Over recent decades a diversified and stimulating scholarship on Walras’ contribution to the history of economic thought has been developed. This scholarship is so extensive it would take more than one volume to critically engage it. In this chapter, in line with the overall theme of this work, we have a very limited focus, namely the philosophy of applied mathematics used by Walras in his unique defence of his theoretical economics. To this end we engage both his Elements and his famous, definitive defence of it in his ‘Economique et Mechanique’ through the lens of Poincaré’s philosophy of applied mathematics. This novel approach was inspired by Walras’ correspondence with Poincaré – the leading French and European mathematician of Walras’ era.
When his programme for the mathematisation of economics received a hostile reception, even among some leading French mathematicians, Walras wrote to Poincaré for support and included with his first letter a copy of his Elements. By contextualising Poincaré’s response in Poincaré’s own philosophy of applied mathematics we offer the reader a more detailed understanding of the originality of Walras’ own philosophy of economic theorising. On the level of critical evaluation, as opposed to the exposition of Walras’ philosophy of economic theorising, we show how Poincaré developed a hierarchy of reservations about Walras’ specific programme culminating in Poincaré’s suggestion (politely expressed) that some of Walras’ economic theses in his Elements went beyond the proper limits of a theoretical mathematical science.
Vis-à-vis Walras’ central defence of his philosophy of economic theorising in his ‘Economique et Mechanique’ (translated by Mirowski and Cook and published in 1990), we focus on Walras’ take on Poincaré’s philosophy of mathematical physics. We show how Walras used Poincaré as a stepping-stone in developing his own (Walras’) scientific realist philosophy of mathematical physics. In his scientific realist philosophy, the principles of mechanics, enunciated in differential equations, are the essential causal mechanisms governing motion in the universe. We show how Walras transfers this scientific realist philosophy to his principles of economics also articulated in differential equations: his principles of economics are the fundamental causal mechanisms governing the economic world. We conclude by arguing that, while Walras was within his rights in using Poincaré’s stature as the leading European mathematician to support the general programme of the mathematisation of economics, he fails to draw the reader’s attention to the fact that Poincaré explicitly rejected the same scientific realist reading of the principles of mechanics adumbrated by Walras in his methodological defence of his economic theorising.
This chapter is structured as follows. We open by contextualising Walras’ novel agenda of the mathematisation of economics in the context of Walras’ own time. Then we introduce what we call three phases in the Walras-Poincaré correspondence where Poincaré raises various reservations about the details of Walras’ defence of his programme. We follow with an elaboration of Walras’ scientific realist reading of theoretical economics in light of his scientific realist reading of the principles of mechanics. We conclude by comparing and contrasting Walras’ scientific realist reading of his principles with Poincaré’s alternative explanation of the robustness of such principles.
As Jaffé pointed out, as early as 1873, in his address to the Académie des sciences morales et politiques in Paris, Walras justified his thesis of the measurability of utility by analogy to mechanics, especially the measurement of the mass of a body. Walras was attacked ‘by the bigwigs of the Académie’, on the grounds that ‘it was “false and dangerous” to treat imponderables mathematically’ (Jaffé 1977b: 301). Taken in the intellectual context of its time, this hostility to the mathematisation of theoretical economics, as correctly suggested by Mirowski and Cook for instance, should not be read ‘as a tale of ignorance and backwardness’ (Mirowski and Cook 1990: 194). We note two influential trends which buttress this suggestion.
Firstly, in so far as theoretical economics was seen as being contained within the ambit of the moral sciences, it would be seen by a considerable number of moral scientists as not being comparable to the physical sciences in general and mechanics in particular. In broad philosophical terms the domain of the moral sciences is human action. A human action has two distinct dimensions: an external behavioural dimension and an internal, private, mental dimension. The behavioural, external dimension is intersubjectively observable and thus in principle may be studied by the method of the physical sciences. However, the internal private mental world of any individual is not intersubjectively observable: each individual has privileged access to the contents of his/her own private mind or stream of consciousness. This private internal dimension, studied in philosophy, is not accessible to the methods of the physical sciences. In this context Walras’ analogy to mechanics is not persuasive: the fundamentals of mechanics, such as distance, velocity and acceleration, are intersubjectively measurable but these, unlike the fundamentals of a private mind, have no internal dimension. For moral scientists with this kind of disposition, Walras’ utility belongs to the private mental sphere and not the sphere of what is measurable à la physics/mechanics.
