Erkenn (2011) 75:67–84 DOI 10.1007/s10670-011-9279-x ORIGINAL ARTICLE
Ontic Structural Realism and the Principle of the Identity of Indiscernibles Peter Ainsworth
Received: 6 August 2010 / Accepted: 10 March 2011 / Published online: 29 March 2011 Springer Science+Business Media B.V. 2011
Abstract Recently, there has been a debate as to whether or not the principle of the identity of indiscernibles (the PII) is compatible with quantum physics. It is also sometimes argued that the answer to this question has implications for the debate over the tenability of ontic structural realism (OSR). The central aim of this paper is to establish what relationship there is (if any) between the PII and OSR. It is argued that one common interpretation of OSR is undermined if the PII turns out to be false, since it is committed to a version of the bundle theory of objects, which implies the PII. However, if OSR is understood as the physical analogue of (sophisticated) mathematical structuralism then OSR does not imply the PII. It is further noted that it is (arguably) a virtue of OSR that it is compatible with a version of the PII for possible worlds.
1 Introduction Recently, there has been a debate as to whether or not the principle of the identity of indiscernibles (the PII) is compatible with quantum physics (see Saunders 2003a, 2006; Muller and Saunders 2008; Dieks and Versteegh 2008; Muller and Seevinck 2009; Ladyman and Bigaj 2010). It is also sometimes argued that the answer to this question has implications for debate over the tenability of ontic structural realism (OSR) (see Morganti 2004; Muller, forthcoming). The central aim of this paper is to establish what relationship there is (if any) between the PII and OSR. Section 2 is an hors d’oeuvre: it contains statements of the various forms of indiscernibility that have been distinguished in the literature, and an investigation into the relationships between them. In Sect. 3 some proposed connections between the PII and OSR are discussed. It is argued that OSR (or rather, one common P. Ainsworth (&) Department of Philosophy, University of Bristol, 9, Woodland Rd, Bristol BS8 1TB, UK e-mail:
[email protected]
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interpretation of OSR) is undermined if the PII turns out to be false, since it is committed to a version of the bundle theory of objects, which implies the PII. There is an interesting parallel here to the debate over the connection between mathematical structuralism and the PII: it has been argued that mathematical structuralism is not tenable because it is committed to a bundle theory of mathematical objects (see, for example, Burgess 1999; Kera¨nen 2001; MacBride 2006) and thus implies that the PII holds for mathematical objects, which it demonstrably does not. This led to the development of a more sophisticated form of mathematical structuralism (or, being charitable, a more careful statement of the original doctrine), which is not committed to a bundle theory of objects (see Shapiro 2008). In Sect. 4 the debate over the relationship between PII and mathematical structuralism is examined, and it is noted that, if OSR is understood as the physical analogue of this sophisticated mathematical structuralism, then OSR does not imply the PII. It is further noted that it is (arguably) a virtue of OSR that it is compatible with a version of the PII for possible worlds.
2 Indiscernibility 2.1 Grades of Discernibility It is usual in the literature to see a distinction drawn between four grades of discernibility: absolute discernibility, relational discernibility, relative discernibility and weak discernibility.1 However, for each of the above grades there are two distinct definitions to be found. For example, some authors (e.g. Muller and Saunders 2008, p. 528; Muller and Seevinck 2009, p. 182; Muller, forthcoming) define two objects (a and b) to be absolutely discernible (or equivalent) just in case there is a monadic predicate, P, such that: :ðPa $ PbÞ
ð1Þ
Other authors (e.g. Quine 1976, pp. 113–114; Saunders 2003a, pp. 4–5, 2006, p. 57; Ketland 2006, p. 306) define two objects (a and b) to be absolutely discernible (or equivalent) just in case there is a formula with one free variable, /(x), such that: :ð/ðaÞ $ /ðbÞÞ
ð2Þ
As Ladyman and Bigaj (2010, p. 125) point out, these definitions are not equivalent. Consider the following structure, for a language with no monadic predicates and one dyadic predicate, R: ðfa; bg; fha; aigÞ 1
ð3Þ
See Quine (1976, pp. 113–114), Saunders (2003a, pp. 4–5; 2006, p. 57), Ketland (2006, p. 306), Muller and Saunders (2008, p. 528), Muller and Seevinck (2009, p. 182), Ladyman and Bigaj (2010, pp. 125–127) and Muller (forthcoming, Section 3). However, not all of these authors distinguish all four grades of discernibility, and not all use the same terminology: Quine distinguishes strong, moderate and weak discriminability (corresponding to absolute, relative and weak discernibility); Saunders distinguishes absolute, relative and weak discernibility; Ketland distinguishes monadic, polyadic and weak discernibility (corresponding to absolute, relational and weak discernibility).
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Clearly a and b are not absolutely discernible according to the first definition, since there are no monadic predicates in the language. But they are absolutely discernible according to the second definition, using the formula AyRxy (since AyRay is true and AyRby is false). In this paper the former type of absolute discernibility is called absolute p-discernibility and the latter absolute /-discernibility. Likewise, there are two types of relational discernibility. In this paper, two objects will be called relationally p-discernible just in case there is an n-ary predicate, R, such that: 9x1 . . .9xn1 :ðRa; x1 ; . . .; xn1 $ Rb; x1 ; . . .; xn1 Þ
or
9x1 . . .9xn1 :ðRx1 ; a; x2 ; . . .; xn1 $ Rx1 ; b; x2 ; . . .; xn1 Þ or. . . 9x1 . . .9; xn1 :ðRx1 ; . . .; xn1 ; a $ Rx1 ; . . .; xn1 ; bÞ
ð4Þ
Two objects will be called relationally /-discernible just in case there is a formula with n free variables, /(x1,…, xn), such that: 9x2 . . .9xn :ð/ða; x2 ; . . .; xn Þ $ /ðb; x2 ; . . .; xn ÞÞ or 9x1 9x3 . . .9xn :ð/ðx1 ; a; x3 ; . . .; xn Þ $ /ðx1 ; b; x3 ; . . .; xn ÞÞ or. . .
