Phi 272 F06, reading guide

In these chapters, Carnap touches on the two other branches of physics that have been of most interest to philosophers—namely, statistical mechanics and quantum theory. Carnap will not address philosophical problems in statistical mechanics but instead use it to introduce the idea of statistical laws and to contrast the significance of its statistical laws with those of quantum theory.

• Carnap gave a brief introduction to explanation by statistical laws in ch. 1 (pp. 8-9). There he takes a very strong position regarding them, saying that such a law can be held to provide at least a weak explanation of even an event it describes as highly improbable. You should look back at what he says and think whether you agree with him; not everyone does.

• Although Carnap does not return to the question of whether statistical laws can provide the basis for an explanation, he does look at them in more detail in this chapter. His concern is mainly to describe their association in much of science with an “ignorance” of the full details of the state of a physical system that is assumed to be perfectly determinate and governed by perfectly deterministic laws. This is probably exhibited best in the example of statistical mechanics that he discusses on p. 280. Think about the reasons such laws are statistical or probabilistic and also about Carnap’s distinction between such reasons and the limitations on certainty forced by the two other reasons, (i) the limited precision and accuracy of measurement and (ii) the idealizations and simplifying assumptions required in order to apply laws to the complexity of real circumstances.

• Carnap will go on to distinguish all of these limitations on certainty from those associated with quantum mechanics, and he sets up that distinction by considering the ideal perfect determinism promised by classical physics: given complete information about a physical system at a given time (i.e., the full details of its state given with perfect precision and comparable information about the state of the universe as a whole) its state at any future time could be predicted with complete accuracy. (He refers you here to his presentation of Laplace’s formulation of this ideal in ch. 22, p. 217—see below for Laplace’s statement of this.)

All events, even those so minute that they seem not to be bound by the grand laws of nature, follow these laws as necessarily as do the revolutions of the Sun.…

…

Thus we should regard the present state of the universe as the effect of its previous state and as the cause of that which will follow. An intellect that, for a given moment, knew all forces by which nature is set in motion, and the positions of the entities that compose it, if it was also vast enough to submit these data to analysis, would embrace in a single formula the movements of the greatest bodies of the universe and those of the slightest atom; nothing would be uncertain for it and the future, like the past, would be present before its eyes.

…

… The curve described by a simple molecule of air or vapor is regulated in as certain a way as are planetary orbits; there is no difference between the two apart from what is put there by our ignorance.

Probability is relative in part to this ignorance and in part to our knowledge.…

Pierre-Simon, Marquis de Laplace, Essai philosophique sur les Probabilitiés (1814), published also as the introduction to the third edition of his Théorie Analytique des Probabilités, which appears in the Œuvres Complètes de Laplace, vol. 7 (Paris: Gauthier-Villars, 1886), pp. vi-vii, viii.

• In quantum theory, the concept of measurement is much more significant than in classical physics. What are classically the various magnitudes describing the state of a system are in quantum theory “observables.” Many of these do not have determinate values in all states of a system, and the nature of measurement is a central theoretical issue. The heart of quantum theory is a description of the way the state of a system (the “wave equation”) evolves over time (this is the “Schrödinger equation”). Measurement has been sometimes treated as a second sort of process (the “collapse” of the wave function), which also changes the state of a system. While the evolution determined by the Schrödinger equation is as deterministic as in classical physics, any change of state associated with measurement is not.

Carnap gives an account of these issues that holds up pretty well, but there is one point that he takes for granted that is no longer a matter of general agreement. The example of a Geiger counter he describes at the bottom of p. 287 is a kind of case that is often discussed with regard to the particular example described by Schrödinger in the following quotation:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The ψ-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

Erwin Schrödinger, “The Present Situation in Quantum Mechanics,” John D. Trimmer (tr.), Proceedings of the American Philosophical Society, vol. 124 (1980), p. 328. (This is a translation of “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften, vol. 23 (1935), pp. 807-812, 823-828, 844-849.)

(The “ψ-function” Schrödinger mentions is the wave function.) Carnap seems to assume (as was widely assumed when he wrote this) that something like the special measurement process is required in order to explain why we do not encounter macroscopic objects with indeterminate properties (e.g., a cat that is indeterminate between being alive and dead). It has since been emphasized that interactions with the environment mean that even a system whose evolution is governed by the Schrödinger equation will appear to exhibit the “classical” sort of behavior we are familiar with. This means that the indeterminism associated with a “collapse of the wave function” seems less central to many now than it did to Carnap.