Problem of Truth in Quantum Mechanics

June 24, 2023 eigentrope1 Comment

This is the introduction of a paper of this name which is in print in the journal Global Philosophy (the article was submitted and accepted in the journal called Axiomathes, but just before the paper was sent back for proof-reading the journal changed its name to Global Philosophy — thus this is where it can be found at http://t https://doi.org/10.1007/s10516-023-09656-4 . A pre-print copy can be found at Academia.com https://www.academia.edu/95833087/The_Problem_of_Truth_in_Quantum_Mechanics

Please go to either of these sources if you wish to see how the argument of the paper progresses.

Abstract

There is a large literature on the issue of the lack of properties (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 (1981), Finkelstein (1981) and the work of Decio 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, 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 §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 §2, by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next section, §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. (end of Abstract)

Pluralism, or at least dualism, about truth has been the unacknowledged orthodoxy for the greater part of the history of philosophy. Most philosophers have recognised ‘truths of fact, or existence’ and those due to the ‘relation of ideas’, to put it in Hume’s way of speaking, synthetic and analytic relations to put it in Leibniz-Kant’s. Why unacknowledged? Perhaps because in thinking about truth it is the relation of the proposition to fact or existence that has dominated the mind, and so most traditional accounts have been cast in terms of agreement with how things are. As the medievals said: Qualitercumque significat esse, ita est (A proposition is true if, howsoever it signifies to be, so it is).(1) This dominance was at the forefront when Russell and Wittgenstein contributed to the unofficial birth of truthmaker theory the idea that truths require states of affairs (or facts, situations, circumstances — der Sachverhalten, in Wittgenstein’s original German) to make them true. What about the relations of ideas? These were consigned to emptiness — or at least mere definition. When Armstrong took up this truthmaker project it was states of affairs that were assumed to be the truthmakers and analytic sentences were for the most part left out of consideration (Armstrong (2004)).(2) But, for all that, truthmaker theorists were often unofficial dualists about truth.

That such a dualism can be carried into a more extensive pluralism has been argued by several authors in the last thirty years, notably Crispin Wright, Michael Lynch and Gila Sher. Here I focus on the version given by Sher (in Sher (2004), (2013)), since Sher sets her pluralism in the general context of the correspondence account of truth. Whereas truthmaker theory has often been accused of not giving an account of certain classes of truths, pluralists seek to leave no class of statement out of consideration. Armstrong was motivated by an ontological parsimony — to see how much one could do with just a physicalist ontology of states of affairs; Lynch is motivated by giving folk attributions of truth a weight equal to the scientific and his alethic pluralism unifies the different folk attributions by their sameness of functional role, this being given by the relation of the purported truths to beliefs and actions, rather than being constrained by ontological considerations.

Sher, by contrast, though motivated by the correspondence intuition, is not wedded to a strict physicalism, but allows higher level abstract structures to be the ita est of the medieval formulation — in short, to be that to which true statements may correspond. This is consonant with the view of truthmaking which is taken in the present paper. The truthmaker idea should not be constrained by an ontological prejudice for what the truthmakers should be, at least not before we know what is. (One finds a different liberalisation in Griffiths (2015).) On the other hand, there should be a clear essence to truth, something that unifies the various plural instances into one clear kind. This is to protect us against including things that merely look like Truth: for example fervent belief.

Giving an account of analytic truths, however, requires that one make the truthmakers something other than states of affairs. I have described this elsewhere but I review it here, keeping the relation between truthmakers and propositions the same as in the standard case. What changes is the truthmaker basis.

What is more, the structure of these two kinds of truthmakers can be seen to exhibit a very general mathematical form: they are both monoids — something resembling a group but with no requirement that there exist inverse elements. Monoids are quite basic in terms of their structure and it is hard to imagine a collection of truthmakers not exhibiting such a structure. If correct this means that we can abstract the idea of truthmaking to a relation between certain monoids and boolean algebras of propositions, where the nature of the mapping is given by the truthmaker relation. We could also say that truth is a representation of a monoid in a boolean algebra. This is its essence: it is a representation of a structure that lacks negative elements in a structure that possesses them. This representation can be understood as a “picture” in the sense of Wittgenstein’s Tractatus.(3) Pictures are Wittgenstein’s (loose) way of speaking of representations.(4)

But ultimately the present reason for wanting to construct a less restrictive truthmaker theory is due to a problem that looms over many theories of truth: the problem of truth in quantum mechanics. Specifically it looms for any theory of truth that depends upon a classical ontology of individualised objects and properties that are then mapped to (correspond with, form sequences that satisfy, etc) assertions, propositions, statements, or sentences. This will include Armstrong’s, Sher’s and Tarski’s theories — but also many others. This is because quantum mechanics has a different mathematical character to that which we find in a classical world view: there are states in a topological vector space and non-commutative sets of operators that act upon those states, with particles that fail to be individualised. How are we to make this into a basis for a truthmaker account?

It is interesting to note that one of the pioneers in the mathematical formulations of quantum mechanics, Hermann Weyl, was also one of the first to properly formulate the notion of truthmaking, some years before Wittgenstein and Russell. Here are his words, written in 1917.

A judgment affirms a state of affairs. If this state of affairs obtains, then the judgment is true; otherwise. it is untrue. States of affairs involving properties are particularly important. (Indeed, logicians often made the mistake of ignoring every other state of affairs.) A judgment involving properties asserts that a certain object possesses a certain property, as in the example: “This leaf (given to me in a present act of perception) has this definite green colour (given to me in this very perception).” A property is always affiliated with a definite category of object in such a way that the proposition “a has that property” is meaningful, , expresses a judgment and thereby affirms a state of affairs, only if a is an object of that category.A judgment corresponds only to a meaningful proposition, a state of affairs only to a true judgement; a state of affairs, however, obtains — purely and simply. (Weyl 1917; 5) (5)

Unfortunately, however, Weyl never returned to this issue following the advent of quantum mechanics in 1925.

The structure of the paper is as follows. In §1 the idea of a monoid is outlined followed by the standard axioms that govern the truthmaker relation from elements of the monoid to propositions. This is followed by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next section 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 a theory of quantum truth.

Notes: (1) 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.

(2) 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.

(3) 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.

(4) 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.

(5) 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.

2 Monoids and Truthmaker Axioms

Our truthmakers will be a collection, denoted

s,s‘,s”,…t,t‘,t”,…,u,vuantum

but also sometimes capitalised. On this collection there is an associative operation ⊕ , i.e.

s⊕(t⊕u) = (s⊕t)⊕u.

This operation will, in the classical case, also be commutative, i.e.

s⊕t = t⊕s

(later this condition will be dropped.) Among this collection there is an element that is an identity for this operation. It is unique. I will denote it as V. Thus, for all s,

s⊕V = s.

With this operation the truthmakers form a monoid. A significant property of monoids is that elements need have no inverses under the given binary operation. A set on which an associative binary operation is defined is called a semigroup. A monoid is a semigroup with an identity element. Thus the theory of monoids is often folded into the theory of semigroups. The first significant results on semigroups did not come until 1928 (see Clifton and Preston (1964)) but since 1940 it has become a rapidly developing field.

