This paper has two components. The first, longer component (sections 1–6) is a critical exposition of Armstrong’s views about the metaphysics of mathematics, as they are presented in Truth and Truthmakers and Sketch for a Systematic Metaphysics. In particular, I discuss Armstrong’s views about the nature of the cardinal numbers, and his account of how modal truths are made true. In the second component of the paper (section 7), which is shorter and more tentative, I sketch an alternative account of the metaphysics of mathematics. I suggest we insist that mathematical truths have physical truthmakers, without insisting that mathematical objects themselves are part of the physical world.
A prime number
But what kind of a thing is a number?
This is a difficult question, but one point at least seems clear:
numbers and other mathematical entities are “abstract,” in the sense
that they have no causal powers and no location in spacetime. We are
told that the number zero was discovered in India, but it would be a
mistake to go to India now to look for it—and not because it
has subsequently been moved. You can’t trip over the number three. The
polynomial
And so, to labour the point, it seems that some abstract entities exist.
And yet David Armstrong began his last book by endorsing what he called “naturalism”:
I begin with the assumption that all that exists is the space-time world, the physical world as we say. […] [This] means the rejection of what many contemporary philosophers call “abstract objects”, meaning such things as numbers or Platonic Forms or classes, where these are supposed to exist “outside of” or “extra to” space-time. (2010, 1)
Despite this naturalism, Armstrong did not reject any part of mainstream mathematics. Indeed, he insisted that the truths of orthodox pure mathematics are necessary and a priori (2010, ch. 12).
In this paper, I explore Armstrong’s attempt to reconcile his denial of abstract entities with his commitment to orthodox mathematics. To be more specific, the paper has three goals.
Armstrong wrote a vast amount on mathematics, and this writing is spread among many papers and books. Some of this work is complex, and Armstrong changed his mind on certain important questions. My first goal in this paper is to describe—clearly, briefly, and in one place—Armstrong’s mature views on the metaphysics of mathematics, including relevant aspects of his work on the metaphysics of modality. To prevent the discussion from sprawling, I focus particularly on Armstrong’s account of cardinal number, as it is presented in his last two books: Truth and Truthmakers (2004, “T&T”) and Sketch for a Systematic Metaphysics (2010, “SSM”). Section 1 and section 2 describe Armstrong’s views about the cardinal numbers; sections 3–6 focus on modality.
My second goal is to present some novel—and, I believe, definitive—objections to Armstrong’s views on the metaphysics of mathematics. These objections are presented in sections 5 and 6.
My third goal is to recommend a different way of thinking about the metaphysics of mathematics—an approach which will, I hope, appeal to people who admire Armstrong’s work. Briefly, I will suggest that we insist that every mathematical truth has a truthmaker in the physical world, without also insisting that mathematical objects themselves are physical things. The proposal is presented in more detail in section 7.
1 Cardinal Numbers as Concrete Entities
The claim that numbers have no spatial location is familiar to metaphysicians, but it sometimes comes as a surprise to students. “But there are three pens on my desk right now!” they exclaim, implying that the number three itself is within arm’s reach. According to Armstrong, the surprised students are on to something.
While he rejected “Platonic forms” which exist outside spacetime, Armstrong did believe that there are properties which exist within the particulars that instantiate them (SSM, ch. 2). For Armstrong, there exists a property is red which exists within London buses, ripe tomatoes, and male cardinals. As he sometimes put it, properties are “immanent” rather than “transcendent.” He inferred that all properties are instantiated. Uninstantiated properties have no place in spacetime, and so no place in Armstrong’s philosophical system (SSM, 15–16). He made the same claim about relations (SSM, 23).
For Armstrong, a cardinal number is a relation between a particular
and a property. Specifically, the cardinal number κ is a relation that a
particular
On Armstrong’s view, then, the surprised student is correct to think that cardinal numbers exist within the “spacetime world.” It turns out, then, that one can coherently maintain that numbers exist and that there are no abstract entities.
