As we have seen in the previous chapter the commitments to rationality and to equilibrium are fundamental to the neoclassical formalisation of economics. In this chapter our aim is to explore how a specific development in the philosophy of pure mathematics at the turn of the twentieth century impinges on various analyses of economic rationality.1 In Chapter Two we interrogated Walras’ exploitation of calculus in his analysis and methodological defence of economic rationality. Without getting involved in the intriguing details of well-developed economic theories of the rational agent including their respective intriguing historical origins, for the purposes of this chapter we sum up as follows: the original Walrasian analysis of the rational agent evolved into that of the self-interested agent who makes his/her choices on the grounds of maximum expected utility.2 The two core mathematical pillars here are the deductive system of pure mathematics and probability theory. Deductive mathematics, especially calculus, is used in the modelling of maximisation, whereas probability theory is exploited in the analysis of expectations and risk.3
The central theme of this chapter is set by Keynes’ General Theory, notably Chapter 12, where he has recourse to convention in his analysis of investment. Keynes maintains that investors’ expectations about the future cannot be adequately understood in terms of the resources of probability theory. The probability calculus enables economists to investigate situations of risk, but investment concerns the radical uncertainty of the economic future. Faced with radical uncertainty the investor has recourse to convention. Since a radical uncertain future cannot be adequately analysed in terms of probability theory, a Keynesian recourse to convention prima facie places it beyond the pale of rationality as represented in neoclassical economics. Thus, as Lawson correctly points out, ‘the question of whether, and if so how, rational conduct is possible in situations characterized by significant uncertainty is central to economic analysis’ (Lawson 1993: 174). This chapter proposes to respond to Lawson’s question by recourse to the philosophy of mathematics.
In this connection we distinguish between the general issue of the rationality of recourse to convention per se in economic decision making from the specific issue of the specific rationale offered for recourse to a specific convention in a specific economic context.4 Vis-à-vis the issue of the rationality of recourse to convention per se, we follow Keynes’ suggestion that economists should look to other domains where conventions are used.5 The two principal philosophical figures who addressed conventions in the social domain are David Hume and, two centuries later, David Lewis. Both have cast their respective shadows over the issue of the rationality of recourse to convention. For those influenced by Hume’s sceptical philosophy, the issue of the rationality of conventions falls into much the same considerations as recourse to induction in the sciences: it is ultimately down to habit. In our view Lewis, by recourse to game theory, which of course was not available to Hume, gives an ingenious solution to the issue of the rationality of recourse to convention by showing how one can reconcile recourse to convention with the neoclassical concept of rationality. In this chapter we introduce the reader to the divergent philosophical contexts of Humean and Lewisian analyses of convention and argue that these analyses do not fully illuminate Keynesian and post-Keynesian recourse to convention. Keynes, like Hume, in recourse to convention is concerned with the spectre of social instability. However, we argue that the Keynesian and post-Keynesian recourse to convention is not only concerned with the establishment and maintenance of social stability; it is also concerned with the proper way to respond to what we call ontological-epistemic indeterminacy. Ontologically the future economic order is open-ended in that some parts of it are not determined by the existing economic order. Epistemologically, our present knowledge does not determine some future rational decisions and probability theory is also ineffective. Rather the rational response is recourse to convention. Given this ontological-epistemic indeterminacy we contend that Lewis’ reconciliation of convention with the neoclassical conception of rationality is of no help to post-Keynesians: it fails to address this basic ontological-epistemic indeterminacy.
In order to address Lawson’s fundamental challenge, viz. the question of whether or not rational conduct is possible in the face of this radical uncertainty now understood as ontological-epistemic indeterminacy, we follow Keynes’ suggestion by looking for some other domain where ontological-epistemic indeterminacy prevails and where recourse to convention is proposed as the rational solution. Our contention is that recourse to convention as the rational response to ontological-epistemic indeterminacy arises in the philosophy of pure mathematics where clearly the issue of social stability does not arise. To anyone not familiar with the history of the philosophy of pure mathematics this claim may seem surprising. What has convention to do with pure mathematics? The answer would appear to be nothing: pure mathematics is a deductive, axiomatic system in which convention plays no substantive role.6 At the turn of the twentieth century this not uncommon philosophical portrait of pure mathematics was challenged by Poincaré.7 Poincaré is famous for his thesis of geometrical conventionalism in the philosophy of pure mathematics.8 As the very name of his thesis suggests, convention is indispensable to pure geometry: mathematicians cannot furnish a complete account of pure geometry without recourse to convention. Moreover the grounds for the recourse to convention is ontological-epistemic indeterminacy. Mathematical rationality, the fine details of which are evident in the various practices of its different research domains, is not adequately conveyed by the simplified image of pure mathematics as an axiomatised deductive system. Recourse to convention in the face of ontological-epistemic indeterminacy is another indispensable dimension of mathematical rationality. In short, just as recourse to convention in the face of ontological-epistemic indeterminacy extends the received notion of mathematical rationality, recourse to convention in post-Keynesian economics, also in the face of ontological-epistemic indeterminacy, similarly extends the received notion of economic rationality.
Under the influence of recent research by economists such as Robert Sugden and H. Peyton Young, there has been a growing interest among economic methodologists in the role of conventions in economic life. This interest has also been added to by the contributions of the ‘école des conventions’ in France, which Latsis interprets as a new heterodox movement within economics which aims ‘to bring economics closer to other social science disciplines such as law and sociology solving a range of problems in neoclassical economics by dropping implicit assumptions and introducing a different style of analysis’ (Latsis 2006: 255). Totally independent of the ‘école des conventions’, Young maintains that ‘conventions regulate much of economic and social life … Conventions with direct economic implications include species of money and credit, industrial and technological standards, accounting rules and forms of economic contract’ (Young 1996: 105). For Sugden the market ‘may ultimately be a form of spontaneous order’, the analysis of which requires recourse to conventions (Sugden 1989: 86).
In much of the methodological literature on conventions, Hume’s A Treatise of Human Nature, first published in 1739, and Lewis’ Conventions, first published in 1969, are frequently cited as key contributions to the philosophical analysis of convention.9 Young is surely correct in maintaining that ‘the key role of convention in defining and maintaining property rights was first articulated by Hume’ (Young 1998: 822). According to Vanderschraaf ‘Hume is rightly credited with giving a brilliant, and perhaps the best, account of justice as convention’ (Vanderschraaf 1998: 215).10 On its publication, however, Hume’s Treatise was not well received. In Hume’s view its hostile reception was largely due to a misunderstanding of his views. Thus in 1740 he published an anonymous pamphlet An Abstract of a Book Lately Published. Entitled A Treatise of Human Nature, &c Wherein The Chief Argument of that Book is further Illustrated and Explained. That pamphlet fell into oblivion until a copy of it was discovered by Keynes in the 1930s. Keynes and Sraffa published it under the title An Abstract of a Treatise of Human Nature, 1740: A Pamphlet hitherto unknown by David Hume. This pamphlet focused on Hume’s theory of knowledge which divides the domain of objective knowledge into a priori truths of reason which convey no information about the world and matters of fact which, grounded in experience, are the sole source of objective knowledge about the world. In light of this theory of knowledge Hume analyses the received notion of causality. Hume identifies three elements: (1) the cause is temporally prior to the effect; (2) cause and effect are spatially contiguous; (3) the effect is said to be necessarily related to the cause. Hume correctly argues that (3) is incompatible with his theory of knowledge: the notion of a necessary connection between events in nature does not stand up to a Humean epistemological scrutiny. What is epistemological justifiable is a constant observed regularity between the events which are illegitimately claimed to be necessarily related. Hume sums up his celebrated analysis of causal reasoning in the sciences as follows: ‘all our reasoning concerning causes and effects are derived from nothing but custom …’ (Hume 1985: 234). This is a source of what various commentators call Hume’s scepticism.
In ordinary usage, the terms ‘custom’ and ‘convention’ are closely related: both mean a social regularity in a population. However, in Book I of his Treatise, where Hume develops his theory of knowledge, he never uses the term ‘convention’ as a stylistic alternative to the term ‘custom’. One may speculate that among the reasons for Hume not using the term ‘convention’ in Book I is that, in addition to conveying a social regularity, the term ‘convention’ also connotes the availability of an alternative option. For instance one community may adopt the convention of driving on the left-hand side of the road where another community could adopt the alternative convention of driving on the other side. However, given Hume’s theory of knowledge, in the case of humanity’s attempts to obtain objective information about the world, there is no such conventional choice available: there is no viable alternative to observation and inductive causal reasoning as explicated by Hume.
Book II of A Treatise, entitled ‘Of the Passions’, continues Hume’s ‘attempt to introduce the experimental Method of Reasoning into Moral Subjects’ (Hume 1985: 323). In Book II Hume develops a scientific portrait of a human being which rejects traditional metaphysical pictures such as those of the Socratic-Platonic and Cartesian traditions where a human being is a thinking being, modelled on mathematical reasoning. In opposition to such metaphysical portraits, Hume’s scientific portrait is grounded in observation and inductive reasoning. Hume’s inductive portrait is of a complex being whose actions are dominated by an intricate interplay of appetites, emotions, passions and sentiments. He opens Book II with a survey of the passions which he divides into direct and indirect. The direct passions include desire, aversion, grief, joy, hope, fear, despair and security. These arise immediately from pain or pleasure. The indirect passions include pride, humility, ambition, vanity, love, hatred, envy, pity, malice, generosity. These also arise from pleasure and pain but in conjunction with other qualities. For instance, beautiful clothes may produce the direct passion of desire. This same quality which produces the direct passion of desire is a necessary condition for the indirect passion of being proud of these beautiful clothes, but that is not sufficient. For the indirect passion of pride to arise the beautiful clothes must also be in some way related to the person who is proud of them (e.g. he/she designed them, or own them, or manufactured them, etc.). Additional pleasure is added by this relationship and this pleasure is conjoined to the original desire giving it additional force.
These passions in turn vary along the spectrum from calm to violent. Moreover each individual has a natural instinct of sympathy. Hume, while attempting to be faithful to his theory of knowledge introduced in Book I (entitled ‘Of The Understanding’), gives the reader a sophisticated account of the interplay between these various passions and instinct and how they are related to pleasure and pain. To illustrate this sophistication, a few words on Hume’s analysis of the abstract idea of pleasure may be useful. Again his analysis is governed by his theory of knowledge. Hume subscribes to Berkeley’s epistemological account of abstract or general ideas. In this account ‘all general ideas are nothing but particular ones, annexed to a certain term, which gives them a more extensive signification and makes them recall upon occasion other individuals which are similar to them’ (Hume 1985: 64). In particular this thesis holds for the ideas of grief, hope, etc. under the rubric of the direct passions and for the ideas of pride, humility, etc. summed up under the rubric of indirect passions and for the idea of pleasure to which all of these are inextricably linked. Vis-à-vis the idea of pleasure ‘
’ts evident under the term pleasure, we comprehend sensations, which are very different from each other, and which have only such a distant resemblance, as is requisite to make them be express’d by the same abstract term. A good composition of music and a bottle of good wine equally produce pleasure; and what is more, their goodness is determin’d merely by the pleasure. But shall we say on that account, the wine is harmonious, or the music is of a good flavour?
(Hume 1985: 523)
Individual pleasures from this infinite array of distinct pleasures motivate individual actions. This scientific portrait of a human being, combined with his theory of knowledge, constitutes the philosophical framework for Hume’s reflections in Book III of his Treatise, entitled ‘Of Morals’. This is where Hume addresses what he calls ‘the foundation of justice’ where he has recourse to convention (Hume 1985: 534).
By the foundations of morality Hume means philosophical foundations, i.e. foundations compatible with the conclusions of Books I and II.11 Hume’s analysis begins with what he calls ‘common experience’ (Hume 1985: 507). This common experience shows that morals influence human actions and affections. Hume’s foundational question is: on what grounds do humans distinguish between vice and virtue, i.e. ‘pronounce an action blameworthy or praiseworthy?’ (Hume 1985: 508). In this connection he opposes various traditions in Western philosophy which locate the foundations of morality in reason. ‘The rules of morality, (therefore) are not conclusions of our reason’ (Hume 1985: 509).12 As established in Book I, reason is confined to the domain of relationships between ideas and thus cannot explain how morality influences actual actions. On the other hand morality does not consist in ‘any matter of fact’ (Hume 1985: 520): moral actions have not distinct empirical moral qualities inherent in them which make them moral. According to Hume an action has both an external and an internal dimension. Moreover the external performance of an action has no moral merit. The moral merit resides in the internal dimension.
If any action be either virtuous or vicious ‘tis only as a sign of some quality of character … We are never to consider any single action in our enquiries concerning the origins of morals; but only the quality or character from which the action proceeded. These alone are durable enough to affect our sentiments concerning the person. Actions are indeed better indications of character than words …
(Hume 1985: 626)
In particular any quality of character which causes love or pride is virtuous and any quality of character which causes hatred or humility is vicious. In short, ‘to have the sense of virtue is nothing but to feel a satisfaction of a particular kind from the contemplation of a character. That very feeling constitutes our praise or admiration’ (Hume 1985: 523). Moreover ‘tis only when a character is considered in general, without reference to our particular interest, that it causes such a feeling or sentiment as denominates it morally good or evil’ (Hume 1985: 524, italics ours).13 In short, ‘the approbation of moral qualities … proceeds entirely from a moral taste, and from certain sentiments of pleasure or disgust, which arise upon the contemplation and view of particular qualities or characters’ (Hume 1985: 632).
This intricate constellation of considerations ranging from an empiricist theory of knowledge, through the differentiation between direct and indirect passions to the analysis of morality in terms of moral taste and the pleasures of the indirect passions constitutes the context for Hume’s analysis of justice in terms of convention. Hume’s foundations of justice hinges on the case he makes for the thesis that some principles of morality are not natural. Fundamentally morality is grounded in a moral sense inextricably linked to feelings and sentiment. However, according to Hume, we cannot remain at this fundamental level. If we did, our morality would consist of an astronomical number of specific precepts for an astronomical number of specific individual actions and it would be impossible to teach such a morality to our children. Hume maintains that
such a method of proceeding is not conformable to the usual maxims, by which nature is conducted, where a few principles produce all the variety we observe in the universe, and everything is carry’d on in the easiest and most simple manner. ‘Tis necessary, therefore to abridge these primary impulses, and find some more general principles, upon which all our notions of morals are founded.
(Hume 1985: 525)
Thus our limited number of moral principles are human-made abridgements of the vast number of more fundamental moral sentiments governing particular actions.14 Hume follows on by immediately asking whether or not these moral principles are natural. In this connection he makes three points. Firstly, if natural is contrasted with the miraculous, then clearly moral principles are natural. Secondly, if natural is contrasted with the unusual and rare, then moral principles are also natural. Thirdly, however, if natural is opposed to artificial then the principles of morality are artificial. After this very long and complex journey Hume finally arrives at his foundations of justice.
Justice, for Hume, operates in a unique moral space. He brings this out by contrasting the good that results from some other moral virtues to the good that derives from justice. For instance, a particular act of generosity is beneficial to the person who is deserving of the generosity shown. This, however, is not always the case with a particular act of justice: a judge may take from a poor man to give to a rich individual or may order the return of a weapon to a vicious person. Such judgements are delivered because justice is not arbitrary, i.e. is not capricious. Thus while such an individual judgement is not beneficial to the poor man in question nor to the potential victim of the vicious person, the system of justice as a whole is advantageous to society. This system is not natural.15 Rather it is a human artifice, which when established ‘is naturally attended with a strong sentiment of morals …’ (Hume 1985: 630).
To identify this specific artifice Hume commences with the thesis that, among all the animals, nature has been most cruel to humans. On the one hand nature has left us with numberless wants and necessities and on the other was miserly in furnishing us with the means of relieving these necessities. The artifice of society enabled man to overcome this cruel situation. In particular in society one can augment one’s very limited physical powers and also, by the division of labour, we increase our abilities. Moreover, in society, by mutual help, we are less exposed to accidents or torture, etc. Thus society becomes advantageous by ‘this additional force, ability and security’ (Hume 1985: 537). However, according to Hume, men ‘in their wild uncultivated state’ could not possibly know or be aware of these advantages (Hume 1985: 537). So how did society originate? Hume’s answer is that the natural appetite between the sexes and their concern for their offspring is ‘the first and original principle of human society’ (Hume 1985: 538). In this vein Hume is opposed to the view of man as a self-interested agent: the emphasis on our selfishness has, in his view ‘been carried too far’ (Hume 1985: 538). Rather for Hume the source of the problem is precisely the generosity and affections among extended families. These, given the scarce resources of nature, give rise to conflict between these groupings.
There are three ‘different species of goods, which we are possess’d of; the internal satisfaction of our minds, the external advantages of our body, and the enjoyment of such possessions as we have acquir’d by our industry and good fortune’ (Hume 1985: 539). Given their scarcity and instability of their possession, the third category, i.e. external goods, are most at risk: they are most exposed to the violence of others and may be easily transferred. ‘In uncultivated nature’ no remedy for this is available (Hume 1985: 539). On the other hand in ‘the golden age’ of the poets where avarice, selfishness, cruelty, etc. does not exist and where nature is lavish with its gifts, the institution of justice would not be required (Hume 1985: 545). Rather as humans ‘acquire a new affection to company and conversation’; and with ‘their early education in society’, they observe
that the principal disturbances in society arises from these goods … (T)hey must seek for a remedy by putting these goods, as far as possible, on the same footing with the fix’d and constant advantages of mind and body. This can be done after no manner, than by a convention entered into by all the members of the society to bestow stability on the possession of those external goods, and leave everyone in the peaceable enjoyment of what he may acquire by his fortune and industry.
(Hume 1985: 541)
This remedy is not derived from nature; rather it is an artifice. Without this convention, society would be inherently unstable.
Hume emphasises the thesis that this convention cannot be analysed in terms of promises. In general promising presupposes conventions. To illustrate how conventions are not promises Hume gives a number of examples of conventions which in his view are in no way related to promises. ‘Two men who pull the oars of a boat do it by agreement or convention, tho’ they have never given promises to each other’ (Hume 1985: 542). His second example is language.16 He points out that both in the case of justice and of language, the convention ‘arises gradually and acquires force by a slow progression and by our repeated experience of the inconveniences of transgressing it’ (Hume 1985: 542). Moreover in the cases of both justice and language ‘the sense of interest has become common to all our fellows and gives us a confidence of the future regularity of their conduct: And ‘tis only on the expectation of this, that our moderation and abstinence are founded’ (Hume 1985: 542). His third example is the conventional use of gold and silver as the common measure of exchange.17
Hume sums up as follows:
And thus justice establishes itself by a kind of convention or agreement; that is by a sense of interest, suppos’d to be common to all, and where every single act is perform’d in expectation that others are to perform the like. Without such a convention, no one wou’d ever have dream’d, that there was such a virtue as justice or have been induc’d to conform his actions to it. Taking any single act, my justice may be pernicious in every respect, and ‘tis only upon the supposition, that others are to imitate my example, that I can be induc’d to embrace that virtue, since nothing but this combination can render justice advantageous, or afford me any motives to confirm myself to its rules.
