James Franklin

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 asserted any of them, but none shows any sign of disappearing from the public consciousness.

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 democratically elected body or to any legal system. Although by and large they have acted for good, the dangers of long-term unaccountability are illustrated by the travesties of justice perpetrated by the International Labour Organisation Administrative Tribunal.

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 the ways in which the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept

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 central issue in the applicable mathematics of the last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics.

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 themes of mathematics, but rarely discussed explicitly.

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 logical probability (or objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case.

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.

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, though, none of these objections has the slightest force, and, moreover, the sampling thesis is undoubtedly true. What we will argue in this paper is that one particular objection that has been raised on numerous occasions is misguided. This concerns the randomness of the sample on which the inductive extrapolation is based

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 explain their relation to reasoning fail, so that ignorance of their nature is profound.

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 rights, while accounting practice has embraced quantification, but has been excessively conservative about what may be counted. The two traditions can be combined, by using some of the ideas economists have devised to quantify difficult-to-measure costs and benefits in environmental accounting.

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 different duties for political leaders, intelligence operatives, opinion leaders and citizens in devising, urging and implementing such plans as transfers of aid with accountability, military interventions in rogue states and limited intakes of refugees. With collectivist excuses for inaction such as sovereignty out of the way, it is possible to take a cautiously optimistic view of the possibility of forceful and morally responsible interventions in the range of major global problems.

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.

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 that many rights are measurable with sufficient accuracy to make them credible and legally actionable.

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 is a major factor in economic exchanges in professional services, such as law and medicine. Finally, we examine the prospects for an economics of gratuitousness at a level higher than the individual, that is, for businesses devoted to social ends more than profit.

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.

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 of two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity.

A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated - what is the truthmaker of such truths? It is argued that Aristotelianism can answer the question, but only a semi-Platonist form of it.

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 denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds.

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' or `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic.

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 criterion with our own brand of metaphysics, 'Aristotelian (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage.