Peter van Inwagen's Extreme Metaphysical Realism

Peter van Inwagen is an American philosopher and John Cardinal O'Hara Professor of Philosophy at the University of Notre Dame. He was also the president of the Society of Christian Philosophers from 2010 to 2013.

Van Inwagen has been a major player in the debate about free will. He introduced the term 'incompatibilism' to stress the position that free will is not compatible with determinism. Van Inwagen has also taken part in the “afterlife debate”. (He wrote a piece called 'I Look for the Resurrection of the Dead and the Life of the World to Come'.) 

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The following piece is primarily a commentary on the 'Objectivity' chapter of Peter van Inwagen's book Metaphysics. 

Orwellian

Perhaps the final paragraph of Peter van Inwagen's chapter 'Objectivity' is what motivated him to adumbrate his positions against anti-realism. He writes:

“Before we leave the topic of Realism and anti-Realism, however, I should like to direct the reader’s attention to the greatest of all attacks on anti-Realism, George Orwell's novel 1984. Anyone who is interested in Realism and anti-Realism should be steeped in the message of this book. The reader is particularly directed to the debate between the Realist Winston Smith and the anti-Realist O’Brien that is the climax of the novel. In the end, there is only one question that can be addressed to the anti-Realist: How does your position differ from O’Brien’s?”

Does van Inwagen see anti-realism as some kind of postmodernist fashion designed to let “anything go”? Or perhaps he sees anti-realism as advancing various political projects (i.e., instead of truth). Another fact (rather than possibility) is that van Inwagen sees idealism as being indistinguishable from anti-realism when it comes to what really matters philosophically. Michael J. Loux (van Inwagen's fellow University of Notre Dame-based metaphysical realist) also sees “subjective idealism” as “the view that we make it all up”. 

If we get back to the van Inwagen passage above. 

Winston Smith is not a realist. Or, rather, he's neither a realist nor an anti-realist. That's because the dispute between anti-realism and realism is largely a 20th century phenomenon within the domain of Anglo-American analytic philosophy.

In addition, anti-realists don't necessarily have specific positions on politics or on anything else (i.e., other than on anti-realism and its relation to realism). Anti-realism can of course be applied to other subjects; though it's not necessarily tied to any other subject. Not only that: it's often said that anti-realists can be anti-realists in one domain and not in other domains. And even when anti-realists do apply anti-realist ideas to a particular domain, they still say different things about other domains.

Metaphysics

It's clear that van Inwagen's take on anti-realism and realism is largely motivated by his position on metaphysics itself.  In his paper, 'The Nature of Metaphysics', he writes:

“We might say that one is engaged in 'metaphysics' if one is attempting to get behind all appearances and describe things as they really are.”

Van Inwagen's definition also ties in with other realist definitions of metaphysics. Take Michael J. Loux again, who tells us that

“[W]e [realists] can, in good conscience, go on believing in a mind-independent reality and go on as well believing that metaphysics gives us access to the nature of being qua being”.

If we concentrate on van Inwagen, it can be said that just as van Inwagen defines the words “objective truth”, “anti-realism” and even “realism” itself in his own individual way (as will be seen later), so too does he do the same with the word “metaphysics”. 

For a start, even if there are philosophers who deny that we can “get behind all appearances” (or whom reject the very notion of appearances), can't we say that they're still doing metaphysics? More importantly, the final clause  (i.e, “describe things as they really are”) begs the question against anti-realism. Thus, from the beginning and even before any analysis/debate begins, van Inwagen is saying that anti-realists (as well as idealists, phenomenalists, Kantians, etc.) can't actually be metaphysicians. Why? It's because they aren't realists in the manner in which van Inwagen himself is a realist.

Van Inwagen also tells us that  

“metaphysics is the attempt to discover the nature of ultimate reality”.

Thus if one doesn't accept that definition (or if one questions the term “ultimate reality”), then one can't be doing metaphysics at all! So it's not a surprise that Inwagen also says the following: 

“It is therefore misleading to think of anti-Realism as a metaphysics... Anti-Realism, rather, is a denial of the possibility of metaphysics...”

It's true that anti-realists emphasise their epistemological approach to metaphysics. Yet that's still an approach to metaphysics. It isn't automatically a denial of the possibility or existence of metaphysics. Or, rather, it is if one accepts van Inwagen's position in full. Indeed it seems that one has to accept van Inwagen's position in full if one wants to continue doing metaphysics. 

Here again van Inwagen assumes that there is only one definition of both “realism” and “metaphysics”:

“And Realism is a metaphysics only in the sense that it is a thesis that is common to all metaphysical theories.”

To repeat: if one doesn't abide by van Inwagen's take on both realism and anti-realism (as well as his take on metaphysics itself), then one can't be doing metaphysics at all. 

Ultimate Reality and Mind-Independence?