This is a striking result—and yet, a problem looms. As I said,
Armstrong insisted that properties and relations exist only when they
are instantiated. On this view, the number
It is tempting to reply to this objection by insisting that space is infinitely divisible. Discussing the problem of size as it arises for Aristotle, Jonathan Barnes writes:
Physical objects are, in Aristotle’s view, infinitely divisible. That fact ensures that, even within the actual finite universe, we shall always be able to find a group of
objects, for any […] If the universe consisted simply of a single sphere, it would also contain two objects (two hemispheres), three objects (three third-spheres) and so on. We shall never run short of numbers of things […]. (1985, 122)
This cannot be considered a satisfactory solution to
Armstrong’s problem, however. For one thing, it is far from
clear that Aristotle was correct in thinking that space is infinitely
divisible: those who study quantum gravity have been known to speculate
that space is in fact discrete. And so the proposed solution is somewhat
“hostage to fortune.” More importantly, even if the proposed account
does secure the existence of large finite numbers such as
In passing, I note that Armstrong faced a problem of size in his
account of set theory too. While Armstrong identified cardinal numbers
with relations, he identified sets with individuals. The central claim
in Armstrong’s account was David Lewis’s “brilliant insight” (T&T,
120): the mereological parts of a set are precisely its non-empty
subsets. For example, Lewis’s claim implies that the mereological proper
parts of
2 Armstrong’s Possibilism
Armstrong was aware of the problem of size. He responded by claiming that larger cardinal numbers exist in posse though not in esse:
A Platonist will solve the problem [of size] by postulating uninstantiated numbers. […] [However,] my own hard choice […] is to accept a deflationary doctrine of what it is for a mathematical entity to exist. Plenty of mathematical structures exist in a straightforward sense, because they are instantiated. We can call them empirical mathematical structures. […] But mathematical existence itself, I suggest, should be reckoned as something less. A mathematical entity exists if and only if it is possible that there should be instantiations of that structure. (T&T, 117)
Armstrong called this possibilism. The doctrine is not easy to interpret. In the above passage, Armstrong seems to suggest that a mathematical entity exists provided that it could be instantiated—even if it is in fact not instantiated. However, in this same passage, Armstrong contrasts his possibilism with the “Platonist” claim that there are uninstantiated mathematical entities.
The following quotation gives us a clue about what Armstrong meant:
We say ‘
’, but this can be rendered more transparently, though more boringly, as necessarily, if there are seven things and five further things, then the sum of these things are twelve things . (T&T, 101)
What this passage suggests is that, for Armstrong, while the sentence
“
Now Armstrong did not develop this proposal systematically; instead, he endorsed Geoffrey Hellman’s modal structuralism:
I recognize, of course, that asserting here this […] doctrine of mathematical existence is to a degree a matter of hand-waving. I have not the logico-mathematical grasp to defend it in any depth. That has been done, in particular by Geoffrey Hellman. (T&T, 117)
The reader who wants a thorough discussion of modal structuralism should consult Hellman (1989). For now, a back-of-the-envelope summary will be sufficient. The modal structuralist claims that the theory of the natural numbers is not a description of some particular sequence of entities; rather, the theory concerns all possible models of the Peano axioms.5
For example, when a mathematician asserts that there are infinitely many prime numbers, what is really meant is something like this:6
It is necessary that, in any model of the Peano axioms, the domain contains infinitely many prime elements.
Notice that this statement does not imply that there is a model of the Peano axioms somewhere in spacetime.
When a modal structuralist mathematician asserts that, necessarily, every model of the Peano axioms contains infinitely many prime elements, she will of course wish to rule out the suggestion that this is true “vacuously”—that is, simply because models of the Peano axioms are impossible. Thus, a modal structuralist mathematician will claim that models of the Peano axioms are possible.
