Malcolm Forster

Kenneth Wilson won the Nobel Prize in Physics in 1982 for applying renormalization group, which he learnt from quantum field theory (QFT), to problems in statistical physics—the induced magnetization of materials (ferromagnetism) and the evaporation and condensation of fluids (phase transitions). See Wilson (1983). The renormalization group got its name from its early applications in QFT. There, it appeared to be a rather ad hoc method of subtracting away unwanted infinities. The further allegation was that the procedure is so horrendously complicated that one cannot see the forest for the trees. The second allegation is justified in the applications that made it famous. But it is not true of the following example, which appears in Chowdhury and Stauffer (2000, 486-488).

Deductive logic is about the validity of arguments. An argument is valid when its conclusion follows deductively from its premises. Here’s an example: If Alice is guilty then Bob is guilty, and Alice is guilty. Therefore, Bob is guilty. The validity of the argument has nothing to do with what the argument is about. It has nothing to do with the meaning, or content, of the argument beyond the meaning of logical phrases such as if…then. Thus, any argument of the following form (called modus ponens) is valid: If P then Q, and P, therefore Q. Any claims substituted for P and Q lead to an argument that is valid. Probability theory is also content-free in the same sense. This is why deductive logic and probability theory have traditionally been the main technical tools in philosophy of science.

What is induction? John Stuart Mill (1874, p. 208) defined induction as the operation of discovering and proving general propositions. William Whewell (in Butts, 1989, p. 266) agrees with Mill’s definition as far as it goes. Is Whewell therefore assenting to the standard concept of induction, which talks of inferring a generalization of the form “All As are Bs” from the premise that “All observed As are Bs”? Does Whewell agree, to use Mill’s example, that inferring “All humans are mortal” from the premise that “John, Peter and Paul, etc., are mortal” is an example of induction? The surprising answer is “no”. How can this be?

Sober (1984) has considered the problem of determining the evidential support, in terms of likelihood, for a hypothesis that is incomplete in the sense of not providing a unique probability function over the event space in its domain. Causal hypotheses are typically like this because they do not specify the probability of their initial conditions. Sober's (1984) solution to this problem does not work, as will be shown by examining his own biological examples of common cause explanation. The proposed solution will lead to the conclusion, contra Sober, that common cause hypotheses explain statistical correlations and not matchings between event tokens

The paper provides a formal proof that efficient estimates of parameters, which vary as as little as possible when measurements are repeated, may be expected to provide more accurate predictions. The definition of predictive accuracy is motivated by the work of Akaike (1973). Surprisingly, the same explanation provides a novel solution for a well known problem for standard theories of scientific confirmation — the Ravens Paradox. This is significant in light of the fact that standard Bayesian analyses of the paradox fail to account for the predictive utility of universal laws like All ravens are black.

The distinction itself is best explained as follows. At the empirical level (at the bottom), there are curves, or functions, or laws, such as PV = constant the Boyle’s example, or a = M/r 2 in Newton’s example. The first point is that such formulae are actually ambiguous as to the hypotheses they represent. They can be understood in two ways. In order to make this point clear, let me first introduce a terminological distinction between variables and parameters. Acceleration and distance (a and r) are variables in Newton’s formula because they represent quantities that are more or less directly measured. The distinction between what is directly measured and what it is not is to be understood relative the context. All I mean is that values of acceleration and distance are determined independently of the hypothesis, or theory, under consideration. I do not mean that their determination involves no kind of inference at all. For instance, acceleration is the instantaneous change in velocity per unit time, and this is not something that is directly determined from raw data that records the position of the moon at consecutive points in time. It is consistent with that raw data that the motion of the moon is actually discontinuous, so that the moon has no acceleration. So, there are definitely theoretical assumptions make about the moon’s motion that are used to estimate the moon’s acceleration at a particular time. But these assumptions are not unique to Newton’s theory. The same assumptions are also made by the rival hypotheses under consideration. In fact, the existence of quantities such as instantaneous acceleration is only called into question by the far more recent theory of quantum mechanics. Likewise, in the case of Boyle’s law, there is no controversy in viewing the volume of the trapped air as being determined in a way that does not make use of the theory that Boyle is introducing.

A and B in signaling games (Lewis 1969). Members of the population, such as our prehistoric pair, are occasionally faced with the following ‘game’. Let one of the players be the receiver and the other the sender. The receiver needs to know whether B is true or not, but only possesses information about whether A is true or not. In some environmental contexts, A is sufficient for B, in others it is not. The sender knows nothing about A or B, but does know that A is sufficient for B in some environments. This is a higher-order signaling game in which both players can benefit from sharing the information that they possess. How does a communication strategy evolve, and is it evolutionarily stable?

Although in every inductive inference, an act of invention is requisite, the act soon slips out of notice. Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception men can no longer easily restore them back to the detached and incoherent condition in which they were before they were thus combined. The pearls once strung, they seem to form a chain by their nature. Induction has given them unity which it is so far from costing us an effort to preserve, that it requires an effort to imagine it dissolved — William Whewell, 1858. (Quoted from Butts (ed.), 1989, p. 143).

The theory of fast and frugal heuristics, developed in a new book called Simple Heuristics that make Us Smart (Gigerenzer, Todd, and the ABC Research Group, in press), includes two requirements for rational decision making. One is that decision rules are bounded in their rationality –- that rules are frugal in what they take into account, and therefore fast in their operation. The second is that the rules are ecologically adapted to the environment, which means that they `fit to reality.' The main purpose of this article is to apply these ideas to learning rules–-methods for constructing, selecting, or evaluating competing hypotheses in science, and to the methodology of machine learning, of which connectionist learning is a special case. The bad news is that ecological validity is particularly difficult to implement and difficult to understand. The good news is that it builds an important bridge from normative psychology and machine learning to recent work in the philosophy of science, which considers predictive accuracy to be a primary goal of science.

Ramsey, Stick and Garon (1991) argue that if the correct theory of mind is some parallel distributed processing theory, then folk psychology must be false. Their idea is that if the nodes and connections that encode one representation are causally active then all representations encoded by the same set of nodes and connections are also causally active. We present a clear, and concrete, counterexample to RSG's argument. In conclusion, we suggest that folk psychology and connectionism are best understood as complementary theories. Each has different limitations, yet each will co-evolve with the other in an overlapping domain of 'normal' psychology.

The Value of Good Illustrative Examples: In order to speak as generally as possible about science, philosophers of science have traditionally formulated their theses in terms of elementary logic and elementary probability theory. They often point to real scientific examples without explaining them in detail and/or use artificial examples that fail to fit with intricacies of real examples. Sometimes their illustrative examples are chosen to fit their framework, rather than the science. Frequently these are non-scientific examples, which distances the discussion from its intended target. In the final analysis, philosophical discussions of explanation, confirmation, scientific realism, and the nature of theories are often too abstract, or too imprecise, or too disconnected with real science, to allow scientists to benefit from the discussion. This is a great loss for both parties. In my experience, working scientists are confronted with philosophical issues not only in their role as researchers, but also in their role as tertiary teachers of science.