(Hume 1985: 549)
In this fashion Hume contextualises his conventional foundations of justice in the details of his epistemology and of his scientific picture of humans as complex passionate and moral animals. This of course does not preclude either economic methodologists or philosophers from having recourse to Hume as a precursor in their game theoretic analysis of convention. Our point is that Hume’s own complex philosophy is not that of a strict game theorist.
The philosophical climate of Lewis’ Convention, first published in 1969, is very different to the philosophical climate of Hume’s Treatise, first published in 1739. An integral dimension of the philosophical climate prevailing in the twentieth century is the extensive expansion of the influence of logic into philosophy: an influential trend in American-British twentieth-century philosophy proposes the adoption of a logical, rather than a psychological, approach to philosophical problems. As we have already seen, at the turn of the twentieth century, at the hands of Frege in Germany and Russell in Britain, traditional logic was utterly transformed. Logical tools, undreamt of at the time of Hume, were for the first time available to philosophers. This approach gave rise to an abstract logico-philosophical view of language. However, under the influence of the later Wittgenstein and numerous others an alternative approach to language emerged which put the focus on natural languages. These two trends in the philosophy of language constitutes the philosophical context of Lewis’ Convention.
In his ‘Languages and Language’, Lewis clearly articulates his central problematic. On the one hand we have the contemporary logico-philosophical view of language where a language
is a function, set of ordered pair of strings and meanings. The entities in the domain of the function are certain finite sequences of types of vocal sounds or types of inscribable marks … The entities in the range of the function are meanings.
(Lewis 1983: 163)18
On the other hand, there is the socio-historical view of languages. In this view a language is
a social phenomenon which is part of the history of human beings; a sphere of human action, wherein people utter strings of vocal sounds or inscribe strings of marks, and wherein people respond by thought or action to the sounds or marks which they observe has been produced.
(Lewis 1983: 164)
Lewis’ central problematic is the connection between these two views.
We know what to call this connection we are after: we can say that a given language £ is used by, or is a (or the) language of, a given population P. We know also that this connection holds by virtue of the conventions of language prevailing in P. Under suitably different conventions, a different language would be used by P. There is some sort of convention whereby P used £- but what is it? … My proposal is that the convention whereby a population P uses a language £ is a convention of truthfulness and trust in £. To be truthful in £ is to act in a certain way: to try never to utter any sentence of £ that are not true in £ … To be trusting in £ is to form beliefs in a certain way: to impute truthfulness in £ to others and thus to tend to respond to another’s utterance of any sentence of £ by coming to believe that the uttered sentence is true in £.
(Lewis 1983: 166–167)
Clearly Lewis is not concerned with issues in economic methodology. His central problematic is in the philosophy of language. To substantiate his ingenious thesis, Lewis needs a clear, precise definition of convention. This definition was worked out in his Convention and modified in his ‘Languages and Language’. In his Convention Lewis has recourse to game theory, especially games of pure co-ordination, to arrive at his definition of a convention. One keynote of conventionality is a certain indifference, where the appropriate sense of indifference is explicated via the existence of multiple-equilibria solutions in games of pure co-ordination. At an equilibrium it is possible that some or all of the agents would have been better off if some or all had acted differently. ‘What is not possible is that anyone of the agents would have been better off if he alone acted differently and all the rest acted as they did’ (Lewis 2002: 8).19 The fact that in general there is not a unique equilibrium solution means that rationally more than one solution is available and thus in principle a genuine choice is available. The emergence of the predominance of one choice over others culminates in conventional behaviour.
However there is much more to Lewis’s definition of convention than the above sense of indifference. Lewis specifies six conditions which are both necessary and sufficient conditions of a convention. A regularity R in action or in action and belief in a population P is a convention if and only if
1 Everyone conforms to R.
2 Everyone believes that others conform to R.
3 The belief that others conform to R gives everyone a good and decisive reason to conform to R himself …
4 There is a general preference for general conformity to R rather than slightly-less-than-general conformity … This condition serves to distinguish cases of convention, in which there is a predominant coincidence of interest, from cases of deadlocked conflict …
5 R is not the only possible regularity meeting the last two conditions. There is at least one alternative R’ such that the belief that the others conformed to R’ would give everyone a good and decisive practical or epistemic reason to R’ likewise … Thus the alternative R’ could have perpetuated itself as a convention instead of R; this condition provides for the characteristic arbitrariness of conventions.
6 Finally the various factors listed in conditions (1) to (5) are matters of common (or mutual) knowledge: they are known to everyone, it is known to everyone that they are known to everyone, and so on …
(Lewis 1983: 165)
Much has been written on the extent to which this rigorous definition of a convention by Lewis is or is not independent of co-ordination game theory. Lewis himself remarks that game theory is merely dispensable scaffolding.20 Without getting involved in this intriguing controversy, we concur with Pettit that ‘Lewis’s (1969) work on convention is often taken as a first rate example of how economic explanation can do well in making sense of a phenomenon outside the traditional economic domain of the market’ (Pettit 2002: 239). According to Pettit’s ‘gloss’ (Pettit 2002: 240) Lewis is neither explaining the emergence nor the continuance of conventions. Rather he is explaining their resilience ‘under various shocks and disturbances’ (Pettit 2002: 238).21 To legitimatise this gloss Pettit draws our attention to Lewis’s response to the counter claim that we actually produce and respond to utterances by habit and not as a result of any sort of strategic reasoning or deliberation which is presupposed in game theory.
An action may be rational and may be explained by the agent’s beliefs and desires, even though that action was done by habit, and the agents gave no thought to the beliefs and desires which were his reason for acting. A habit may be under the agent’s rational control in this sense: if the habit ever ceased to serve the agent’s desires according to his beliefs, it would at once be overridden and corrected by conscious reasoning. Action done by a habit of this sort is both habitual and rational.