Hellman discusses in some detail how to extend this approach beyond
number theory, and into applied mathematics. We need not look into these
details. For our purposes, the key point is that the appeal to modal
structuralism allows Armstrong to say that “
The attractions of the approach are obvious, but Armstrong’s possibilism brings with it a new problem. Armstrong was a truthmaker maximalist—he believed that every true proposition has a “truthmaker,” that is, an entity in the spacetime world which is sufficient (and perhaps more than sufficient) to explain the truth of the proposition.8 Armstrong was thus stuck with the formidable task of identifying truthmakers for complex modal truths like those described above. It is my contention that Armstrong did not succeed at this task, as I shall explain in the next three sections.9
3 Armstrong’s Entailment Principle
Before we look at Armstrong’s discussion of truthmaking and modality,
we must consider his Entailment
Principle, which is crucial to his account. Some notation will be
helpful: I will put a sentence between angled brackets to represent the
corresponding proposition. For example,
Entailment Principle (SSM
Version). If
For example, since
Disjunction Principle.
To see the conflict, consider the following argument:
Let
This little argument appears to show for any two true sentences
In Truth and Truthmakers, Armstrong gives a more sophisticated version of the Entailment Principle which is not subject to the same objection:
Entailment Principle (T&T
Version). If
Here, entailment* is some non-classical entailment relation, to be
specified. By insisting that
We have seen that the SSM Version of the Entailment Principle conflicts with the Disjunction Principle, and that one can maintain the Disjunction Principle by endorsing the T&T Version instead. If a truthmaker theorist wishes instead to maintain the simpler, SSM Version of the Entailment Principle, she may choose to reject the Disjunction Principle—and indeed philosophers have presented independent reasons for rejecting this principle.11 We need not settle this dispute here. Suffice it to say that appeals to the SSM Version of the Entailment Principle are subject to dispute.12
4 Armstrong on Truthmaking and Possibility
Having briefly looked at the Entailment
Principle we are ready to consider Armstrong’s account of
truthmaking and modality. Let’s start with possibility. Suppose that the
sentence
Possibility Principle. If
The principle is not attractive on its face. As Pawl (2010) has
pointed out, Armstrong’s being legged is a truthmaker for
(1) | (Assumption) | |
(2) | (Assumption) | |
(3) | (From (2), and the nature of the contingency of propositions) | |
(4) | (From (1), (3) and the Entailment Principle) |
As Armstrong later recognized, this argument is fallacious. The error
is in step (3): in no standard modal logic is
it true that
So the Possibility Principle is implausible on its face, and the argument Armstrong gave for it is unconvincing. I think we should conclude that the principle should be rejected.14
Happily, Armstrong also offered another and more attractive account of truthmaking and possibility. The idea is this. Suppose that we have (separately) two slices of bread, fifteen slices of cheese, and two slices of tomato. These things could have constituted a cheese and tomato sandwich—although in fact they don’t. Plausibly, they together form a truthmaker for the proposition that a cheese and tomato sandwich could exist. Armstrong wrote:
Consider, in particular, the cases where the entities in question do not exist, where they are mere possibilities. It is, let us suppose, true that
it is possible that a unicorn exists . What then is a minimal truthmaker for this truth? The obvious solution is combinatorial. The non-existent entity is some non-existent (but possible) combination out of elements that do exist. The phrase “non-existent combination” may raise eyebrows. Am I committing myself to a Meinongian view? No, I say. The elements of the combination are, I assert, the only truthmakers that are needed for the truth that this combination is possible. (T&T, 91–92)
I think that this is a very attractive account of what the
truthmakers are for some truths about possibility (including
truths about unicorns and tomato sandwiches).15
However, it is doubtful that the combinatorial approach provides us with
sufficient truthmakers for all the possibility claims made by the modal
structuralist mathematician. The modal structuralist will assert that
second-order ZFC could have had a model, but it seems unlikely that such
a model could be created by recombining physical objects, because it
seems unlikely that there are enough physical objects to go
around. If, for example, there are only
5 Armstrong on Truthmaking and Necessity (Part 1: Truth and Truthmakers)
Let’s turn to Armstrong’s discussion of propositions about necessity. Since our concern is Armstrong’s philosophy of mathematics, we need not discuss all of Armstrong’s views about truthmaking and necessity. Instead, we’ll focus on what he had to say about truthmakers for the theorems of mathematics. Armstrong’s discussions of this topic in Truth and Truthmakers and Sketch for a Systematic Metaphysics are very different. In this section, I’ll consider chapter eight of Truth and Truthmakers, leaving the later book until section 7.