Some familiar examples of monoids are: the natural numbers N with addition as the binary operation and 0 as the identity; also the natural numbers with multiplication as the operation and 1 as the identity element; the powerset P(S) of a non-empty set S with ∪ as the binary operation and the empty set ∅ as the identity element. A simple example of a non-commutative monoid would be strings, i.e. a set of symbols/letters under the operation of concatenation. (Jacobson (2009) ch. 2.)

Significantly, the operators in quantum mechanics form a monoid, as noted in von Neumann (1929) which can then also be made into a regular ring (Murray and von Neumann (1936) and von Neumann (1936)). This leads to the operator algebra point of view, to give it its more modern name.

Additional relevant aspects of monoids will be gathered together in an appendix at the end of the paper.

As noted above, the usual idea in the literature is that truthmakers are, in Wittgenstein’s original term, der Sachverhalten — in translation: facts, or circumstances, or states of affairs.(6) This is the idea we start with. We begin by describing the relationship between the set of truthmakers and propositions.

Truthmaker Defn:

• If s is a truthmaker for proposition A, designated s ⇝ A, then it is necessarily the case that if s obtains then A is true.

The embedded conditional in the Truthmaker Definition is intended to be a natural conditional. Note that it cannot be strengthened to an ‘if and only if’ conditional, because A may also be true due to the obtaining of some other truthmaker. The existence of the truthmaker s may be said to be a sufficient condition for A to be true. This is often summarised in a slogan: truth depends upon being. This is slightly misleading, in that it suggests that we have here the dependence of one thing, Truth, upon another thing, Being. In fact it is the dependence of the obtaining of a property of a thing (a proposition’s truth) on another complex thing (usually a state of affairs).

We now have the main claim of truthmaker theory — a claim that may, as Russell said, seem so obvious that it is hard to imagine anyone denying it. (Russell 1956).

Primary Truthmaker Axiom:

• For any true proposition A, and no untrue proposition, there is an s such that s ⇝ A. Conversely, for any s there is a proposition A that it makes true.

In other words, no true proposition exists that is true independently of the truthmakers that obtain, nothing is true ‘all by itself’, no truth is an island. David Armstrong calls this principle truthmaker maximalism and assumes it — the present paper attempts to make it more plausible. On the other hand false propositions are false due to the absence of anything that might have made them true — they are islands. There are no “falsemakers”. And just as true propositions require truthmakers so also truthmakers require propositions that they make true: there are no truthmakers that are idle. (It is also worth noting that a proposition’s being true is also a state of affairs and thus a higher level truthmaker.)

We now relate the operation ⊕ to the Boolean structure of the propositions.

Co-obtaining:

• If s ⇝ A and t ⇝ B then s⊕t⇝A & B where s⊕tis the agglomeration (or mereological sum) of the truthmakers s and t, i.e. the co-obtaining of both truthmakers.

Closure:

• If s and t are both truthmakers then s⊕tis also a truthmaker.

One-Sided Disjunction:

• If s ⇝ A then s⇝A ∨ B, for any B. (The converse does not hold — though of course this does not mean that true disjunctions have no truthmakers, they do.)

In older formulations of truthmaker theory it was supposed that truthmaking should be transmitted by classical logical entailment from one proposition to another (see Heathcote (2003) (2006) and Read (2000)). However this led to undesirable consequences, since any proposition entails a necessarily true proposition and thus any true proposition does also; this would mean that all of the truthmakers will become truthmakers for all and any necessary propositions. (This has been called truthmaker monism collapse.) This is not a consequence of the present view, for we explicitly reject the idea that truthmaking is transmitted by classical entailment. (Armstrong (2004) pp. 10–12 wanted to keep in some limited form of this entailment principle, but no specific proposal was made)

The monoid with the operation ⊕ has an identity element V (for Void) which has the property that for any s we have

s⊕V= s.

This is analogous to the empty set, which is the identity element in the set-theoretic example of a monoid given on p. ? Co-obtaining has some interesting consequences. Since our monoid is commutative there exists a pre-ordering ≤ on its elements that gives us the intuitive idea of smaller and larger truthmakers, given by

s≤s‘ if there exists some t such that s⊕t=s‘.

This allows us to speak of truthmakers that are parts of other truthmakers.

Parts of truthmakers:

• A truthmaker s is a part of a truthmaker t if and only if s⊕tis just t.

Enlargement:

• If s is a truthmaker for A and s is a part of t then t is also a truthmaker for A.

The final consequence is the existence of a Universe.

Universe:

• The Universe (or totality) U of truthmakers is that for which, given any truthmaker s, s⊕U= U. It can be regarded as the agglomeration of all of the truthmakers and so by closure a truthmaker itself — thus all truthmakers are parts of U. (See Jacobson (2004).) (7)

As a consequence of these axioms we can also say what it is for a complement to a state of affairs s to exist. Thus s‘ is a complement of s if and only if (1) no part of s‘ (resp. s) is a part of s (resp. s‘); and (2) s‘ ⊕ s =U. It does not follow from this that every state of affairs has a complement.

From these axioms it follows trivially that negative existentials have truthmakers. Let p be ‹ there exists a Unicorn ›: p is false and so has no truthmaker. ∼p is true and so has a truthmaker. Since U has all truthmakers as parts, U is a truthmaker for ∼p. This was one of Wittgenstein’s insights in the Tractatus. At 2.05 he says, ‘The totality of existing states of affairs also determines which states of affairs do not exist.’ (8) Very often when we speak of the truthmaker for a proposition A we mean a minimal truthmaker — this can be defined as a truthmaker for which there are no proper parts that are also truthmakers for A. With this in mind we can note that even though U will always be a truthmaker for negative existentials it may not be the minimal truthmaker. Thus consider ‹ There are no Arctic penguins ›. If Arctic penguins existed then they would have to exist in the Arctic, or at least on Earth. So now repeat the above argument replacing U by the set of existing states of affairs constituting the Arctic (or Earth). The Arctic is then the truthmaker for the truth that there are no Arctic penguins. Other things are such that they will inevitably require all of U. There are no 10 GeV Higgs Bosons anywhere and so we can’t consider a state of affairs smaller than U itself.

The example of the Arctic penguins is reminiscent of an example that Russell tried out on Wittgenstein. Before Wittgenstein formulated what we can now see is the correct view he held that negative existentials are one and all meaningless. On the 1st of November 1911 Russell in a lecture tried to get Wittgenstein to agree with the statement ‘There is no rhinoceros in this room’. Wittgenstein refused to agree. Russell responded by looking all around the room including under the desk while Wittgenstein said nothing. But then Russell in his lectures on logical atomism advocated negative facts (see Russell 1956) as a solution. (We don’t know if he had this view in 1911). This idea has been restated with more sophistication in Priest (2000) and Beall (2001). When he revived the notion of truthmaking David Armstrong advocated for a higher-order totalising fact (Armstrong (1997) and (2004)). Others had advocated absences (Martin (1997)). In this way truthmaker theory came to seem encumbered by an indenumerable number of phantom, causally inert, states of affairs. Metaphysics run rampant! The problem for these views was put sharply by George Molnar in his (2001).