(Lewis 1983: 181)22
Our concern in this chapter is not with the possible extensions of economic explanation into other domains of the social sciences. Our primary concern is with the mathematical theorising of economic rationality. Among our central questions is the issue of the implications of Lewis’s analysis of convention for the rationality of the post-Keynesian account of recourse to conventions in the face of radical uncertainty. Our thesis is that, if Lewis’s analysis/definition of convention is comprehensive and correct, Keynesian radical uncertainty is no threat to orthodox economic rationality. Indeed one can go much further and agree with Pettit that Lewis ‘reconciles’ (Pettit 2002: 241) recourse to conventions in coordination situations with the strategic reasoning of orthodox economics as explicated in game theory.23 In game theory strategic reasoning culminates in multiple-equilibria solutions: no single unique solution is guaranteed. However, by various means, including Schelling’s salience, one of these equilibria solutions gains conventional precedence in a population P. In such a conventional equilibrium situation, there is no gain to be attained by the unilateral deviation on the part of an individual in P. Now suppose this convention comes under severe threat from external shocks, such that the convention is no longer seen to serve the interests of the members of P. In these circumstances the previous established conventional equilibrium will give way and be replaced by another equilibrium solution which in turn becomes a different convention. In this sense, convention, for a time, guarantees the persistence of one particular rational solution and whenever that convention is undermined by external shocks, strategic thinking leads to other multiple-equilibria solutions and again by convention one of these becomes established in that community. In this way convention and orthodox rationality are reconciled.24 Thus Lawson’s problem posed in the introduction of this chapter, viz. whether, and if so how, rational conduct is possible in situations characterised by radical uncertainty (Lawson 1993: 174) is resolved in favour of orthodox economic reasoning. In short if co-ordination in the social domain is the one and only key determining factor in post-Keynesian analysis and if Lewis’s analysis of co-ordination in terms of convention is correct, then post-Keynesian rationality is utterly compatible with the orthodox theory of rationality.
In the following sections we take up the Davidsonian theme of the ontological foundations of post-Keynesian radical uncertainty by showing how this theme is connected to conventions where (a) the concept of convention, unlike Lewis, has no connection whatsoever to game theory and (b) conventional choice is fundamentally rational, but not in the sense of rationality in mathematicised orthodox economics. We contend that a primary source of conventional decision-making in post-Keynesian economics arises from ontological-epistemic indeterminacy. Moreover this ontological-epistemic indeterminacy is not unique to the economic world. This ontological-epistemic indeterminacy also holds in the domain of pure geometry, where considerations of social instability à la Hume and considerations of social co-ordination à la Lewis are ipso facto excluded. In pure geometry ontological-epistemic indeterminacy alone necessitate recourse to convention. Thus contrary to what is tacitly assumed, conventional choice is indispensable to mathematical rationality. In this fashion the rationality of recourse to convention per se in situations of ontological-epistemic indeterminacy is guaranteed without recourse to either Humean inductive considerations on the one hand or to orthodox economic rationality à la Lewis on the other.
As already indicated, conventions and conventional behaviour made their appearance initially in Chapter 5 but more extensively in Chapter 12 of Keynes’s General Theory. Here Keynes analysed investors’ long-term expectations of prospective yields in situations where they are unable to compute definite probabilities of the outcomes of their investment decisions arising from the prevailing uncertainty. Keynes’s considered answer to this situation was that:
In practice we have tacitly agreed, as a rule, to fall back on what is, in truth, a convention. The essence of this convention – though it does not, of course, work out quite so simply – lies in assuming that the existing state of affairs will continue indefinitely, except in so far as we have specific reasons to expect a change. This does not mean that we really believe that the existing state of affairs will continue indefinitely.
(Keynes 1936: 152, emphasis in original)
In The General Theory, uncertainty, or what post-Keynesians would later call radical uncertainty, was the centre of Keynes’s analysis of long-run expectations and the estimates of prospective yields from long-run investments. In a telling passage Keynes wrote:
The outstanding fact is the extreme precariousness of the basis of knowledge on which our estimates of prospective yield have to be made. Our knowledge of the factors which will govern the yield of an investment some years hence is usually slight and often negligible. If we speak frankly, we have to admit that our basis of knowledge for estimating the yield ten years hence of a railway, a copper mine, a textile factory, the goodwill of a patent medicine, an Atlantic liner, a building in the City of London amounts to little or sometimes to nothing; or even five years hence.
(Keynes 1936: 149–150)
In the following year, 1937, Keynes returned to the topic of uncertainty in his Quarterly Journal of Economics (QJE) article. Here Keynes clearly identified the mainstream position with respect to uncertainty with the capacity of mathematically calculable probabilities to describe current and future events within a standard and unified framework, namely, the calculus of probability. As he argued in the 1937 article: the calculus of probability, though mention of it was kept in the background, was supposed to be capable of reducing uncertainty to the same calculable status as that of certainty itself (Keynes 1937: 112–113). However, this was not a position that Keynes could subscribe to, arising from his belief that in many instances our expectations about the future could not be accommodated within the framework of the calculus of probability. Consequently for Keynes, with respect to these matters, ‘there is no scientific basis upon which to form any calculable probability whatever. We simply do not know’ (Keynes 1937: 114). While the relative merits of this position remain contentious for many, there is no ambiguity about the fact that a clear distinction between risk as calculable probability and radical uncertainty was defended by Keynes, as it had been earlier by Frank Knight (1921) in his celebrated Risk, Uncertainty and Profits. This distinction has become central to post-Keynesianism.
In the face of this analysis, Keynes, in the 1937 article, poses the question, ‘How do we manage in such circumstances [of radical uncertainty] to behave in a manner which saves our faces as rational, economic men?’ In response to his own question he provided what he described as ‘a variety of techniques’, comprised of the following three:
1 We assume that the present is a much more serviceable guide to the future than a candid examination of past experience would show it to have been hitherto. In other words we largely ignore the prospect of future changes about the actual character of which we know nothing.
2 We assume that the existing state of opinion as expressed in prices and the character of existing output is based on a correct summing up of future prospects, so that we can accept it as such unless and until something new and relevant comes into the picture.
3 Knowing that our own individual judgment is worthless, we endeavour to fall back on the judgment of the rest of the world which is perhaps better informed. That is, we endeavour to confirm with the behaviour of the majority on average. The psychology of a society of individuals each of whom is endeavouring to copy the others leads to what we may strictly term a conventional judgment.
(Keynes 1937: 214, emphasis in original)
The above quotations from both The General Theory and in particular the 1937 Quarterly Journal of Economics article represent the most extensive account of convention provided by Keynes in what is termed his mature economic writings, with the QJE article addressing the specific circumstances of highly volatile financial markets. What unfolded in the wake of Keynes’s contributions was an increasing focus on the implications of uncertainty for both economic theory and policy issues. Two early contributors who identified uncertainty as fundamental to economics were George Shackle and Paul Davidson.
While Shackle’s unique and highly individualistic contributions never forged any alliance with the emerging post-Keynesian school, the writing of Paul Davidson over the last forty years established him as a most prolific and pivotal figure in the analysis of uncertainty. After Sydney Weintraub’s death, Davidson emerged as the intellectual leader of American post-Keynesianism and has been a major influence on international post-Keynesianism (Davidson 1972, 1982–1983, 1988, 1991, 2003). Central to his analysis of uncertainty is his rejection of the ergodic hypothesis, which for Davidson postulates:
In an ergodic environment, knowledge about the future involves the projection of calculated averages based on the past and/or current cross section and/or time-series data to forthcoming events. The future is merely the statistical reflection of the past. Economic activities are timeless and commutable.
