•  1368
    • It would be a moral disgrace for God (if he existed) to allow the many evils in the world, in the same way it would be for a parent to allow a nursery to be infested with criminals who abused the children. • There is a contradiction in asserting all three of the propositions: God is perfectly good; God is perfectly powerful; evil exists (since if God wanted to remove the evils and could, he would). • The religious believer has no hope of getting away with excuses that evil is not as bad as it …Read more
  •  419
    The Epistemology of Geometry I: the Problem of Exactness
    with Anne Newstead
    Proceedings 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
  •  287
    Global and local
    Mathematical 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
  •  271
    Perceiving Necessity
    with Catherine Legg
    Pacific 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
  •  223
    Uninstantiated Properties and Semi-Platonist Aristotelianism
    Review of Metaphysics 69 (1): 25-45. 2015.
    Once the reality of properties is admitted, there are two fundamentally different realist theories of properties. Platonist or transcendent realism holds that properties are abstract objects in the classical sense, of being nonmental, nonspatial, and causally inefficacious. By contrast, Aristotelian or moderate realism takes properties to be literally instantiated in things. An apple’s color and shape are as real and physical as the apple itself. The most direct reason for taking an Aristotelian…Read more
  •  215
    Proof in Mathematics
    Quakers Hill Press. 1996.
    A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
  •  195
    Randomness and the justification of induction
    with Scott Campbell
    Synthese 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
  •  192
    Non-deductive logic in mathematics
    British Journal for the Philosophy of Science 38 (1): 1-18. 1987.
    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 Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' …Read more
  •  181
    Thomas Kuhn's irrationalism
    New Criterion 18 (10): 29-34. 2000.
    Criticizes the irrationalist and social constructionist tendencies in Kuhn's Structure of Scientific Revolutions.
  •  156
    Evidence gained from torture: Wishful thinking, checkability, and extreme circumstances
    Cardozo Journal of International and Comparative Law 17 281-290. 2009.
    "Does torture work?" is a factual rather than ethical or legal question. But legal and ethical discussions of torture should be informed by knowledge of the answer to the factual question of the reliability of torture as an interrogation technique. The question as to whether torture works should be asked before that of its legal admissibility—if it is not useful to interrogators, there is no point considering its legality in court.
  •  148
    Are dispositions reducible to categorical properties?
    Philosophical Quarterly 36 (142): 62-64. 1986.
    Dispostions, such as solubility, cannot be reduced to categorical properties, such as molecular structure, without some element of dipositionaity remaining. Democritus did not reduce all properties to the geometry of atoms - he had to retain the rigidity of the atoms, that is, their disposition not to change shape when a force is applied. So dispositions-not-to, like rigidity, cannot be eliminated. Neither can dispositions-to, like solubility.
  •  145
    Discrete and continuous: a fundamental dichotomy in mathematics
    Journal 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
  •  141
    Indispensability Without Platonism
    In Alexander Bird, Brian Ellis & Howard Sankey (eds.), Properties, Powers, and Structures: Issues in the Metaphysics of Realism, Routledge. pp. 81-97. 2012.
    According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this …Read more
  •  130
    The late twentieth century saw two long-term trends in popular thinking about ethics. One was an increase in relativist opinions, with the “generation of the Sixties” spearheading a general libertarianism, an insistence on toleration of diverse moral views (for “Who is to say what is right? – it’s only your opinion.”) The other trend was an increasing insistence on rights – the gross violations of rights in the killing fields of the mid-century prompted immense efforts in defence of the “inalien…Read more
  •  126
    Aristotelianism in the philosophy of mathematics
    Studia 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
  •  119
    Resurrecting logical probability
    Erkenntnis 55 (2): 277-305. 2001.
    The logical interpretation of probability, or ``objective Bayesianism''''– the theory that (some) probabilitiesare strictly logical degrees of partial implication – is defended.The main argument against it is that it requires the assignment ofprior probabilities, and that any attempt to determine them by symmetryvia a ``principle of insufficient reason'''' inevitably leads to paradox.Three replies are advanced: that priors are imprecise or of little weight, sothat disagreement about them does no…Read more
  •  113
    International compliance regimes: a public sector without restraints
    Australian 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
  •  111
    Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few who have deni…Read more
  •  106
    Mathematical necessity and reality
    Australasian 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
  •  105
    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
  •  105
    Two caricatures, I: Pascal's Wager (review)
    International Journal for Philosophy of Religion 44 (2). 1998.
    Pascal’s wager and Leibniz’s theory that this is the best of all possible worlds are latecomers in the Faith-and-Reason tradition. They have remained interlopers; they have never been taken as seriously as the older arguments for the existence of God and other themes related to faith and reason.
  •  99
    The formal sciences discover the philosophers' stone
    Studies 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.
  •  93
    Caritas in Veritate: Economic activity as personal encounter and the economy of gratuitousness
    Solidarity: 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. Finally, we examine the pr…Read more
  •  83
    Review of N. Wildberger, Divine Proportions: Rational Trigonometry to Universal (review)
    Mathematical Intelligencer 28 (3): 73-74. 2006.
    Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.
  •  79
    Science by Conceptual Analysis: The Genius of the Late Scholastics
    Studia 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
  •  78
    Accountancy as Computational Casuistics
    Business 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
  •  75
    Symbolic connectionism in natural language disambiguation
    with James Franklin and S. W. K. Chan
    IEEE Transactions on Neural Networks 9 739-755. 1998.
    Uses connectionism (neural networks) to extract the "gist" of a story in order to represent a context going forward for the disambiguation of incoming words as a text is processed.
  •  73
    Immigration vs democracy
    IPA Review 54 (2): 29. 2002.
    Democracy has difficulties with the rights on non-voters (children, the mentally ill, foreigners etc). Democratic leaders have sometimes acted ethically, contrary to the wishes of voters, e.g. in accepting refugees as immigrants
  •  71
    Probability Theory: The Logic of Science (review)
    Mathematical Intelligencer 27 (2): 83-85. 2005.
    A standard view of probability and statistics centers on distributions and hypothesis testing. To solve a real problem, say in the spread of disease, one chooses a “model”, a distribution or process that is believed from tradition or intuition to be appropriate to the class of problems in question. One uses data to estimate the parameters of the model, and then delivers the resulting exactly specified model to the customer for use in prediction and classification. As a gateway to these mysteries…Read more