Armstrong (T&T, 99, 111) suggested that the numbers themselves
constitute truthmakers for some arithmetical truths. For example, seven,
five and twelve may together form a truthmaker for
This is an important idea, and I will return to it in section 6. But this is not on its own a complete
solution to the problem at hand. Consider for example the proposition
Later in the chapter, Armstrong discussed analytic truths. He wrote:
A traditional view, which has many supporters, is that [analytic] truths are true solely in virtue of the meanings of the terms in which they are expressed. (T&T, 109)
Armstrong went on to say that “[t]he phrase ‘in virtue of’ inevitably
suggests truthmakers.” So Armstrong proposed that if a sentence S is
analytic, the proposition it expresses is made true by the meanings of
the words in S. For example,
Now Armstrong suggested—somewhat tentatively—that statements in mathematics about what is necessary are analytic.18 On this view, the meanings of mathematical terms make true all such statements.
I do not dismiss completely the claim that mathematical truths are
analytic.19 However, Armstrong’s version of
this thesis is insufficient to solve the problem at hand. Let
6 Armstrong on Truthmaking and Necessity (Part 2: Sketch for a Systematic Metaphysics)
By the time he wrote Sketch for a Systematic Metaphysics, Armstrong had decided to reject his earlier suggestion that mathematical truths are analytic, saying that such a view implies that mathematics is “too arbitrary or conventional” (SSM, 91). But he suggested an alternative approach, which we will now consider.
We should begin by looking at Armstrong’s metaphysics of law.
Armstrong claimed that a law is a relation between properties (SSM, 35).
Here is a toy example. Suppose that it is a law that being dehydrated
causes headaches. For Armstrong, this means that a certain relation
(viz.
Now Armstrong claimed that laws have “instantiations.” Our law, for example, is instantiated whenever someone is dehydrated and, consequently, has a headache. A law, on this view, is itself a property. And we have already seen that Armstrong was happy to posit “immanent” properties. This led Armstrong to the view that every law is instantiated. He wrote:
If laws are a species of universal, then, according to me at least, they have to be instantiated at some place and time. Well, we talk of laws being instantiated, do we not? (The points where the laws are ‘operative’.) So this instantiation of laws is the instantiation of a special sort of universal. (Note that this would require every law to be somewhere instantiated in space-time.) […] One consequence of this is that there cannot be laws that are never instantiated. (SSM, 41)
Now Armstrong suggested that it is certain mathematical and logical laws which make true the necessities of mathematics.
He began his discussion of this proposal by appealing to his Entailment Principle, arguing that truthmakers for the axioms of a mathematical theory must also be truthmakers for the theorems (SSM, 90). This manoeuvre is suspect. While it is known that the theorems of orthodox mathematics are entailed by the axioms (that’s what makes them theorems, after all) it is far from clear that the axioms entail* the theorems.22
And it gets worse. To complete his account, Armstrong still needed to specify truthmakers for the axioms of our mathematical theories. To do this, he appealed to his theory of laws:
We do, of course, have to recognize that introducing the Entailment Principle drives us back to consider the axioms from which mathematical systems are developed. […] I suggest that we should postulate laws in logic and mathematics (non-contradiction, excluded middle in logic, Peano’s axioms for number, or whatever laws logicians and mathematicians wish to postulate). In the light of the nature of proof just argued for we might suggest that such laws might be all we needed to postulate in the way of an ontology for logical and mathematical entities. (SSM, 90–91)
It is not credible, as Armstrong suggests here, that the Peano axioms are “laws” in Armstrong’s sense. For example, one of the Peano axioms states that the natural numbers are unending in the sense that every natural number has a successor.23 For Armstrong, this statement may not be true when taken at face value. For Armstrong, as we’ve seen, the existence of very large natural numbers is doubtful, and it is at least possible that there is a largest natural number, which has no successor. To circumvent this point, Armstrong will presumably insist on a modal reinterpretation of the axiom. On this view, the axiom, properly interpreted, states that, necessarily, every model of the Peano axioms is unending.
