James Franklin

Defends the arguments for the irredicibility of dispositions to categorical properties in "Are dispositions reducible to categorical properties?" (Philosophical Quarterly 36, 1986) against the criticisms of D.M. Armstrong (Philosophical Quarterly 38, 1988). Laws that involve an irreducibly dispositional element, such as rigidity, contrast with laws that contain no such element and are absolutely necessary, such as "All bodies symmetrical about bom a horizontal and a vertical axis are also symmetrical about the point of intersection of the axes."

Eugene Wigner’s 1960 article on the “unreasonable effectiveness of mathematics” used the word “miracle” of the fit between abstract mathematics and physical reality. William Lane Craig has developed a theistic argument from Wigner’s hints, claiming that the best explanation of the “miraculous” fit is divine creation. It is argued that this argument does not succeed. An Aristotelian realist philosophy of mathematics renders the applicability of mathematics to physical reality unmysterious by showing that mathematics, like any other science, is a study of certain aspects of reality, hence there is no miracle of fit. However, that does not preclude other arguments for the existence of God involving mathematics, for example design arguments from the elegance of the universe’s structure, fine-tuning arguments or ones from the nature of mathematical understanding.

These interlinking essays are connected by a core theme: the necessary structures in reality that allow certain knowledge of absolute truths. Franklin’s Aristotelian realist philosophy of mathematics shows how mathematical truths are directly about physical reality, and at the same time certainly and provably true. Ranging from mathematics to evidence evaluation to ethics, his philosophy of probability sees the relation of evidence to hypothesis, such as in science and law, as purely logical, hence necessary. Across ethics and the philosophy of religion, the theme of necessity is repeated: basic ethical truths (such as the worth of persons and the wrongness of murder) are shown to have the same certainty as mathematics. Focus on the history of ideas connects the philosophical work in the present with the medieval scholastic tradition, which defended similar necessities but is now neglected. Here is an up-to-date introduction to Franklin’s overall perspective. Recalling Western philosophy to its roots, it reveals the way absolute necessities are discoverable across the abstract fields of mathematics, logical evidence and ethics.

In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis

Eugene Wigner’s 1960 article on the “unreasonable effectiveness of mathematics” used the word “miracle” of the fit between abstract mathematics and physical reality. William Lane Craig has developed a theistic argument from Wigner’s hints, claiming that the best explanation of the “miraculous” fit is divine creation. It is argued that this argument does not succeed. An Aristotelian realist philosophy of mathematics renders the applicability of mathematics to physical reality unmysterious by showing that mathematics, like any other science, is a study of certain aspects of reality, hence there is no miracle of fit. However, that does not preclude other arguments for the existence of God involving mathematics, for example design arguments from the elegance of the universe’s structure, fine-tuning arguments or ones from the nature of mathematical understanding.

The most successful science on the Aristotelian model was geometry in the style of Euclid. As advocated in the Posterior Analytics, Euclid’s Elements laid out geometry as a structure of theorems deduced from definitions and axioms that were evident to reason. However, geometry deals with the category of quantity, whereas Aristotelian definitions are paradigmatically in the category of substance. This chapter argues that definitions in the category of quantity have fulfilled well the Aristotelian ideal of stating the essence of ‘what a thing is’, in such a way as to support a superstructure of demonstrations about that thing. Euclid’s definition of circle is a prime example: it is philosophically satisfying and from the first theorem of the Elements proved itself very productive of theorems. As this chapter demonstrates, the definition of ratio was more difficult. Euclid stands at the end of a long search for a definition of ratio, one of the most central notions of mathematics, that is both philosophically satisfying and mathematically productive. The search was not wholly successful.

Readers of “lives” of the famous know well the tendency of biography, and especially autobiography, to become steadily less interesting as the subject grows older. A predictable record of challenges met, enemies shafted, honours received and great men encountered often succeeds an account of a childhood that is a highly-coloured and unique emotional drama. Often the best pages are those on the subject’s schooldays, when the personality first tangles with the public realm. As Barry Oakley says of school in a piece quoted in the book’s preface: “Like the stage, it’s an image of life: life accelerated, life concentrated, life more formidable.” The project of selecting just the highlights of all the stories of Australians’ schooldays promises, then, a high payoff if it is well done. It is a high-risk enterprise, though: a pointillist canvas brilliant in each fleck may easily look like mud from a distance. There are well over a hundred authors here, with only three pages or so each to paint a vignette of school. In fact, the result is an enormous success. The editors have a sure eye, and they and their research assistant, Pamela Williams, have put in the work to find the goods. Almost every piece is gripping, and quite different from the others. The total effect is additive, and is an unexampled insight into how the Australia we know came into being. The classics are there: Henry Lawson and Patrick White, Seven Little Australians and The Getting of Wisdom, Donald Horne, Barry Humphries and Clive James. So are the many unknowns whose recollections take us into obscure corners. If there is one overall theme, it is that of sameness, difference and “fitting in”. The effect of the accumulated evidence is rather more subtle than the received ideas on “identity and difference”, multiculturalism and so on. School is where the strangeness of one’s own family, or of one’s own personality, meets the social world – itself perhaps no less weird, objectively speaking, but possessed of resources for ensuring conformity.