A challenge to the second law of thermodynamics from cognitive science and vice versa

Abstract

We show that the so-called Multiple-Computations Theorem in cognitive science and philosophy of mind challenges Landauer’s Principle in physics. Since the orthodox wisdom in statistical physics is that Landauer’s Principle is implied by, or is the mechanical equivalent of, the Second Law of thermodynamics, our argument shows that the Multiple-Computations Theorem challenges the universal validity of the Second Law of thermodynamics itself. We construct two examples of computations carried out by one and the same dynamical process with respect to which Landauer’s principle implies contradictory predictions concerning the entropy increase. Our two examples are based on a weak version of the Multiple-Computations Theorem, which is quite uncontroversial, and therefore they amount to a clear refutation of the universal validity of Landauer’s Principle. We consider some responses to this argument that do not attempt to single out one computation over the others, and we show that they do not work. We further consider ways out of the argument by externalist approaches supporting the computational theory of the mind who propose that the interaction of a computing system with the environment is enough to select a single computation over the others. We show on physical grounds that this approach fails too. We then reverse the direction of our challenge and formulate a dilemma for supporters of the computational theory of the mind: (i) they must reject (or amend somehow) the causal closure of physic; or else (ii) they must accept on a priori grounds that Landauer’s Principle and the Second Law of thermodynamics are not universally valid. Finally, we present our version of a type–type mind-brain identity theory called Flat Physicalism, which is based on the paradigm case of statistical mechanics, and we show that it circumvents the challenge from Landauer’s Principle and the Multiple-Computations Theorem and does not fall prey to our dilemma.

Appendices

Appendix 1: Landauer’s principle

Landauer’s Principle is widely accepted as a universal theorem of statistical mechanics. Since Bennett (1982) it stands at the heart of the attempts to exorcise Maxwell’s Demon in statistical mechanics (both classical and quantum; see e.g. Leff and Rex 2003). Here is the prevalent line of thinking about it (but there are various versions of this idea). Consider Fig. 1 which depicts the state space of a physical system that implements logical computations. The computation is implemented by the “information bearing degrees of freedom”, which are the parts (or aspects of properties etc.) of the system for which there is a value assignment (they can be voltage values, positions, etc.). They are represented here by the horizontal axis. In addition, the system has non-information bearing degrees of freedom that are here represented by the vertical axis in Fig. 1. Figure 1 describes a dynamics in which if the system starts out in state 1X, it ends up in 0Y, and if it starts out in 0X, it ends up in 0Z. This dynamics implements the function of erasure: 1 → 0 and 0 → 0, along the information bearing degrees of freedom. (Liouville’s theorem and all other mechanical constraints are satisfied).

Fig. 1

Erasure along information-bearing d.o.f with finer-grained measuring device

Consider now Fig. 2, which describes the same system, the same dynamics, and the same implemented function, but with one difference: it is seen by a different measuring device or a different observer. Both devices can distinguish between 0X and 1X, but whereas the first device, the fine device, can distinguish between 0Y and 0Z, so that Fig. 1 is the way things seem to it, the coarse device cannot distinguish between 0Y and 0Z, and as far as it is concerned things look as in Fig. 2. Because of this, while for the first device the process is logically reversible—the input can be inferred from the output—for the second device this is not the case. We stress that these facts about the measuring devices are as objective as anything can be: for example, some voltmeters are finer and some coarser, and this fact doesn’t introduce any deep sense of relativity or subjectivity to the ontology.

Fig. 2

Same dynamics, same implemented function, coarser-grained measuring device

Now, in (Boltzmanninan) statistical mechanics (which is the relevant framework for discussing Landauer’s Principle), entropy is measured by the size of the region corresponding to the state: it is klogW, where W is the Lebesgue measure of the region representing the macrostate. For the fine device entropy is conserved: it is klogW both before and after the operation, while for the coarse device it increases from klogW to klog2W. Figures 1 and 2 and their analysis so far illustrate that different partitions, that in suitable cases are understood as different value assignments, give rise to different degrees of entropy increase, and all of them are objectively relative to a measuring device. This line of thinking, in which entropy changes are relative to the degree of coarse graining (which is the physics underlying value assignment, as we saw), may seem, on first sight, as radically different from the standard understanding of entropy, but we think that it does correspond to the empirical predictions, not only in statistical mechanics but also in thermodynamics, if one considers the nature of measuring devices. We address this important point later in Sect. 7 of the paper.

Notice that on this sort of relativized-externalist view of Landauer’s Principle, whether or not a computation in K is actually carried out and which computation is carried out depends on the physical matters of fact outside K, i.e. outside the degrees of freedom that carry out the computation. These matters of fact are fixed in the measuring devices in the environment of K. This view (we think) is different from Landauer’s: he seems to have thought that whether or not a computation (of some given logical function) is carried out by a computing system K depends entirely on the physics of the so-called information-bearing degrees of freedom in K that carry out the computation. The question of the entropy increase is different. According to Landauer, the minimum dissipation, say, that necessarily accompanies the computation of a logically irreversible function, might and usually will occur outside K, i.e. in degrees of freedom other than the information-bearing ones. One could still say, of course, that this externalist strategy retains Landauer’s core idea that logical irreversibility has a physical effect of thermodynamic irreversible behavior that’s manifested in some (minimum) dissipation in the vicinity of K, relative to the measuring device correlated with the corresponding value assignment.

