-
456Science by Conceptual AnalysisStudia Neoaristotelica 9 (1): 3-24. 2012.The late scholastics, from the fourteenth to the seventeenth centuries, contributed to many fields of knowledge other than philosophy. They developed a method of conceptual analysis that was very productive in those disciplines in which theory is relatively more important than empirical results. That includes mathematics, where the scholastics developed the analysis of continuous motion, which fed into the calculus, and the theory of risk and probability. The method came to the fore especially i…Read more
-
52Risk-driven global compliance regimes in banking and accounting: the new Law MerchantLaw, Probability and Risk 4 (4): 237-250. 2005.Powerful, technically complex international compliance regimes have developed recently in certain professions that deal with risk: banking (the Basel II regime), accountancy (IFRS) and the actuarial profession. The need to deal with major risks has acted as a strong driver of international co-operation to create enforceable international semilegal systems, as happened earlier in such fields as international health regulations. This regulation in technical fields contrasts with the failure of an in…Read more
-
26Is philosophy irrelevant to science?Philosopher's Zone (ABC Radio National) 0-0. 2009.Scientists get on with the job – they do stuff with test tubes or with computers – but can philosophers help them? Do they need help and, if so, do they think they need help? This week, we examine what philosophers of science talk about and what effect it might have on what scientists actually do.
-
12Thomas Kuhn's irrationalismNew Criterion 18 (10): 29-34. 2000.Criticizes the irrationalist and social constructionist tendencies in Kuhn's Structure of Scientific Revolutions.
-
123Aristotle on Species VariationPhilosophy 61 (236). 1986.Explains Aristotle's views on the possibility of continuous variation between biological species. While the Porphyrean/Linnean classification of species by a tree suggests species are distributed discretely, Aristotle admitted continuous variation between species among lower life forms.
-
2Species in AristotlePhilosophy 64 (247). 1989.Reply to H. Granger, Aristotle and the finitude of natural kinds, Philosophy 62 (1987), 523-26, which discussed J. Franklin, Aristotle on species variation, Philosophy 61 (1986), 245-52.
-
40The Sokal hoaxThe Philosopher 1 (4): 21-24. 1996.Describes the Sokal hoax and defends it against attacks by postmodernists.
-
65The renaissance mythQuadrant 26 (11): 51-60. 1982.THE HISTORY OF IDEAS is full of more tall stories than most other departments of history. Here are three which manage to combine initial implausibility with impregnability to refutation: that in the Middle Ages it was believed that the world was flat; that medieval philosophers debated as to how many angels could dance on the head of a pin; that Galileo revolutionised physics by dropping weights from the Leaning Tower of Pisa. None of these stories is true, and no competent historian has asserte…Read more
-
57International compliance regimes: a public sector without restraintsAustralian Journal of Professional and Applied Ethics 9 (2): 86-95. 2007.Though there is no international government, there are many international regimes that enact binding regulations on particular matters. They include the Basel II regime in banking, IFRS in accountancy, the FIRST computer incident response system, the WHO’s system for containing global epidemics and many others. They form in effect a very powerful international public sector based on technical expertise. Unlike the public services of nation states, they are almost free of accountability to any de…Read more
-
150The Epistemology of Geometry I: the Problem of ExactnessProceedings of the Australasian Society for Cognitive Science 2009. 2010.We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing th…Read more
-
856Discrete and continuous: a fundamental dichotomy in mathematicsJournal of Humanistic Mathematics 7 (2): 355-378. 2017.The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a centr…Read more
-
11Structure and domain-independence in the formal sciencesStudies in History and Philosophy of Science Part A 30 721-723. 1999.Replies to Kevin de Laplante’s ‘Certainty and Domain-Independence in the Sciences of Complexity’ (de Laplante, 1999), defending the thesis of J. Franklin, ‘The formal sciences discover the philosophers’ stone’, Studies in History and Philosophy of Science, 25 (1994), 513-33, that the sciences of complexity can combine certain knowledge with direct applicability to reality.
-
9Homomorphisms between Verma modules in characteristic PJournal of Algebra 112 58-85. 1988.The composition series of Verma modules and homomorphisms between Verma modules in the case of a complex semisimple Lie algebra were studied by Verma and by Bernstein, Gelfand and Gelfand. The author studies homomorphisms between the Verma modules in characteristic p.