The above Wittgensteinian solution to the negative existentials problem was restated in 2003. It was proposed in Heathcote (2003) and then in nearly the same terms by Lewis and Rosen (2003) with respect to Lewis’ own qua strategy; it was again put forward in 2004, in Niiniluoto (2004); and was defended in Cheyne and Pigden (2006) and again more recently in Griffith (2015). However, it seems to have escaped the notice of many critics, for it has not to my knowledge been either accepted, and the problem laid to rest, or refuted. (That said, a rather similar proposal was made by Cameron in his (2008), but adding the idea that U — the World, in Cameron’s terminology — necessarily lacks unicorns. He says: ‘Given truthmaker necessitarianism this commits me to the claim that the world is essentially lacking in unicorns, talking donkeys etc.’ (p. 413) I think this is a mistake, and it is certainly not part of the traditional view. Given the world as it is ‹ there are no unicorns ›is necessitated to be true; so if ‹ there are no unicorns ›had been true the world would have had to be different, in that there would then have been unicorns. That is all that follows — U is not essentially unicornless, it is contingently unicornless. Thus it should be noted that the states of affairs that make up U are for the most part contingent — thus forcing U to be contingent, for if one contingent state of affairs were different then U would be different.)(9)

This gives us the basis to speak of truthmaking for propositions including negative existentials. To extend it to sentences S in a given language L we have:

Truthmaker Axiom for Sentences:

• i) If there exists an s such that s ⇝ S then S is true; ii) For any true sentence S that expresses a true proposition A, if s is such that s ⇝ A then s ⇝ S when S is so understood.

We do not assume that for every true proposition A there will be some sentence S that expresses it, simply because sentences are finitary objects. Ambiguous sentences may have a truthmaker on one resolution of their ambiguity but not on another.(10)

One might ask: what is it about monoids that make them particularly suitable for characterising states of affairs? The answer is that in a monoid there is no need for there to be any negative elements, no inverses relative to the binary operation ⊕ — there is no requirement that they exist. So there need be no members of the monoid which “negate” or “undo” other truthmakers under the application of the operation, as one would have to have if the structure were a group — just as in the monoid of the natural numbers under the operation of addition there are no additive inverses, no negative numbers (which is why N does not form a group under either addition or multiplication). So we also don’t have the following situation with truthmakers, with t being the negative of s:

s⊕t = V.

If this situation were to occur the mere aggregation of the two truthmakers s and t would cancel one another to yield the Void. (V be it noted is a positive state of affairs.) Being is only positive, it is. It is propositions that bring in negation.(11) Thus we agree with the Tractatus at 2.04: ‘The totality of existing states of affairs is the world.’ Also, as already noted, agreed with is Tractatus 4.1: ‘Propositions represent the existence and non-existence of states of affairs’. Unknowingly, but presciently, Wittgenstein was characterising a commutative monoid.

The introduction of U, the universe of states of affairs, may tempt someone to the view that one could make do with U as a single truthmaker for all truths, particularly in the light of the Enlargement principle above (see for example, Shaffer (2010)). But simplicity is not everything and there are very good epistemological reasons for our needing to consider more fine-grained truthmakers — such as in addressing the Gettier problem (see Heathcote (2006) (2012) (2014)).

In the next section we take up the derivation of necessary truths from a different monoidal representation, before moving on to the problem of extending truthmaking to quantum claims.

Notes: (6) 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.

(7) 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.

(8) 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 .

(9) 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.

(10) 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) and (2009), I reject a generalised entailment principle — for discussion see Jago (2009) and López de Sa (2009).

(11) 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.

(Continued in the full pdf mentioned at the outset.)

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Review of Demopoulos 2022

June 23, 2023 eigentropeLeave a comment

(Printed version is on Metascience doi: 10.1007/s11016-022-00807-8; an unedited pre-print is on Academia.com: https://www.academia.edu/86788063/Traces_in_Probability_Space )

Traces in Probability Space

William Demopoulos’ On Theories: Logical Empiricism and the Methodology of Modern Physics, Cambridge: Harvard University Press. xxiv + 247 pp.

William Demopoulos died in May 2017, altogether too young, and with a manuscript nearing completion that needed some tidying editorial attention. This is the volume under review, and Demopoulos’ long-time friend and collaborator, Michael Friedman, has admirably completed the necessary editorial tasks. In addition he has contributed an Editor’s Foreword and Afterword to help give the chapters more continuity — and, more importantly in the case of the Afterword, to trace Demopoulos’ complex path in interpreting quantum theory.

Since the quest to understand quantum mechanics spanned Demopoulos’ career and since nearly half of the present book is given over to it, we focus on this.

In the late 1970s Demopoulos was one of the leading figures in the ‘Western Ontario’ school of quantum logic. It was an exciting period, even if one was unconvinced by the overall plan. (It was during this period that this reviewer, then a graduate student, became familiar with his work; one never had cause to regret reading his essays, which were always sharp and to the point.) By 1982 Demopoulos had become disenchanted with quantum logic, as he had realised that there were classical contradictory propositions that were nevertheless true in quantum logic! Unable to reconcile himself to this, for the next 22 years he turned away from quantum logic to focus on other, unrelated, issues.

He returned to quantum theory only around 2004, around the same time as Itamar Pitowsky published his paper interpreting quantum probabilities through Dutch book arguments (Pitowsky 2003). They were clearly in close communication and the association informs the discussion in the present book.

The basic idea behind the proposal is as follows. Consider Minkowskian space-time and its group of rotations, the Lorentz group. The introduction of this structure takes a phenomenon that looks to be dynamical, namely Lorentz contraction, or time dilation, and reinterprets it as a kinematical phenomenon. It is nothing more than the result of a rotational change of frame. This might be called a structural explanation or, less happily, a principle explanation. In it, something that looks dynamical receives a purely geometrical explanation. We have a similar case, not mentioned by Demopoulos but worth having in front of us, in the Pauli exclusion principle. This originally also looked dynamical — a strange “force” that kept electrons apart from one another — and was then, almost immediately, given a geometrical interpretation: it was due to the reduction of the tensor product space to just the antisymmetric quotient (for fermions), which is now identified as the exterior power, a Grassmann algebra. Or we could consider the way that gravitation became nothing more than a kinematical phenomenon on a curved space-time. Such structural explanations have had considerable success in 20th Century physics!

But how can we understand the baffling probabilities that occur in quantum theory, both for measurement outcomes and the distant correlations? These also have a faux dynamical character and thus look fit for a structural explanation. The crucial theorems for Demopoulos are those of Gleason and Kochen-Specker (KS) — the latter following from the former plus a (logical) compactness claim — and the relevant structure the projection lattice of subspaces of a Hilbert space. Gleason’s theorem tells us that a generalised measure defined on such a lattice of a Hilbert space (with dim ≥ 3) extends uniquely to a linear functional on all bounded operators. In quantum mechanical terms this means that it extends from pure states to apply also to mixed states. This probability measure is the trace of a density matrix. The pure states, the extremal points of the convex set of all states, are in 1 : 1 correspondence with the subspaces of a Hilbert space. This justifies in a striking way the Born Rule of probabilities for pure and mixed states: it ‘drops out’ of the formalism in a natural way. (In the exceptional case of two dimensions it is possible to have a 0 : 1 measure on the basis vectors that can be consistently continued to the whole space, called the ‘Bloch sphere’ — this is sometimes described as ‘colouring the sphere’ with just two colours.)

