The main question this essay will address is: What, if anything, makes mathematical propositions true or false?
Mathematical propositions and knowledge seem particularly special compared to ordinary empirical propositions and knowledge. For example, mathematical truth is necessary rather than contingent, and mathematical knowledge appears to be uniquely certain. This makes for an interesting philosophical investigation.
The typical philosophical explanation of mathematical truth is to propose the existence of abstract mathematical objects. Positing such objects has many advantages. In addition to the semantic convenience of having uniform truth conditions for both ordinary and mathematical language, mathematical objects have strong explanatory power in accounting for how mathematics can be used to accurately predict the physical world. The fact that mathematics does provide such accurate predictions would seem, by itself, a compelling reason to conclude that mathematical objects exist, but there are two seemingly insurmountable problems for this view. Firstly, it seems impossible that there could be a non-arbitrary way of distinguishing each distinct mathematical object from another. Secondly, the nature of the proposed mathematical objects seems to exclude the possibility of humans acquiring knowledge of them. The fact that there are such persuasive arguments for both sides of the debate makes this a puzzling philosophical problem as, if one must choose between these two options, whichever side one chooses will be subject to the apparently conclusive counter-arguments of the other. This essay attempts to synthesise both sides by advocating a position which accounts for the positive arguments made in favour of mathematical objects without postulating mathematical objects and, with them, the problems they face.
Contents
Section II: Platonism, the Thesis
Section III: Nominalism, the Antithesis
Section IV: Structuralism, the Synthesis
Section I: Introduction
The purpose of this essay to resolve an antinomy as to the status of mathematical truth. There are compelling considerations in favour of both Platonism, the thesis, and nominalism, the antithesis. I intend to expound the strongest of these arguments in detail before proposing a synthesis, structuralism, which gives an explanation of mathematical truth that is unaffected by the considerations of both sides. I begin with the arguments for Platonism, namely Gödel’s Incompleteness Theorem and, most importantly, the indispensability argument. In a later section, however, it transpires that the indispensability argument is not sufficient to establish Platonism specifically, only realism regarding mathematical truth. The arguments for nominalism considered in the third section are the apparent arbitrariness of identifying numbers with specific set-theoretic definitions, and the epistemological issues with knowledge of Platonistic mathematical objects, given their purported nature. In the fourth section, I take the nominalist’s arguments to prove that mathematical objects do not exist1, but also the ‘Platonist’ indispensability argument of the second section to prove that mathematical propositions are objectively true. These two positions are reconciled by dispensing with Platonist mathematical objects and proposing that mathematical truth comes in virtue of the mathematical structure. In the final section I summarise the arguments on both sides and how structuralism accounts for them.
Section II: Platonism, the Thesis
There are compelling arguments in favour of mathematical Platonism and in this section I intend to expound two such arguments as well as provide an explanation of what Platonism is. The first argument considered is Gödel’s Incompleteness Theorem. Secondly, and most importantly, I defend the Quine/Putnam indispensability argument. These arguments have typically been taken to prove the existence of mathematical objects. In a later section, however, I intend to argue that, though these arguments support a realist conception of mathematics, they do not necessarily support this stronger claim that Platonistic mathematical objects exist.
The most common manifestation of mathematical realism is Platonism. Platonism, so named because of the resemblance of mathematical objects to Plato’s ‘Forms’2, is the view that numbers (and other mathematical concepts) exist as objects. These objects have the same metaphysical status ordinary objects are taken to have on a realist view in that they exist independently of whether we think about them. A theory that posits mathematical objects of this kind is the obvious strategy for a realist conception of mathematics because it creates a uniform semantics between ordinary and mathematical propositions. Mathematical propositions are capable of being true or false in virtue of whether they accurately describe the external realm of mathematical objects in the same way an empirical proposition, for example ‘the Eiffel Tower is in Paris’, is true if it accurately describes the external world (i.e. if the Eiffel Tower is, in fact, in Paris). Unlike empirical objects though, mathematical objects are thought to be abstract, lacking any causal power and located outside of space and time, which explains the differences between empirical and mathematical propositions. Because we are naturally inclined to think of truth as we do for basic empirical propositions, and because it is natural to want to say mathematical propositions are true or false, Platonism is often described as the ‘default view’ of mathematics.
Though the Platonist statement ‘numbers exist’ may seem philosophically strange prima facie, ordinary mathematical statements, for example ‘there is at least one perfect number greater than 30’, do not tend to strike one as quite so odd. However, ‘numbers exist’, or ‘(Ǝx)(Nx)’, follows from ‘there is at least one perfect number greater than 30’, which can be symbolised as ‘(Ǝx)(Nx·Px·Gx)’. (Putnam, 1979a: p349) This makes it difficult to deny ‘(Ǝx)(Nx)’ and maintain ‘(Ǝx)(Nx·Px·Gx)’ – as the mathematician who is new to philosophy may initially claim – without a suitable semantics for the language of mathematics. ‘Notice that the apparent grammatical and logical forms of mathematical existence-claims are the same as those of more mundane existence-claims.’ (Resnik, 1999: p41) The lack of any obvious difference between the two kinds of existence claims is illustrated by Resnik in the following example sentence: ‘The solutions to some of the problems involved numbers exceeding the capacities of some pocket calculators.’ (Resnik, 1999: p41) This sentence makes and mixes two existential claims. Firstly, that numbers exist which solve certain problems and secondly, that there exist pocket calculators that cannot handle these numbers. There is no obvious grammatical difference between the two claims and so, it seems, the semantics of mathematics is the same as the semantics of ordinary empirical propositions.