The corresponding Armstrongian law would then have to be:
But this is problematic, because Armstrong believed that all laws are instantiated in the physical world—and it is far from clear that this law is instantiated. It may be that the physical world is finite. However, every model of Peano arithmetic is infinite. And so it may be that there are no physical models of Peano arithmetic, in which case the above-mentioned law is uninstantiated.
We might be able to avoid this problem by arguing on empirical
grounds that there are infinitely many physical objects. For example, we
might appeal to the common (though admittedly contested) assumption in
physics that space is infinitely divisible. However, the problem that I
have just described will reassert itself when we turn our attention from
Peano arithmetic to other branches of mathematics which posit a greater
number of entities. The most extreme case is set theory. Any model of
second-order ZFC would have to have a truly vast domain, containing
To avoid these problems, an Armstrongian would have to list a number of basic principles for mathematics which express laws that are instantiated in the physical world, and argue that they entail* the truths of mathematics. I don’t know that this is impossible, but it is far from obvious that it can be done. And even if it could be done, the problem of identifying truthmakers for facts about what is possible would remain.
7 An Alternative Approach
Let’s review. Armstrong believed that mathematical entities are located within the physical world. For example, wherever there is a pair of things, there is the number two. However, Armstrong realized that the physical world is not large enough to accommodate all the entities posited by modern pure mathematics. So he adopted a modal interpretation of mathematics. For Armstrong, pure mathematics tells us not about what is, but about what could be and what must be. However, Armstrong believed that every truth has a truthmaker within the physical world, and so he was left with the unenviable task of identifying truthmakers for modal truths within the physical world. I have argued that he did not succeed.24 In this final section, I would like to put forward an alternative approach—an approach which will, I hope, appeal to those impressed by Armstrong’s metaphysical system.25
Armstrong accepted a version of the methodological principle known as Occam’s razor. He rejected mathematical Platonism largely for this reason. A “Platonic realm of numbers,” he wrote, is an “ontological extravagance” (T&T, 100).26 However, Armstrong did not use his razor to excise supervenient entities. Supervenient entities, he thought, are an “ontological free lunch.” For example, he did not think that universalism in mereology is objectionably unparsimonious:
Whatever supervenes or, as we can also say, is entailed or necessitated, is not something ontologically additional to the subvenient, or necessitating, entity or entities. […] The terminology of “nothing over and above” seems appropriate to the supervenient. […] If the supervenient is not something ontologically additional, then this gives charter to, by exacting a low price for, an almost entirely permissive mereology. Do the number 42 and the Murrumbidgee River form a mereological whole? […] The whole, if it exists, is certainly a strange and also an uninteresting object. But if it supervenes on its parts, and if as a consequence of supervening it is not something more than its parts, then there seems no objection to recognizing the whole. So in this essay permissive mereology, unrestricted mereological composition, is embraced. (1997, 12–13)
On an uncharitable interpretation of this passage, Armstrong’s view
was that if the existence of
Cameron (2008) has suggested a more promising way of developing Armstrong’s idea that supervenient entities are “free.” To put it briefly, Cameron’s proposal is as follows. Compare the following two propositions:
The former proposition is made true by certain patterns of human
activity—involving perhaps Ashni, Ben, a registrar, some pieces of
paper, and some metal rings. Ashni and Ben’s marriage is a
derivative entity: its existence is explained by facts about
things other than itself. The electron
More generally, Cameron’s proposal is this. When
Cameron adds that it is derivative entities in this sense that are an “ontological free lunch,” to use Armstrong’s phrase. In effect, Cameron replaces the familiar slogan “Do not multiply entities beyond necessity” with a variant: “Do not multiply fundamental entities beyond necessity”. Since mereological compounds are non-fundamental, Cameron infers, mereological universalism is not objectionable on grounds of parsimony.30
Cameron briefly suggests an application of this idea to impure set theory. He proposes that an impure set is “nothing over and above” its elements, so there is no objection on grounds of parsimony to positing all those impure sets that can be built up from basic elements whose existence can already be established. On this view, there is no need to re-interpret set theory in a “possibilist” manner. We maintain that all the sets posited by set theorists really do exist, although they are not fundamental. Let’s develop this Cameronian proposal in more detail.
