathematical structuralism.)

On the surface, mathematical structuralism appears to be radically at odds with Platonist (or realist) conceptions of numbers. That is because numbers in the Platonist scheme are (as it were) free-standing. That is, they exist apart from their relations to other numbers. Thus any relations numbers do have only occur after the fact. This means that any (necessary) relations between a given number and other numbers only come about because of the prior nature of the original number. That original number's nature makes those relations to other numbers possible.

Thus it's extremely hard to even conceive of what it would mean for numbers (or the number n) to have no relations to other numbers.

Nonetheless, it's entirely possible that a Platonist needn't necessarily believe that numbers are in fact free-standing (in either a basic or derivative sense of that word). Another way to put this is to say that numbers can still be seen as independent entities, yet they also have necessary relations to other numbers. In other words, even if number n is independent, it may still have necessary relations to other numbers.

In terms of specific Platonist positions, mathematical structures are deemed to both abstract and real. This position is classed as ante rem (“before the thing”) structuralism.

The Platonist position on structures can be characterised as the position that structures exist before they are instantiated in particular systems. The Aristotelian position on structures, on the other hand, has it that they don't exist until they are instantiated in systems.

The Platonist position can also be expressed by analysing the grammar of mathematical statements. Take the statement: 5 x 5 = 25. In this case, the numerals '5' and '25' refer to abstract objects. In other words, they are like (or even are) proper names.

To explain the Platonist position one can use Stewart Shapiro's own analogy. In his view, mathematical structures are akin to offices. Different people can work in a particular office. When one office worker is sacked or leaves, the office continues to exist. A new person will/can take his/her role in the office. Thus offices are like mathematical structures in that different objects can take a role within a given structure. What matters is the structure – not the objects within that structure.

Nonetheless, the people who work in offices are real. The idea of an office which is divorced from the people who work in it is, of course, an abstraction. Thus one may wonder why the office/structure is deemed to be more ontologically important than the persons/objects which exist in that office/structure. Surely it should be the other way around.

Aristotelian Structuralism

We also have Aristotelian structuralism. 

This is an in re ("in the thing") structuralism. Here structures are only “exemplified” in particular systems. That is, uninstantiated structures have no kind of existence.

A traditional problem (at least if it's seen as a problem) arises for this type of mathematical structuralism. Platonists claim that Aristotle's account of universals is problematic in that there may well be universals which have never been instantiated. Similarly with mathematical structuralism: there may well be bone fide structures which have never been exemplified or concretised (if we can use that latter term in this context) in a mathematical “system”. One reason cited for this possibility is that the world (or a part thereof) need not be tied to every mathematical structure. Therefore structures may exist and still not yet have been exemplified.

The earlier reference to universals isn't simply analogical or comparative. For example, just as the universal RED is to a particular red rose, so Stewart Shapiro (for example) believes that a universal STRUCTURE' is to a particular mathematical system. Thus traditional universals are instantiated by particulars; whereas a universal mathematical structure is exemplified by a mathematical system.

Paul Benacerraf's Structuralism

Finally we have Paul Benacerraf's position: post rem ("after the thing") structuralism.

In this case, abstract objects are completely rejected. This is why this position is sometimes classed as “eliminative structuralism”. That is, if “progressions” and relations are everything, then doesn't that mean that we can get rid of numbers altogether? (Or at least wouldn't it be possible to do so?) We may indeed have nodes (as it were) which exist within progressions and which plot relations and whatnot - though need they be numbers (as such)? That, of course, would depend on what we take numbers to be. And that's exactly the problem these philosophers are attempting to solve.

Benacerraf's position is also nominalist in nature. That is, even though it can be accepted that different structures have features in common, that commonality doesn't exist apart form its instances or exemplifications. Thus as with the nominalist position on red things: all red things don't share an identical property (or universal) that is RED (which, to Platonists, need not be instantiated). Instead the only things in common between different red things are their mutual similarities (which are taken to be “unanalysable” or “primitive”) and the fact that they're all classed as “red”. In other words, there is no universal RED or abstract mathematical STRUCTURE' biding its time in a Platonic domain waiting to be exemplified or concretised.

Nominalism?

Benacerraf's nominalist position is stressed by his reference to “notation” in the following:

“If what we are generating is a notation, the most natural way for generating it is by giving recursive rules for getting the next element from any element you may have...”