One explanation of mathematical language which does not presuppose mathematical objects would be to define mathematical truth as provability. In this case, ‘there is at least one perfect number greater than 30’ is true because it can be proved (constructed, demonstrated) that a number defined in a certain way, within that system, meets the requirements of the proposition. Though this upsets the uniform semantics described earlier, this is not a compelling argument for the existence of mathematical objects, as it would be wrong to suppose that our linguistic practices have the power to bring objects into existence to satisfy a certain notion of truth to correspond to these practices. This would have the consequence that it would be wrong to describe mathematical propositions as ‘traditionally’ true, if it was in fact the case that mathematical truth = provability. However, Kurt Gödel’s first Incompleteness Theorem shows that the notion of truth is broader than provability and so this conception of mathematical language is mistaken.
Gödel numbering is a process of generating a unique number for every meta-mathematical symbol, formula, or set of formulas and means that meta-mathematical statements, for example ‘x is a variable’, can be represented solely within the ‘object-language’ of arithmetic. Below is a table showing how each constant sign of the meta-language can be assigned a unique number:
| ~ | 1 |
| V | 2 |
| → | 3 |
| Ǝ | 4 |
| = | 5 |
| 0 | 6 |
| s | 7 |
| ( | 8 |
| ) | 9 |
| , | 10 |
Further, each numerical variable is assigned to a prime number > 10:
| x | 11 |
| y | 13 |
| z | 17 |
Sentential variables are assigned to the square of a prime number > 10:
| p | 11² |
| q | 13² |
| r | 17² |
Predicate variables are assigned to the cube of a prime number > 10:
| P | 11³ |
| Q | 13³ |
| R | 17³ |
Using this numbering system, formulas – such as ‘(Ǝx)(x=sy)’ – can be assigned a unique Gödel number based on the numbers of the individual symbols. For example:
| ( | Ǝ | x | ) | ( | x | = | s | y | ) |
| 8 | 4 | 11 | 9 | 8 | 11 | 5 | 7 | 13 | 9 |
Each symbol has been assigned a unique number.
To get the unique number of the whole formula, we raise each prime number (in ascending order) to the power of the Gödel number (GN) of the relevant symbol and multiply them all:
(2^8) x (3^4) x (5^11) x (7^9) x (11^8) x (13^11) x (17^5) x (19^7) x (23^13) x (29^9) = GN of the formula ‘(Ǝx)(x=sy)’.
Further to the elementary symbols above there are two functions: ‘Sub’ and ‘Dem’. Both are expressible solely using the elementary symbols, though I will not show this. ‘Dem (x,y)’ says: The sequence of formulas with Gödel Number x provides a proof of the formula with Gödel Number y. ‘Sub (y,13,y)’ describes the Gödel number of the formula that is obtained from the formula with Gödel number y, by substituting for the variable with GN 13 the numeral for y.3
Gödel uses this numbering system to give the formula ‘(x)~Dem (x, Sub(y,13,y))’ – which says ‘the formula with Gödel Number ‘Sub (y,13,y)’ is unprovable’ – a unique number n. If we substitute, in the sequence of formulas with Gödel number y, this numeral n in place of the variable represented by the Gödel number 13, the following formula results:
(x) ~Dem (x, Sub(n,13,n))
This ‘Gödel sentence’, G, says of itself ‘I am unprovable’. A proof of G would thus amount to ~G and vice versa, which is a contradiction. This shows that G must be unprovable within the arithmetical system from which G is constructed if that same system is consistent. Though G is undecidable within this system it can be shown that G is true on the assumption that the system is consistent because of the contradiction that results if G is false. ‘It should be noted, however, that we have established an arithmetical truth, not by deducing it formally from the axioms of arithmetic, but by a meta-mathematical argument.’ (Nagel & Newman, 1958: p72) A similar Gödel sentence can be created from any formal system capable of representing arithmetic and so no single axiomatisation4 will be able to prove every truth of that system. This shows that truth cannot be understood as provability, as may be claimed to avoid positing mathematical objects, because, as demonstrated by Gödel’s first Incompleteness Theorem, there are more arithmetic truths than there are proofs.
Gödel’s theorem shows that, if we are to say mathematical propositions are true, the semantics of mathematical truth has nothing to do with provability. A further, and perhaps the most important, argument for the Platonist’s positive argument for the existence of mathematical objects is the Quine/Putnam indispensability argument. It is called the ‘indispensability’ argument because it reasons from the relatively uncontroversial premise that mathematics is indispensable to science to the conclusion that we have at least as much reason to believe in mathematical entities, at least those which are practically applied, as we do theoretical, but unobserved, entities such as electrons and quarks due to the accuracy with which they can be used to make predictions of observable entities.
The indispensability argument can be summarised in the following way:
- We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
- Mathematical entities are indispensable to our best scientific theories.
- Therefore we ought to have ontological commitment to mathematical entities.
(Colyvan, 2011: §1)
In order for any given physical law, for example Newton’s Second Law of Motion5, to be true the entities it refers to, namely force, mass and acceleration, must exist. In the language of the above argument, because science is our best way of understanding the physical world, we ought to have an ontological commitment to these entities. This vindication of the first premise is relatively uncontroversial if one accepts that science is, or at least aspires to be, true6.