Why does the set
(1) If
So much for propositions about the existence of sets. But a complete truthmaker theoretic account of the sets will also include an account of what the truthmakers are for other propositions, including propositions about the identity and distinctness of sets, and propositions about what is an element of what.
Let’s start with identity. Suppose that someone asks us why Joe is
identical to Joe—that is, we are asked why Joe is identical with
himself. This is a very peculiar question. The best answer to it that I
can come up with goes like this. For Joe to bear the identity relation
to himself, it suffices that he exists. Self-identicality is not some
additional characteristic that requires further explanation. Joe exists,
and so he is self-identical. And that is that. If this is right, I
suggest, any truthmaker for the proposition
Something similar is plausible in the case of non-identity. If,
bizarrely, we are asked why it is that Jill is not identical with Joe—if
we are asked why they are two people and not one—all we can say in reply
is that to be non-identical the two Bidens need only exist. Jill exists.
She is one person. Joe exists too. He is another. And that is all. There
is nothing extra that Jill and Joe need to do or to be in order to be
distinct—existing is enough. And so, I suggest, any truthmaker for
I want to recommend a similar treatment of the relations of
membership and non-membership. If we are asked why Joe is an element of
his singleton, there is nothing we can say except that, for this to be
so, it suffices that Joe and his singleton exist. No more is needed. And
if we are asked why Joe is not an element of
Let me put all of this in a rather different way. Let’s say that a
relation
(1) If
(2) If a relation
(3) The relations of identity, non-identity, membership and non-membership are strongly internal.
Let’s take this further. Suppose for example that
By (1) again,
By (2) and (3),
We can go further still, into the uncountable. Given our account,
And what of Armstrong’s claim that all entities exist “somewhere, somewhen” (SSM, 15)? Well, some readers may find it edifying to insist that a set is located wherever its elements are.35 On this view, you are co-located with your singleton, and its singleton, and its singleton, and so on ad infinitum. I offer no objection to this proposal. But I find it hard to see how to justify the claim that sets have spatial locations, and more importantly it seems to me that we need not endorse this claim to earn the title “naturalist.” We insist that all fundamental objects are physical, and that all truths have physical truthmakers—and this is naturalism enough.
Back to the cardinal numbers. According to the current proposal, even
if the fundamental objects are rather few, nevertheless the sets are
fantastically numerous. This allows us to maintain Armstrong’s original
account of cardinal number without having to worry about the problem of
size, and without recourse to possibilism. Given the current proposal,
for example,
Of course, a thorough truthmaker-theoretic account of mathematics would also cover functions, complex numbers, matrices, ordinal numbers, graphs, and all the other mathematical creatures. You will probably be relieved to hear that I don’t intend to deal with all these topics now. It’s time for a cup of tea, after all. But I hope that my discussion of sets and cardinals is sufficient to motivate cautious optimism about Armstrongian naturalism—despite the errors of detail that we have identified in Armstrong’s discussions of the metaphysics of mathematics.
Thomas M.E. Donaldson
0000-0001-6497-7038
Simon Fraser University, Canada
tmdonald@sfu.ca
Acknowledgements
I would like to thank two reviewers at Dialectica for their hard work, and for their thorough and enormously helpful comments. I would also like to thank Jennifer Wang, Bahar Raiszadeh, Haktan Akcin, and Dans Waxman and Marshall for comments on drafts. (I’ve been working on this paper on and off for a long time; I apologize if I’m missing some acknowledgements!)
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