SIMPLICITY AS AN EPISTEMIC STRATEGY
Finally, I will consider a view according to which simplicity should be understood as part of a strategy for modifying theories as new data emerge. If such a strategy can be demonstrated most likely to eventually lead to the truth in a maximally effective way, then simplicity is shown to be an epistemic virtue.
At least that is the claim. An idea of this sort was suggested years ago by Hans Reichenbach,[87] and more recently by Kevin Kelly.[88]For Reichenbach, the aim of induction is to infer the probability of certain types of events, which for him means inferring the limit of the relative frequency of events of that type in an infinite sequence of such events. Reichenbach advocates using the “straight rule” of induction for this purpose. This rule is to infer that the limit of the relative frequency of a type of event E in an infinite series of events is the same as the observed relative frequency of events of type E in that series. In short, infer that the “population” will match the observed “sample.” Reichenbach's famous justification for this rule (his “justification of induction”) is that if you continue to use this straight rule, and if there is a limit, then as you observe more and more events in the series, there will come a point after which the observed relative frequency of Es will be, and stay, within any desired margin of the correct limit of the relative frequency. In other words, if you continue to use the inductive straight rule to infer limits, and if there is a limit, your inferences will eventually get as close as you like to the true limit.
As Reichenbach himself noted, however, there are infinitely many inductive rules about which the same can be said. Let m/ n be the observed relative frequency of an event of type E at a point n in the sequence. Consider some function f(n) which is such that as n approaches infinity, f(n) approaches 0.
Consider rules of inference of the form:If the observed relative frequency of E is m/ n, then infer that the limit of the relative frequency of E is (m/ n) + f(n).[89]
'1 his yields an infinite set of “inductive” rules, each of which will permit your series of inferences to “converge” to the correct limit, if one exists, but each of which will yield different predictions about the limit along the way. The straight rule is the special case in which f(n) = 0 for all n. Why choose this special case, as Reichenbach does?
Well, you might say, the straight rule is the simplest: it doesn't add a “corrective” factor to the observed relative frequency. But then the question arises: Is this epistemic simplicity, or is it merely pragmatic? That is, is the choice of the simpler “straight rule” based on an assumption such as: the continued use of the straight rule is likely to lead to the correct limit faster than the others? Or is the choice of the straight rule based on convenience? Reichenbach's response sends a mixed message. He admits that choosing some corrective factor f(n) other than 0 could hasten the convergence to the correct limit. But it could also delay the convergence; and he adds, “we know nothing about the two possibilities.” So, he concludes, choosing f(n) = 0—i.e., the straight rule— is to choose “the value of the smallest risk; any other determination may worsen the convergence. This is a practical reason for preferring the inductive principle.”[90] The claim that the simple straight rule gives the “smallest risk” suggests an epistemic defense of this simple rule over other, more complex convergent rules. But Reichenbach also claims that the fact that it gives the smallest risk is a practical reason for choosing the rule. Perhaps, then, it is supposed to give both an epistemic and a practical reason.
The main problem I have with Reichenbach's approach is this: Suppose you choose the corrective factor f(n) = 0, saying that it is the simplest choice.
And suppose, following Reichenbach, you argue that if you continue to use this simple straight rule to infer probabilities, then as more and more evidence is gathered, your simple inductive conclusions about the probability of some type of event will at some point converge to the correct probability, assuming there is a limit to the series. Finally, suppose you claim that this is both an ep- istemic and a practical reason for using the simple straight rule. Is this enough to show that simplicity is an epistemic virtue? If so, it is a far cry from the original Epistemological Claim or even to the substantially modified Bayesian version. Let me say why.To confine our attention just to inductive inferences of the sort Reichenbach is concerned with, suppose, using the simple straight rule, I make an inductive inference from the fact that the observed relative frequency of type E events is m/ n to the claim that the limit of the relative frequency is m/ n. Reichenbach is not telling me that the fact that I am using his simple straight rule to infer this when I have determined that m/ n of the observed members of the series are of type E provides a good reason, whether epistemic or practical, to believe that my conclusion is correct—that the limit of the relative frequency is m/ n. Nor is he telling me that the fact that I am using his straight rule to infer this constitutes a better reason than before. What he is telling me is that if I continue to use this strategy of following the simplest inductive rule, and if there is a true conclusion of the sort I am seeking, then eventually I will arrive at a conclusion that is as close to the true one as I would like. Furthermore, I will never know whether there is a true conclusion here (i.e., whether there is a limit of the relative frequency), and if there is, whether I have reached a point of convergence. Finally,
he is telling me that (practically or epistemically speaking) continuing to follow the straight rule is better at getting to a point of convergence, if there is one, than continuing to follow more complex rules.
This defense of simplicity provides little if any reason for Newton to appeal to simplicity in making his inference to the law of gravity. It would be telling Newton: “Well, Sir Isaac, simplicity cannot justify making the inference you did—i.e., inferring that all bodies obey the law of gravity on the grounds that all the ones that have been observed do. (Simplicity cannot justify inferring that the limit of the relative frequency of bodies that obey the law of gravity is 1, from the fact that the observed relative frequency is 1.) But as you observe more and more bodies, and continue to make inferences from the observed relative frequency to the limit of that frequency using the simple straight rule, then, if there is a limit of relative frequency here, you will eventually get closer and closer to it, and do so in a better manner than with more complex rules, although you will never know if there is a limit, or when you have reached a point of convergence.”
I think Sir Isaac would be unimpressed, and with good reason. He wants simplicity to justify the inference he in fact made—which Reichenbach does not do—not a strategy to continue to make such inferences.
Kevin Kelly offers an epistemic justification for simplicity that is similar in certain respects. Like Reichenbach, Kelly is concerned with situations in which you are receiving new empirical data over time, and you are inferring and modifying hypotheses that go well beyond the data. In the light of new data, how should you do so? Kelly provides a formal strategy that proceeds in the simplest way. It minimizes the number of revisions or retractions you make on your road to truth. And he claims that the formal account of simplicity he provides is epistemic, not pragmatic, because “retraction-minimalization (i.e., optimally direct pursuit of the truth) is part of what it means for an inductive inference procedure to be truth-conducive, so retractions are a properly epistemic consideration.”[91]
Without examining the formal details, suffice it to say that Kelly, like Reichenbach, does not provide an epistemic role for simplicity of the sort I have been discussing.
To do so, it needs to be shown that the fact that a theory is simple provides a good reason for believing it is true or empirically adequate (or at least a better reason than without that fact). But even for Kelly, the fact that some modification of a theory in the light of new data is simple, or the simplest one possible, does not by itself give you any reason for believing that it is true. He admits that his simplicity strategy “cannot point at or indicate the true theory in the short run and... alternative methods [could] have converged to the truth eventually.”[92] What Newton, Einstein, and other epistemic simplicity theorists want, or at least need, is an epistemic justification not for continuing to modify theories in the simplest way in the light of new data but for taking the simplicity of a particular theory to be at least some epistemic reason to believe it to be true or empirically adequate.[93]12.