Two years ago, I wrote a short essay on Kuhn entitled "On How to Study the History of Science Philosophically" for a Japanese journal.
I opened the essay with the question: "how should a student of science decide which paradigm to adopt?" I argued that giving a philosophically satisfying answer to this question requires some appeal to truth. That is, appealing merely to "social circumstances," "individual research interests," and "the effectiveness of the paradigm in solving contemporary scientific puzzles" do not suffice to convince a student that this is the paradigm within a given field to follow. I here have in mind cases where the student has the option to choose from a variety of social circumstances, individual interests, and puzzles. Such cases are not all that rare -- for example, a student may just as well decide to join a physics program dominated by QBists as opposed to another program dominated by GRW theorists. These two programs each put the student in very different social circumstances, allow the student to pursue very different interests, and directs the attention of the student to very different puzzles. Therefore, I argued, the student can only be convinced to side with one program rather than another if a convincing case can be made that one program represents the truth more so than the other. For example, maybe there is stronger scientific evidence, or a stronger philosophical argument, that GRW theory represents the truth, whereas QBism misses some of the elements of that truth.
Based on the above premise, I then continued by claiming that Kuhn's way of studying the history of science did not provide the normative grounds for guiding such a student. This is because Kuhn tries to explain the transition from one paradigm to another without appealing to normative ideas such as truth. I use the term "normative" here in a strong and restricted sense, or, for those with Kantian inclinations, in a transcendental sense. That is to say, an idea is normative only if that idea is a necessary presupposition for the student to do science at all, regardless of which social group he or she decides to join, or which puzzle he or she happens to be interested in.
By comparison, I argued that Hegel and Collingwood's way of studying the history of science is more properly philosophical, because both philosophers argue that a given concept in science (such as that of force or of causation) is added or eliminated from a particular field because that concept succeeds or fails to represent the truth of the subject-matter. This, however, is a digression in the context of this blog entry.
While writing this essay, I deliberately avoided the topic of incommensurability for one simple reason - at least judging from Kuhn's Structure of Scientific Revolutions, it was not clear to me what the criteria were for distinguishing mere difference or inconsistency from incommensurability. After reading Richard Grandy's otherwise illuminating essay, and after discussing the topic in class, I am still not entirely clear on what the aforementioned criteria are.
Grandy does make it clear that incommensurability obtains not through piece-meal scientific progress, but only through full-blown revolutions. However, when it comes to what such "full-blown revolutions" are, Grandy has little more to say than to point out that the nature of such changes are more likely to be discovered through a study of small-scale changes rather than large-scale ones (1982, p.14). Nevertheless, Grandy does clarify the basic notion of "incompatibility" between two truths. To paraphrase Grandy, if T1 is true in language L1 and T2 is true in language T2, and if there is a metalanguage M such that both T1 and T2's truth relative to their respective languages can be stated in M, then T1 and T2 are incompatible iff in M T1 and T2 cannot both be true relative to L1 and L2 (1982, p.19-20). The seemingly convoluted formulation is necessary, because Grandy is at pains to save the relativity of the truth of T1 and T2 while also proposing a way of comparing the two through a metalanguage M.
The strength of Grandy's account is that it shows how incommensurability does not necessarily imply incomparability, or, more simply, does not necessarily imply that the users of the two languages have not way of communicating to one another. All it takes for incommensurability to obtain is that if T1 is true relative to L1 then T2 is false relative to L2 or vice versa.
At this level of abstraction, this is all well and fine. However, is this account true? In other words, are there concrete historical examples which plausibly fit Grandy's account? This was the big question which I had when going into Week 4's class.
Several concrete examples were mentioned in class.
One was from astronomy, where the existence of a satellite was assumed in order to explain the behavior of mercury. Later on, according to one of my peers, relativity theory allowed astronomers to explain the behavior of mercury without assuming the existence of the satellite. Moreover, the satellite was never observed in the place where it was assumed to be. However, it was not clear how this example is relevant to Grandy's discussion. This example seems to be a case of piecemeal scientific progress - after all, the satellite hypothesis was falsified by experiment. It was not like two explanations were both true relative to two different equations. Therefore, this example does not support Grandy's claim.
Another example was put forth from physics, according to which Newton's concept of time was said to be so different from Einstein's that the two were "incommensurable." Although it was not clarified in class, I take it that the difference referred to here boils down to whether there is an absolute time or not. Newton clearly states that there is absolute time, whereas Einstein clearly states that there isn't. If this is what the example is intended to point to, then it is clearly a better example compared to the astronomy case. I assume that if Newton's equations are true relative to Newton's concept of absolute time, then some of Einstein's equations are (presumably) false relative to Einstein's denial of absolute time. (I am not an expert in relativity theory, hence, I am only assuming that this is how the example is supposed to play out.) Whatever that means, the question still remains as to whether this is a case of incommensurability as defined by Grandy. I have heard and read physicists state that there are a number of observations which support Einstein's relativity theory over Newton's mechanics. If this is the case, then the deciding factor seems to be, quite simply, empirical evidence. That is, Newton's equations are not true even if one is a Newtonian, because empirical evidence contradicts them.
Now, maybe the above account is false, and maybe Newton's axioms do allow him to make predictions through equations which are then justified by the exact same observations which justify Einstein's equations. If this is the case, then this is truly a case of incommensurability. However, I am not competent enough to say something constructive on this problem any further.
Finally, there was the suggestion that Euclidean geometry and non-Euclidean geometry are incommensurable. The claim was that Euclidean geometry presupposes "Euclidean space" whereas non-Euclidean geometry presupposes "Riemannian space." Again, I do not know what a "Riemannian space" is, and the nature of this space was not explained in class. Hence, I cannot really comment on this example. I can, however, substitute space with parallel lines and see if that works. In Euclidean geometry, the parallel postulate is, well, just a postulate, not an axiom. This is because it might be possible to draw more than one line parallel to a given line. Various attempts have been made in the history of geometry to prove that only one parallel line can be drawn to a given line, but the various tweaking of the wording of the parallel postulate, various additions of additional axioms, etc. did not work. In the end, the parallel postulate remains a postulate. (Or so goes the account given in Sommerville's introduction to non-Euclidean geometry.) In comparison, here is how Sommerville explains how parallel lines work in hyperbolic and elliptic geometries:
Source: D. Sommerville, Elements of Non-Euclidean Geometry, p.29-30.
The question then is, are these definitions of parallel lines incommensurable? My intuition is to answer that they are merely different, but not incommensurable. Why? Because, within a metalanguage M (which, at least in this case, can just be plain English), all hypotheses concerning parallel lines can come out as true relative to each of the respective definitions. Thus, according to Grandy's definition, these definitions of parallels are not incommensurable - or so it seems, at least to me. Maybe a person more versed in geometry might have more refined and possibly different intuitions.
These were the examples that were brought up in class, and I have explained why I think that each example either fails to provide support for Grandy's claim or at least fails given the insufficient level of detail in the presentation of the example. Thus, after class, I was still unable to decide whether there really is such a thing as incommensurability, assuming that Grandy's definition is good (it certainly is better than Kuhn's vague usage of the term!)