Whereas Gleason’s theorem shows the global failure of a 0-1 measure, the KS theorem constructs quite small sub-algebras in which the failure is made manifest (Thm. 1 of Kochen and Specker 1967)). Over the years both Gleason’s and the KS theorem have been made progressively more tractable. For KS and a Hilbert space of dim = 4, the number of basis sets needed is only 9, with 20 vectors (q.v. Kernaghan (1994)). Even in dim = 3 it is possible to use only 31 vectors — down from the KS original 117. A consequence of Gleason’s theorem that is stronger than the KS theorem is this: if two pure states a and b are a part of a set Γ of pure states and if they are not orthogonal, then there is no probability measure on Γ that assigns both a and b a probability of zero or one (a truth value) unless they are both assigned 0, i.e. are both false. This is Pitowsky’s Indeterminacy thesis (Pitowsky 1998) as used by Demopoulos.

In the final section of the KS paper they proved that there were classical tautologies that were quantum-false. Even the seemingly harmless associativity law will be satisfiable only under restricted circumstances. proved the dual of this in (1977; 86). He argued that there were classical contradictions that were true. In particular he showed that a finite partial Boolean subalgebra of subspaces can be represented by a Boolean polynomial in 86 variables — less now of course in the light of Kernaghan (1994). For every assignment of either 0 or 1 this Boolean polynomial takes the value 0 – thus it is a classical contradiction. And yet it is satisfiable by some assignment of co-measurable quantum variables and so can be ‘true’. The conclusion Demopoulos reached was to believe that the semantic determinacy of quantum logic cannot be maintained — there have to be disjunctions that are true but where none of the disjuncts are true. And this has led him to the conclusion that the interpretation of the elements of the projection lattice as propositions about some particle must be false, for if it were not, quantum truth would have as a subset some classical contradictions. As he said: ‘Hence, if we think of the laws of logic as “laws of truth” — as a guide to our concept of truth — the quantum logical interpretation’s concept of truth cannot be our classical concept.’ (2010; 380). It would be a step into paraconsistency that Demopoulos is unwilling to take. (This is all, of course, in contrast to intuitionist truth that is a subset of the classical truths.)

This is apparently why Demopoulos abandoned quantum logic for more than 20 years.

But since 2004 he has returned, obviously having thought deeply on these matter in the intervening years. Quantum logic, as originally conceived, cannot work; quantum probability can work, provided that the probabilities are not understood as a measure on a lattice of subspaces conceived as propositions. The problem then is: how should the states be conceived? His answer is that they should be conceived as effects and such effects have probabilities of occurring. An effect is an event, the result of an interaction with another system. Demopoulos says of effects that they ‘are induced by an interaction with an experimental set-up and registered in the experimental apparatus.’ (Demopoulos 2010; 382) If there is no interaction Demopoulos avails himself of the memorable saying of Asher Peres, ‘Unperformed experiments have no results’. The probabilities are interpreted by Demopoulos in a Bayesian fashion using the notion of a Dutch book, but the underlying space is not the standard Boolean event space but the non-Boolean projection lattice and presumably its continuation to all density operators. This idea (Quantum Bayesianism, or “QBism”) was also set out in Pitowsky (2003) but it goes back to the beginning of quantum mechanics and one can find many comments in von Neumann’s work, including his letters, which would support it (albeit inconsistently). The idea is that we have here a structural explanation: the structural change is to the underlying event space.

An obvious question must arise at this point in the mind of any reader: what does this say about the EPR-type distant correlations? How does a Bayesian probability work in this case, where the correlated probabilities seem perfectly objective? This question is taken up in the book under review, based on the ideas we’ve already seen. Drawing on the analogy with the structural role of Minkowski space-time, Demopoulos puts it as follows: ‘… the probabilities exhibited by the EPR correlations are understood as a consequence of the Hilbert space structure of the properties of physical systems, rather than the effect of unknown local causes’ (p. 183). Underneath this there is a reliance on an unpublished formulation of Bell’s theorem and KS due to Pitowsky. On this formulation, if we have a finite set F of Hermitian operators, a state s and a probability distribution P_s on all of the eigenvalues for the Hermitian operators in F, and agreeing with QM probabilities for commuting members in F, then Bell’s theorem says that there is an s (the EPR state) and an F (which in this case are local operators) for which there is no P_s; and KS says that there is an F such that for any s the P_s does not exist. Thus KS is not dependent on the choice of state and Bell’s theorem follows from KS and so also from Gleason’s theorem. then deploys this against Einstein’s ‘criterion of reality’ and on behalf of Bell’s analysis of the prospects for a completion of QM along the lines Einstein envisaged. All of this one can accept. But I still don’t quite see how the Bayesian framework plus the non-Boolean character of the quantum probability space explains the distant correlations. In the analogous case of Minkowski space-time there is a clear, provable, sense in which it explains a Lorentz contraction and time dilation. In the current case, Minkowski space-time both forbids causality and allows correlations. Does this suggest that another group is needed? Another space for the probabilities?

The cost of making probabilities Bayesian and personal is that one needs a rich theory of the mutual ‘effect’ of a quantum system on a measurement device — the objective causal event we are betting on. The problem here is that there is no such causal theory on offer. Moreover there may be good reasons why nothing is forthcoming. To describe such an interaction you need, at the very least, two Hilbert spaces under a tensor product. But even this simple requirement has posed a great obstacle for quantum lattices and their logics. This was the problem uncovered by Aerts, Foulis and Randall, Pulmannová, in the early 80s and discussed by Boris Ischi more recently. Under the product, one of the lattices is forced to be commutative. The underlying problem is that a tensor product is not a Cartesian product but is vastly larger. And then there is an additional problem that this reviewer has not seen addressed anywhere: we are really interested in the quotient spaces of the tensor product: the symmetric power (for bosons) and the exterior power (for fermions); where is the lattice product corresponding to these? Where the logics and the probabilities? And what about these quantum probabilities? It has been argued (by Gromov and Hrushovski) that probability, measure theory, has a monadic presupposition, and that this naturally creates a ‘space’ that is imposed from above on the algebra of sets. Relations between things are not fundamental, rather they are functions on this space. In a sense our spatial metaphysics makes everything ‘loose and separate’ from the outset, by assumption. The problem has been put this way:

Thus pure probability logic sees, very precisely, the monadic space of the theory; and conditional probabilities of basic relations as functions on this space. Beyond this, or relative to it, the universe is boring; everything is locally statistically independent, no special features can be made out, and no new relations can be described. Hrushovski (2020; 393)

What is missing is a probability theory that puts the right relations in from the get-go. Unrepresented structures, we might say, imitating Peres, have no interpretation.

Unfortunately we do not have either Demopoulos or Pitowsky — who died within 7 years of one another — to address these issues.

But a further interesting aspect of Demopoulos’ discussion in the book has to do with understanding what Einstein believed — and then, by contrast, with what Bohr believed. Commenting in 1949, in the Schilpp volume, Einstein noted that the best explanation of the EPR situation given by the ‘orthodox’ interpretation [the view that the ψ-function is complete] comes from Bohr:

If the partial systems A and B form a total system which is described by its ψ-function ψ/(AB), there is no reason why any mutually independent existence (state of reality) should be ascribed to the partial systems A and B viewed separately, not even if the partial systems are spatially separated from each other at the particular time under consideration. Schilpp (1949)

Demopoulos doubts that this is Bohr’s view. I think he is right about this. But even though Bohr in his published remarks never said anything resembling this (though von Neumann and did) we have good reasons to believe it to be true: the reduced density matrices of A and B are maximally mixed states, so there is a good sense in which no ‘independent existence (state of reality) should be ascribed to the partial systems’, no ‘being-thus’.

