Science and the Philosophy of Science

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Science is commonly defined as the systematic study of the universe—through observation and experiment—in the pursuit of knowledge that allows us to generalize. In A Treatise of Human Nature, David Hume stated that ‘Even after the observation of the frequent conjunction of objects, we have no reason to draw any inference concerning any object beyond those of which we have had experience’ (Hume 1739–40), and the futility of bias-free learning has since been proven (Mitchell 1980, Schaffer 1994, Wolpert 1996). This implies that the scientific method involves making intelligent assumptions and practicing self-consistent reasoning in the face of uncertainty. Self-consistent reasoning involves avoiding a Dutch book, treating cases of symmetry as symmetrical, being self-consistent across time and respecting truly random processes.

If an individual is not susceptible to a Dutch book, their previsions are said to be coherent. A set of betting quotients is coherent if (Ramsey 1926, de Finetti 1937, Shimony 1955) and only if (Kemeny 1955, Lehman 1955) they satisfy the axioms of probability. A good exposition of Bayesian probability is given in Lindley (1972). Symmetrical cases should be treated symmetrically, this is known as exchangeability (de Finetti 1937). One's current beliefs must equal one's current beliefs about one's future beliefs, which is known as the reflection principle (van Fraassen 1984, 1995). When truly random processes are involved (i.e. only phenomena which are aspects of quantum mechanics), the principal principle (Lewis 1980) should be observed. Intelligence is the ability of an individual to perform a novel cognitive task (derived from Carroll (1993) p. 16), so intelligence should be positively correlated with one's ability to assign a realistic prior probability to a novel hypothesis. An intelligent individual should exhibit superior calibration.

In other words, science is intelligent applied subjective Bayesian analysis constrained by exchangeability, the reflection principle and the principal principle. Reassuringly, over the past fifty years subjective Bayesianism has become the dominant theory of scientific method (see Forster 2001 p. 87 and Nola and Sankey 2007 p. 186). For more on the subjective Bayesian approach to science, see Lad (1996) and Howson and Urbach (2005).

Where H = hypothesis, D = data and B = background information, Bayes' theorem tells us that

P(H|DB) = P(H|B)P(D|HB)/P(D|B).

P(D|B) is independent of any hypothesis, so may be ignored when comparing hypotheses. Enumerative induction seeks to maximize P(H|DB) directly, Popper's falsification generalizes to maximizing P(D|HB), whilst the aim of abduction (also known as inference to the best explanation) is to maximize P(H|B)P(D|HB) where D consists of facts. As explained above, Bayesian inference is necessary for science. On that basis, induction and abduction are in essence scientific, whilst Popper's falsification is an incomplete notion of science.