ð5Þ
9x1 . . .9xn1 :ð/ðx1 ; . . .; xn1 ; aÞ $ /ðx1 ; . . .; xn1 ; bÞÞ Likewise, there are two types of relative discernibility. In this paper, two objects will be called relatively p-discernible just in case there is a dyadic predicate, R, such that: :ðRa; b $ Rb; aÞ
ð6Þ
This means that two objects are relatively p-discernible just in case there is a relation that is asymmetric with respect to these two objects (i.e. it is satisfied by the objects in one order only). Two objects will be called relatively /-discernible just in case there is a formula with two free variables, /(x, y), such that: :ð/ða; bÞ $ /ðb; aÞÞ
ð7Þ
Likewise, there are two types of weak discernibility. In this paper, two objects will be called weakly p-discernible just in case there is a dyadic predicate, R, such that: ðRa; b v Rb; aÞ & ð:Ra; a & :Rb; bÞ
ð8Þ
This means that two objects are weakly p-discernible just in case there is a relation that is satisfied by the objects (in either order), but that is irreflexive with respect to these two objects (i.e. it is not satisfied by either object with itself). Two objects will be called weakly /-discernible just in case there is a formula with two free variables, /(x, y), such that: ð/ða; bÞ v /ðb; aÞÞ & ð:/ða; aÞ & :/ðb; bÞÞ
ð9Þ
These definitions are summarised below: 1.
a and b are absolutely p-discernible if and only if there is some monadic predicate, P, such that :ðPa $ PbÞ.
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2.
a and b are absolutely /-discernible just in case there is a formula with one free variable, /(x), such that :ð/ðaÞ $ /ðbÞÞ. a and b are relationally p-discernible just in case there is an n-ary predicate, R, such that 9x1 . . .9xn1 :ðRa; x1 ; . . .; xn1 $ Rb; x1 ; . . .; xn1 Þ or. . . a and b are relationally /-discernible just in case there is a formula with n free variables, /(x1,…, xn), such that 9x2 . . .9xn :ð/ða; x2 ; . . .; xn Þ $ /ðb; x2 ; . . .; xn ÞÞ or. . . a and b are relatively p-discernible just in case there is a dyadic predicate, R, such that :ðRa; b $ Rb; aÞ. a and b are relatively /-discernible just in case there is a formula with two free variables, /(x, y), such that :ð/ða; bÞ $ /ðb; aÞÞ. a and b are weakly p-discernible just in case there is a dyadic predicate, R, such that ðRa; b v Rb; aÞ & ð:Ra; a & :Rb; bÞ. a and b are weakly /-discernible just in case there is a formula with two free variables, /(x, y), such that ð/ða; bÞ v /ðb; aÞÞ & ð:/ða; aÞ & :/ðb; bÞÞ.
3. 4.
5. 6. 7. 8.
In the following subsections the relationships between these grades of discernibility will be examined. 2.2 Relationships between the Grades of /-Discernibility Firstly, absolute /-discernibility implies relative /-discernibility: (i) Suppose a and b are absolutely /-discernible. (ii) So there is a formula with one free variable, /(x), such that :ð/ðaÞ $ /ðbÞÞ. (iii) So, either /ðaÞ and :/ðbÞ or :/ðaÞ and /ðbÞ. (iv) Let wðx; yÞ ¼ /ð xÞ & :/ð yÞ. (v) If /(a) and :/ðbÞ then w(a, b) is true and w(b, a) is false. (vi) If :/ðaÞ and /(b) then w(a, b) is false and w(b, a) is true. (vii) In either case, :ðwða; bÞ $ wðb; aÞÞ is true. (viii) So a and b are relatively /-discernible. But, as Quine has noted, the converse is not true: ‘‘All ordinals are [relatively discernible], since any two of them satisfy the open sentence ‘x \ y’ in one order and not the other. Yet they are not all [absolutely discernible]. If they were, then […] each ordinal would be uniquely determined by the set of open sentences that it satisfies. Sentences, being finite in length, are denumerable, so the number of sets of sentences is the power of the continuum. Thus if the ordinals were [absolutely discernible] they would be limited in number to the power of the continuum.’’ (Quine 1976, pp. 113–114). Secondly, relative /-discernibility implies relational /-discernibility: (i) Suppose a and b are relatively /-discernible. (ii) So there is a formula with two free variables, /(x, y), such that :ð/ða; bÞ $ /ðb; aÞÞ. (iii) So, either /(a, b) and :/ðb; aÞ or :/ða; bÞ and /ðb; aÞ. (iv) Suppose /(a, b) and :/ðb; aÞ (the other case is analogous). (v) Either /(a, a) or :/ða; aÞ.