I have one further caveat: I think Demopoulos goes too far in trying to enlist Bohr as an anticipator of this kind of structural view he favours. Bohr showed no interest in the algebraic structure of quantum mechanics, preferring instead his own talk of ‘complementarity’, which had no algebraic interface. Bohr’s form of words that was meant to prove that the EPR argument is unsuccessful is, to put it politely, less than transparent: ‘But even at this stage [the stage at which the final measurement is made on a subsystem] there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behaviour of the system.’ Bohr (1935; 700) What is influencing what here, and when? Just as with the above QBism, we are missing a theorem — and Bohr showed no interest in proving such.

This issue aside, Demopoulos has crafted a thoughtful and interesting interpretation of quantum mechanics that completes his earlier work of the mid-70s; it must be hoped that it gets the attention it deserves. (Potential readers are recommended to have Demopoulos (2010) and (2012) on hand.) I turn now to a brief account of the remaining chapters.

The theme of scientific theories and how theoretical terms should be understood runs through the book and its basic tenets are outlined in the introduction. The stress here is on the importance of theory-mediated measurement, something first noted by Newton in the 3rd edition of his Principia, as against the partial interpretation of theories under the logical empiricists, Carnap in particular. The second chapter lays out this argument in detail. It threads through a number of familiar needles: the Putnam model-theoretic argument; the significance of Ramsey sentences; van Fraassen’s constructive empiricist view, as outlined in The Scientific Image; the Newman argument against Russell’s ‘causal theory of perception’ in Analysis of Matter — as resurrected in the 80s by Demopoulos and Friedman. The criticism of constructive empiricism has its roots in a trenchant 1982 review from Demopoulos of The Scientific Image and was continued in a 2003 article. The argument about unobservable entities is continued here with Demopoulos arguing for the inadequacy of a response to him by van Fraassen. This argument with van Fraassen is continued in chapter 2 on Jean Baptiste Perrin. The M.H.A. Newman 12-page argument against Russell is fascinating: Demopoulos has developed it over a number of publications since 1985 in such a way that it has come to seem like one of the central works of 20th century philosophy. It is a new point here that Frank Ramsey developed his idea of Ramsey sentences as a result of being confronted by early chapters of Russell’s Analysis of Matter. It is one of Demopoulos’ major contributions to our understanding.

All in all, this posthumous publication is a wonderful tribute to a very significant philosopher.

References

Bohr, N. 1935. Can Quantum-Mechanical Description of Physical Reality be considered Complete?, Physical Review, 48, 696–702.

Demopoulos, W. (1977). Completeness and realism in quantum mechanics. In: Butts, R., Hintikka, J. (eds.) Foundational Problems in the Special Sciences, 81–88. Reidel, Dordrecht.

Demopoulos, W. (2010). Effects and Propositions. Foundations of Physics, 40(4), 368–389. doi: 10.1007/s10701-009-9321-x

Demopoulos, W. (2012). Generalized Probability Measures and the Framework of Effects. Ch. 13 of Probability in Physics ed. Yemima Ben-Menahem and Meir Hemmo, The Frontiers Collection, Berlin: Springer-Verlag.

Hrushovski, E. (2020). On the Descriptive Power of Probability Logic, in M. Hemmo, O. Shenker (eds.), Quantum, Probability, Logic, Jerusalem Studies in Philosophy and History of Science. doi: 10.1007/978-3-030-34316-3_17

Kernaghan, M. (1994). Bell-Kochen-Specker theorem for 20 vectors. Journal of Physics A: Mathematical and General, 27(21), 829–830. doi: 10.1088/0305-4470/27/21/007

Kochen, S. and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87. doi: 10.1512/iumj.1968.17.17004

Pitowsky, I. (1998). Infinite and finite Gleason’s theorems and the logic of indeterminacy. Journal of Mathematical Physics, 39, 218–228. doi: 10.1063/1.532334

Pitowsky, I. (2003). Betting on the outcomes of measurements: a Bayesian theory of quantum probability. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34(3), 395–414. doi: 10.1016/s1355-2198(03)00035-2

Schilpp, P. A. (ed.) 1949. Albert Einstein, Philosopher-Scientist, Library of Living Philosophers Vol. VII, N.Y.: MJF Books.

The Sleep of Reason Produces Monsters: A review of a review by Tim Maudlin of two books in the philosophy of science

July 15, 2018August 11, 2018 eigentrope6 Comments

Can one review a review? Well, perhaps one can if the review is over 7,000 words long and has a wide focus. Tim Maudlin’s review of two books, Adam Becker’s What is Real?: the Unfinished Quest for the Meaning of Quantum Physics, and Errol Morris’ The Ashtray: (or the Man who Denied Reality) is the basis for an extended look at two, alleged, sources of modern irrationalism — one says ‘alleged’ because there is no attempt to make a causal connection from the figures under discussion, Neils Bohr and Thomas S. Kuhn, to the modern “post-Truth” world, and even a cursory look at discussions on social media would suggest that the sources are quite distant from quantum mechanics and Bohr’s idea of complementarity. (Kuhn, however, is another story, as I will argue.) Still, as one who is as worried at the furious irrationalisms of the modern world as I think Maudlin is, this is a discussion worth having.

I must begin by saying that Maudlin writes so well and so persuasively and the areas of agreement are so large that any small matter of disagreement or addition is likely to mislead. Nevertheless, some disagreements are worth recording — because I want to indicate that there are interconnections that have been overlooked.

With respect to Bohr there is no doubt of his Kantian education, nor of his creating an almost religious sense of mystification about Quantum Mechanics. His response to the Einstein, Podolsky, Rosen paper of 1935 is a staggering example of throwing up meaningless Kantian phrases to hide the vacuity of his response. He seems not even to grasp the significance of non-locality, nor of its potential for being in conflict with Special Relativity. And yet, as Maudlin notes, physicists bought it, hook, line and sinker; the Kantian phraseology was sufficient to allow them to resume their dogmatic slumbers. The EPR argument was ignored, as was Schrödinger’s contemporaneous paper that introduced entanglement. Here was a tremendous opportunity that went begging — for decades. And all because physicists were lulled by a philosophical gibberish by which philosophers would not have been fooled.

Modern physicists today like to disparage their philosophical colleagues — they appear to think that philosophy is an easy subject and something they can turn their hands to with no training whatsoever. Yet most of their forays into philosophy are staggeringly naive, embarrassing even. And this naivety was fully in evidence in the disputes with Einstein and Schrödinger in the 1930’s and 40’s, and the consequent attempts, in the 1950’s, to undermine their reputations. But it was not Bohr who was most to blame for this outrage — Bohr managed to maintain cordial relations with Einstein throughout his life. The real culprits in this were Heisenberg and Robert Oppenheimer — and despite being German, Heisenberg showed less allegiance to Kant than did Bohr. Oppenheimer was another story.