(Davidson 1994: 90, emphasis in original)
If, however, as Davidson contends, some economic process are not ergodic,
then conditional expectations based on past distribution functions can persistently differ from the probabilities that will be generated as the future unfolds and becomes historical fact. If people believe that the economic environment is uncertain (non-ergodic) then it is perfectly sensible for them to disregard past and present market signals. The future is not statistically calculable from past data and therefore is uncertain.
(Davidson 1994: 90)
Davidson is also at pains to establish a methodological basis for post-Keynesianism by grounding his analysis in the reality of historical time, uncertainty, expectations, and political and economic institutions, all of which for Davidson ‘represent fundamental characteristics of the world we inhabit – the real world’ (1981: 171, emphasis in original). Davidson is here establishing an ontological foundation for post-Keynesian radical uncertainty. Economic agents for Davidson are not able to predict the future arising from either Shackle’s lack of imaginative abilities or Herbert Simon’s limited cognitive or information-processing capacities on the part of economic agents, and the ontological indeterminism of the future. Predicting the future is not due merely to epistemic limitations, it also arises from the in-built non-ergodic characteristics of the real economy: ‘For Keynes and the post-Keynesians, long-run uncertainty is associated with a non-ergodic and transmutable reality concept’ (Davidson 1996: 492).
Davidson went on to enshrine non-ergodicity, along with the nonneutrality of money and the absence of gross substitutability between money and other goods, as the three foundational characteristics of post-Keynesian economics that differentiates it from mainstream neoclassical economics.25 In the following section we consider Davidson’s ontological separation of the non-ergodic from the ergodic in the context of the philosophy of mathematics.
In Chapter Two we contrasted Walras’s scientific realist reading of the principles of mechanics (which we saw is a methodological cornerstone of his philosophy of mathematical economics) with Poincaré’s conventionalist reading of these same principles. The grounds for Poincaré’s conventionalist reading of the principles of mechanics lies in their origins in observed regularities which are open to further empirical investigation. The principles of mechanics are creative conventional constructs from these observed regularities which (a) given their small number, serve the relatively stable, systematisation of the ever increasing vast flux of factual knowledge and (b) simultaneously enhance the inherited scientific conceptual-linguistic framework used for the accurate articulation of scientific facts and regularities. If we compare Lewis’s definition of a convention with Poincaré’s conventionalist reading of the principles of mechanics, we are struck by the contrast between Lewis’s focus on behavioural regularities in communities and Poincaré’s focus on the correct epistemic analysis of a small group of scientific sentences which are claimed to be the fundamental principles of the discipline of mechanics. Poincaré is not concerned with regularities in patterns of behaviour of the members of the community of mathematical physicists. Rather he is addressing the epistemic role of the principles of mechanics within the extensive corpus of objective knowledge conveyed in the mathematical physics. For instance, Poincaré is claiming that, contrary to Walras, the principles of mechanics cannot serve in any scientific causal explanation precisely because, as conventions, they do not convey any information about the world. Poincaré’s point is that within mathematical physics, as well as scientific facts, regularities based on observation and experimentation and extensive applied mathematical reasoning, conventions, i.e. sentences which are neither true nor false and result from a genuine choice and are not capricious, i.e. arbitrary, also play an indispensable role. This indispensable role of conventions in mathematical physics was largely ignored by the scientific community and philosophers of science prior to Poincaré. As Carnap, one of the most outstanding philosophers of science of the twentieth century, remarks, ‘Poincaré can be called a conventionalist only if all that is meant is that he was a philosopher who emphasized more than previous philosophers the great role of convention’ (Carnap 1966: 59, italics ours).26 Thus the laurels for highlighting the indispensable role of conventions in our mathematico-scientific knowledge undoubtedly goes to Poincaré: this constitutes a part of his major contribution to philosophy of applied mathematics. By focusing on this role, Poincaré expands our inherited conception of scientific rationality.
In addition to the indispensable role of conventions in mathematical physics, Poincaré also draws our attention to the indispensable role of convention in pure geometry. Here we focus on Poincaré’s case for the existence/indispensable role of conventions in pure geometry because (a) in this instance, unlike mathematical physics, ontological-epistemic indeterminacy is clearly isolated as the sole grounds for the pure geometer’s recourse to convention and (b) we want to argue that the same kind of ontological-epistemic indeterminacy is an integral component of the post-Keynesian non-ergodic world. In other words given Davidson’s division of the post-Keynesian world into the ergodic and non-ergodic, we focus on this non-ergodicity. The non-ergodicity of the future economic world includes ontological-epistemic indeterminacy and that indeterminacy rationally necessitates recourse to convention. However, unlike the case of pure geometry, other constraints also impinge of the specific chosen convention: these constraints include social stability and coordination. Thus while social stability and coordination are more than demanding constraints on any post-Keynesian decision about future investment, a fundamental source of recourse to convention in post-Keynesian rational decision-making also lies in the ontological-epistemic indeterminacy of the long-run future of the economic world.
By the concluding decades of the nineteenth century the subject matter of geometry was dramatically enhanced since its origins in Euclid. For instance, by recourse to Cartesian co-ordinates, geometry was opened up to algebraic representation and analysis. This in turn led to the study of spaces of more than two or three dimensions – n-dimensional space. While it is extremely difficult, if not impossible, to imagine a space of four dimensions, the conceptualisation of such a space is clearly articulated in n-dimensional geometry. Moreover, the challenge to Euclid’s parallel postulate, namely through a point outside a straight line, one and only one line can be drawn parallel to the given line, resulted in the birth and development of non-Euclidean geometries. Also interest in projective geometry grew at a phenomenal pace in the nineteenth-century. This geometry is not primarily concerned with issues of distance. Rather it studies the projections of figures in a plane from a fixed point outside the given plane by projecting rays on to another plane. Finally the geometry of topology emerged: this geometry studied the properties of continuous manifolds, without any reference to straightness, angle or distance. Philosophically many of these developments were seen as mere logical curiosities.27 With the success of Newtonian mechanics, the Newtonian picture of space dominated much of science.28 In Newtonian mechanics physical space is a kind of container (like a bucket) which extends in three dimensions to infinity and physical bodies are placed in this container. Thus the container space is independent of the material bodies contained in it. Moreover, this container space has an intrinsic metric: its geometry is Euclidean. Later Kant and numerous post-Kantians privileged Euclidean geometry by arguing that it is synthetic a priori. As such there is no way of challenging its truth. This is a very brief summary of the mathematico-philosophical context of Poincaré’s reflections on space and geometry.