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(vi) If /(a, a) then 9xð/ða; xÞ & :/ðb; xÞÞ ðsince :/ðb; aÞÞ. (vii) So 9x:ð/ða; xÞ $ /ðb; xÞÞ, so a and b are relationally discernible. (viii) On the other hand, if :/ða; aÞ then 9xð/ðx; bÞ & :/ðx; aÞÞ ðsince /ða; bÞÞ. (ix) So 9x:ð/ðx; aÞ $ /ðx; bÞÞ, so a and b are relationally discernible. (x) So in either case a and b are relationally discernible. But, as Ladyman and Bigaj (2010) have noted, the converse is not true: ‘‘If our language was limited to those dyadic predicates that are symmetric in a given domain (for instance, if L contained identity as the sole predicate), then it still would be possible to discern relationally, for instance, with the help of the formula x ¼ x & :ðx ¼ yÞ, and yet no formula would be satisfied by pairs of objects in one order only’’(Ladyman and Bigaj 2010, p. 126). Thirdy, as Ladyman and Bigaj (2010) have noted, relational /-discernibility is equivalent to weak /-discernibility. Relational /-discernibility implies weak /-discernibility: (i) Suppose a and b are relationally /-discernible. (ii) So either 9x1 . . .9xn1 :ð/ða; x1 ; . . .; xn1 Þ $ /ðb; x1 ; . . .; xn1 ÞÞ or. . . (iii) Suppose, without loss of generality, that 9x1 . . .9xn1 :ð/ða; x1 ; . . .; xn1 ÞÞ $ /ðb; x1 ; . . .; xn1 Þ. (iv) Let wðx; yÞ ¼ 9x1 . . .9xn1 :ð/ðx; x1 ; . . .; xn1 Þ $ /ðy; x1 ; . . .; xn1 ÞÞ (v) So w(a, b) and :wða; aÞ and :wðb; bÞ. (vi) So a and b are weakly /-discernible. Weak /-discernibility implies relational /-discernibility: (i) Suppose a and b are weakly /-discernible. (ii) So there is a formula with two free variables, /(x, y), such that ð/ða; bÞ v /ðb; aÞÞ & ð:/ða; aÞ & :/ðb; bÞÞ. (iii) Suppose, /ða; bÞ & ð:/ða; aÞ & :/ðb; bÞÞ (the other case is analogous). (iv) So /ða; bÞ & :/ða; aÞ. (v) So 9xð/ðx; bÞ & :/ðx; aÞÞ. (vi) So 9x:ð/ðx; aÞ $ /ðx; bÞÞ. (vii) So a and b are relationally /-discernible. 2.3 Relationships between the Grades of p-Discernibility The relationships between the different grades of p-discernibility are not completely analogous. Firstly, absolute p-discernibility does not imply relative p-discernibility. Consider a language with a single monadic predicate P and no relation predicates. Consider the structure: ðfa; bg; fagÞ
ð10Þ
The structure makes Pa true and Pb false, so a and b are absolutely p-discernible. But they are not relatively p-discernible, since there is no dyadic predicate in the language. (Moreover, relative p-discernibility does not imply absolute p-discernibility. Consider a language with a single dyadic predicate R and no monadic predicates.
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Consider the structure ({a, b}; {ha, bi}). Here a and b are relatively p-discernible but not absolutely p-discernible.) Secondly, relative p-discernibility does not imply weak p-discernibility.2 Consider a language with a dyadic predicate relation R. Consider the structure: ðfa; bg; fha; bi; ha; ai; hb; bigÞ
ð11Þ
Here a and b are relatively p-discernible but not weakly p-discernible. (Moreover, weak p-discernibility does not imply relative p-discernibility. Consider a language with a single dyadic predicate R. Consider the structure ({a, b}; {ha, bi, hb, ai}). Here a and b are weakly p-discernible but not relatively p-discernible.) Thirdly, relational p-discernibility does not imply weak p-discernibility. Consider a language with a single dyadic predicate R. Consider the structure: ðfa; bg; fha; ai; hb; bigÞ
ð12Þ
a and b are relationally p-discernible, since 9x:ðRa; x $ Rb; xÞ, but a and b are not weakly p-discernible. However, weak p-discernibility does imply relational p-discernibility: (i) Suppose a and b are weakly p-discernible. (ii) So there there is a dyadic predicate, R, such that ðRa; b v Rb; aÞ & ð:Ra; a & :Rb; bÞ. (iii) Suppose, for reductio, that a and b are not relationally p-discernible. (iv) So, 8zððRaz $ RbzÞ & ðRza $ RzbÞÞ. (v) So, ðRaa $ RbaÞ & ðRaa $ RabÞ (vi) So :Rba and :Rab (since from (ii) we know that :Raa). (vii) So :ðRab v RbaÞ. (viii) But (vii) contradicts (ii), so a and b are relationally p-discernible. 2.4 Relationships between the Grades of /-Discernibility and the Grades of p-Discernibility It is trivial that absolute p-discernibility implies absolute /-discernibility, as the former is a special case of the latter, and it has already been noted (in Sect. 2.1) that the converse does not hold. Similarly, it is trivial that relative p-discernibility implies relative /-discernibility, as the former is a special case of the latter. And the converse does not hold. Consider a language with a single monadic predicate P. Consider the structure: ðfa; bg; fagÞ
ð13Þ
a and b are relatively /-discernible: let /ðx; yÞ ¼ Px & :Py. Then /(a, b) and :/ðb; aÞ. So :ð/ða; bÞ $ /ðb; aÞÞ. But a and b are not relatively p-discernible, since there is no dyadic predicate in the language.