Of course not all French moral scientists were resolutely committed to this philosophical conception of the domain of the moral sciences. Some, under the influence of Auguste Comte, would be sceptical about a separate domain of philosophy, which smacked of metaphysics. However, while sharing Comte’s suspicion of metaphysics, they retained a Comteian suspicion vis-á-vis the application of mathematics to the moral sciences. Comte viewed the sciences in a hierarchical order of abstractness and complexity, ranging from mathematics and astronomy to sociology. As Porter, for instance, points out, Comte ‘firmly refused to privilege mathematics or measurement as motors of the progress of science’ (Porter 2001: 5). This hostility was buttressed by various scandalous attempts at the application of probability theory – then in its infancy – to specific, concrete issues in the moral sciences.
One notorious scandal was the use of probability theory in the infamous Dreyfus affair which polarised French citizens and rocked the Third Republic. Captain Dreyfus was a Jewish army officer, convicted in the 1890s of spying for Germany and imprisoned in Devil’s Island. The prosecution used probability theory to buttress its contention that the handwriting on the central piece of evidence was that of Dreyfus. Eventually a group of experts, led by Poincaré, who was correctly perceived as the leading French probability theorist of the time, evaluated the evidence.1 After exposing various errors in the use of probability theory by the ‘experts’ for the prosecution, Poincaré, who authored the report, sums up as follows.
However, a priori, probability in questions such as the one in which we are engaged is simply made up of moral elements which are totally excluded from calculation … Thus Auguste Comte has quite rightly stated that the application of probability calculation to the moral sciences was the scandal of mathematics. Wishing to eliminate moral elements by substituting numbers is as dangerous as it is useless.
Simply put, the calculus of probability is not, as one would appear to believe, a marvellous science which exempts the learned (savant) from having common sense.
This is why one should never apply calculations to moral issues; if we do this it is us who are constrained by it.
(Poincaré, Darboux and Appell 1908: 503 as quoted in Rollet 1997: 88, italics added)
While the Dreyfus affair exemplified the specific difficulties with the application of probability theory when one attempts to infer from a given effect to its cause in the domain of the moral sciences, it also clearly indicates the influences exerted by Comteian scepticism towards the mathematisation project for the moral sciences. Needless to say, Walras was not impressed by any such scepticism. He dismisses his hostile reception in the Académie in his preface to the fourth edition of his Elements as follows:
I grieve for this learned body … So far as I am concerned the cold reception I had from the Académie actually brought me good luck, for since that time the doctrine I espoused twenty seven years ago has gained wide acceptance both in form and content.
Everyone competent in the field knows that the theory of exchange … which was evolved almost simultaneously by Jevons, Menger and myself and which constitutes the very foundation of the whole edifice of economics, has become an integral part of the science in England, Austria, the United States, and wherever pure economics is developed and taught.
(Walras 2003: 44)
Vis-á-vis those opposed to his project on philosophical grounds Walras adopted the strategy of ignoring them. Thus in concluding his preface to the fourth edition of his Elements he says:
As for these economists who do not know any mathematics, who do not even know what is meant by mathematics and yet have taken the stand that mathematics cannot possibly serve to elucidate economic principles, let them go their way repeating that ‘human liberty will never allow itself to be cast into equations’ or that ‘mathematics ignores frictions which are everything in social sciences’ and other equally forceful and flowery phrases. They can never prevent the theory of the determination of prices under free competition from becoming a mathematical theory.
(Walras 2003: 47)
Twenty-seven years later, however, all was not well for Walras’ programme of the mathematisation of ‘the very foundation of the whole edifice of economics’ among various French mathematicians, particularly those open to the use of mathematics in economics. The distinguished French mathematician Hermann Laurent had an extensive correspondence with Walras. In 1898 he described himself as one of Walras’ ‘most devout (plus fervent) followers’ (Jaffé 1965, 3: letter 1383). Laurent, to what must have been the utter surprise and disappointment of Walras, at a meeting in June 1900 of the L’Institut des Actuaires Francais – which according to Laurent had extended its interests to the issue of furnishing a real scientific basis for political economy – attacked Walras’ efforts to construct a theory of price determination by making utility measurable. While Laurent praised Walras for ‘opening up new horizons’ and, with Pareto, for ‘supplying a corner stone in the edifice’, he posed the rhetorical question ‘how can one accept that satisfaction be measured? No mathematician will ever consent to that’ (Jaffé 1965, 113, letter 1448, note 2). Not surprisingly, as Jaffé notes, Walras was cruelly shaken (Jaffé 1977b: 301). In September of the following year Walras wrote to Poincaré and included a copy of the fourth edition of his Elements. In this letter he points out that he has devoted thirty years improving it in four successive editions. It is clear from Walras’ letter that he was fully aware of the mathematical genius of Poincaré: he states that his own (Walras’) mathematical ability is modest, at least in comparison with his own economic competence. He identifies for Poincaré the key issue which is ‘crucial for both scientific and social progress’, namely ‘the cause of the application of the mathematical method to political economy in a country as conservative and authoritarian as ours’ (Jaffé 1965, 3: 158, letter 1492).