Poincaré was not a professional philosopher. Rather he was an exceptionally talented mathematician who, as it were, shared his philosophical reflections on his research with the informed public of his time. Included in his various contributions to mathematics, one finds his pioneering research into the relatively new geometrical discipline of topology.29 In topology space is as it were stripped down to its bare bones: fundamentally from the mathematical point of view of topology space is a continuous manifold. Given the post-Kantian climate at the time it was assumed that such a space would have an intrinsic metric which is Euclidean. The creativity of Poincaré was to challenge that assumption. Poincaré showed that this assumption is incorrect: topological space has not an intrinsic metric. On the contrary it is metrically amorphous. Ontologically, topological space is metrically indeterminate.30 There is nothing in the nature of a continuous manifold which privileges say, a Euclidean metric over, say, a Riemannian metric or vice versa. More precisely there is nothing in the nature of topological space which privileges any metrical geometry whatsoever. When it comes to the ascertainment of distance, both the nature of a continuous manifold and mathematicians’ complete knowledge of such a continuous manifold are no guide whatsoever in how to progress. The only rational recourse is convention.31 A choice of metric must be made: this is the only route into considerations of distance. Without a conventional choice, there is no metrical future for mathematicians. In this way Poincaré places convention alongside traditional axiomatisation at the core of geometrical and therefore mathematical rationality.
We now turn to post-Keynesian conventions and, to use Pettit’s term, our ‘gloss’ on post-Keynesian radical uncertainty is as follows. We concur with Davidson that post-Keynesian radical uncertainty has an ontological dimension. The long-term future of an economy is ontologically indeterminate. Economies exist in a socio-historical time framework and history teaches us that the long-term future is ontologically indeterminate.32 This is not just an epistemic indeterminacy. The long-run economic future is also ontologically indeterminate. Some may very well wish that the economic future will be the statistical reflection of the past while others may not. The point is that (a) neither the short-term nor the long-term future exist; (b) the long-term future of an economy is not historically determined; and (c) major, non-routine, economic or economico-political or political decisions or events can have major unintentional and unforeseeable consequences. In the face of this kind of ontological-epistemic indeterminacy, as shown by Poincaré any rational response must necessarily involve recourse to some convention or other.
Vis-à-vis Lawson’s ‘central question of economic analysis’ viz. ‘whether and if so how, rational conduct is possible in situations characterized by radical uncertainty’ (Lawson 1993: 174) our answer is that rational conduct is possible. As to how this is possible is established by recourse to Poincaré: convention in the face of radical uncertainty understood as ontological-epistemic indeterminacy is not unique to economics; it also, as shown by Poincaré, occurs in the domain of pure mathematics. Poincaré has demonstrated that mathematical rationality, which is a paradigm of rationality, includes recourse to convention. Thus we concur with O’Donnell that in post-Keynesian economics conventions ‘are defined as practical measures for resolving undeterminate decision problems’ (O’Donnell 2003: 99). For us the source of these undeterminate decision problems lies in the ontological-epistemic indeterminacy of the long-run future. Economists, like pure geometers, face undeterminate decision problems and, like pure geometers, recourse to convention is an indispensable component of any proposed solution. Moreover O’Donnell is correct in emphasising the practicality of the conventional measures in post-Keynesian economics. In the case of pure geometry these practical concerns do not arise. Thus, for instance, the practical concerns of socio-economic stability and of co-ordination do not arise in the case of pure geometry.33 Clearly these practical concerns also operate as major constraints on any conventional choice made by post-Keynesians in the face of the ontological-epistemic indeterminacy of the long-run future.
In this chapter, by recourse to Hume, Lewis and Poincaré, we identified three distinct and intriguing ways in which recourse to convention emerged in the philosophical literature. Hume in his quest for the foundation of justice highlights the issue of social instability and the recourse to the conventions of justice to gain stability. Lewis is primarily concerned with language and has recourse to co-ordination game theory to elucidate the conventional nature of language. Poincaré is primarily concerned with a totally different domain, namely the relationship between the geometry of topology and metrical geometry and the indispensable role of convention in that relationship. To date the post-Keynesian debate on conventions has focused on Hume and Lewis. With the exception of Boylan and O’Gorman (2013), Poincaré has not been addressed in the post-Keynesian debate on the issue of the rationality of conventions. This is regrettable. Poincaré – recognised as the father of conventionalism – in his reflections on the indispensable role of convention in pure geometry highlighted the ontological-epistemic, metrical indeterminacy of topological space. The rational response in pure mathematics to this ontological-epistemic indeterminacy is recourse to convention. Thus mathematical rationality, contrary to popular opinion, contains an indispensable conventional component. We contend that an identical ontological-epistemic indeterminacy holds in the non-ergodic dimension of the economic world. Given that convention in the face of ontological-epistemic indeterminacy is rational in pure mathematics – the paradigm of a rational discipline – recourse to convention in the face of ontological-epistemic indeterminacy in post-Keynes economics is equally rational. There is more to economic rationality than what falls within orthodox formalisations of rationality.
1 In the following chapter we address the issue of the mathematical modelling of equilibrium in the neo-Walrasian programme.
2 This view, it could be argued, is complemented by or incorporated into game theory. Game theory focuses on interactive situations in which the outcomes are determined by deliberating agents and where the concept of a Nash equilibrium is taken as ‘the embodiment of the idea that economic agents are rational: that they simultaneously act to maximize their utility’ (Aumann 1985: 43). As Skyrms puts it ‘a simultaneous choice of acts by all players is called a Nash equilibrium if no player can improve his or her payoff by a unilateral defection to a different act. In other words, at a Nash equilibrium, each player maximizes his or her utility conditional on the other player’s act’ (Skyrms 1990: 13).
3 More advanced deductive pure mathematics is required to prove the existence of a Nash equilibrium in finite n-person zero sum games with mixed strategies. As Skyrms points out, Nash used Brouwer’s fixed point theorem to prove the existence of equilibrium. We will see in the next chapter why one may have legitimate reservations about recourse to Brouwer’s fixed point theorem in theoretical economics!
4 The specific rationale for a specific convention may prima facie be plausible. However, recourse to the theory of rationality could undermine this prima facie case. Hence our concern with the rationality of recourse to convention per se.