By the late 1950’s Oppenheimer was, in the minds of many, the epitome of the distinguished scientist. Yet he had harassed Einstein while the head of the Institute of Advanced Study, and he was, as Maudlin notes, the person responsible for setting the FBI on David Bohm — to cover up, it has been said, the fact that he himself was, all along, a secret member of the Communist Party. Then, during the war, Oppenheimer’s secret lover, the ardent Communist Jean Tatlock, died under suspicious circumstances — some months after he visited her, without security clearance, while leading the Manhattan Project. She was found drugged and drowned in her bath. Coincidentally, earlier, in the 1930’s, Oppenheimer had attempted to murder his PhD supervisor at Cambridge, by poisoned apple — an event that was hushed up only with the intervention of his parents. Then he attempted to strangle his best friend when the latter announced his engagement.

And yet he became — was allowed to become — the representative for many of the distinguished intellectual-scientist. It was under his stewardship that the negative picture of Einstein hardened: he was a dinosaur, out of touch with modern physics, had done nothing for the last 20 years of his life. When the idea was mooted that there should be a publication of Einstein’s collected works, it was Oppenheimer who personally intervened to stop it. Bohr and Heisenberg may have crafted and weaponised the Copenhagen Interpretation but it was Oppenheimer who used it to the lethal administrative effect.

Kuhn enters the story at this point. What is omitted from Maudlin’s review is that Kuhn was very influenced by the Copenhagen Interpretation and by Bohr’s philosophy in particular. Kuhn’s denial of truth and his denial of the idea that science progresses toward the truth, are really his application of Bohr’s philosophy to science as a whole. One can tell this only by the fact that to a large extent Kuhn leaves Bohr out of his exposition and instead uses as his example that of Einstein’s theory of gravitation replacing Newton’s. This is baffling until one realises that this is yet another hit at Einstein. When Einstein’s theory replaced the Newtonian theory of gravitation this was not progress, this was one paradigm replacing another, but the paradigms were just different ways of seeing the same facts. It was as though one fashion had been replaced by another fashion: flares are no more a progress to the truth than is a straight leg.

So why was Quantum Mechanics not his prime example, why was that also not one fashion replacing another? The reason is that the Copenhagenists had enshrined the One True Philosophy into their account of physics — in fact this is where Kuhn probably had learned the philosophy in the first place — and the One True Philosophy was that there was no truth, there was only ‘perspective’, of complementary (i.e mutually incompatible) kinds: complementary, that is, in just the way Kuhn’s paradigms were.

The one who was supposed not to have understood all of this, who was too much of a dinosaur to get it, was Einstein. So Einstein’s pretensions of saying something about the real world, of there being an objective world of which something could be said, had to be punctured. If he couldn’t be refuted then he had to be ridiculed. Einstein was wrong that Quantum Mechanics wasn’t perfected by Bohr and Heisenberg because he didn’t understand the truth about truth that Bohr preached; he was wrong even where he might most have seemed to be right, in his theory of General Relativity. But it was just a fashion choice.

This “no progress” picture of scientific progress needed one element to be complete: it needed to be married to an inductive scepticism. Fortunately for Kuhn this had already been made popular by Karl Popper, who endlessly repeated the falsehood that David Hume had proven inductive scepticism in the 18th Century — endlessly repeated and endlessly refuted, this canard lives on to this day.

But whereas America — at least in its philosophy of science departments — had a strong probabilistic tradition and could therefore resist these arguments of Popper and Kuhn, other places with a weak background in mathematical probability crumbled. Thus it was that there were “philosophy of science” departments that grew up in Britain and Australia, in the latter case without exception, that taught this anti-truth, anti-science credo, with a dogmatic fervour. And once it is in place it takes no leap to join it with “social theory” and SJW rhetoric to condemn science as imperialistic, patriarchal — or any of the other ready-made jibes that fit comfortably on a placard or in a twitter post.

The modern world was baked in the oven provided by Kuhn, and ultimately Kant — or perhaps one could say “half-baked”.

Kant’s role in all this was far more than providing a framework for Bohr’s or Kuhn’s self-contradictions. He created an environment in which to speak an unintelligible nonsense could — like the Emperor’s New Clothes — be made to pass for the profoundest speech: Kant as Cant. Whereas philosophers before this had striven to be clear and where to write unclearly was a sign of failure, Kant reversed it. Kant wrote as though he could only be understood by the initiate, the one who had earned the right to kneel in the presence of his Mysteries. Kant replaced the task of making sense of the world with that of making sense of him. Good anglophone philosophers like Bertrand Russell were able to see though the pretensions, but only one who knew German culture intimately was able to properly skewer Kant. That was Nietzsche. Here is part of his denunciation, in The Antichrist, section 10.

Among Germans I am immediately understand when I say that philosophy has been corrupted by theologian blood. The protestant pastor is the grandfather of German philosophy. Protestantism itself is its peccatum originale. Definition of philosophy: the half-sided paralysis of Christianity — and of reason. . . One has only to say the words ‘College of Tubingen’ to grasp what German philosophy is at bottom — a cunning theology. . . The Swabians are the best liars in Germany, they lie innocently. . . . Why the rejoicing heard throughout the German academic world — three-quarters composed of the sons of pastors and school-masters — at the appearance of Kant? Why the German’s conviction, which still finds an echo even today, that with Kant things were taking a turn for the better? The theologian instinct in the German scholar divined what was henceforth possible once again. . . . A back staircase to the old ideal stood revealed, the concept “true world”, the concept of morality as the essence of the world (— these two most vicious errors in existence!) were once more, thanks to a crafty-sly scepticism, if not demonstrable, yet no longer refutable. . . . Reason, the prerogative of reason, does not extend so far. . . . One had made of reality an ‘appearance’; one had made a completely unintelligible world of beings into a reality. . . . Kant’s success was merely a theologian’s success: German integrity was far from firm, and Kant, like Luther, like Leibniz, was one more load upon it. — —

This idea, that Kant’s obscurantism was theology for people who felt that they’d gotten beyond religion explains so much of the worst parts of what is called “Continental Philosophy”, where practitioners act like members of a Pentecostal church speaking and writing in tongues. The effort is entirely to write so that one may not be understood, but to use select phrases so there is a something that teases the listener’s understanding, to make them think that something important is being said. It’s a sham performance. University lecturers get away with this only by first making sure that their students are so browbeaten and intimidated that they won’t — and by the end of it can’t — speak up against it. (And it only adds to the shamefulness of this situation that the students are actually paying to be bullied thus into a cognitive paralysis, paying to the point of long term bankruptcy — even in the Soviet Union students were not required to pay for their own mental hobbling.)

Kantian philosophy is called “critical” and the word itself has spread like a virus. It is a word that presents a deceptively positive image: it suggests that here is a philosophy that is open to logical argument, reason, and criticism, that it will consider all positions fairly. But that is the opposite of the truth. In fact “critical philosophers” write and act as though they are contemptuous of logic and reason — in fact of all that “analytical philosophers” do. “Critical” in fact means to be closed to logical persuasion — just as Bohr’s notion of complementary meant exactly the opposite of what it seems to mean. Instead the critical philosopher is one who sees through things like logic, to its “presuppositions” — a very Kantian term — and who, having seen through them is instantly beyond them. This works with any and all social institutions: to apply a “critique”, and to question presuppositions is all that is required for the institution to be instantly up for replacement. The game of issuing social critiques is easy because it brooks no back-and-forth questioning of the value of what is being questioned. One repeats the magic incantation, rolls one’s eyes back in one’s head, and the job is done. Nor can it be undone with the same method of critique — one could not, for example, question Feminism with the same naively simplistic trick. The method is inherently asymmetric — a door that you can go through but cannot return through.