Does anyone really know what the word “ultimate” means in the context of the often-used phrase “ultimate reality”? 

Van Inwagen, for one, tells us that 

“there is such a thing as ultimate reality, a reality that lies behind all appearances”. 

Now even if the word “ultimate” simply means reality as it is regardless of minds, we can still ask why the word “ultimate” has been used. 

In addition, many anti-realists will say that even if there is a reality that “lies behind appearances”, we can still never know what that reality is (or what it's like). So why use the adjective “ultimate” at all? Can x be an ultimate anything if it can never be known, seen, experienced or whistled? And how, exactly, does reality “lie behind” anything? What does it mean for reality to lie behind appearances? Indeed it can't it be said that appearances also belong to reality? What else can appearances belong to?

Van Inwagen is also most certainly wrong when he claims that the “anti-realists say that nothing is independent of the mind”. Or, at the very least, I can't imagine any anti-realist making that claim. Van Inwagen writes:

“The anti-realist who says that nothing is independent of the mind, however, really does mean something very much like this: the collective activity of all minds is somehow determinative of the general nature of reality.” 

As stated, I don't suppose for one second that most (or even any) anti-realists believe that “nothing is independent of the mind”. There are lots of things which are independent of the mind. However, once we speak of them, describe them, or have knowledge of them, then they become (by definition) dependent on minds. These things aren't literally created by minds - though they do come to be known or experienced by minds. 

Objective Truth

It's strange that van Inwagen takes the term 'objective truth' literally in that he says 

“our beliefs and assertions is therefore 'objective' in the sense that truth and falsity are conferred on those beliefs and assertions by their objects, by the things they are about”.

Of course no one would doubt or deny that statement S is about the things it is about. The problem is how we make philosophical sense of that claim.

Van Inwagen also assumes too much when he talks about what he calls “objective truth”. 

Firstly he tells us exactly what he takes objective truth to be. And then he says that when his definition is rejected or denied, then objective truth is automatically rejected or denied too. Yet we needn't accept van Inwagen's take on objective truth.

Van Inwagen tells us that 

“those statements would be objectively true that correctly described the ultimate or context-independent reality”.

In other words, van Inwagen is telling us that we have to accept both ultimate reality (as well as his philosophical take of it) and context independence before we can claim objective truth. However, that isn't a logical claim. It's not an epistemological or metaphysical claim either. Instead, it has all the hallmarks of a stipulative definition of the two words “objective truth”. A philosopher can easily argue that objective truth has nothing to do with “ultimate reality” or “context independence”. Sure, he'd need to argue his case and perhaps his arguments wouldn't be very convincing. However, this option is on the table; yet van Inwagen seems to assume that it simply doesn't even exist. 

Take just one example. Hartry Field has written an entire book called Truth and the Absence of Fact; as well as a paper called 'Mathematical Objectivity and Mathematical Objects' (in which mathematical objects are denied or rejected). Now Hartry Field's arguments may be airtight or they may be rubbish. But at least his option is in the marketplace. And there are many other similar options in the marketplace too.

We can go into more detail here. 

How would we know that statements “correctly described the ultimate or context-independent reality”? This too makes an obvious assumption. In order to know that our statements correctly describe ultimate reality, we'd already need to know ultimate reality in order to be sure that the matches between our statements and ultimate reality are correct. We'd also need to explain how statements (as bits of a natural language) could match things that aren't themselves statements (or bits of a natural language). In addition, does the tie between true statements about ultimate reality itself also partake in ultimate reality? For example, is the correspondence between the statement “Snow is white” and snow's being white itself an aspect of ultimate reality? What's more, do natural-language statements about ultimate reality themselves belong to “appearances”? And if not, why not?

As for context-independent reality: how can we gain access to it? What does it look like? Or, perhaps more correctly, what is it like? Can a context-free reality even be described or whistled?

Mount Everest

Van Inwagen puts the realist case in the simplest terms by invoking the nature of Mount Everest. He writes:

“.... if there had never been any intelligent beings on the earth, Mount Everest would, despite the absence of intelligence from the terrestrial scene, have exactly the size and shape it has in fact.” 

We can accept (provisionally) that Mount Everest would have “exactly the size and shape it has [] if there had never been any intelligent beings on the earth”. But what does that claim amount to? Van Inwagen is saying: 

Whatever x is, x will be as x is regardless of human minds. 

But that's to say very little. Almost nothing. All we have is what people have said about Mount Everest. The claim that minds don't change the nature of Mount Everest doesn't get us anywhere. It doesn't get us anywhere practically. More relevantly, it doesn't get us anywhere philosophically either. It's a claim that has very little substance. Despite that, it purports to tell us something profound about the force of metaphysical realism. 