General statements about these entities are not enough for a law to be scientifically useful however as the law must be applicable. To apply the law there needs to be a way of representing the relevant information so as to use it to make accurate predictions. Returning to Newton’s Second Law, if one wanted to calculate whether a given force was greater than another, we would need a language that could represent a greater than relation and a multiplication relation. The relationship between the mathematical representation and the physical facts would thus correspond to one another so that the greater force would correspond to a larger number when represented mathematically (Brown, 1999: p47). This simple example does not do justice to quite how closely linked mathematics and science are and to the extent to which mathematics is used in science. In the above example the mathematical model corresponds to the physical data. However, in theoretical physics for example, mathematics makes predictions that are then later observed to be true and so we see that the empirical world corresponds to the mathematical world. One particularly useful illustration of this is the ‘EPR paradox’ from quantum mechanics. Written in 1936, the EPR7 paper claims that the mathematical predictions of quantum mechanics must be mistaken, due to the following counter-intuitive predictions:
‘In an EPR-type set-up… a decay process gives rise to two photons moving in opposite directions toward detectors at either end of a room. The detectors include polaroid filters which can determine whether the incoming photons have the so-called property spin-up or spin-down. Both theory and experience tell us that the two outcomes are always correlated: one photon is up and the other down… A direct causal connection [between the two photons to give them opposing properties] would have to be faster than light, something ruled out by special relativity, and a common cause in the past… would amount to a local hidden variable, something ruled out by the Bell results.’
(Brown, 1999: p16-17)
The prediction of a perfect correlation without a causal explanation was considered to undermine quantum mechanics. However, when the appropriate scientific apparatus became available to actually test these predictions, experiments showed that it was our intuition, and not quantum mechanics, that was mistaken (Fine, 2011: §3.1 & §3.2). This shows how mathematics not only explains, but can also predict the empirical world, illustrating just how closely interlinked the two are. ‘We require, if we are to use the law, a language rich enough to state not just the law itself, but facts of the form [‘force = x’, ‘mass = y’, etc]’ (Putnam, 1979a: p338) What language, other than a mathematical one, could serve this purpose? This is what it means for mathematics to be indispensable but further, if this language does not tell us anything true about the physical world, why does it produce such accurate predictions? In the absence of an explanation of this, we must be as committed to the ontological reality of the mathematics used in science as the science the mathematics is used in.
Section III: Nominalism, the Antithesis
Gödel’s incompleteness theorem shows that mathematical truth cannot suitably be defined as provability. The indispensability argument shows that mathematical propositions can be objectively true as when they are used to make accurate scientific predictions of the empirical world. This commits us to belief in the existence of mathematical entities at least insofar as these scientific theories are true, or, that we believe them to be true. The indispensability argument, as described previously, is clearly valid and none of the premises are obviously false. However, there are equally strong arguments against the existence of mathematical objects, and it is the purpose of this section to defend two such objections. Firstly, I intend to argue that, if there were mathematical objects, they would be best understood as sets. However, if numbers are sets, they can be defined in a number of possible ways and so, whatever the actual nature of mathematical numbers as sets is, it would be an arbitrary distinction as they could be defined, without any difference in consequences, another way. The second objection I will defend is that, if mathematics is true in virtue of Platonist mathematical objects, then mathematical knowledge would be impossible to attain.
If there really were mathematical objects, apart from being abstract, located outside of space and time, and so on, what would they be like? What makes the object that is the reference of ’17’ distinct from the object referred to by the sign ‘347’, for example? The sign we choose to assign to a number is an arbitrary one. 5 can be represented as ‘5’, ‘five’, ‘cinq’, ‘fünf’, ‘V’, or whatever, but what is it that is common to all these words but that distinguishes it, say, from the meaning of ‘6’, ‘six’, and so on? Set theory provides the simplest account of the nature of mathematical objects independent of the signs that reference them. A set is a collection of objects and, as such a basic concept, cannot be reduced to anything more fundamental. ‘Using the basic construction principles, and assuming the existence of infinite sets, one can define numbers, including integers, real numbers and complex numbers, as well as functions, functionals, geometric and topological concepts, and all objects studied in mathematics.’ (Jech, 2011: section 1) Set theory thus appears to provide necessary and sufficient conditions for defining numbers in as basic terms as possible and so, if mathematical objects do exist, set theory appears the best way of understanding the nature of specific mathematical objects.
