Soshichi Uchii, Kyoto University

10. The Case for or against Reductionism?

Now, after this long review of the development of kinetic theory, heavily depending on Dr. Tomonaga's analysis, we can come back to our initial problem of reductionism. As I said at the beginning, the kinetic theory is often referred to as one of the best examples of one theory being reduced to another. But after taking a look at what was going on in the kinetic theory, we have to be more careful. First of all, what does "reduction" mean? Second, what is allleged to be reduced to what?

Let us begin with the first question. I understand that, according to the standard view, one theory T1 is reduced to another theory T2 if and only if the following three conditions hold:

(1) The basic concepts of T1 are all definable in terms of concepts of T2.

(2) All of the basic laws of T1 can be translated, by means of such definitions, into laws of T2, which are of course derivable within T2.

(3) The concepts and laws of T2 are more basic, in some sense, than those of T1.

Now, taking this definition of "reduction" for the moment, what is alleged to be reduced to what?---this is our second question. There may be several alternatives, but simplifying the matter, we will consider only two.

(A) Suppose Thermodynamics is alleged to be reduced to the Newtonian mechanics. Then, clearly, this allegation is false, because the concept of (epistemic or inductive) probability is not contained in the Newtonian mechanics. As Dr. Tomonaga has made clear, we need not only the probability concept as time-average (which may be regarded as a mechanical concept) but another concept independent of the mechanics, in order to do justice to the second law of thermodynamics. Thus even the first condition (1) fails.

(B) Suppose, then, Thermodynamics is alleged to be reduced to the full kinetic theory, which assumes the statistical point of view as well as the mechanical point of view, including all the basic concepts associated with them. Then we may admit that the conditions (1) and (2) can be satisfied. But this time, the condition (3) is dubious, to say the least. Take the concept of irreversibility or entropy from the thermodynamics, and the concept of (epistemic or inductive) probability from the kinetic theory. Can we say that the latter is more basic than the former? Or in what sense is it more basic? Honestly, I am not sure; I really do not know. But certainly not "more basic" in the ontological sense, as some atomist wants to say; because probability does not seem to be an ontological concept.

Without getting involved in this sort of messy questions, I think it is more sensible to say that the word "reduction" is inappropriate here. In the course of the development of the kinetic theory, thermodynamics is integrated within a larger framework, eventually giving birth to a new branch of physics called "statistical mechanics." Whether or not its foundations are more basic, more secure, I really do not know. But one thing seems to be clear. The kinetic theory or statistical mechanics has certainly wider applications, richer in its theoretical content. Thus it seems to me that the word "extension" is more appropriate here. That is to say, Thermodynamics is extended, by adding new concepts and new point of view, into a more powerful theory.

If you are not convinced of this, take an analogous case from mathematics. Almost everyone knows that Frege-Russell's logicism failed. They tried to reduce mathematics into logic. But this program did not work well, primarily because they had to introduce the concept of set when they wanted to go into arithmetic or mathematics. The crucial problem for them turned out to be this: how can we define the concept of set which is rich enough to reproduce all of mathematics, but which is, at the same time, basic or simple enough to be called "logical"? All of their heroic attempts failed, and the upshot is that we can indeed reconstruct mathematics, if we add to logic the concept of set and a number of basic assumptions about sets (axiomatic set theory). That is to say, mathematics is not reducible to logic, but it can be reconstructed as an extension of logic.

I claim that the situation is quite similar in the kinetic theory. Thermodynamics can indeed be reconstructed within the kinetic theory or statistical mechanics; but the latter theory is an extension of the Newtonian mechanics and of the original thermodynamics as well. And, of course, the key concept which has enabled this extension is the concept of probability. It plays a somewhat similar role as that of set in Frege-Russell's logicism.

Thus I wish to conclude that the logicism and the kinetic thery are a typical example in which the original or alleged attempt of reduction has failed and, ironically, ended up with an extension of the original system. However, this conclusion does not mean that "reductionism as a methodology" (or a "research program") is useless; on the contrary, that methodology was quite fuitfull in the case of kinetic theory, giving birth to a new branch of physics, i.e. statistical mechanics. Thus, although I think the kinetic theory failed as a literal reductionism, it has certainly succeeded as a fruitful methodology.

Bibliography / Appendix

9. Probability Applicable when we are Ignorant

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March 1, 1999; last modified March 4. (c) Soshichi Uchii