So an anti-realist doesn't need to deny that any x is “entirely independent of all human mental activity”. However, he does have a problem with the prefixed statement: “the fact F is entirely independent of all human mental activity”. That's because it includes the predicate symbol F and also uses the word “fact”. It makes little sense to say that facts are “mind-independent”. 

It's certainly true that many philosophers have deemed propositions, universals and numbers to be mind-independent. However, there aren't many who've believed that facts are mind-independent too. Van Inwagen does. He writes:

“If there were no beings with minds, there would be no one to observe or grasp or be aware of this fact, but the fact would still be there.”

So, in this context, van Inwagen believes that the world is cut up into “sentence-shaped objects”, as Peter Strawson once put it. That means that not only is the mind-independence of the world important to van Inwagen, so too are the many (or infinite) sentence-shaped chunks of the world - those bits which precisely match true statements.

Yet x doesn't tell us what to say about it. However, we can state the following:

x [an obviously unquantified variable, in this case] is how it is regardless of minds. 

But what is x? -

x is independent of minds. What we say about x isn't.

Thus x's ontological purity is “collapsed” (to steal a word from quantum mechanics) by minds once things are said about it. Or, to use Kantian phraseology, when x is talked about, it immediately moves from being a noumenon to being a phenomenon.

Idealism

Van Inwagen also conflates idealism with anti-realism. 

Firstly, he asks the following question:

“Is not idealism essentially the thesis that there is no mind-independent world 'out there' for our sensations to be correct or incorrect representations of?”

And then he asks:

“And is not anti-realism the thesis that there is no mind-independent world 'out there' for our sensations to be true or false statements about?"

Idealism and anti-realism are very different. Despite what van Inwagen says, anti-realism does not state that there is no mind-independent world “out there for our sensations to be true or false about”. It says that there is one; and it's causally responsible for our sensations and our true and false statements. However, as it is “in itself” (i.e., in its mind-independent state) is of no meaning or purpose. It's a difference which quite literally doesn't make difference.

Donald Davidson (not usually deemed to be an anti-realist) summed up this position very well. However, in the following passage (from his paper 'A Coherence Theory of Truth and Knowledge') he talked primarily about “belief” and “sensation”, not about statements and the mind-independent world. He wrote:

“The relation between a sensation and a belief cannot be logical. Since sensations are not beliefs or other propositional attitudes. What then is the relation? The answer is, I think, obvious: the relation is causal. Sensations cause some beliefs and in this sense are the basis or ground of those beliefs. But a causal explanation of belief does not show how or why the belief is justified.” 

So it's worth rewriting this passage for clarification and putting it within the context of anti-realism and realism. Thus:

The relation between true or false statements and a mind-independent world cannot be logical. Since true or false statements are not the mind-independent world. What then is the relation? The answer is, I think, obvious: the relation is causal. The mind-independent world causes some true or false statements and in this sense are the basis or ground of those true or false statements. But a causal explanation of true or false statements doesn't show how or why those statements are true or false of the mind-independent world.

As hinted at, it can't be said that this rewriting works in all respects. That's primarily because Davidson was talking about the justification of beliefs, not the relation of statements to the world. Nonetheless, the form of the argument still stands and the central point about causation does too.

Another passage from the same paper by Davidson is even more apposite in this context. He wrote:

“Accordingly, I suggest that we give up the idea that meaning or knowledge is grounded on something that counts as an ultimate source of evidence. No doubt meaning and knowledge depend on experience, and experience ultimately on sensation. But this is the 'depend' of causality, not of evidence or justification.” 

Here again we can rewrite Davidson in the context of van Inwagen's take on realism and anti-realism. Thus:

I suggest that we give up the idea that true or false statements are grounded on something that counts as an ultimate reality. No doubt true or false statements ultimately depend on the mind-independent world. But this is the 'depend' of causality, not mind-independent truth or fact.

So, to repeat, anti-realists don't reject a mind-independent world (though some brands of idealism do). Instead, anti-realists simply has a different take on the mind-independent world. In Davidson's terms, many of our true and false statements causally “depend” on that mind-independent world. However, that mind-independent world doesn't and can't in and of itself guarantee us truth.

Bishop Berkeley

Since we've just discussed idealism, it's worth commenting on something else that van Inwagen writes in this respect.

Van Inwagen offers us an “argument without force” to show us that anti-realism fails.  His argument is about “form[ing] an image” of an event which “no one is observing”. He states:

“Isn’t it [the anti-realist position] like saying that a painter can never paint a picture of someone who is alone, since any attempt to do so represents the figure in the painting as being observed by someone who is occupying a certain point of view—the point of view that the viewer of the painting is invited, in imagination, to share?”

Firstly, van Inwagen simply assumes that the position above is wrong without actually saying why. What he appears to be saying is that since the person painted is painted as being alone, then the actual painting itself must somehow represent that person's genuine aloneness. In other words, the subject has been painted form literally no “point of view”. But clearly all paintings of people ostensibly on their own are painted from a point of view. There's not even a possibility of painting a lone person from no point of view – not even conceivably (except, perhaps, in some very abstract piece).