2
The analogue does hold in the case of /-discernibility, since relative /-discernibility implies relational /-discernibility and relational /-discernibility is equivalent to weak /-discernibility.
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Weak discernibility
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Relational discernibility
Relational pdiscernibility
Relative discernibility
Absolute discernibility
Absolute pdiscernibility
Relative pdiscernibility
Weak pdiscernibility
Fig. 1 The relationships between the eight grades of disernibility
Finally, it is trivial that relational p-discernibility implies relational /-discernibility, as the former is a special case of the latter. And in this case the converse does hold. This can be proved by contraposition: (i) Suppose a and b are not relationally p-discernible. (ii) So there is no n-ary predicate, R, such that 9x1 . . .9xn1 :ðRa; x1 ; . . .; xn1 $ Rb; x1 ; . . .; xn1 Þ or. . . (iii) So for every n-ary predicate, R, 8x1 . . .8xn1 ðRa; x1 ; . . .; xn1 $ Rb; x1 ; . . .; xn1 Þ and. . . (iv) So for every n-ary relation, R*, and every (n-1)-tuple of elements ha1,…, an-1i, the n-tuple ha, a1,…, an-1i [ R* if and only if hb, a1,…, an-1i [ R* and the n-tuple ha1, a, a2,…, an-1i [ R* if and only if ha1, b, a2,…, an-1i [ R* and… (v) So for any formula with n free variables, /(x1,…, xn), and every (n-1)-tuple of elements, ha1,…, an-1i, the n-tuple ha, a1,…, an-1i satisfies /(x1,…, xn) if and only if hb, a1,…, an-1i satisfies /(x1,…, xn) and the n-tuple ha1, a, a2,…, an-1i satisfies /(x1,…, xn) if and only if ha1, b, a2,…, an-1i satisfies /(x1,…, xn) and… (vi) So, for any formula with n free variables, /ðx1 ; . . .; xn Þ; 8x2 . . .8xn ð/ða; x2 ; . . .; xn Þ $ /ðb; x2 ; . . .; xn ÞÞ and. . . (vii) So a and b are not relationally /-discernible. These relationships are summarised in Fig. 1.3
3
Another intuitively plausible definition of indiscernibility, would classify a and b as indiscernibile (in a structure) if and only if there is an automorphism of the structure that maps them onto each other. If two objects are not relationally discernible then they are indiscernible in this sense: (i) Suppose a and b are not relationally p-discernible. (ii) So there is no n-ary predicate, R, such that 9x1 . . .9xn1 :ðRa; x1 ; . . .; xn1 $ Rb; x1 ; . . .; xn1 Þ or… (iii) So for every n-ary predicate, R, Vx1…Vxn-1(Ra, x1,…, xn-1 $ Rb, x1,…, xn-1) and…
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3 The Principle of the Identity of Indiscernibles and Ontic Structural Realism 3.1 The Principle of the Identity of Indiscernibles The PII is often formulated in second-order logic as follows (see, for example, Forrest 2006, section 2): 8PðPx $ PyÞ ! ðx ¼ yÞ
ð14Þ
In standard second-order semantics this is trivially true, since the second-order variable ranges over every set in the power set of the domain, so every object has a unique property. However, it is not trivially true in Henkin semantics, where the second-order variable ranges over only a subset of the power set of the domain. If the second-order variable ranges over only those properties that are the extensions of the monadic predicates of a given language (P1, P2,…, Pn) then the principle can also be stated in first-order logic as: ððP1 x $ P1 yÞ & ðP2 x $ P2 yÞ. . . & ðPn x $ Pn yÞÞ ! x ¼ y
ð15Þ
Implicitly, this assumes that a and b are indiscernible if and only if they share all the same monadic properties. Two objects are indiscernible in this sense just in case they are not absolutely p-discernible. However, while this version of the PII is arguably the version originally proposed by Leibniz it is now widely accepted that this version is false: in both classical and quantum physics there can be two distinct objects that share all the same monadic properties.4 Obviously, one can devise a number of different versions of the PII, corresponding to the different grades of (in)discernibility. The weakest version of the principle holds that any ‘‘two’’ objects that are not relationally discernible are identical (or, equivalently, that any ‘‘two’’ objects that are not weakly /-discernible are identical). Or, to put it the other way round, any two distinct objects are Footnote 3 continued (iv) So for every n-ary relation, R*, and every (n-1)-tuple of elements ha1,…, an-1i, the n-tuple ha, a1,…, an-1i [ R* if and only if hb, a1,…, an-1i [ R* and the n-tuple ha1, a, a2,…, an-1i [ R* if and only if ha1, b, a2…, an-1i [ R* and… (v) So the function f (f(a) = b, f(b) = a and f(x) = x for all other x) is an automorphism of the structure. However, the converse does not hold. Consider the structure: ({a, b}; {ha, bi, hb, ai}) The function f (f(a) = b, f(b) = a) is an automorphism of this structure, but a and b are relationally discernible (in fact, they are even weakly p-discernible). 4
In fact, as James Ladyman has pointed out to me, this is probably not even an accurate characterisation of the version of the PII that Leibniz had in mind. Leibniz (arguably) thought that no two distinct objects share all the same intrinsic properties. But whether or not there are monadic predicates for all and only the intrinsic properties depends on the language in question. In English, for example, there are monadic predicates for what appear to be relational properties (e.g. ‘‘uncle’’) and it seems likely that there exist intrinsic properties for which there are currently no predicates in English (even within the last 100 years physicists have discovered new intrinsic properties and have had to invent new predicates for them (charm, strangeness, etc.): it would be rash to suppose that this will not happen again).
