Tag Archives: quantum mechanics

Immanence and Objectivity

In her 1979 paper “Cognitive Repression in Contemporary Physics”, Evelyn Fox Keller describes the scientific viewpoint associated with classical (i.e. Newtonian) mechanics as based on a pair of conjoined assumptions. Firstly, that the subject of scientific knowledge is strictly separable from the possible objects of such knowledge; and secondly, that it is possible to establish a direct correspondence between what is known of each such object and its actuality. According to this viewpoint, “nature” is both objectifiable and ideally knowable by a scientific subject which stands apart from that nature in order to observe it.

We have here an operation of dividing-and-regluing very like that described by Laruelle as characteristic of philosophy (and, indeed, Laruelle would likely describe Fox Keller’s account as true of a certain philosophy-of-science, or philosophical epistemology, rather than of science itself as a practical stance). The immanent Real is split into an objectifiable domain of distinct entities, and a transcendental order of knowledge which proposes to organise those entities into a world (“the scientific worldview”, let’s say). Rather than thinking “according to the Real”, or from the premise that both “knower” and “known” are immanent to the same reality (and thus share a fundamental identity), the stance Fox Keller describes is “decisional” in Laruelle’s sense: it begins by making a cut, and by giving itself the authority to repair that cut.

Fox Keller observes that the principles of objectifiability and knowability break down in the face of quantum mechanical phenomena: they cannot be maintained simultaneously, and every attempt to do so produces metaphysical monsters in the guise of “interpretations” of quantum mechanics (as Derrida once put it: coherence in contradiction indicates the force of a desire). Her Piagettian reading of the resistance to non-classical epistemology in terms of affective positions is suggestive (in that smug “clever men in white coats are really just big babies” sort of way that has never quite seemed to go out of vogue, for reasons I could probably venture some pseudo-psychological explanations for myself), but doesn’t particularly help us resolve the problem of how to develop such an epistemology.

Susskind describes the situation as follows: in a classical system, we are confident of always being able to make a “gentle enough” measurement that the system being measured is not perturbed: the apparatus is able to determine how the system would behave if the apparatus itself were not present. This is, in fact, perfectly possible a lot of the time. Within a quantum system, however, measurement is carried out by means of operations* which participate in the total behaviour of the system itself, such that we are always observing the outcome of what I will call an effectuation. Any such effectuation is the effectuation both of a measurement and of a new state of the system. Both (classical) objectifiability and (classical) knowability are untenable under these conditions; the former because the apparatus of measurement is not strictly separable from the system being measured, and the latter because the process of obtaining information about one aspect of the system may render information about another aspect inaccessible.

This is helpful, but doesn’t go quite far enough. If we had not measured the particle’s position, we should have been able to measure its momentum (and vice versa); but this does not mean that, at some moment prior to measurement, the particle necessarily had both position and momentum (i.e. possessed some complete, if hidden, state in which both position and momentum were simultaneously inscribed). Rather, measurement-of-position and measurement-of-momentum are distinct operations that effectuate one value at the expense of being able to effectuate the other.

A thoroughgoingly immanent account of how science proceeds will be one which sees scientific theory-building, measurement and knowledge-formation as effectuations of the same Real, rather than the work of one kind of agent – the detached scientific knower – upon one kind of patient – “Nature”, etherised upon a table. That is one way of looking at the problematic with which Laruelle is engaged, and of understanding why the “quantum” has taken on such a totemic significance for him in his later work.


  • As Susskind also reminds the reader, an operator is a mathematical entity, which acts on state vectors to produce new state vectors. It does not, however, change physical reality – rather, it describes how a real-valued measurement is derived from a state vector in a quantum system. How the state of that system changes in the process of carrying out the measurement is a quite different matter. Accordingly, I have corrected “operator”, where it appears above, to “operation”. Caveat lector, as always when I’m trying out new stuff.

Solace of Quantum

Early on in Leonard Susskind and Art Friedman’s Quantum Mechanics: The Theoretical Minimum, the authors illustrate a difference between the logic of predicates and sets, and the logic of observations at quantum scale. One way to look at it is in terms of modelling.

If the state space of a classical system is the set of all the states the system can be in, then a proposition about the system corresponds to a predicate P that picks out certain states as possible states of the system if that proposition is true. Susskind’s example is a single roll of a six-sided die. Let the “state” of the die be the number that is facing upwards after the roll – the state space S is then the set {1, 2, 3, 4, 5, 6}. To the proposition “the value of the die is even” corresponds the predicate P that picks out the subset of the state space S' = {2, 4, 6}. We have S, the state space, P, the predicate, and S' = {x ε s & P(x)}, the subset of the state space which satisfies the predicate.

The state space is thus a model for propositions about the system, which means that we can translate statements about propositions into statements about predicates and subsets. If two propositions about the system are true at once, then the states the system can be in are those which are picked out by both of their corresponding predicates. For example, take the propositions “the value of the die is even”, and “the value of the die is greater than three”. Considered separately, these correspond to two predicates, P1 and P2, which pick out two different subsets of S, S1 = {2, 4, 6} and S2 = {4, 5, 6}. The state subset corresponding to the proposition “the value of the die is even, AND is greater than three” is the intersection of the subset S1 picked out by P1 and the subset S2 picked out by P2: {4, 6}. The subset corresponding to the proposition “the value of the die is even, AND/OR is greater than three” is the union of S1 and S2: {2, 4, 5, 6}. There is thus a “primitive” or “direct” set-theoretic interpretation of propositions concerning the state of the system, based on Boolean algebra.

An important characteristic of classical systems is that they “hold still” while being observed: the state space is a static image of possible measurement results within a given reference frame, and we can combine measurements freely without that image shifting beneath our feet. You can measure both the position and the velocity of a billiard ball, in whichever order you like, since there exists a state of the billiard-ball-system that captures both of these properties simultaneously.

The way we model the state of a quantum system is necessarily different, because the state of such a system is not independent of observation: to every measurement of the truth of a proposition about a quantum system corresponds a new distribution of possible states of the system. We can lose information by measuring (for example, if we observe the velocity of a particle, we lose information about its position). The order in which observations are made is therefore significant: O1 followed by O2 may well get you a different result to O2 followed by O1, which means that the algebra of quantum observation – unlike the Boolean algebra of classical systems – is noncommutative.

Susskind makes the following eyebrow-raising statement – “the space of states of a quantum system is not a mathematical set; it is a vector space” – and then qualifies it in a footnote: “To be a little more precise, we will not focus on the set-theoretic properties of state spaces, even though they may of course be regarded as sets” (italics mine). This is a subtle distinction. A vector space is a set, or at least is not not a set – working in vector spaces doesn’t in any sense take you out of the set-theoretic mathematical universe. But the way states “fit together” in a quantum system – their way of being compossible – is not readily definable in terms of primitive set-theoretic operators like union and intersection: you need the language of linear algebra, of orthonormal bases and inner products, to make sense of it.

In the light of this, it becomes a little clearer why Laruelle mobilises a metaphorics of “quantumness” – of vectors and matrices, superpositions and complex conjugates – as a way of undermining or circumventing the static philosophical architecture of “Being”: the claim he’s trying to establish is that non-standard philosophy is in some sense to standard philosophy as quantum mechanics is to classical mechanics…