The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)

The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

Leibniz's best-of-all-possible worlds solution to the problem of evil is defended. Enlightenment misrepresentations are removed. The apparent obviousness of the possibility of better worlds is undermined by the much better understanding achieved in modern mathematical sciences of how global structure constrains local possibilities. It is argued that alternative views, especially standard materialism, fail to make sense of the problem ofevil, by implying that evil does not matter, absolutely speaking. Finally, itis shown how ordinary religious thinking incorporates the essentials of Leibniz's (...) view. (shrink)

The essential idea of Leibniz’s Theodicy was little understood in his time but has become one of the organizing themes of modern mathematics. There are many phenomena that are possible locally but for purely mathematical reasons impossible globally. For example, it is possible to build a spiral staircase that is rising at any given point, but it is impossible to build one that is rising at all points and comes back to where it started. The necessity is mathematically provable, so (...) not subject to exception by divine power. Leibniz’s Theodicy argues that God could improve the universe locally in many ways, but not globally. This paper defends Leibniz, giving positive reasons for believing that there are so many necessary interconnections between goods and evils that God is faced with a choice like the classic Trolley case, where all of the scenarios that could be chosen upfront contain evils, but some more than others. Local changes for the better seem easy to imagine, but a proper understanding of global constraints undermines the initial impression that they can be done without global cost. The paper concludes by explaining how the context of the Leibnizian argument makes it reasonable to pursue the issue of whether there are no worlds better than this one. (shrink)

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 objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

A polemical account of Australian philosophy up to 2003, emphasising its unique aspects (such as commitment to realism) and the connections between philosophers' views and their lives. Topics include early idealism, the dominance of John Anderson in Sydney, the Orr case, Catholic scholasticism, Melbourne Wittgensteinianism, philosophy of science, the Sydney disturbances of the 1970s, Francofeminism, environmental philosophy, the philosophy of law and Mabo, ethics and Peter Singer. Realist theories especially praised are David Armstrong's on universals, David Stove's on logical probability (...) and the ethical realism of Rai Gaita and Catholic philosophers. In addition to strict philosophy, the book treats non-religious moral traditions to train virtue, such as Freemasonry, civics education and the Greek and Roman classics. (shrink)

Professor Franklin is correct to say that there are significant areas of agreement between his account of formal science (Franklin, 1994) and my critique of his account. We both agree that the domain-independence exhibited by the formal sciences is ontologically and epistemically interesting, and that the concept of ‘structure’ must be central in any analysis of domain-independence. We also agree that knowledge of the structural, relational properties of physical systems should count as empirical knowledge, and that it makes sense to (...) talk about an empirical ‘science’ of structure. Where we disagree is over the frequency of occasions where ‘practical certainty’ is actually attained: Franklin argues that practical certainty is not uncommon in the formal sciences, while I argue that there are barriers to practical certainty that Franklin fails to appreciate. In Franklin’s response, he briefly presents and offers criticisms of my two main arguments against his central thesis. Franklin asserts that the flaw in my first argument is that it does not appreciate that modern mathematical models exhibit structural stability, and thus that many of their predictions are robust under small variations in input or parameter values (and hence, insensitive to the inevitable gaps between estimates and actual values). Nevertheless, what I had in mind in using the term ‘realistic modelling situations’ were models of fairly complex natural systems, such as the dripping faucet, spruce budworm and ecosystem ecology examples. My objection is not that mathematical models of such systems cannot legitimately and accurately describe structural properties that are genuinely predicable of real-world systems, but simply that for.. (shrink)

James Franklin has argued that the formal, mathematical sciences of complexity — network theory, information theory, game theory, control theory, etc. — have a methodology that is different from the methodology of the natural sciences, and which can result in a knowledge of physical systems that has the epistemic character of deductive mathematical knowledge. I evaluate Franklin’s arguments in light of realistic examples of mathematical modelling and conclude that, in general, the formal sciences are no more able to guarantee certainty (...) than the natural sciences. Yet the formal sciences are characterized by a ‘domain-independence’ that is philosophically interesting, and I argue that it is this property that Franklin actually employs to distinguish the formal from the natural sciences. I use Einstein’s ‘principle’/‘constructive’ theory distinction to contrast the domain-independence of physical theories with the domain-independence of formal mathematical theories, and show how both kinds of domain-independence function to generate the domain-independence that is observed in the complex systems sciences. © 1999 Elsevier Science Ltd. All rights reserved. (shrink)

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.