Because mathematics can be understood solely within the language of set theory, we could imagine someone whose mathematical education consisted in learning set-theoretic concepts first and ‘vulgar’ mathematical language second, as Paul Benacerraf does in What Numbers Could Not Be. This person may derive their ‘theorems’ as to what the vulgar mean by number words from set theoretic axioms in the following way: ‘[O]ne determines that a set has k elements by taking (sometimes metaphorically) its elements one by one as we say the natural numbers one by one.’ (Benacerraf, 1983b: p275) This person, then, would derive the theorem as to what the number we call ‘3’ is by taking a particular set and counting a one to one correspondence between each member of that set and each natural number8 of vulgar, ordinary mathematical language. ‘3’, in set theoretic language, would thus be the set which has a member for each number 1, 2, and 3. For example, the first three numbers can be defined thusly:
- [0]
- [0,[0]]
- [0,[0],[0,[0]]]
Here, 0 represents the empty set, the set with no members, and the brackets represent sets. All the natural numbers, to infinity, can be defined in this way. These definitions not only serve all the purposes of the natural numbers within mathematics, but they can also be used to prove theorems about the natural numbers. An example of such a theorem that can be derived from these definitions is ‘for any two numbers, x and y, x is less than y if and only if x belongs to y and x is a proper subset of y.’ (Benacerraf, 1983b: p278)
Multiple definitions of the natural numbers can be given in set theory, similar to above, that serve all the uses of the natural numbers within mathematics. We might also define the first three as follows:
- [0]
- [[0]]
- [[[0]]]
This is not a problem within arithmetic as both these definitions will yield the same mathematical theorems. However, if set theory represented the true nature of mathematical objects, then, unless there were a ‘correct’ definition, we could derive contradictory meta-mathematical theorems about the natural numbers. The above definitions yield the theorem that ‘given two numbers, x and y, x belongs to y if and only if y is the successor9 of x.’ (Benacerraf, 1983b: p278) Application of these two theorems in answering ‘does 3 belong to 17?’ yield the answer ‘yes’ on the first definitions and ‘no’ on the second definitions.
If ordinary Platonism is correct then there must be a definitive answer as to whether 3 belongs to 17. However, given that set theory is as fundamental as the analysis of number appears to go, what would a proof of one definition of number over another look like? Surely, one would think, such a proof is impossible. Benacerraf argues that ‘if such a question has an answer, there are arguments supporting it, and if there are no such arguments, then there is no “correct” account that discriminates among all the accounts’ (Benacerraf, 1983b: p281). What Benacerraf means by ‘correct’ here is slightly unclear. He appears to be saying either that there is no reason to justify identifying a particular number with a particular set or, more strongly, that the lack of arguments in favour of one definition over any other means there is no true account. This latter option appears to be what Benacerraf is saying when he writes: ‘But if the number 3 is really one set rather than another, it must be possible to give some cogent reason for thinking so; for the position that this is an unknowable truth is hardly tenable.’ (Benacerraf, 1983b: p284) Quite why this position is untenable is not clear. It is obvious, at least within the scope of this essay, that something could be true but unknowable and, if this is the case with the set-theoretic nature of numbers, then this might just be an unfortunate fact. Just because we cannot know which set is the correct definition of 3, for example, it does not follow, if this is Benacerraf’s argument, that 3 cannot, in fact, be identified with any set.
Despite this apparent ambiguity Benacerraf does raise a devastating point as it does appear, not only that there are no reasons to prefer one set-theoretic definition over another, but that there could not be any reason that numbers would be any particular sets. ‘There is no way connected with the reference of number words that will allow us to choose among them, for the accounts differ at places where there is no connection whatsoever between features of the accounts and our uses of the words in question.’ (Benacerraf, 1983b: p285) Because there is no such way of choosing between one account of number over another and further, that choosing one account makes no difference to our use of number words, Benacerraf’s argument can be likened to Wittgenstein’s beetle. In section 293 of Philosophical Investigations Wittgenstein imagines a society where everyone has a box with something inside, which they call a ‘beetle’. No one can look in anyone’s box but their own but each says they know what ‘beetle’ means only by peering into theirs. ‘Here it would be quite possible for everyone to have something different in his box. One might imagine such a thing constantly changing.’ (Wittgenstein, 2009: p106e) If so, this would have no effect on the meaning of ‘beetle’. Similarly, identifying ‘3’, as traditionally used in mathematics, with [[[0]]] makes no difference to its meaning as if it were identified with [0,[0],[0,[0]]], or any other definition. From this, as Benacerraf argues, ‘there is little to conclude except that any feature of an account that identifies 3 with a set is a superfluous one’ (Benacerraf, 1983b: p285).
Not only does it appear impossible to know which set defines a certain number but lack of evidence in favour of any conception suggests there is no ‘correct’, or even true, definition. The epistemic problem of mathematical knowledge is not limited to knowing which set defines a certain number either. Given the description of mathematical Platonism in the previous section, and the proposed nature of platonic mathematical objects, we might wonder how it is possible to acquire mathematical knowledge. Though mathematical propositions, on the Platonist view, are true in the same way empirical propositions are, our epistemic access to them is somewhat mysterious. One knows ‘grass is green’ and ‘the Eiffel Tower is in Paris’ to be true from perhaps direct observation, hearing evidence, or else some other perception but this method is not obviously analogous to the acquisition of mathematical knowledge. Firstly, mathematical objects exist outside of space and time. Second, and perhaps most importantly, mathematical objects are said to be acausal. In Mathematical Truth, Benacerraf argues that the impossibility of mathematical knowledge, given the Platonist10 account of mathematical truth, is a fatal argument against Platonism. Kurt Gödel’s attempt to unite Platonistic mathematical truth and epistemology is used as an example to highlight this gulf and illustrate the problem with positing mysterious abilities to acquire knowledge of these objects:
‘But despite [mathematical objects’] remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuitions, than in sense perception’
(Gödel, 1983: p483-484)
Because, on the Platonist view, mathematical propositions are true in a similar way to empirical propositions it might be argued, as Gödel appears to here, that mathematical knowledge is similarly analogous to empirical knowledge. However, to state merely that ‘the axioms force themselves upon us as being true’ does nothing in accounting for the extraordinary fact that we spatio-temporal beings have epistemic access to another, non spatio-temporal, realm of causally inert objects. The very nature of mathematical objects is such that it appears to make knowledge of them impossible and so, if we are supposed to accept that our knowledge is of these objects, one appears well within his rights to ask how this is possible, rather than taking the fact that we do have mathematical knowledge, and thus that it is possible, as proving the existence of the objects in question. This is precisely what Gödel’s, and other Platonist accounts, lack.