Thus all that matters (in this case) are “recursive rules”, not abstract or even concrete objects. In other words, if it's primarily about notation, then instead of using number symbols such as '1', '2' or '1001', we can use any symbol – such as 'cat', 'shlimp' or 'x*$*'. Indeed nominalists must go beyond the issue of the symbols we can use: the objects they refer to mustn't matter either. Or, rather, there are no objects at all – abstract or concrete. (In that sense, the symbols or numbers used are basically autonyms.)

Benacerraf puts this himself when he tells us that

“any recursive sequence whatever would do suggests that what is important is not the individuality of each element but the structure which they jointly exhibit”.

He continues:

“One would be led to expect from this fact alone that the question of whether a particular 'object' for example, [[[∅]]] – would do as a replacement for the number 3 would be pointless in the extreme, as indeed it is. 'Objects' do not do the job of numbers singly; the whole system performs the job or nothing does.”

We do have problems here. Firstly, surely we want to “generate a notation” for reasons which go beyond the notation itself. That is, we don't want to generate a notation solely for the sake of generating a notation. And, in this case, the notation is meant to capture what happens when we apply “recursive rules” to “elements” in order to get more elements. But then we're left with two questions:

i) What are recursive rules?

ii) What is an element?

For a start, recursive rules must be applied to (or operate upon) things which aren't themselves recursive rules. Benacerraf (in this instance at least) calls these things “elements”. So recursive rules have no meaning without these elements. (Just as Benacerraf elsewhere argues that numbers have no meaning outside of “progressions” or “structures”.)

In more detail, a recursive rule is displayed in the formal definition of the natural numbers according to Peano's axioms. Thus:

0 is a natural number, and each natural number has a successor, which is also a natural number.

This can be put more formally in the following manner:

0 is in N.

If n is in N,

then n + 1 is in N...

This means that we have the base case (i.e., “0 is a natural number”) and a recursive rule (“each natural number has a successor, which is also a natural number”). From that base case and recursive rule we can produce the set of all natural numbers.

However, we both start with numbers and end with numbers. That is, the base case is a number (i.e., 0) and the recursive rule both operates upon and produces numbers. Thus the recursive rule operates upon 0 and gets 1. Or can we say, instead, that the recursive rule actually creates the number 1? Perhaps we can. So does it create the number 0 too? Not if we're sticking to natural numbers because nothing comes before 0 - so no recursive rule could have created 0. And if the recursive rule somehow creates such numbers, it's still the case that it operates upon numbers too. So this is a two-way street we're discussing here.

Peano's axiom (at least grammatically) also seems to assume the prior existence of numbers. That is, the clause “0 is a natural number” and the clause “each natural number has a successor” seem to assume the prior existence of numbers. If that is correct, then recursive rules can be said to discover (not create) numbers; or, at the least, to codify and explain them.

Conclusion

Perhaps all this structure-object talk is an example of a “binary opposition”. Think here about the ontological debate about objects and events. Donald Davidson picks up on this in the following comment on P.F. Strawson's position:

“What does seem doubtful to me is Strawson's contention that while there is a conceptual dependence of the category of events on the category of objects, there is not a symmetrical dependence of the category of objects on the category of events.”

Put simply, this ontological hierarchy can easily be reversed (at least in most cases) without making any difference. In other words, there may well be no hierarchy at all: both objects and events are dependent on each other. Thus let's rewrite that passage from Davidson:

While there is a conceptual dependence of the category of numbers (as objects) on the category of structure, there is not a symmetrical dependence of the category of structure on the category of number.

In turn, does it really matter if we reverse that passage in this way? -

While there is a conceptual dependence of the category of structure on the category of numbers (as objects), there is not a symmetrical dependence of the category of numbers on the category of structures.

In addition to that, many mathematical structuralists seem to commit a logical sin that philosophers are always spotting in all sorts of others areas: that of assuming x in the very definition/description of x. In this case, they assume numbers in their definitions (or descriptions) of numbers. (Take as a comparison the notion of metaphysically-realist truth as seemingly presupposed in epistemological and coherentist accounts of truth.)

This can also be seen in something that Stewart Shapiro writes.

In defence of the structuralist position, Shapiro argues that “in the system of Arabic numerals, the symbol ‘2’ plays the two-role”. He also states that “anything at all can play the two-role in a natural number system”. Here the word “two” has been used to describe a structural “role” which is meant to provide the meat as to what a number is. It's true that Shapiro uses the word “two” rather than the symbol “2” - but does that make a difference? We can of course argue that the word “two” is a contingent natural-language expression; whereas the number 2 is an abstract object. Yet even when it comes to using the word “two”, the number 2 is still both presupposed and tacit.

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