Poincaré is quick in responding. He informs Walras that he has read his work and immediately adds that he is ‘not, a priori, hostile to the application of mathematics to the science of economics provided one does not go beyond certain limits’. He concludes by saying that Walras’ efforts ‘have led to interesting results’ (Jaffé 1965, 3: 160–161, letter 1494). Certainly Poincaré’s response is brief and to the point, namely a priori he has no objection to using mathematics in economic theorising, on condition that certain limits are not transgressed. This may appear surprising in view of how Poincaré used the Comteian thesis as noted in the previous section.2 Poincaré, however, was there concerned with the difficulties in exploiting probability theory in applied areas of the moral sciences. Walras’ Elements concerns theoretical, as distinct from applied, economics and thus Poincaré would not necessarily read it through Comteian eyes.
Moreover, we know that Poincaré supervised Louis Bachelier’s thesis Théorie de la Spéculation, which exploits probability theory and which today is acknowledged by some as a ‘pioneering analysis of the stock and options market’ and ‘the birthdate of mathematical finance’ (Courtault et al. 2000: 341). As Bru points out Bachelier’s thesis was out of the ordinary for its time: ‘it was a mathematical physics thesis (Poincaré held the chair of mathematical physics and the calculus of probability) but since it was not physics, it was about the Stock Exchange, it was not a recognised subject’ (Taqqu 2001: 6). In his report in March 1900, Poincaré alludes to this and points out that Bachelier’s thesis ‘focuses on the application of Probability Theory to the Stock Exchange’. Echoing Comteian reservations, Poincaré continues ‘one might fear that the author has exaggerated the applicability of Probability Theory as has often been done. Fortunately this is not the case … he strives to set limits within which one can legitimately apply this type of calculation’ (Courtault et al. 2000: 348. The authors have added Poincaré’s report as an appendix to their research, italics added). Once again we see Poincaré’s preoccupation with the issue of limits to the legitimate application of mathematics in the human sciences. This issue is a core theme of subsequent sections of this chapter. For the moment, we see that in writing to Poincaré, Walras was not only addressing an internationally recognised mathematical genius, he was also writing to an open-minded academic who was willing to encourage theoretical innovation and to push out the boundaries of theoretical research. It is the same Poincaré who once remarked that ‘thanks to the education it has received, our imagination, like the eagle’s eye the sun does not dazzle, can look truth in the face’ (Poincaré 1958: 88).
Vis-à-vis Walras’ modest competence at mathematics,3 this was not a problem for Poincaré. According to Poincaré there are ‘two entirely different kinds of minds’ in the mathematical sciences (Poincaré 1958: 15). The first is preoccupied with logic and the quest for the highest standards of rigour possible. The second kind of mind is ‘guided by intuition and at first stroke makes quick but sometimes precarious conquests, like bold cavalrymen of the advance guard’ (Poincaré 1958: 15). Both kinds of minds are equally necessary. ‘Each is indispensable. Logic, which alone can give certainty is the instrument of demonstration, intuition is the instrument of invention’ (Poincaré 1958: 23).
According to Poincaré ‘there are many kinds of intuition’ in the mathematical sciences (Poincaré 1963: 20). In particular intuition is indispensable in applied mathematics: ‘without intuition (young minds) would never become capable of applying mathematics’ (Poincaré 1958: 21). Of course intuition is fallible: expert pure and applied mathematicians make mistakes, they ‘are not infallible’ (Poincaré 1952: 47). In Poincaré’s eyes ‘an imperfect solution may happen to lead us towards a better one’ (Poincaré 1956: 38). Thus one may reasonably speculate that Poincaré viewed Walras’ Elements as a piece of applied mathematics4 in its infancy and which would require much more work before attaining anything comparable to the status of mathematical physics. Indeed Walras was fortunate in appealing to an intuitive mathematical mind like that of a Poincaré, rather than the strictly rigorous mathematical mind of a Hilbert, who would have no tolerance for lack of expertise in mathematics.
Be that as it may, we now turn to Poincaré’s reservation: a priori he has no objection to the use of mathematics in economic theorising, on condition that one remains within certain limits. Unfortunately he gives no indication to Walras of what these limits might be. As we have already noted, we know from his report on Bachelier’s thesis, written eighteen months prior to his response to Walras, he was concerned about ‘exaggerations’ which result from the abuse of probability theory in the moral sciences and he praises Bachelier for acknowledging ‘the limits within which one can legitimately apply this type of calculation’ (Courtault et al. 2000: 348, italics added).