Defenders of Platonism attack this argument by claiming it relies on a causal theory of knowledge and Benacerraf himself explicitly states this assumption in his argument for the impossibility of mathematical knowledge (Benacerraf, 1983a: p412). Of course, if we need to be causally connected to an object to have knowledge of it, and mathematical objects are causally inert, then mathematical knowledge is impossible, but why grant the nominalist these epistemological assumptions? It may be objected, firstly, that this treats mathematical objects as though they were physical objects. This would, in some sense, make mathematical study the same, or closely similar to, ordinary empirical science but the very nature of platonic mathematical objects is such that they are not like physical objects. It is thus mistaken to subject them to the same epistemological standards as the physical sciences demand and will inevitably result in the seeming impossibility of mathematical knowledge. Further, causal theories of knowledge are widely regarded as flawed due to counter-examples such as the following:
‘Henry identifies various objects on the landscape as they come into view… [He says]“that’s a barn,” etc. Henry has no doubt about the identity of these objects; in particular, he has no doubt that the last-mentioned object is a barn, which indeed it is… Suppose we are told that, unknown to Henry, the district he has just entered is full of papier-mache facsimiles of barns… They are so cleverly constructed that travellers invariably mistake them for barns. Having just entered the district, Henry has not encountered any facsimiles; the object he sees is a genuine barn. But if the object on that site were a facsimile, Henry would mistake it for a barn. Given this new information, we would be strongly inclined to withdraw the claim that Henry knows the object is a barn.’
(Goldman, 1983: p175-176)
Here Henry has a causally justified (the visual perception of the actual barn he sees) true belief that does not count as knowledge. This example shows that a definition of knowledge as causally justified true belief is insufficient. One way the nominalist may respond to this would be that, though not a sufficient account, an appropriate causal relation is a necessary condition of knowledge. Though this is plausibly true, I do not intend to pursue this point as it requires a detailed theory of knowledge that gives an account of what ‘appropriate causal relation’ means without excluding actual cases of knowledge11 or including the very sorts of Gettier cases12 the causal theory is a response to as knowledge. ‘The causal theory of knowledge has gradually lost favour in the years since the appearance of Benacerraf’s article, while the sentiment that there is a persuasive Benacerraf-style argument against Platonism remains strong.’ (Maddy, 1990: p42) Instead, I intend to propose one such ‘Benacerraf-style argument’ against Platonism that doesn’t assume any particular epistemology.
Rather than challenging the ability to justify beliefs about Platonistic objects as Benacerraf does, Hartry Field’s challenge to the Platonist is for him to explain the reliability of his mathematical beliefs. The Platonist would, presumably, accept the following to be true: ‘If mathematicians accept ‘p’ then p… holds in nearly all instances, when ‘p’ is replaced by a mathematical sentence.’ (Field, 1989: p26) This is the rather uncontroversial claim that, more often than not, widely accepted mathematical propositions are true propositions. Though there may be contested propositions in the suburbs of the metaphorical city of mathematical knowledge13, the majority of accepted propositions in the mathematical community are true. Further, assuming the Platonist accepts this, and, given his account of the nature of the objects of mathematical knowledge, there would need to be an explanation of the ‘reliability thesis’ described above. To accept this correlation as brute fact would, as Field describes, be ‘rather as if someone claimed that his or her belief states about the daily happenings in a remote village in Nepal were nearly always disquotationally true, despite the absence of any mechanism to explain the correlation between those belief states and the happenings in the village.’ (Field, 1989: p26-27) Surely, just as one would demand a more basic explanation of the accuracy of a person’s beliefs regarding the Nepalese village, so too would it be reasonable to demand an explanation of the accuracy of mathematical belief. If anything, without an explanation of the reliability thesis, the Platonist’s claim is even more absurd than the case in Field’s analogy. The knowledge of the remote Nepalese village, though massively unlikely, could be put down to a bizarre coincidence or, even less likely, there could be some long-winded causal process that explains it – but neither of these options is plausible for the Platonist. Firstly, the scale of the coincidence regarding mathematical knowledge is far greater, given the number of mathematical beliefs and the number of mathematicians who believe them. That all these people happen to have almost perfectly reliable beliefs, with no explanation of this reliability whatsoever, is so absurd as to provide a reductio of Platonism. Any sort of causal explanation is out of the question too, given the nature of the proposed mathematical objects. At this point, Benacerraf’s original objections to the possibility of mathematical knowledge become relevant once again, as any reason to doubt this also appears to undermine any explanation of the reliability thesis. The problem resurfaces and seems just as much a problem as ever. ‘[T]he causal inactivity and impassivity of abstracta seems to rule out the possibility of an sort of explanation of the reliability thesis… what could a non-causal explanation of the reliability of synthetic beliefs about abstracta conceivably be like?’ (Burgess&Rosen, 1997: p42) This question is one that seems impossible for the Platonist to respond to.