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relationally discernible (or, equivalently, any two distinct objects are weakly /-discernible). This principle appears to be true in classical physics. It also appears to be true of fermions in quantum physics (see Muller and Saunders 2008). Whether or not it is true of bosons is a matter of debate (Muller and Seevinck 2009, argue that it is, but Ladyman and Bigaj 2010, question this result.)5 But what significance is it to find that any version of the PII appears to be true (or false) according to the lights of our current theories? Some people seem to have a very strong intuition that the PII must be true of physical objects. Saunders, for example, suggests that if we were to discover that some purported objects do not obey the PII then we should infer that they are not after all genuine objects: ‘‘free photons are certainly a counterexample […] to the PII.6 Does it follow that the principle should be abandoned? But the argument can be turned on its head […] The number of elementary bosons [e.g. photons] all in exactly the same state may be better thought of as the excitation number of a certain mode of a quantum field [rather than as a number of genuine objects],’’ (Saunders 2003a, p. 6, footnote added). But even if one does not share this intuition it is still worth investigating whether or not the PII is true or false. One philosophically significant consequence is in relation to the bundle theory of objects. According to this theory objects are just bundles of properties and relations. If an object is just a bundle of properties and relations then, on the face of it at least, any ‘‘two’’ objects that have all the same properties and stand in all the same relations must in fact be the same object, i.e. on the face of it, the PII must be true. So, if the PII appears to be false according to the lights of our current theories then either our current theories are false or the bundle theory of objects is false.7 It has also been argued (by Morganti 2004 and Muller, forthcoming) that the question of whether or not the PII holds is relevant to the debate about OSR. 3.2 The Principle of the Identity of Indiscernibles and Ontic Structural Realism Morganti (2004) claims that the PII, ‘‘is crucial for the whole argument [for OSR]’’ (Morganti 2004, abstract). The argument for OSR he has in mind here is the ‘‘underdetermination’’ argument (first proposed by Ladyman 1998). Roughly speaking, this argument runs as follows: 5
If one is not familiar with this debate then one might think that our scientific theories could not possibly suggest that the weakest version of the PII was false, because, on the face of it, it seems there could be no empirical grounds to postulate the existence of two distinct objects that share all the same properties and relations. But this is not correct. We would have empirical grounds to postulate the existence of two distinct but indiscernible objects if (for example) the properties of a system consisting of two such objects were not identical to the properties of a system consisting of just one of the objects (for example, the former system might have twice the mass of the latter). So even though the objects are individually indiscernible, there is a discernible difference between a system that consists of one such object and a system that consists of two such objects.
6
In the light of the arguments of Muller and Seevinck (2009), it is no longer so clear that free photons are a counterexample to all forms of the PII.
7
However, Rodriguez-Pereyra (2004) has argued that the bundle theory does not imply the PII.
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Premise If we take the basic ontology of quantum mechanics to be one of objects, properties and relations then the physics does not determine whether quantum objects are individuals or non-individuals. Conclusion OSR.
We can (and should) avoid this underdetermination by moving to
It is certainly possible to raise objections to this argument (see Ainsworth 2010, pp. 52–53) but it is not obvious that the argument requires the PII. Morganti seems to suggest that the premise of this argument is itself arrived at with the help of the PII, via the following argument: Premise No two distinct individuals share all the same properties and stand in all the same relations (a version of the PII). Premise Quantum mechanics suggests that in some cases two quantum objects do share all the same properties and stand in all the same relations. Conclusion So either: (i) such quantum objects are not individuals or; (ii) they are individuals, and they differ with respect to some (perhaps non-empirical) property or relation that is not included in their quantum mechanical description. So the physics does not determine whether quantum objects are individuals or nonindividuals. But the PII is really doing no work here. Indeed, if we do not assume it, then the underdetermination is, in a sense, worse: we still have the first two options but we also have a third option, viz: (iii) the objects are individuals, and they do not differ in any way. French (2010) also criticises Morganti’s reconstruction of the underdetemrination argument, and claims that, ‘‘there is no argument [for OSR] in French and Ladyman’s papers on the basis of the application of the PII’’ (French 2010, p. 95). Muller (forthcoming) also thinks that a version of the PII plays a role in the argument for OSR. Muller claims that, ‘‘One of the reasons provided for the shift away from an ontology for physical reality of material objects & properties towards one of physical structures & relations (Ontological Structural Realism: OntSR) is that quantum–mechanical description of composite physical systems of elementary particles entails they are indiscernible’’ (Muller, forthcoming, abstract, original emphasis). How does the alleged fact that quantum objects are indiscernible provide an argument in favour of OSR? Muller claims that ‘‘indiscernible objects are no objects at all’’ (a version of the PII) so, ‘‘elementary particles as material objects whither away’’ and we should conclude that, ‘‘At the truly and uniquely fundamental level of reality there are only structures’’ (Muller, forthcoming, section 1, original emphasis). Muller goes on to refine this argument, in effect by replacing ‘‘indiscernible’’ by ‘‘not absolutely discernible’’. The argument then seems to run as follows: Premise PII).
