James Franklin

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.

Australian women who are university graduates have fewer children than non-graduates. In most cases this appears to be the result of circumstantial pressures not preference. Long years of study fill the most fertile years of women students and new graduates need further time to establish their careers. The chance of medical infertility increases with age so, for some, this means that childbearing is not postponed but ruled out. Graduates who do make the transition from university to professional work find that working hours are long and that professional occupations are now both highly demanding and insecure. Women who take time off to care for young children must depend on one insecure income (their partner’s) rather than two, and their return to work is uncertain. These difficulties of time, money and insecurity are compounded by problems in finding a suitable partner. They are magnified by the enduring tendency of women to marry up. Thus it can be more difficult for women graduates to find husbands than it is for women who are non-graduates.

Some courses achieve existence, some have existence thrust upon them. It is normally a struggle to create in a scientific academic community a course on the philosophical or social aspects of science, but just occasionally a confluence of outside circumstances causes one to exist, irrespective of the wishes of the scientists. It is an opportunity, and taking advantage of it requires a slightly different approach from what is appropriate to the normal course of events, where a “social” course needs a fight to establish it, and faces a struggle for more than marginal existence.

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning.

The winning entry in David Stove's Competition to Find the Worst Argument in the World was: “We can know things only as they are related to us/insofar as they fall under our conceptual schemes, etc., so, we cannot know things as they are in themselves.” That argument underpins many recent relativisms, including postmodernism, post-Kuhnian sociological philosophy of science, cultural relativism, sociobiological versions of ethical relativism, and so on. All such arguments have the same form as ‘We have eyes, therefore we cannot see’, and are equally invalid.

The imperviousness of mathematical truth to anti-objectivist attacks has always heartened those who defend objectivism in other areas, such as ethics. It is argued that the parallel between mathematics and ethics is close and does support objectivist theories of ethics. The parallel depends on the foundational role of equality in both disciplines. Despite obvious differences in their subject matter, mathematics and ethics share a status as pure forms of knowledge, distinct from empirical sciences. A pure understanding of principles is possible because of the simplicity of the notion of equality, despite the different origins of our understanding of equality of objects in general and of the equality of the ethical worth of persons.