Section IV: Structuralism, the Synthesis
So far we have seen conclusive arguments on both sides of the Platonist/nominalist debate. If these two options exhausted the domain of options then, it seems, one would find himself in a contradictory position as to the status of mathematical truth. Indeed, this is often how the debate is framed: one must either be a Platonist or a nominalist, there are no other options. Evidence of this false dichotomy is seen in, for example, Benacerraf’s Mathematical Truth where the history of the debate is posed as a battle between the standard view, which, as mentioned, is synonymous with Platonism and the combinatorial view, which is synonymous with nominalism. Further, anti-Platonist arguments, such as those described in the nominalism section above, are often seen to establish nominalism as though refuting Platonism proves, by disjunctive syllogism, nominalism to be the correct position and vice versa. Instead, I propose, a more apt disjunction would be either mathematical realism or mathematical anti-realism. ‘A realist (with respect to a given theory or discourse) holds that (1) the sentences of that theory or discourse are true or false; and (2) that what makes them true or false is something external.’ (Putnam, 1979b: p69-70) This means that true mathematical statements, for the realist, are objectively true in virtue of something external regardless of whether we can know them to be true. ‘Notice that, on this formulation, it is possible to be a realist with respect to mathematical discourse without committing oneself to the existence of ‘mathematical objects” (Putnam, 1979b: p70) The purpose of this section is to propose a structuralist interpretation of mathematics which says that mathematical propositions have objective truth values in virtue of an abstract mathematical structure, thus making it a realist account which satisfies the indispensability argument, but to show how this interpretation avoids the fatal problems directed at Platonism described in section III.
Section II of this essay was presented as arguments for Platonism. However, though such arguments are typically used by the Platonist to support his position, they are not sufficient for the purpose of establishing the existence of mathematical objects. The Incompleteness Theorem, for example, says only that mathematical truth cannot be identified with provability and, though we may have the intuition that for a proposition to be true requires the appropriate objects to make it so, as in empirical propositions, this assumption is unjustified. Structuralism is a realist interpretation of mathematical truth, and thus identifies truth with something external and objective, but does not presuppose the existence of any mathematical objects. Rather than considering the essence of the number 3 in isolation, for example, as Platonism does, Structuralism takes the ‘objects’ of mathematics to have no properties other than their place in a larger structure: ‘The structuralist vigorously rejects any sort of ontological independence among the natural numbers. The essence of a natural number is its relations to other natural numbers.’ (Shaprio, 2000: p258) It is in virtue of the abstract structure of mathematics that mathematical propositions are true or false. This means that the truth of mathematical propositions are in no way dependent on our beliefs and, as an external, though abstract, entity, it means that the semantics of mathematics is uniform with ordinary, empirical, semantics:
‘The suggestion here is that sometimes competent speakers of English treat the positions of a mathematical structure as objects, at least when it comes to surface grammar. Some structuralists… take this to give the underlying logical form of mathematical language. That is, sentences in the language of arithmetic, such as ‘7+9=16’ and ‘for each natural number n, there is a prime number m>n’ are taken literally to refer to the places of the natural number structure.’
(Shapiro, 2000: p269)
The arguments of section II are only arguments for the objectivity, and so realism, of mathematics and so support structuralism as much as they do Platonism. The arguments of section III are arguments against Platonism and so, not only do they not affect structuralism, but, in the case of the following argument, actually provide support for structuralism.
Structuralism is said to trace its lineage back to Benacerraf’s What Numbers Could Not Be14 and so his argument in that paper is an apt place to start in reconciling mathematical truth with the considerations of the previous sections. It explains the arbitrariness of identifying individual numbers with individual sets because, as described, structuralism rejects the idea that numbers have ontological independence and thus are identifiable with any set:
‘Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role – not by being a paradigm of any object which plays it, but by representing that relation that any third member of a progression bears to the rest of that progression… On this view many things that puzzled us in this paper seem to fall into place. Why so many interpretations of number theory are possible without any being uniquely singled out becomes obvious: there is no unique set of objects that are the numbers. Number theory is the elaboration of the properties of all structures of the order type of the numbers. The number words do not have single referents.’
(Benacerraf, 1983b: p291)
It also provides an elegant rebuttal and account of the language of the ‘(Ǝx)(Nx·Px·Gx) ∴ (Ǝx)(Nx)’ argument described in the first section because, on the structuralist view, ‘there are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20.’ (Benacerraf, 1983b: p294). Because structuralism ‘vigorously rejects’ an account of number by itself, but not an account of number relative to its place in the number structure, there is no problem with affirming ordinary mathematical statements, for example ‘there is at least one perfect number greater than 30’, but denying the existence of numbers in themselves.
If, as the above argument suggests, mathematical propositions describe relations among a larger mathematical structure we can ask whether this structure is objective15 or a subjective creation by humans. Given the indispensability argument described in section II, the latter option is unsatisfactory as it would not explain how mathematical modelling can be used to make accurate predictions of the empirical world. However structuralism is well equipped to account for the objectivity, and thus indispensability to science, of mathematics. ‘It is easy to see why on this view mathematics is applicable to the non-mathematical realm: mathematics describes the structure or pattern, and the structure is present in the physical system itself.’ (Brown, 1999: p57) The empirical world ultimately has an underlying mathematical structure and it is in virtue of this abstract structure, not in virtue of abstract mathematical objects, that mathematical propositions are true or false. Any collection of empirical objects, a system16, will exhibit some kind of structure among them and mathematics is the study of this structure and so systems in their most general form, independent of particular instances. ‘An infinite string of stars, for example, has the same underlying structure as an infinite sequence of moments in time, or as an infinite string of strokes, | | | | | | |….’ (Brown, 1999: p57) ‘Define a pattern or structure to be the abstract form of a system’ (Shapiro, 2000: p259) and it is no surprise that analysis of the abstract structure, as in the mathematics of quantum physics, yields accurate predictions of particular systems in the empirical world. This explains why mathematical modelling is so useful in science and thus why mathematics is indispensable to it. Because empirical systems necessarily adhere to this abstract mathematical structure, as evidenced by the indispensability argument, it cannot be said that this structure is a subjective creation by humans who perceive patterns that are not really there. Mathematical propositions are thus just as objective as they would be on the Platonist view.
