James Franklin

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.