Abstract
There is a large literature on the issue of the lack of properties (i.e. accidents) in quantum mechanics (the problem of “hidden variables”) and also on the indistinguishability of particles. Both issues were discussed as far back as the late 1920’s. However, the implications of these challenges to classical ontology were taken up rather late, in part in the ‘quantum set theory’ of Takeuti (Curr Issues Quant Logic 303–322, 1981), Finkelstein (in Beltrametti EG, Van Fraassen BC (eds) Current issues in quantum logic. Plenum, New York, 1981) and the work of Décio Krause (1992)—and subsequent publications). But the problems created by quantum mechanics go far beyond set theory or the identity of indiscernibles (another subject that has been often discussed)—it extends, I argue, to our accounts of truth. To solve this problem, i.e. to have an approach to truth that facilitates a transition from a classical to a quantum ontology one must have a unified framework for them both. This is done within the context of a pluralist view of truthmaking, where the truthmakers are unified in having a monoidal structure. The structure of the paper is as follows. After a brief introduction, the idea of a monoid is outlined (in Sect. 1) followed by a standard set of axioms that govern the truthmaker relation from elements of the monoid to the set of propositions. This is followed, in Sect. 2, by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next Sect. 3, then applies these ideas to quantum mechanics. It gives an account of quantum states and shows how these form a monoid. The final section then argues that quantum logic does not, despite what one might initially suspect, stand in the way of an account of quantum truth.
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Notes
Moody (1953) p. 102. It has to be said that, strictly, this formulation does not say anything as to what the proposition signifies to be and it may just as well be how ideas relate to one another as what the states of affairs are, even if the latter are what one tends to think of as intended.
This is not to say that Armstrong did not give a lot of space to necessary truths, including the mathematical, indeed he devoted a third of his book to the subject, chapters 7–10.
In the Tractatus, all the 2 and 3 series of propositions, e.g. 2.173: ‘A picture represents its subject from a position outside it. (Its standpoint is its representational form.) That is why a picture represents its subject correctly or incorrectly.’ Also Tractatus 4.1: ‘Propositions represent the existence and non-existence of states of affairs’. This last could almost be a motto for this paper.
For algebraic objects like groups and monoids, in fact associative algebras in general, there is a mathematical notion of representation where the algebra is represented by symmetries in a vector space (usually) [see Steinberg (2016), or for a general introduction Gowers (2009, IV.9) or Derksen and Weyman (2005)]. This is not quite the sense of ‘representation’ used above, but the underlying principle is the same: one structure is represented in a different structure. It cannot be ruled out that the fuller notion may enter into the picture in the future. Into which structure we make the representation is something that can be adjusted to purpose.
Weyl acknowledges the prior work from 1900-1 of Edmund Husserl in the Logical Investigations, and Husserl there relied heavily on the idea of states of affairs. The publication of Weyl’s work was delayed by the war.
Armstrong characterises the relationship between truthmakers in terms of mereology (and others have followed suit). But mereological fusion generates a monoid structure; thus we lose nothing in speaking of the latter notion, which is accepted algebra.
Such an element is also called an absorbing element or an annihilating element—though this is only suggestive when the operation is analogous to multiplication. Wittgenstein also spoke of a ‘totality of facts’, at Tractatus 1.1.
Wittgenstein (1921). However from the part relation one can also see that there is no sense in which all states of affairs can be regarded as independent or separate from one another—thus the Tractatus’ proposition 2.061 cannot be accepted: ‘States of affairs are independent of one another.’ But the complement to a state of affairs s could be said to be independent of s .
It has to be noted that Cameron is trying to square his view with Lewisian metaphysical claims, such as Humean Supervenience and this may have caused him to adopt an account of truthmaking which is different to that found in the authors mentioned in footnote 6 below.
See Heathcote (2003). Many of these axioms go back in some form or other to Mulligan et al. (1984). See also Armstrong (2004) and Read (2000)—with useful clarification on half-disjunction in the latter paper. See 4 below. The same denial of a full disjunction thesis goes back to Aristotle in On Interpretation. With Rodriguez-Pereyra (2006, 2009), I reject a generalised entailment principle—for discussion see Jago (2009) and de Sa and Dan (2009).
As long as we have a monoid with no negative elements we should not get paradox. It is reasonable to conjecture that contradictions such as the Liar paradox arise because there are negative elements in the truthmaker ground (in this case the monoid of sentences) as well as the objects (also the sentences) that we are mapping to. We leave this suggestive idea hanging.
At 4,4611 he likens tautologies and contradictions to the symbol “0” of arithmetic. But this may strike us as odd—surely both can’t be analogous to the digit for zero.
In fact Armstrong does embrace necessary states of affairs, as well as the addition of different kinds of truthmakers. His strategy could be described as pluralist—as was Wittgenstein’s. Armstrong’s strategy is to enlarge the set of truthmakers to include specific objects, such as numbers, but to construe these in a possibilist fashion [taking the lead from Hellman (1989)].