5 Keynes in a letter to Townsend in 1938 draws our attention to the role of conventions in other non-economic domains. The relevant passage from this letter is quoted in Lawson (1993).
6 In an axiomatic system, nominal definitions which are conventions are extensively used. These conventions, however, can in principle be eliminated. See Boylan and O’Gorman (2013) for more details.
7 We will see in the next chapter that Brouwer also challenges this axiomatic portrait. Brouwer’s challenge is more radical than that of Poincaré.
8 Much of the secondary literature focuses on Poincaré’s conventional of applied geometry. See O’Gorman (1977).
9 Lewis’ Convention is frequently presented as a neo-Humean analysis.
10 Vanderschraaf argues that Hume’s account of convention ‘can plausibly be interpreted as an informal game theory which predates the first formal theory of games presented by von Neumann and Morgenstern (1944) by more than two centuries’ (Vanderschraaf 1998: 216).
11 ‘I am not, however, without hope, that the present system of philosophy will acquire new force as it advances; and that our reasonings concerning morals will corroborate whatever has been said concerning the understanding and the passions. Morality is a subject that interests us above all others …’ (Hume 1985: 507).
12 Reason plays a merely instrumental role in morality. It may be used to discover that a passion is based on a false judgement, e.g. when the object of one’s fear is shown by reason to be non-existent. It may also be used to show that a chosen means is not adequate to the task at hand. However, reason is not the philosophical foundation of morality.
13 Hume admires the moral qualities on an enemy, even though the enemy has not Hume’s interests at heart.
14 This thesis does not receive an extended elaboration by Hume.
15 Justice, for Hume is not grounded in a universal sentiment of concern for the whole of humanity. While sympathy is universal, there is no universal sense of public interest.
16 In Hume’s epistemology language is conventional. Knowledge begins with sensory perceptions. From these, images are derived and ideas are but faint images. Then each individual idea is annexed to a linguistic term. Thus a Frenchman uses a French term whereas an Englishman uses an English term. In this way French and English are conventional regularities.
17 Vanderschraaf (1998) gives an excellent game-theoretic analyses of Hume’s examples. However, as is evident in our very brief synopsis of Hume’s reflections, morality is a key interest of Hume. Indeed ‘it is a subject which interests us above all others …’ (Hume 1985: 507). For various reasons we are not convinced that Hume anticipated the strategic notion of a Nash equilibrium in this analysis. If morality were taken out of Hume’s deliberations – which would be like Hamlet without the Prince – then of course one would be free to give his examples a strategic game-theoretic analysis. For Hume, much of practical rationality is morality-laden whereas the strategic rationality of game-theory in orthodox economics is explicitly morality-neutral.
18 The meaning of a sentence is in turn another function from possible worlds to truth-values.
19 Lewis does not use the expression Nash equilibrium. In line with various commentators we are assuming that Lewis’s notion is the same as a Nash equilibrium. However, one should recall that Nash’s focus is on non-cooperative games, while, given his interest in language, Lewis’s focus is primarily on games of coordination.
20 For various views on this see Topoi (2008).
21 Pettit’s gloss is at variance with other interpretations of Lewis. For instance according to Skyrms Lewis is responding to Quine’s challenge, in Quine’s famous 1936 piece ‘Truth by Convention’. In that piece Quine raised the issue of how convention without communication is possible? This question ‘has two parts: (1) How can convention without communication be sustained? And (2) How can convention without communication be generated? Lewis gave the answer to the first question in terms of equilibrium (or stable equilibrium) and common knowledge. His discussion of the second question – following Schelling (1960) – is framed in terms of salience …’ (Skyrms 1990: 54). In our gloss Lewis is responding to all of these issues but that the issues of continuance and of resilience are much more important to him than the issue of origins.
22 For instance one might use Lewis’s thesis to analyse the demise of the Irish language. After the external shock of colonisation the population of Ireland over a period of time, on the grounds of conscious strategic thinking, abandoned its conventional Irish language and embraced the conventional language of its English colonial power. We doubt that historians of the colonisation of Ireland would concur with this analysis. However, Lewisians might reply that this account is the logic of the situation which should inform any historical account.
23 We hope that this is not ‘a gloss’ on Pettit’s own ‘gloss’!, i.e. that it accurately represents a significant aspect of Pettit’s thesis.
24 A number of epistemological issues arise here.
(1) Can or should the concept of convention be defined in terms of necessary and sufficient conditions?
(2) Is Lewis’s definition merely persuasive: is he merely prescribing how we should use the term convention?
(3) If the concept of convention can be defined in terms of necessary and sufficient conditions has Lewis correctly identified these? After all he changed his own mind on these. Thus much debate has occurred around, for instance, his condition of common knowledge.
(4) What presuppositions underlie his recourse to game theory? See Skyrms (1990) for some of these.
25 See Boylan and O’Gorman (2013) for a very brief summary of other developments.
26 Carnap’s intention is to undermine the view that Poincaré was a radical conventionalist who held that even the laws of science are conventions. ‘Poincaré also has been accused of conventionalism in this radical sense but that, I think, is a misunderstanding of his writings. He has indeed often stressed the important role conventions play in science, but he was also well aware of the empirical components that come into play. He knew that … we have to accommodate our system to the facts of nature as we find them …’ (Carnap 1966: 59).
27 A notable exception was the young Bertrand Russell. In his early writings he read projective geometry as synthetic a priori.
28 The famous exception here of course is Leibniz.
29 See Stirwell (1996).
30 Grünbaum’s (1964) was the first to fully appreciate the ontological dimension of Poincaré’s geometrical conventionalism.
31 Because Newtonian space is mathematically a continuous manifold and because of the dualism of space and matter in Newtonian mechanics, Poincaré also argued for the conventionalism of applied geometry (see O’Gorman 1977). In Einstein’s general theory of relativity this dualism no longer holds. Poincaré had died before the publication of Einstein’s general theory. However much has been written on Poincaré’s anticipation, as it were, of Einstein’s special theory (see Miller 1996).
32 Poincaré was among the first to identify a conventional dimension in the mathematisation of time in mechanics (see Poincaré 1958: Chapter 2). The issue of the simultaneity of events separated by astronomical distances requires recourse to convention. The issue of simultaneity is not relevant here.
33 As we pointed out in Boylan and O’Gorman (2013), we disagree with O’Donnell in identifying this rationality as weak. In our analysis recourse to convention is strongly rational. The reason for this is that the same kind of recourse to convention also occurs in pure mathematics which is the par excellence rational discipline.