Any two (distinct) individuals are absolutely discernible (a version of the
Premise Quantum mechanics suggests that in some cases two (distinct) quantum objects are not absolutely discernible.
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Quantum objects are not individuals.
Conclusion
OSR provides the correct ontology for quantum mechanics.
Before turning to consider objections to this argument, it is worth noting that it is in almost direct conflict with the underdetermination argument for OSR. The underdetermination argument claims that we should move to OSR because quantum mechanics does not determine whether quantum objects are individuals or not individuals. Muller claims that we should move to OSR because quantum mechanics determines that quantum objects are not individuals. There are a number of objections that one might raise to Muller’s argument. Firstly, the conclusion of this argument is only really warranted if (i) OSR can make sense of the fact that quantum objects are not individuals and (ii) no (available) alternative ontology can make sense of this. But it is not really clear that either of these is true. Whether or not (i) is true depends on how one interprets OSR. Muller interprets OSR as the claim that ‘‘Elementary particles are structures.’’ (Muller, forthcoming section 1, original emphasis).8 But why should we be prepared to accept that there can be non-individual ‘‘objects’’ if these are really structures but not prepared to accept that there can be non-individual objects that are really ontologically primitive objects? Secondly, to borrow Saunders’ (2003a, p. 6) expression, the argument can be turned on its head. From the fact that in some cases two (distinct) quantum objects are not absolutely discernible, and the assumption that any two (distinct) individuals are absolutely discernible, Muller reaches the conclusion that quantum objects are not individuals. But surely it would be more natural to go from the fact that in some cases two (distinct) quantum objects are not absolutely discernible and the assumption that quantum objects are individuals to the conclusion that not all (distinct) indivduals are absolutely discernible. When Max Black (1952) claimed that there could be a world with two (distinct) indiscernible iron spheres, he did not take this to support the view that iron spheres are not individuals, but rather the view that not all (distinct) individuals are discernible. The case of quantum mechanical indiscernibles is analogous. In fact, Muller’s interpretation of OSR as the view that objects are structures is somewhat unusual. OSR is more usually understood as the claim that objects are secondary to the relations in which they stand.9 Understood this way there does appear to be a connection between OSR and the PII: OSR appears to be a variant on the bundle theory of objects, in which case it seems to imply some form of the PII.10 8
As structures themselves contain objects one might think that this does no more than assert that ‘‘elementary’’ particles are themselves composed of more fundamental objects, and this does not appear to be a form of OSR at all. Possibly, Muller has in mind the view that there is no lowest level of objects: any objects we find will themselves turn out to be structures on further analysis. Cf. Saunders (2003b), Stachel (2006, p. 54), Ladyman (2007a) and Ladyman and Ross (2007).
9
This is the version of OSR that French (2010) calls ‘‘eliminativist’’ OSR. French characterises this as the claim that, ‘‘the very constitution (or essence) of the putative objects is dependent on the relations of the structure’’ (French 2010, p. 106). See also Ainsworth (2010).
10 Cf. Dieks and Versteegh (2008) who state that, ‘‘The form taken by the ‘bundle idea’ in the case of relational properties is actually much discussed in present-day philosophy of science, and known as a from of structuralism. […] This position fits PII very well’’ (Dieks and Versteegh 2008, p. 927).
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And if we turn Muller’s argument on its head, and infer from the fact that in some cases two (distinct) quantum objects are not absolutely discernible, the conclusion that not all (distinct) individuals are absolutely discernible (i.e. that at least one version of the PII is false) then this might be an argument against OSR (although it might not be, because even if OSR implies some form of the PII, it may not imply the version that appears to be falsified by quantum mechanics). There are thus three very different proposed connections between OSR and the PII: Morganti (2004) argues that the PII is a premise used to argue that we should move to OSR because the question of whether or not quantum objects are individuals or non-individuals is not resolved by quantum mechanics. Muller (forthcoming) uses the PII as a premise to argue that we should move to OSR on the grounds that quantum mechanics determines that quantum objects are not individuals. It has been suggested here that OSR, understood as the claim that objects are secondary to relations, implies some form of the PII.
4 Lessons for Ontic Structural Realism from Mathematical Structuralism Following Shapiro’s Philosophy of Mathematics: Structure and Ontology (1997), there was a debate over the question of whether or not mathematical structuralism is committed to a form of bundle theory for mathematical objects (and thus to the PII for mathematical objects) and consequently over the question of whether or not some form of the PII was tenable for mathematical objects (see Burgess 1999; Kera¨nen 2001; Button 2006; Ketland 2006; MacBride 2006; Leitgib and Ladyman 2008; Shapiro 2008). It is often supposed that there is a close connection between OSR and mathematical structuralism (see, for example, Ladyman 2007b, pp. 38–40 and French 2010, section 5). So, we might hope to learn about the relationship between OSR and the PII by examining the debate over the relationship between mathematical structuralism and the PII. 4.1 Mathematical Structuralism and the Principle of the Identity of Indiscernibles A concrete structure is a collection of objects (e.g. bricks) that have a variety of properties (particular masses, colours, etc.) and stand in various relations to one another (on top of, heavier than, etc.). An abstract structure also consists of a collection of objects that have a variety of properties and stand in various relations Footnote 10 continued The connection between OSR and the bundle theory of objects has also been noted by French (‘‘in the absence of further metaphysical explication of the notion of structure itself, it is not yet clear whether or not such an approach [i.e. OSR] collapses into another form of the well-known conception of objects as bundles of properties’’ (French 2006, pp. 10–11)), Pooley (‘‘the idea of the independent existence of structures suggests an obvious comparison, viz. with the view that physical objects are nothing but bundles of collocated properties (‘bundle theory’)’’ (Pooley, 2006, p. 93)) and Ainsworth (‘‘[the OSRist’s] claim that relata are secondary to relations is not as revolutionary as it might at first appear: a similar claim is put forward in the so-called ‘bundle theory’ of objects’’ (Ainsworth 2010, p. 52)).
