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 mathematics. A typical mathematical truth is (...) that there are six different pairs in four objects: Figure 1. There are 6 different pairs in 4 objects The objects may be of any kind, physical, mental or abstract. The mathematical statement does not refer to any properties of the objects, but only to patterning of the parts in the complex of the four objects. If that seems to us less a solid truth about the real world than the causation of flu by viruses, that may be simply due to our blindness about relations, or tendency to regard them as somehow less real than things and properties. But relations (for example, relations of equality between parts of a structure) are as real as colours or causes. (shrink)

Don Garrett, Cognition and Commitment in Hume's Philosophy, New York and Oxford, Oxford University Press, 1997, pp. xiv + 270, Hb 40.00 ISBN 0-19-509721-1.

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)

In the Treatise of Human Nature, David Hume mounts a spirited assault on the doctrine of the infinite divisibility of extension, and he defends in its place the contrary claim that extension is everywhere only finitely divisible. Despite this major departure from the more conventional conceptions of space embodied in traditional geometry, Hume does not endorse any radical reform of geometry. Instead Hume espouses a more conservative approach, claiming that geometry fails only “in this single point” – in its purported (...) proofs of infinite divisibility – while “all of its other arguments” remain intact. -/- In this paper, after laying out the prima facie case for Hume’s radical challenge to traditional geometry, I consider five strategies for blocking the arguments for infinite divisibility while conserving most of geometry. I show that each of these interpretive strategies suffers from serious substantive problems, and so none of them delivers an interpretation of Hume’s account that provides him with a way of blocking the geometric arguments for infinite divisibility while sustaining his geometric conservatism. (shrink)

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.

The leaders of the Scientific Revolution were not Baconian in temperament, in trying to build up theories from data. Their project was that same as in Aristotle's Posterior Analytics: they hoped to find necessary principles that would show why the observations must be as they are. Their use of mathematics to do so expanded the Aristotelian project beyond the qualitative methods used by Aristotle and the scholastics. In many cases they succeeded.

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. (shrink)

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 (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

This paper considers whether elements of T 1.2 Of the Ideas of Space and Time in Hume’s Treatise is inconsistent with skepticism regarding the external world in T 1.4.2 Of Scepticism with regard to the Senses. This apparent tension vexes commentators, and efforts to resolve it drives the recent scholarship on this section of Hume’s Treatise. To highlight this tension I juxtapose Hume’s “Adequacy Principle” with what I call his “skeptical causal argument” in T 1.4.2. The Adequacy Principle appears to (...) state that we can form conclusions about objects through the comparison of our ideas of them, while Hume’s skeptical causal argument asserts that we cannot form any conclusions about objects using ideas. Is Hume being inconsistent? I argue that he is not. My paper has three parts. First, I give a general reading of the Adequacy Principle in light of Hume’s peculiar claim that it is “the foundation of all human knowledge.” I then explain why the Adequacy Principle itself is not inconsistent with Hume’s skeptical causal argument by explaining how the term “object” should not be strictly read as “external” object but merely the “the object of inquiry” or even “the matter under consideration.” Second, I sketch Hume’s employment of the Adequacy Principle in his Divisibility Argument, explaining how this is also not at odds with his skeptical causal argument. In the third and final part I explore various interpretations in the literature and consider whether “finite extensions” need necessarily be read as external objects. (shrink)

The main problem of this study is David Hume’s (1711-76) view on Metaphysical Realism (there are mind-independent, external, and continuous entities). This specific problem is part of two more general questions in Hume scholarship: his attitude to scepticism and the relation between naturalism and skepticism in his thinking. A novel interpretation of these problems is defended in this work. The chief thesis is that Hume is both a sceptic and a Metaphysical Realist. His philosophical attitude is to suspend his judgment (...) on Metaphysical Realism, whereas as a common man he firmly believes in the existence of mind-independent, external, and continuous entities. Therefore Hume does not have any one position; accordingly, a form of “no one Hume” interpretation (Richard Popkin, Robert J. Fogelin, Donald L.M. Baxter) is argued for in the book. The key point in this distinction is the temporal difference between Hume’s philosophical and everyday views. It is introduced in order to avoid attributing a conscious contradiction to him (a problem which has not attracted enough attention in the literature). The method of the work is modelled on Peter Millican’s work on Hume and induction. The approach to the main problem is to study the two “profound” arguments against the senses that Hume presents in the Section 12 of An Enquiry concerning Human Understanding (1748). These arguments are first reconstructed in detail resulting in Millican-type diagrams of them and then Hume’s endorsement of them is established on the basis of the diagrams. The first profound argument concludes that Metaphysical Realism and thus any Realistic theory of perception is unjustified as well as the existence of God and the soul. The second argument goes further having first conceptual conclusion: the very notions of Real entitity, material substance, and bodies are completely out of the reach of the faculty of understanding. Therefore they ought to be rejected according to Hume. This is a consequence of the consistent use of the Humean faculty of reason: idea-analysis and inductive inference. The second profound argument thus concludes that believing in Metaphysical Realism is inconsistent with the rational attitude that is to refrain from this belief. Hence, if we attributed both of them to Hume, we would end up with a great philosopher who embraces a manifest contradiction. The study is finished by arguing that this sceptical and Metaphysically Realistic interpretation concurs well with (1) Hume’s professed Academical philosophy and (2) project of the science of human nature. (1) According to Hume, Academical philosophy is in the first place diffidence, modesty, and uncertainty including suspension on certain issues. Secondly, it is restriction of the range of topics for which experience can provide a standard of truth. This kind of empiricist epistemological realism is coherent with the sceptical attitude on Metaphysical Realism because the latter does not rule out inter-subjective consensus on what we experience. (2) Suspension of judgment on Metaphysical Realism coheres with the mind-dependency of the objects of Hume’s science of human nature: the understanding, passions, morals, aesthetics, politics, and the human culture in all of its manifestations. Although the study takes the first Enquiry to be Hume’s authorised word on the understanding, his juvenile work A Treatise of Human Nature (1739-40) is argued to support this “no one Hume” interpretation. Hume’s other works are also discussed when needed. (shrink)