However it would be quite wrong to lay all of the blame for the modern world’s irrationalism at Kant’s door; this would be to overlook a far more insidious culprit much closer to home. As Nietzsche said: opening up a back staircase for religion may have been Kant’s underlying purpose, but he was not alone in having that purpose. William James, with his Pragmatism — as expressed in his essay ‘The Will to Believe’ — was attempting to make it reasonable to believe in a God under the very loose conditions of there being no definitive proof against the existence of God, and of that belief “working” for the believer. (This was much to the horror of his friend, the philosopher C. S. Peirce, who could see the disaster looming in such a permissive criteria, and it caused Peirce to dissociate himself from James’ Pragmatism and to call his own view Pragmaticism.)

Nothing placed lying so at the heart of American Letters as effectively as James’s Pragmatism. Eventually, once it had sunk in, the criterion of believability would become the simple does it work for you? Of course it would have been naive to think otherwise: of course lies work for liars, that is why they tell them. But to have James’s personal blessing — that was more than could have been wished for! Of course this was not James’s intention, and no one could have suggested that it was — but philosophical ideas can have unintended consequences just as much as actions do.

William James was a distinguished psychologist, a man who had the ear of presidents, but he could not have created this disaster alone. Help came to him from John Dewey, who had done his PhD on Kant, considered himself a follower of Hegel — he wrote like Kant-from-Vermont — and who was an indefatigable promulgator of Pragmatist-like ideas. (One has to say “Pragmatist-like” because Pragmatists were very unclear what this idea of ‘beliefs working’ actually amounted to — but one can say that it was Dewey who emphasised that this “working” meant in a social context, in the furtherance of liberal democracy.)

Dewey had very little effect outside America but within America his influence was profound — in part simply from his having lived so long a life and written such an extraordinary number of books. He became America’s leading public intellectual. Yet it was through his disciples that he exerted his greatest influence very near the time of his death in the 1950s. The idea that beliefs can be maintained come what may — that is, despite the fact that they are manifestly false — was held by W.V.O Quine (in his essay ‘Two Dogmas of Empiricism’) in one of the most notorious passages of contemporary philosophy: ‘Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. Even a statement very close to the [sensory] periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws.’ And also later in the same essay:

But in point of epistemological footing the physical objects and the gods differ only in degree and not in kind. Both sorts of entities enter our conception only as cultural posits.

The phrase can be held true is a devious alternative to speaking of what is true — but it really just means believed to be true. However the point is clear: beliefs about the gods can be held true if one is prepared to be pig-headed in the assessment of evidence. The delusional, the liars, the conspiracy-theorists, can take great comfort in all of this. Pragmatism is a gift of irresponsibility that was set as a curse upon an entire nation.

One may object that these words are too severe, too unkind to American philosophers, who for the most part have used their irresponsibility fairly responsibly. (It has to be stressed that there are many, many analytic philosophers who have not succumbed to this view and who work to maintain high standards.) But the Pragmatists were themselves very adept at mud-slinging with a distinctly militaristic tenor. William James likened those who didn’t believe in his religious beliefs to cowards — according to him it was cowardice in the face of the enemy not to risk your cognitive life by refusing to believe ridiculous things. (Yet who on earth was the enemy in this case?) F. C. S. Schiller, the Columbia and Oxford-based Pragmatist, likened his opponents to defenders of the city of Jericho, for whom it self-crumbled without even the need of a trumpet. Having described his own view in glowing terms we have, on the other hand

… opposed to it on every point, an old metaphysic of tried and tested sterility, which is condemned to eternal failure by the fundamental perversity of its logical method.

And this was Naturalism!

Pragmatism won whatever following it had by insult and denigration — and that is how it has continued to the present day. It declared the correspondence theory of truth refuted with not a single non-question begging argument anywhere in sight. And that lie also has continued to be told — repeated in one classroom after another across America.

But it was this philosophy that, in a sense, became the Copenhagen interpretation in physics. When Feynman issued his declaration: just shut up and calculate! it was because all that mattered were the empirical predictions that could be wrung out of quantum theory. Who cared whether the theory had two dynamical processes and no clear rule as to when one takes over from the other? American physicists did not repeat Bohr’s Kantian obfuscations, they ignored them — they repeated Pragmatism’s injunctions to just look at whether the theory worked — and they are still doing that to this day.

But because they were in this Pragmatist coma, physicists missed what may be the greatest discovery of the last two thousand years: entanglement. And they only began to take notice of this when money began to be thrown at anything that could be called quantum computing. In effect, yet more Pragmatism — more of that same dull mixing of the businessman’s idea of things working — of a theory paying off — and a more commonplace empiricism.

In doing all of this the Pragmatist may well have come close to enshrining a logical fallacy as a central tenet of American intellectual life: from, if p then q, and q, infer p. Dewey and F. C. S. Schiller often seemed to be committed to something even more dangerous: from if p then q, and q is unknown, infer p is unknown and in fact meaningless. (What is often taken to be Wittgenstein’s signature doctrine that meaning is use is in fact a commonplace Pragmatist idea, attributable originally to Sidgwick.) The corollary of Pragmatism’s emphasis on the importance of things working — for someone, somewhere, in whatever dubious way — is that if they don’t work they are meaningless.

The wreckage that is America’s current political and social discourse — on both the left and the right — possesses a toxicity that has spread itself around the world, due in no small part to the philosophical doctrines that are so peculiarly American: Protagorean relativism, Pragmatism, and the concept of a uni-directional, unanswerable, social “critique”: all a kind of political midden that has replaced rational discussion and an openness of debate with a rancorous hatred at even the smallest trace of disagreement. In this one can see the signs of a timidity that is afraid that its convictions will not stand up under scrutiny — that the lie that keeps one drugged and unable to see things as they are will wear off, that it will no longer work.

What will happen to America, and the Western world in general, if these trends continue unabated? Chaos of belief is the precondition for, and the first step towards, a true chaos. One cannot have a country where the people cannot think, cannot discuss, and cannot disagree without violence and mutual hatred, and it is a disgrace — and that word cannot encompass the true magnitude of the offence — that this situation should have been abetted and the groundwork laid by philosophers.

Pragmatism in the development of America’s intellectual life — if such a life deserves that adjective — shows the truth of the old proverb: Crooked logs make straight fires. And large fires at that.

Review of Jeff Bub’s Bananaworld: Quantum mechanics for Primates

January 9, 2018January 9, 2018 eigentropeLeave a comment

Jeffrey Bub is one of the most distinguished philosophers of quantum mechanics writing today. His first book on the subject, The Interpretation of Quantum Mechanics, was published more than forty years ago, and he has written many works on the subject since, including the Lakatos Prize winning Interpreting the Quantum World in 1997. His latest book presents a wealth of material that has emerged in the last fifteen years related to the explanation of the central aspect of quantum mechanics as we understand it today: entanglement. A novel pedagogic device in the book is provided by an analogy to two different ways of peeling a banana, with a consequent difference in how it tastes. On the basis of this analogy there are some wonderful illustrations of a Carrollian bent — Tenniel himself would have been proud! I’ll come back to the analogy in the course of the essay.