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to one another, but in this case the objects are intrinsically featureless and the properties and relations are purely extensional. Roughly speaking, mathematical structuralism is the view that mathematics is the study of abstract structures. In itself, this thesis does not seem to have any implications for the ontological status of mathematical objects. The idea that mathematical structuralism amounts to a form of bundle theory for mathematical objects does however, seem to be suggested by passages like these: By the ‘structuralist view’ of mathematical objects, I mean the view that […] objects have no more to them than can be expressed in terms of the basic relations of the structure, (Parsons 1990, p. 303). The essence of a natural number is its relation to other natural numbers […] there is no more to the individual numbers ‘in themselves’ than the relations they bear to each other (Shapiro 1997, pp. 72–73). As noted, it seems, on the face of it, that the bundle theory implies some form of the PII. So if mathematical structuralism is a form of bundle theory then it seems that mathematical structuralism is committed to a form of the PII. Shapiro has since (2008) explicitly stated that he is not committed to any form of the PII for mathematical objects. And he had better not be, because it appears that all forms of the PII are demonstrably false when it comes to mathematical objects. For example, consider the following set-theoretic structure: ðfa; bg; fa; bgÞ
ð16Þ
(The second {a, b} here is the extension of a monadic predicate.) In this structure a and b have all the same properties and are, on any account, indiscernible. So, according to any form of the PII, a = b. But this is absurd: there are two objects in the domain of this structure. To take a slightly less trivial example, consider a variant on the ‘‘dumb-bell’’ structure (discussed in this connection by Ketland 2006, p. 309): ðfa; bg; fa; bg; fha; ai; ha; bi; hb; ai; hb; bigÞ
ð17Þ
In this structure a and b have all the same properties and stand in all the same relations to every object in the structure and are, on any account, indiscernible. So, according to any form of the PII, a = b. Again, this is absurd. If mathematical structuralism is not committed to any form of the PII then there are two options: either it is not a form of the bundle theory or the bundle theory does not, after all, imply the PII. Shapiro takes the first option. He explicitly states that for the mathematical structuralist mathematical objects are not bundles of universals. He characterises mathematical objects rather as ‘‘components of universals’’ (Shapiro 2008, p. 302, original emphasis). The universals being referred to here are structures. It is often the case that if a is a component of b, then b is built out of a or a is ontologically prior to b, in the sense that a could exist without b, but not vice versa. For example, hydrogen atoms are components of water molecules, and it is both the case that water molecules are built out of hydrogen (and oxygen) atoms and the case
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that hydrogen atoms are ontologically prior to water molecules, in the sense that hydrogen atoms could exist without water molecules, but not vice versa. Thus the claim that objects are components of structures seems, on the face of it, to be a rather strange claim for a structuralist to make. It seems to be the sort of claim that an advocate of an object based ontology would endorse: for the advocate of such an ontology, objects are components of structures rather as hydrogen atoms are components of water molecules. Objects are the building blocks out of which structures are made and are ontologically prior to the structures of which they are components. But it is not always the case that if a is a component of b, then b is built out of a or that a is ontologically prior to b. Holes are (at least in some sense) components of cribbage boards, but cribbage boards aren’t built out of holes, and the ontological priority is rather the other way round: it would be meaningless to talk about the holes without the board, but not meaningless to talk about the board without the holes. The holes are really features of the board. Likewise Shapiro’s suggestion is perhaps that although there is a sense in which mathematical objects are components of structures, structures are not built out of mathematical objects and mathematical objects do not exist independently of structures: they are really features of structures, so the ontological dependence is really the other way round. This seems to be suggested when Shapiro claims that, ‘‘The structure is prior to the mathematical objects it contains, just as any organization is prior to the offices that constitute it,’’ (Shapiro 1997, p. 78). What then of the question of ontological priority between mathematical objects and relations? As noted, structuralism is often understood as the claim that objects are ontologically secondary to the relations in which they stand, which seems to imply the bundle theory of objects. As noted, Shapiro explicitly rejects this view. His view then seems to be that (i) structure is prior to objects but (ii) relations are not prior to objects. This is not an obviously inconsistent view. But can we say more? It is useful here to be a little clearer about what it means for one thing to be prior to another. To this end the following definitions will be adopted: Existential dependence exist without b.
a is existentially dependent on b if and only if a could not
Terminological dependence a is terminologically dependent on b if and only if we would not call something ‘‘a’’ unless b existed. Independence a is existentially (terminologically) independent of b if and only if neither is existentially (terminologically) dependent on the other. Interdependence a is existentially (terminologically) interdependent with b if and only if each is existentially (terminologically) dependent on the other. Priority a is existentially (terminologically) prior to b (and b is existentially (terminologically) secondary to a) if and only if b is existentially (terminologically) dependent on a and not vice versa.