If one puts this in terms of possible worlds: the particular truthmaker that makes the proposition true here may not obtain in another world, and so the proposition as a whole will be made true by a different truthmaker, say that of Socrates being alive, or Paris being in the Netherlands.
This is notable as Armstrong wishes to explain arithmetic by postulating natural numbers as truthmakers for arithmetic truths. [see Armstrong (2004), pp. 96–104] In effect this would have been to postulate another monoidal model.
It is interesting to note that this view of the experimental character of the a priori appears also to have been the view of C. S. Peirce.
It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science. Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts (Peirce 1885, p. 182).
It may be problematic for a quite different reason but I come to that shortly.
The phrase echoes a similar comment in von Neumann’s article on the nature of mathematics, in his (1947). ‘I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But once they are so conceived, the subject begins to live a peculiar life of its own . . .’
To be clear: the following argument is not that set theory is not ever available or applicable: it is that it is not always available. In particular, it is not available when we are considering the fundamental particles of physics.
What I am outlining here is often called the Received View. This is not to say that there has not been push back against it, for example Muller (2015); Dorato and Morganti (2013); Bueno (2014) among others. In fact French and Krause (2006), though they discuss the Received View, do not entirely hold to it: they believe that QM is underdetermined by the evidence and that it can also support a view of particles as individuals. For pushback against the pushback, see Huggett and Norton (2013) and Heathcote (2020, 2021, 2021b).
Here is one such argument that appeared originally in Fraassen et al. (2008) p. 23. Suppose you have a set S containing three indistinguishable particles. Then what it means for the cardinality of that set to have a cardinality of three is that there is a function f from the entities of S to the ordinal set \(\{ 0, 1, 2 \}\) but then \(f^{-1} (0) \ne f^{-1} (1) \ne f^{-1} (2)\) : thus the entities of S are distinguishable after all. The cardinality of the set seems unquestionable: three electrons have three times the mass and charge of one, so what must be rejected is the existence of the function to the ordinals. Applying this problem to Sher’s model-theoretic concept we can see that it manifests in the requirement that we have a function which orders entities in sequences: such sequences being required in the definition of satisfaction of a formula. These sequences are functions mapping a set of entities to ordinals—precisely what we have seen reason to doubt.
The problem that indistinguishability creates for set theory has been studied over a number of years beginning with Parker-Rhodes (1981) Takeuti (1981); Finkelstein (1981); and then the papers Dalla Chiara et al. (1985, 1995); Schlesinger (1999), and nearly simultaneously Da Costa and Krause (1994), Krause (1992), Da Costa and Krause (1994, 1999); Krause and Arenhart (2018). More papers can be found in the reference sections of these articles.
This account of states was given in Weyl (1931) and Von Neumann (1932)—in both cases the ideas were first developed in 1927 in German publications—and have been given in many textbook accounts since, for example Emch (1984), Takhtajan (2000) or Bengtsson and Życzkowski (2017). A notable account was also given in Husimi (1940). See also Mielnik (2012). The Hilbert space gives a representation space to the pure states of \(\mathscr {S}\) but not to the mixed states and thus from early on came, to some, to seem anomalous and possibly unnecessary. It was done away with in I. E. Segal’s more general set of postulates for quantum theory. There are non-realist views of the states but here we take a realist view.
It is self-adjoint as a consequence of being positive.
It would be possible, and perhaps more desirable, to build up the set of states \(\mathscr {S}\) simply from a locally convex, Hausdorff, topological vector space since the Krein-Milman theorem states that if we have \(\mathcal {C}\), a compact convex set of elements of such a space, then \(\mathcal {C}\) is the convex hull of its extreme points \(\mathcal {P}\) (Krein and Milman 1940). This generalises the original theorem of Minkowski to infinite dimensional topological vector spaces. (If the space is Hausdorff but not locally convex, then the theorem does not go through.) Likewise there is a partial converse due to Milman: suppose A is a compact subset of a locally convex topological space and the closed compact hull \(\mathcal {C}\) of A is compact then all the extreme points of \(\mathcal {C}\) are in A [see (Schaefer 1971, p. 67)].
This implies that the set of states \(\mathscr {S}\) is, in addition to the above, a self-dual convex set. Duality (or topological duality), means that there is a convex set \(\mathscr {S} ^{*}\) such that there is an inner product \(<\, ,>\) defined on \(\mathscr {S}\) and \(\mathscr {S} ^{*}\), which is positive.
$$\begin{aligned} \mathscr {S} ^{*} = \{ \sigma : \forall \rho \in \mathscr {S}, <\sigma \, , \rho >\;\, \ge 0 \}. \end{aligned}$$Self-duality then implies:
$$\begin{aligned} \mathscr {S} ^{*}\; =\; \mathscr {S}. \end{aligned}$$States, as Penrose once noted, can be regarded as the square root of a probability. The self-duality of \(\mathscr {S}\) seems fit to formulate that idea (see Bengtsson and Życzkowski (2017), ch. 8.6, for more on this).