Perhaps the only real challenge for structuralism from the preceding arguments is to explain how it is possible to acquire mathematical knowledge on this view. It may be objected that knowledge of an abstract structure is equally mysterious as knowledge of mathematical objects is. Like the objects posited by Platonism, the mathematical structure is abstract and, in itself, causally inert. This seems, at first, to make structuralist mathematical epistemology subject to the same criticisms as Platonism. There is, however, a key difference between Platonism and structuralism that vanquishes the main force of these concerns.
On the Platonist view, the mathematical realm is separate and distinct from the empirical, non-mathematical realm. When considering the nature of the objects of the mathematical world, as described multiple times in previous sections, it seems impossible to account for how facts about it could permeate the empirical world and come to be grasped by spatio-temporal human beings. Though the abstract structure of mathematical reality, according to structuralism, shares similarities with this view, there is only need for one realm of reality. Our knowledge of mathematics is not of a separate abstract world but is knowledge of our own, empirical, world but in its most general form. As mentioned two paragraphs back, all physical systems will have some kind of structure, and mathematical knowledge is knowledge of the necessary principles these systems must adhere to. This, already, makes the possibility of mathematical knowledge, on a realist view, far more palatable because, though the structure is abstract, its instances can be grasped in ordinary perception of systems. This is not yet a full account however.
‘A structuralist might begin with the thesis that one can apprehend some structures via pattern recognition. Of course, pattern recognition is a deep and challenging problem in cognitive psychology, and there is no accepted account of the underlying mechanisms. Nevertheless, pattern recognition is not philosophically occult, as, say, Gödelian intuition is supposed to be.’ (Shapiro, 2000: p276) The point here is a reiteration of the above. Though we’ve yet to fully explain how humans acquire mathematical knowledge, such an explanation at least seems possible according to structuralism. Mathematical knowledge on this view is far more palatable compared with Platonism which needs to invoke the ‘occult’, supernatural and unscientific methods which, as seen in Benacerraf’s criticism of Gödel’s mathematical intuition, do not satisfactorily explain how Platonistic objects can affect humans. Some structuralists have even attempted to give an empirical account of the origins of mathematical knowledge of structures.
Stewart Shapiro speculates as to the origins of mathematical knowledge by beginning with small cardinal numbers (Shapiro, 2000: p277). It is not difficult to imagine, for example, teaching a child the ‘4-pattern’ by pointing to particular examples: members of the Beatles, legs of a dog, walls of a typical room, and so on. Eventually the child will abstract the concept of four independently of any concrete instantiation of it. Similarly other small cardinalities can be taught and, once taught enough of these basic number concepts, reflection on them, and their relations to each other, could cause this individual to recognise the possibility of continuing the sequence beyond the specific cardinalities counted. If one can accept this then it is not hard to imagine, given further reflection and use of larger and larger number concepts, that this person could form the concept of any number or even of infinity. ‘A related possibility is that humans have a faculty that resembles pattern recognition but goes beyond simple abstraction. The small finite structures, once abstracted, are seen to display a pattern themselves… We then project this pattern of patterns beyond the structures obtained by simple abstraction.’ (Shapiro, 2000: p279)
Michael Resnik’s Mathematics as a Science of Patterns is another example of an attempt to give a more detailed and full account of the origins of mathematical knowledge of structures. In the case of geometry, he begins with the uncontroversial claim that basic recognition of the benefits of certain arrangements comes from experience and that ‘because of their practical importance, we find ourselves driven to invent a vocabulary to refer to things exhibiting important patterns.’ (Resnik, 1999: p226) Use of this vocabulary to talk of ‘how things are shaped, arranged or designed’, he imagines, could be fairly comprehensive without the need to introduce any abstract entities at this point. He continues the account:
‘But we can also use the medium [talk of patterns] (for constructing templates of a given kind) to construct configurations without any representational role, such as random doodlings on blueprint paper. Surely, the ancients did this too. We have thus advanced the ancients from the barest recognition of the practical importance of certain shapes and arrangements to representational systems for designing previously unseen concreta, and thence to playful and creative attempts to explore possibilities.’
(Resnik, 1999: p227)
The ‘playful and creative’ exploration of possibilities described is essentially what geometry is. Whether or not either of these provides an accurate historical or psychological account of mathematical knowledge – but both have prima facie plausibility, particularly when compared to Gödelian intuition and other Platonist epistemologies – is not wholly important to this essay. Whatever the facts about the origins of mathematical knowledge, an explanation that appeals to empirical experience is preferable to an account that appeals to mystical abilities, and structuralism allows for such a plausible explanation by bringing the objects of mathematics into the empirical world. It then becomes a matter for psychologists and other such scientists to explain the ordinary, non-occult, way humans grasp mathematical knowledge from experience.