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In general, mathematical objects are existentially prior to mathematical relations. Shapiro himself discusses what he calls ‘‘finite cardinality structures’’: structures that have a domain of objects but no relations (Shapiro 1997, chapter 4, 2008, p. 287). These structures demonstrate that objects can exist without relations. But we cannot have a structure that contains relations but no objects, so relations cannot exist without objects. Hence objects are existentially prior to relations. However, it is plausible to say that, in the context of a given structure, particular mathematical objects are terminologically dependent on particular mathematical relations. For example, it is plausible to say that, in the context of the natural number structure, we would not call an object ‘‘5’’ if it did not stand in the ‘‘\’’relation to 6 (and it could not stand in the less than relation to 6 if this relation did not exist). On the other hand, in the same context, we would also not call something ‘‘the less than relation’’ if 5 did not stand in that relation to 6. In this case then, 5 and the less than relation are terminologically interdependent.11 So it seems that what the mathematical structuralist ought to say is that, in general, mathematical objects are existentially prior to mathematical relations, although, in the context of particular structures, particular objects can be terminologically interdependent with (and possible secondary to) particular relations. 4.2 Mathematical Structuralism and Ontic Structural Realism French (2010) draws a comparison between mathematical structuralism and OSR. He says that the mathematical structuralist holds that, ‘‘a number is a place in the number structure’’ (French 2010, p. 98) and that the OSRist holds that, ‘‘an electron is a node in the electron structure’’ (French 2010, p. 98). The key difference, according to French, is that while mathematical structuralists think that mathematical structures exist, ‘‘independently of any exemplifying concrete system’’ (French 2010, p. 98) OSRists do not of course think that the physical structures that they are interested in exist independently of any exemplifying concrete system: they think they are concrete systems. The comparison between mathematical structuralism and OSR, together with the discussion of mathematical structuralism in the previous subsection, suggests a version of OSR that is a rather different to those discussed in Sect. 3. Two main versions of OSR were discussed there: (i) the view that objects are secondary to relations and, (ii) the view that objects always turn out to be structures on further analysis.12 The sophisticated mathematical structuralist would not endorse either of these claims. His view is that structures are prior to objects but that relations are
11 We need to restrict ourselves to the context of a particular structure here. In the context of the structure we get if we remove the less than relation from the natural number structure 5 is not terminologically dependent on the less than relation, since we would still call an object 5 even though the less than relation does not exist (in that structure). Likewise, in the context of the structure we get if we remove 5 from the natural number structure the less than relation is not terminologically dependent on 5. 12
There are other forms of OSR in the literature. For example, Esfeld (2004) and Esfeld and Lam (2008) develop a form of OSR according to which objects and (multi-place) relations are interdependent.
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not. And he is certainly not committed to the view that objects always turn out to be structures on further analysis. Suppose we construe OSR along these lines. What then is the relationship between OSR and the PII? As noted, the mathematical structuralist is not committed to the PII. And there is no reason why an OSRist of this sort need be committed to it either, because an OSRist of this sort does not hold objects to be secondary to relations. This brand of OSR is consistent with the PII being true in the actual world, but it does not imply that the PII is necessarily true. However, we could also consider a version of the PII that applies to possible worlds, rather than to individual objects. This would state that there cannot be two distinct but indiscernible possible worlds.13 Let’s call this the PIIW. Esfeld and Lam (2008) argue (in effect) that it is a virtue of OSR that it allows one to accept the PIIW. The argument arises in the context of general relativity. Given some model of the field equations of general relativity one can generate (infinitely many) novel models by (roughly speaking) permuting the space-time points in the original model. If we take the view that the objects in these models (space-time points) are the fundamental building blocks of reality then these models seem to represent infinitely many distinct but indiscernible possible worlds. Thus we must reject the PIIW. The OSRist, however, can accept the PIIW, because he does not treat objects as the fundamental building blocks of reality. For the OSRist an object is a place in a structure. This implies that there is no real difference between the models: it makes little sense to say that we can permute space-time points, i.e. put a in the place of b, if space-time points just are places in structures. All we could do would be to swap the names of the places (cf. permuting the holes in a cribbage board). What is the significance of this? If one is not a realist about possible worlds, then the question of whether or not there can be distinct but indiscernible worlds seems to be of little importance, and it seems likely that the answer will depend on how one cashes out talk about possible worlds. However, if one is a realist about possible worlds, then the question seems more important. The claim that there cannot be distinct but indiscernible worlds amounts to the claim that the multiverse of possible worlds does not contain duplicates.
5 Summary of Conclusions 1.
2. 3.
If OSR is understood as the claim that objects are secondary to relations then it appears to imply that (some form of) the PII is true. Whether or not some form of the PII is true of physical objects is an open question. If OSR is understood as the physical counterpart to sophisticated mathematical structuralism, then OSR is not committed to the PII. It is arguably a virtue of OSR (at least if one is a realist about possible worlds) that (arguably unlike object based ontologies) it is compatible with the truth of the PIIW.
13 It is thus the denial of the claim that there can be two distinct but indiscernible possible worlds, a doctrine sometimes known as haecceitism.
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Acknowledgments I gratefully acknowledge the financial support of the Leverhulme foundation. This paper was inspired by discussions in the Bristol structuralism group and I owe a huge academic debt to everyone there. I’m especially indebted to James Ladyman, who made some helpful comments on an early draft of this paper. I’m also indebted to the anonymous referees of the journal, both of whom made a number of useful comments.
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