The cases of Figs. 1 and 2 are in line with Landauer’s Principle: irreversibility goes hand in hand with entropy increase, of klog2 per lost bit of information, given the dynamics and partition to macrostates depicted in the figures. However, Landauer’s Principle is provably not a theorem of fundamental physics.²⁹ Here is a brief argument, by an example.

Consider Fig. 3, which depicts the state space of a physical system with dynamics that is different from the one of Figs. 1 and 2, and still that implements the function of erasure: 1 → 0 and 0 → 0 along the information bearing degrees of freedom, as before. And consider now the two observers that we considered before: the fine device that can distinguish between Y and Z and the coarse device that cannot distinguish between them. For the coarse device, the entropy increases as before from klogW to klog2W, in line with Landauer’s Principle.

Fig. 3

Same implemented function, different dynamics, finer-grained measuring device

Now consider the fine device, under the dynamics of Fig. 3. Here, the state after the erasure has been completed is either (0,Y) or (0,Z), and therefore the entropy is a klogW: it is conserved. Nevertheless, due to the mixing nature of the dynamics (or “blending”, not to commit ourselves to any particular kind of dynamics³⁰), the input cannot be retrodicted from the output, and therefore the operation is logically irreversible. Here we have a non-dissipative erasure, which is a counter-example for Landauer’s Principle.³¹

Appendix 2: Multiple realization of computational states

How can the computation associated with the mind be determined in a physical way, if one allows for multiple realization of the facts that determine which physical states are assigned which computational symbols (1 or 0)? In mainstream philosophy of mind and cognitive science, non-reductive views such as functionalism (of all sorts, computational and also causal) assume some sort of supervenience of mental (or computational kinds) on physical kinds, and this supervenience requirement is usually taken to be the hallmark of physicalism. So, as a working hypothesis, let us assume that these facts supervene on the physics of the brain, but nonetheless they are not identical to physical facts (in or outside the brain). This means that the computational kinds are multiply realizable by physical kinds. Importantly, according to the multiple realizability thesis, computational or mental kinds (and in general special sciences kinds) are realizable by heterogeneous physical kinds, where “heterogeneous” means that the physical (token-)realizers of the same mental or computational kind can radically differ physically, so that the realization of the same mental kind cannot be accounted for by any physical facts pertaining to the physical realizers.³² That is, in these approaches the physical realizers are taken to have nothing physical about them in virtue of which they all realize the mental kind, and in this sense the set of realizers can only be given by a disjunction, possibly an open disjunction, of physical kinds, no matter how deep one goes into the details of the physics of the realizers (see e.g., Fodor 1997). Putnam (1975) already saw that, insofar as the metaphysical theses of multiple-realizability and functionalism are concerned, the realizers may not even be physical, so that (he thought) that these theses are compatible with dualism.³³ We argue below that even if supervenience is assumed, functionalism and multiple realizability are not only compatible with dualism, but entail dualism. That is, genuine multiple realizability (regardless of whether or not supervenience holds!) implies that there are non-physical facts over and above the physical facts about each and every token-realizer of a multiply-realized kind. Moreover, we will show that, if for example our mental states (e.g., pain) are indeed multiply realized, as non-reductive physicalism has it, then the very fact that we are in these mental states implies that we have direct and immediate access to these non-physical facts.

The argument is this (for more details see Hemmo and Shenker 2015, 2019b, 2020b). Since the crucial feature of multiple realizability is that there is no shared macrovariable (or physical kind) in all the micro-realizers of a high-level kind, two questions arise: (i) What facts make it the case that a given microstate realizes (or is a token of) one high-level property L rather than another? (ii) How can the high-level kind L be identified, for instance, by us? In other words, given a token state of the high-level kind L, how can we sense or have experience of the high-level kind to which it belongs?

Let us begin by supposing that we have a list consisting of the disjunction of all possible physical microstates of the entire universe (physical tokens) that realize anywhere in the universe the high-level kind L (say, pain). If there is some macrovariable, however complex, anywhere in the universe (jn our brains, or elsewhere, e.g., in some measuring devices sensing pain in our environment, or in the brain of a physician who examines us when we are in pain), which is shared by all L-microstates, then, by definition, the high-level kind L is not multiply realizable. So let us assume that this is not the case, as envisioned by non-reductive views such as functionalism: namely, the microstates that may appear in this list do not share any (relevant) physical macrovariable of the universe in virtue of which they belong to the L-list.

Consider now question (i): which physical facts about the universe make this list the list of all (and only) realizers of L? Which physical facts make it the case that each and every microstate in the L-list (and no others) realizes L? There can be only two answers to this question.