Section V: Conclusions
Section II of this essay demonstrated conclusive evidence for mathematical realism, i.e. the position that mathematical propositions are capable of being objectively true. The most important such evidence for realism is the undeniable usefulness and accuracy of predictions of applied mathematics in science (the indispensability argument), which seems impossible to explain unless we accept that mathematical statements can be objectively true. Though there are other supporting arguments for realism, such as Gödel’s Incompleteness Theorem, the indispensability argument remains the most powerful to conclude that mathematical truth is objective. Having established that mathematical truth is objective, assumptions about how language works may tempt us to accept mathematical Platonism. Our ordinary semantics for empirical propositions, for example, typically assumes what Wittgenstein calls an ‘Augustinian’ conception of language, described thusly: ‘In this picture of language we find the roots of the following idea: Every word has a meaning. This meaning is correlated with the word. It is the object for which the word stands.’ (Wittgenstein, 2009: p5e. Italics added) By positing mathematical objects analogous to empirical objects, Platonism creates a uniform semantics for both empirical and mathematical truth.
The Platonist theory of mathematical truth leads to two insurmountable problems, however, as described in Section III of this essay. Firstly, assuming that each mathematical object would have some set-theoretic nature, it is impossible to specifically explain what this nature would be as there are many ways each mathematical object could be defined in set theory without affecting its use in mathematics. The arbitrariness of choosing one definition over another suggests that no such definition could accurately encapsulate the essence of any mathematical object and thus that mathematical objects cannot be identified with sets. Further, there seems to be no other way of defining numbers in isolation other than with set theory. The second problem for Platonism is that it would seemingly make mathematical knowledge impossible. The acausal nature of the mathematical objects posited by Platonism is such that it would be a total mystery why mathematicians’ beliefs about mathematics are almost always true.
However, although the arguments in Section III refute mathematical Platonism, these arguments do not refute mathematical realism. Section IV demonstrates that there does not need to be individual objects correlated to number words – as Platonism contends – for mathematical statements containing these number words to be true. Section IV describes how mathematical structuralism can provide a realist account of mathematics, and with it a suitable mathematical semantics, without positing mathematical objects and falling foul of the nominalist’s objections to Platonism. By describing the nature of number only in relation to other numbers and not in isolation, Structuralism avoids the issue of identifying numbers with particular set-theoretic definitions while still maintaining a realist account of mathematical truth that can account for why mathematics is indispensable to science. Finally, though I did not give an in-depth psychological or historical account of how mathematical knowledge is attained on the structuralist view, I gave reasons why it should not be difficult to explain – at least in comparison to Platonism, for which the task is seemingly impossible. In summary, structuralism is supported by the arguments for mathematical realism while avoiding the nominalist’s objections to Platonism.
1 Obviously it is difficult, if not impossible, to prove the non-existence of anything, for example unicorns or, to use Bertrand Russell’s famous example, a china teapot orbiting the sun between Earth and Mars. What I take the nominalist’s arguments to show is that, if there are mathematical objects, then they are irrelevant to the reference of mathematical language and our mathematical knowledge.
2 ‘And we say that the particulars are objects of sight but not of intelligence, while the forms are the objects of intelligence but not of sight.’ (Plato, 2007: p233. Italics added) ‘Particulars’ here are instances of perception of empirical objects which, according to Plato, can only give rise to opinion (Plato, 2007: p234, p236 (divided line) & pp. 240-248 (simile of the cave)). ‘Knowledge’ is reserved for the intelligible realm of eternal and unchanging objects, the Forms, which cannot be perceived through the senses. Mathematical knowledge is knowledge of the intelligible realm.
3 This modified version of Gödel numbering is that found in Nagel & Newman, 1958, pp.53-65
4 Axiomatisation: A process of reducing all truths to a single set of axioms and inferences.
5 Force = Mass x Acceleration.
6 This is contested by some philosophers, for example, Laudan 1981 and Popper 1972 & 1985. Due to essay constraints however, I will not defend this claim in any greater detail.
7 EPR: Named after the last initials of its writers: Albert Einstein, Boris Podolsky and Nathan Rosen.
8 Natural number: Ordinary, whole, ‘counting’ numbers.
9 Successor: ‘The next biggest number’ in vulgar mathematical language. See Benacerraf, 1983b: p273.
10 Benacerraf describes ‘the standard view’ which, for the purposes of this essay, is synonymous with mathematical Platonism.
11 E.g. knowledge of the future. See Burgess and Rosen, 1997, p39-40
12 Counter-examples to the ‘justified true belief’ account of knowledge. E.g. A man watching television whose phone, unbeknownst to him, is on silent. A phone rings on the television and this man believes this to be his phone that’s ringing but, by sheer coincidence, his own phone is actually ringing (though it is on silent). His belief that ‘my phone is ringing’ is justified and true but is not knowledge.
13 To borrow a metaphor from Wittgenstein’s Philosophical Investigations, §18
14 Brown, 1999, p57.
15 There are disagreements among structuralists as to its ontology and whether mathematical structures exist independently of the systems that instantiate them. Ante rem structuralism holds that these structures are independent and in rebus structuralism maintains that mathematical propositions are only true if there exists a structure that exemplifies the particular proposition. Due to essay constraints, and the fact that the difference between these two options has little, if any, impact on its conclusion, I do not advocate either position here.
16 System: ‘Define a system to be a collection of objects with certain relations among them.’ (Shapiro, 2000: p259)
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