There are two properties that quantum mechanics (henceforth QM) satisfies: 1) there is no superluminal signalling (NS); and 2) the observables can be contextual (C). Combined with the well known fact — a result of Bell’s theorem — that the predictions of QM cannot be reproduced by a non-contextual hidden variable theory (NCHV) and it follows that QM is non-local. From non-locality and (NS) it follows that QM cannot be fully deterministic. But — and this is the first surprise — principles (NS) and (C) do not uniquely delimit the set of correlations to just those predicted by QM. There are supra-quantal non-local correlations that are non-physical (as far as we know) which satisfy (some neutrally formulated) version of (NS) and (C). Thus (NS) and (C) are necessary but not sufficient for QM. The task then, as it is now formulated, is to find the underlying principles that distinguish QM not just from classical physics, but also from supra-quantal ‘physics’.

The upper bound of QM has been known for a long time, since 1980: it is called the Tsirelson bound (from Cirel’son (1980)). In 1994 Sandu Popescu and Daniel Rohrlich devised a set of correlations that exceed the Tsirelson bound but that satisfy (NS) and non-locality (Popescu and Rohrlich (1994)). This showed that what was above the Tsirelson bound was at least something that could be described consistently. But were there principles that would naturally rule out such supra-quantal correlations as unphysical, and does any of this shine a brighter light on QM itself? Why is our world not more non-local than it is? Or more indeterministic? These are the questions with which Bub’s book is concerned….

[To read the rest go to the full text on academia.com ]

Who discovered entanglement I

January 7, 2018January 7, 2018 eigentropeLeave a comment

One answer to this question — probably a common one — is that entanglement was discovered by Einstein in the Einstein-Podolsky-Rosen paper of the 15th of May, 1935: `Can Quantum-Mechanical Description of Physical Reality be Considered Complete?’. It was discovered only to be instantly repudiated! The non-locality implicit in a change in a wave function at place A leading to a change in a wave function at place B seemed to Einstein to be inconsistent with relativity and the assumption that A was separable from B was one of his assumptions of reality.

That definition of separability was:

Separability: Whatever we regard as existing (real) should be localised in time and space (space-time). Or, more weakly, if two dynamical systems are space-like separated then each system can be characterised by its own properties, independently of the properties of the other system.

Since this separability was the denial of entanglement and so EPR discovered entanglement — or so the story might go — by denying its possibility.

However, even though Einstein et al were the first to publish on this the better answer as to who discovered entanglement was Schrödinger. Schrödinger published the paper `Discussion of probability relations between separated systems’ in the Proceedings of the Cambridge Philosophical Society for 1935, in fact in August of that year, only a few months after the Einstein et al paper, and he and Einstein had been in correspondence on this for several years, both with the idea that the measurement process suggested something deeply problematic about quantum theory. But Schrödinger is better at drawing out the consequences and indicating that it arose from the nature of the tensor product itself. What he says on this is far clearer than what is in the Einstein et al paper — and Einstein would probably have agreed, as he was known to have disliked how Rosen and Podolsky (mostly the latter) wrote up the idea after their talks together.

Whereas, by 1935 Einstein was well-settled in Princeton, Schrödinger was unsettled in Oxford. He had just had a baby daughter by his mistress, Hilde March, and he had just won the Nobel Prize (in 1933), but he had no permanent position and he was unhappy with the home he had been given by ICI who were paying his stipend — a stipend that he also didn’t feel was enough. Nor did Oxford take entirely to Schrödinger: Frederick Lindemann, the head of physics, strongly objected to Schrödinger’s ménage à trois and wanted to get rid of `this bounder’. And yet out of this chaos Schrödinger managed to write, in lucid prose, an account of the characteristic trait of quantum mechanics.

Here is Schrödinger’s conclusion:

When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or $\psi$-functions) have become entangled. To disentangle them we must gather further information by experiment, although we know as much as anyone could know about all that happened. Of either system, taken separately, all previous knowledge may be entirely lost, leaving us but one privilege: to restrict experiments to one only of the two systems. After reestablishing one representative by observation, the other one can be inferred simultaneously. In what follows the whole of this procedure will be called \textit{disentanglement}. Its sinister importance is due to its being involved in every measuring process and therefore forming the basis of the quantum theory of measurement, threatening us thereby with at least a regressus in infinitum, since it will be noticed that the procedure itself involves measurement.

We might note that Schrödinger distinguishes between the state and its representative; why does he do this? The reason is that the state is not Lorentz invariant, so it changes as we change Lorentz frames. So Schrödinger is careful not to speak of the state, as though it were something absolute — which would be as bad a mistake as speaking of the velocity of an object in the light of Galilean relativity. For Schrödinger the quantum state is something rather mysterious, and he thought that others were far too free with the notion, and too incurious as to what lay behind it. But though we cannot speak of the state we can speak of its \textit{representative} in a frame. It should be noted that this frame relativity does not mean that entanglement is also relative: it is frame invariant (because the fact that the state for the joint system is a pure state is an invariant).

The entanglement that we have here was called, by Schrödinger, in his German publications, Verschränkung — cross-linking.

He continues:

Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated and therefore virtually capable of being “best possibly known”, i.e. of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself.

Attention has recently been called to the obvious but disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter’s mercy in spite of his having no access to it. — Schrödinger 1935, p. 556.

It might seem as though everyone would pay attention to this startling prediction. But as it happened, it was almost universally ignored. For decades it lay dormant.

Introduction

January 1, 2017January 2, 2017 eigentropeLeave a comment

The purpose of this blog is to say some things that don’t fit into the format of a traditional academic paper. There are many such things, and they would, in the past, have gone unsaid, or have been said only in the context of a discussion among academics over coffee. To the outside world such things are invisible — but they constitute the living world in which academic debates and disagreements exist and they constitute the invisible 0.9 iceberg as compared to the published tip (as it were). It is the existence of this larger context that makes it virtually impossible for anyone who is not within a University to understand what is going on in a particular discipline — and I would say this is especially true for philosophy. For example, just browsing a philosophy bookshelf in your local bookstore would give you no idea what it is philosophers do or what they are interested in — and this would be true even if the bookstore is well-stocked with recent publications. But this has the consequence that no one who is not in philosophy can really understand the subject, though many people will be deceived into thinking that they can and do.

This is a great pity. But it is more than that — it has divorced philosophy from the culture that surrounds it and consequently left that culture philosophy-less, clutching on to fragments of misunderstood eastern mysticism and wisdom literature, blended together with cat-memes and the saloon-bar sawdust of political outrage. I can’t possibly hope to reverse the damage that has already been done. But it may do something (an epsilon of something) to say a few things that are fragments of that wider philosophical discussion, not behind closed doors but out in the open. However I can only do that from within my own areas of interest, which are Philosophy of Physics, Philosophy of Mathematics, Epistemology and Metaphysics. Perhaps the best that could be hoped for is that people will understand their debt to Plato or Aristotle — or have a little more knowledge of what they were saying, rather than at present, where it is vanishing to zero.

As you can see, I’m a pessimist even in my optimism.

The Philosphere

The Blog of Adrian Heathcote

Category: Philosophy of Physics