-
151Artifice and the natural world: Mathematics, logic, technologyIn Knud Haakonssen (ed.), The Cambridge history of eighteenth-century philosophy, Cambridge University Press. 2006.If Tahiti suggested to theorists comfortably at home in Europe thoughts of noble savages without clothes, those who paid for and went on voyages there were in pursuit of a quite opposite human ideal. Cook's voyage to observe the transit of Venus in 1769 symbolises the eighteenth century's commitment to numbers and accuracy, and its willingness to spend a lot of public money on acquiring them. The state supported the organisation of quantitative researches, employing surveyors and collecting stat…Read more
-
628Global and localMathematical Intelligencer 36 (4). 2014.The global/local contrast is ubiquitous in mathematics. This paper explains it with straightforward examples. It is possible to build a circular staircase that is rising at any point (locally) but impossible to build one that rises at all points and comes back to where it started (a global restriction). Differential equations describe the local structure of a process; their solution describes the global structure that results. The interplay between global and local structure is one of the great …Read more
-
192Non-deductive Logic in Mathematics: The Probability of ConjecturesIn Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics, Springer. pp. 11--29. 2013.Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of l…Read more
-
30Aristotelian realismIn A. Irvine (ed.), The Philosophy of Mathematics (Handbook of the Philosophy of Science series), North-holland Elsevier. 2009.Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity, or numerosity, or symmetry. Let us start with an example, as Aristotelians always prefer, an example that introduces the essential themes of the Aristotelian view of m…Read more
-
133Randomness and the justification of inductionSynthese 138 (1). 2004.In 1947 Donald Cary Williams claimed in The Ground of Induction to have solved the Humean problem of induction, by means of an adaptation of reasoning first advanced by Bernoulli in 1713. Later on David Stove defended and improved upon Williams’ argument in The Rational- ity of Induction (1986). We call this proposed solution of induction the ‘Williams-Stove sampling thesis’. There has been no lack of objections raised to the sampling thesis, and it has not been widely accepted. In our opinion, t…Read more
-
129Arguments Whose Strength Depends on Continuous VariationInformal Logic 33 (1): 33-56. 2013.Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and …Read more
-
478The formal sciences discover the philosophers' stoneStudies in History and Philosophy of Science Part A 25 (4): 513-533. 1994.The formal sciences - mathematical as opposed to natural sciences, such as operations research, statistics, theoretical computer science, systems engineering - appear to have achieved mathematically provable knowledge directly about the real world. It is argued that this appearance is correct.
-
123Mathematical necessity and realityAustralasian Journal of Philosophy 67 (3). 1989.Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics
-
12Accountancy and the quantification of rights: Giving moral values legal teethCentre for an Ethical Society Papers. 2007.If a company’s share price rises when it sacks workers, or when it makes money from polluting the environment, it would seem that the accounting is not being done correctly. Real costs are not being paid. People’s ethical claims, which in a smaller-scale case would be legally enforceable, are not being measured in such circumstances. This results from a mismatch between the applied ethics tradition and the practice of the accounting profession. Applied ethics has mostly avoided quantification of…Read more
-
43Global justice: an anti-collectivist and pro-causal ethicSolidarity 2 (1). 2012.Both philosophical and practical analyses of global justice issues have been vitiated by two errors: a too-high emphasis on the supposed duties of collectives to act, and a too-low emphasis on the analysis of causes and risks. Concentrating instead on the duties of individual actors and analysing what they can really achieve reconfigures the field. It diverts attention from individual problems such as poverty or refugees or questions on what states should do. Instead it shows that there are diff…Read more
-
136Proof in Mathematics: An IntroductionQuakers Hill Press. 1996.A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. The book expounds the traditional view of proof as deduction of theorems from evident premises via obviously valid steps. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
-
327An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structurePalgrave MacMillan. 2014.An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and are…Read more
-
17Accountancy as Computational CasuisticsBusiness and Professional Ethics Journal 17 (4): 21-37. 1998.When a company raises its share price by sacking workers or polluting the environment, it is avoiding paying real costs. Accountancy, which quantifies certain rights, needs to combine with applied ethics to create a "computational casuistics" or "moral accountancy", which quantifies the rights and obligations of individuals and companies. Such quantification has proved successful already in environmental accounting, in health care allocation and in evaluating compensation payments. It is argued …Read more
-
75Caritas in Veritate: Economic activity as personal encounter and the economy of gratuitousnessSolidarity: The Journal of Catholic Social Thought and Secular Ethics 1 (1). 2011.We first survey the Catholic social justice tradition, the foundation on which Caritas in Veritate builds. Then we discuss Benedict’s addition of love to the philosophical virtues (as applied to economics), and how radical a change that makes to an ethical perspective on economics. We emphasise the reality of the interpersonal aspects of present-day economic exchanges, using insights from two disciplines that have recognized that reality, human resources and marketing. Personal encounter really …Read more
-
835Perceiving NecessityPacific Philosophical Quarterly 98 (3). 2017.In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or l…Read more
-
113Mathematics, The Computer Revolution and the Real WorldPhilosophica 42 (n/a): 79-92. 1988.The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.
-
837Aristotelianism in the Philosophy of MathematicsStudia Neoaristotelica 8 (1): 3-15. 2011.Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio …Read more
Sydney, New South Wales, Australia
Areas of Specialization
Applied Ethics |
Science, Logic, and Mathematics |
Philosophy of Mathematics |
Interpretation of Probability |
Areas of Interest
Philosophy of Mathematics |
General Philosophy of Science |
PhilPapers Editorships
Mathematical Aristotelianism |