Thus Bishop Berkeley was perfectly justified (at least at first) in stating the following:

"But, say you, surely there is nothing easier than for me to imagine trees, for instance, in a park... and nobody by to perceive them... The objects of sense exist only when they are perceived; the trees therefore are in the garden... no longer than while there is somebody by to perceive them."

So when one imagines a tree (or person to be painted) without anyone looking at it, one actually imagines someone looking at a tree (or looking at the person to be painted) – except that the observer (or painter) isn't supposed to be there. 

Now of course realists, anti-realists or idealists/phenomenalists don't need to rely on this argument about our imaginative limitations to establish their positions. Nonetheless, Berkeley was still at least partly justified in saying what he said.

Despite that, van Inwagen may be correct to argue that “mind-independence [] does not require those to whom the argument is addressed form a mental image”. That's certainly the case if one is talking about “unobserved geological processes”. But it doesn't follow from this that what's been said also applies to a painter painting a supposedly unobserved person. Nonetheless, if mental images are out of the question for ancient geological processes, then so too are “certain verbal descriptions of those [geological[ processes”. That's because even without mental imagery, verbal descriptions are still at least partly – as well as obviously - a result of minds. Regardless of mental images, those descriptions will include contingent natural-language words, concepts, and whatnot. To put that another way: these seemingly mind-independent geological processes could be described in other ways – indeed in an infinite number of other ways!

Self-Referential Self-Destruction?

Van Inwagen attempts to sum of the anti-realist position in a single slogan. This:

“Objective truth and falsity do not exist.”

You can see what's coming now. What we have here is a self-referential self-destruction... Or so it seems. Van Inwagen continues:

“[The above] is a statement about all statements, and it is therefore a statement about itself. What does it say about itself? Well, just what it says about all other statements: that it is neither objectively true nor objectively false. And, of course, it follows from this that it is not objectively true. If it is not objectively true, if it is not true in virtue of corresponding to a reality that is independent of human mental activity, what is it - according to the anti-Realists? What status do they accord to it?”

Van Inwagen made exactly the same argument (in another article) about the position advanced by logical positivist at one point in the 1930s. Firstly he expresses their position thus:

“The meaning of a statement consists entirely in the predictions it makes about possible experience.” 

And then van Inwagen gleefully notes its self-referential flaws:

“Does this statement make any predictions about possible experiences? Could some observation show that this statement is true?... It would seem not... And, therefore, if the statement is true it is meaningless; or, what is the same thing, if it is meaningful, it is false.” 

Let's stick to anti-realism. 

Van Inwagen assumes that anti-realists deny or reject objective truth. Not all of them do. He also assumes that everyone must accept his own definition of the words “objective truth”. This is problematic because if they do so, then van Inwagen's conclusions about self-referential self-destruction would indeed be correct. So, again, anti-realist needn't deny the notion of objective truth and they certainly don't need to accept van Inwagen's own personal definition of what constitutes objective truth.

The other problem with van Inwagen's analysis is that even if anti-realists accepted the statement “objective truth and falsity do not exist”, what van Inwagen says about this statement may not be the case. Anti-realists could take the statement “objective truth and falsity do not exist” as a second-order (or a meta) statement. Either that or as a principle (normative or otherwise). In other words, it's a statement about statements, not a metaphysical statement. That is, it's not a statement about the nature of the world: it's a statement about statements about the world. Another way of putting that is to say that it's an epistemological take on statements about the world.

The failure to make this kind of distinction is summed up by the science journalist John Horgan when he recalls an interview with Karl Popper. He writes:

“'The first thing you do in a philosophy seminar when somebody proposes an idea is to say it doesn't satisfy its own criteria. It is one of the most idiotic criticisms one can image!'... Falsification itself is 'decidedly unempirical'; it belongs not to science but to philosophy, or 'metascience', and it does not apply to all science. Popper was admitting... that his critics were right: falsification is a mere guideline, a rule of thumb, sometimes helpful and sometimes not.” 

Conclusion

Peter van Inwagen isn't very fair to anti-realists - and that's apart from the fact that he never quotes a single one of them. Nor does he paraphrase particular arguments (or statements) from any anti-realists. (Though he does quote his very own fictional anti-realist.) Instead, he tells us what he thinks anti-realists believe. More importantly:

i) Peter van Inwagen misrepresents the positions of most anti-realists. 

ii) He at least partly conflates anti-realism with idealism. 

iii) And he defines terms such as “metaphysics”, “objective truth” and "anti-realism" in such a way that renders all his opponents straw targets or self-contradictory. 

Finally, van Inwagen assumes that ant-realism is a much wider philosophy than it actually is. 

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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