Apart from the real, complex and quaternionic Hermitian operators there are Jordan algebras called Spin Factors \(J_{2} V_{n}\) that use a Minkowski metric and whose positive cones are the future light cones in a Minkowski space of dimension \(n + 1\). These appeared already in the original classification of Jordan et al. (1934).
Jordan Banach algebras are the proper analog in the operator algebra domain.
Weyl stated these relations but Von Neumann then proved them in his (1932, ch. VI).
Notice that there are exceptional cases where disjunction does distribute: for example where we have \(s \leadsto (A \vee A)\) then this reduces to just \(s \leadsto A\).
The arguments above had been given in an earlier unpublished paper (by the present author) with an example which was not particularly good. But Read (2000) then came up with a better example and this is the one used in the text. It is interesting that this modal point was something neither Wittgenstein nor Russell considered. The preceding arguments were given in Heathcote (2003).
Von Wright’s preface has it: ‘It was always thought that modality is akin to probability . . .Yet it was a surprise to me, when I first noted that a modal logic could be constructed which contains an analogue to the multiplication principle of probability theory.’ In von Neumann’s Collected Papers volume 4 there is a short mimeographed note from 1937 on ‘Quantum Logics (Strict and Probability Logics)’ that discusses the introduction of probability as a logical operator. See also Birkhoff (1958).
The presence of the above suggestive analogy might make some philosophers wish that Birkhoff and von Neumann had given some attention to Hertz and Gentzen’s slightly earlier work on natural deduction systems as a model for their quantum logic, rather than the less obviously logical algebraic approach. Hermann Weyl, whose work we have already mentioned, was the doctoral advisor for Gentzen in the early 1930s; he tried, but failed, to get him to the IAS.
It is worth noting in this context that von Neumann himself, when he came to the problem of constructing a quantificational generalisation of quantum logic—and therefor a system for multiple lattices in a tensor product—found it impossible, describing the difficulties, over years and after several serious attempts, as ‘exceedingly great’. This should give us more than momentary pause. See his letters written to F. B. Silsbee from 1944, in particular the letter of July 2\(^{\textrm{nd}}\) 1945 in Rédei (2005).
If Wittgenstein or Russell had known of the idea of group representations which had come from Frobenius, Burnside and Schur they would have had a clearer model on which to formulate their nascent ideas.
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Appendix on Monoids
Appendix on Monoids
A semigroup \((\mathscr {M}\,,\, \cdot )\) is a set of elements \(\mathscr {M}\,\) with an associative binary operation \(\cdot\). If the operation is commutative, so that
then it is called a commutative semigroup. A semigroup that has an identity element e is a monoid \((\mathscr {M}\,,\, \cdot \; ,\; {\textit{e}})\), with, for all a,
This identity element is unique. It is a commutative monoid if the operation is commutative. An absorbing element z is an element such that, for all a
If an absorbing element exists then it is unique. An idempotent element is an element u such that
Obviously the identity element is an idempotent element. A semilattice is a partially ordered set in which any pair of elements a and b have a meet, or greatest lower bound, \(a \wedge b\). Every semilattice is a commutative semigroup under the operation \(\, \wedge\). Commutative semigroups in which every element is an idempotent element are semilattices. If it is a monoid then it is a semilattice where the identity element is the least element.
A commutative semigroup, or monoid, is said to have the cancellative property when one has the following implication: for all elements of \(\mathscr {M}\,\)
As an example, the non-negative integers under either addition or multiplication form a cancellative monoid. A cancellative semigroup can contain at most one idempotent element, namely the identity element.
There is a theorem that is attributed variously to Ore, Dubreil, or Grothendieck, to the effect that a commutative semigroup can be embedded in a group (in the sense that there is an isomorphic map into a subset of a group) if and only if it is cancellative. (For a non-commutative semigroup the conditions are more complex, but it is the commutative case that concerns us here.) In fact if a cancellative semigroup is finite then it is already a group. There is an explicit procedure for constructing the group completion of a monoid. For example, to construct the integers from the natural numbers \(\mathbb {N}\) one forms the negative integers by considering, for all n , \(- {\textit{n}}\, := \, 0 - {\textit{n}}\). (See Grillet 2001, ch. 1), or Clifford and Preston (1964, ch. 1)
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Heathcote, A. The Problem of Truth in Quantum Mechanics. glob. Philosophy 33, 9 (2023). https://doi.org/10.1007/s10516-023-09656-4
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DOI: https://doi.org/10.1007/s10516-023-09656-4