Answer 1

(e.g., Fodor 1974, 1997, and many others): since there is no shared physical fact about the microstates in the list that makes them (and only them) possible realizers of L, the fact that makes this set of microstates (rather than another) the set of all realizers of L is by itself a brute fact about the universe. (Since this fact is brute (or primitive) it is unexplained, or even inexplicable by science (!) and God knows why this fact holds; see Fodor’s 1997, “molto misterioso” approach.)

Answer 2

There is some other matter of fact or thing outside the universe that determines the list. (Since it is outside the universe, perhaps we may assume also here that God knows which fact it is; see Fodor’s 1997, “molto misterioso” approach.)

The astonishing point is that there are no other answers to this question in the literature.

In both answers the fact alluded to (the brute fact and the fact outside the universe) to account for the L-list is a non-physical fact over and above the facts described by physics. Here is why. According to all the contemporary theories of physics (e.g., classical and quantum mechanics) the microstate of the universe at every moment of time (which is of course different in these theories) is a shorthand for describing the list of all microphysical facts about the universe at that time. In other words, according to contemporary physics the microstate of the universe gives the complete and exhaustive description of all the microphysical facts occurring in the universe at the time in question. And then the equations of motion in each of these theories describe the way in which this complete microphysical state changes over time (given some parameters and boundary conditions), so that we have the entire complete full-blown microphysical history of the universe. As we said, we assume that all the high-level facts and laws described by the higher-level special sciences supervene on the microphysical facts and laws as described by fundamental micro-physics. But what happens when we add to the picture multiple-realizability?

Before we go on, let us take off the table another third answer to question (i) (in addition to the brute and external facts answers, 1, 2 above) that might seem at first sight quite natural and attractive for non-reductive physicalism in capturing its central non-reductive idea. The idea is roughly that multiple realizability only expresses the non-reductive nature of the high-level facts: that although all high-level facts are at bottom physical facts, since they supervene on the microphysics, they are not reducible to the micro-physical facts. And the thought is that this idea is perfectly consistent with physicalism for the following reason.

Answer 3

It is true (by the definition of multiple realizability) that the microstates in our L-list of all micro-realizers of the high-level kind L do not share any micro-physical property. In this sense L is multiply realizable. But this only (and benignly) means that L is a kind (or a property) that is emergent from the micro-structure, or is non-reducible to the micro-structure. It does not imply that what makes a token-microstate be of kind L is an additional fact over and above the facts of physics, nor that it is outside physics.

However intuitive this answer may seem, it is a non-starter. There are only two options here: either L is identical with a single macrovariable (which is a function of all the microstates of the universe that belong to the L-list) in which case the existence of this macrovariable (or function) implies that there is no multiple realizability; or else the kind L is identical with a function of the microstates in the list, which is itself a disjunction of such macrovariables. In this latter case, L is indeed multiply realizable (as required) but question (i) is not answered: which facts make it the case that the disjunctive list of macro-realizares defining L consists of the disjunction of these specific macrovariables rather than any others?

So, as we said, we are left with only answers 1 and 2 to question (i), both of which introduce facts over and above the facts of physics. Answer 1 claims that the fact that makes each and every microstate (call it X) in the list be a realizer of L is a brute fact, but since there is no single physical macrovariable, which is a function of X and which is shared by all the other microstates in the list, this means that nothing in the complete micro-physical state of the universe in each occurrence determines whether or not L occurs. This is token-dualism, because the ‘decision’ as to whether or not L should obtain when X obtains must be made on each individual occurrence of a microstate in L in addition to the occurrence of X. Answer 2 boils down to assuming that some other matter of fact or thing outside physics determines the set of micro-realizers of L. This fact or thing is by construction outside physics.

Note with respect to both answer 1 and 2 that Laplace’s demon, for example, who by assumption has access to all possible fundamental physical facts, will never be able to know the brute fact or the fact outside physics. This means that there is no way in which the Demon can come up with the list. create the list. That is, if multiple realizability is assumed, the Demon will not be able to partition the state space into the sets corresponding to the L-kinds; in this sense it will be completely blind to the special sciences.

So the conclusion is that both answers presuppose some non-physical facts in or outside the universe and therefore are incompatible with physicalism! They are rather dualism in disguise.³⁴

Finally, one might want to say that it is also a brute fact that although the list of realizers cannot be learned by looking into the physics of the micro-realizers—because they don’t share any macrovariable, the L-list can still be learned by experience.³⁵ However, this makes the situation even worse for non-reductive physicalism. According to contemporary physics all the physical matters of fact that actually occur at all times are exhausted by the sequence of microstates given by the equations of motion.³⁶ Therefore, if the fact that a microstate belongs to the L-list is not physical (i.e., if realizing the L-kind is not fixed by something physical in this microstate, as we just argued), there is no physical way in which we could ever experience this fact. And so if multiple-realizability holds, the fact that we do experience high-level L-kinds not only implies that there is some non-physical feature in each and every realizer of L, as we argued above (we called this result toke-dualism), but also that this non-physical feature is the one that we experience when we experience L! So it follows that non-reductive approaches and in particular functionalism imply that we must have access (causal or otherwise) to these non-physical facts.