We also know from Poincaré’s reflections on other mathematical disciplines that he was concerned about misconceptions/exaggerations about what is achievable in these disciplines. For instance, while acknowledging the contribution of Russell’s logic,5 Poincaré was concerned about the exaggeration in Russell’s logicist programme which, in Poincaré’s eyes, erroneously attempted to reduce pure mathematics to logic. Any such programme fails to do justice to mathematical practice and thereby ignores the intuitive-creative dimension of pure mathematics. Was Poincaré signalling to Walras that, while he had attained interesting results, his thesis that his Elements ‘constitutes the very foundation of the whole edifice of economics’ (Walras 2003: 44) was an unjustifiable exaggeration? We will return to this issue in a later section. What we do know is that Poincaré’s qualification rang alarm bells for Walras, in particular, it flushed out Walras’ more specific concern, i.e. the cardinal measurability of utility, raised by Laurent.
In his response to Poincaré, Walras gets directly to the point. Basically he tells Poincaré that Laurent – the Vice-President of l’Institut des Actuaires – accused Walras of transgressing proper limits in propounding the measurability of utility.
I have supposed rareté (or intensity of the last want satisfied) to be ‘a decreasing function of the quantity consumed of a commodity’ and I added that while this rareté is not a measurable magnitude, it suffices to think of it as such in order to obtain the principal laws of political economy from the fact that rareté decreases.
(Jaffé 1965, 3: 161–162, letter 1495)
This is what Laurent perceived as going beyond proper limits. Walras justifies his thesis by analogy to celestial mechanics, in particular, the indispensable role which the concept of the mass of a body plays in that mathematical science. He draws Poincaré’s attention to Poinsot’s La Statique, where the mass of a body is defined as ‘the number of molecules of a body’ or ‘the amount of matter it contains’ (Jaffé 1965, 3: 162, letter 1495).6 Clearly the notion of the mass of a body is central to the success story of mathematical physics and the articulation of the laws of mechanics. According to Walras this central notion is treated as ‘an appreciable magnitude’ in mechanics, but it is not so, since ‘no one has ever counted the number of molecules in any body’ (Jaffé 1965, 3: 162, letter 1495).
Unfortunately Walras does not expand on this central point. We take him to mean something along the following lines. The concept of the mass of a body is unquestionably a central concept in mechanics/mathematical physics. However, when scientists come to define this notion they end up with something like Poinsot’s definition: the amount of matter in a body – a rather metaphysical definition based on the metaphysical notion of matter – or the number of molecules in a body, a number which no one has counted! Whichever definition of mass one prefers, being ‘an appreciable magnitude’ is not explicitly contained in its definition. Yet the success of mechanics depends on the assumption that mass is an appreciable magnitude. Similarly with the concept of rareté: in theoretical economics it is taken to be an appreciable magnitude, even though that is not part of its conventional definition. In other words, just as the concept of mass, taken as an appreciable magnitude, is indispensable to the proper articulation of the laws of mechanics, the concept of rareté, as a measurable magnitude, is indispensable to the proper articulation of the fundamental laws of economics.
In his reply, which is much longer than his first letter, Poincaré raises a number of key issues, the first being Walras’ defence of the cardinal measure of utility. Vis-à-vis Laurent’s critique of Walras, Poincaré says ‘You misunderstood my thoughts. I never wished to say that you have gone beyond proper limits’ (Jaffé 1965, 3: 162, letter 1496). Laurent’s accusation, in Poincaré’s eyes, is not justified. Without reading any further, Walras must have been relieved. Poincaré was not siding with Laurent on this crucial issue. Indeed Poincaré immediately adds that ‘your definition of rareté seems to me to be legitimate’ (Jaffé 1965, 3: 162, letter 1496) – again music to Walras’ ears. Poincaré, however, proceeds to legitimate Walras’ definition of rareté as the intensity of the last want satisfied, without any reference to the notion of the mass of a body in mechanics – a cornerstone of Walras’ justification. Whatever difficulties one may encounter vis-à-vis the proper definition of the notion of the mass of a body – the notion of the mass of a body is what is frequently called a cardinal magnitude concept.
For any cardinal magnitude concept, e.g. weight, one has a unit, e.g. kilogram, which is used to determine how much greater/smaller or how many times greater/smaller the weight of one body is relative to another. If body A weighs 88 kilos and body B weighs 22 kilos the following inferences make empirical sense: (a) A is heavier than B, (b) A is four times heavier than B, (c) if any other body C weighs 88 kilos it is equal in weight to A. Moreover, if we change the unit of measurement to, say, a pound weight, the above relationships (a), (b) and (c) still hold. According to Poincaré, Walras’ notion of utility/rareté is not a cardinal magnitude concept. If cardinality, thus understood, is taken as a defining characteristic of a measurable magnitude, then satisfaction, in Poincaré’s terminology, is a ‘non-measurable magnitude’ (Jaffé 1965, 3: 162, letter 1496).
Poincaré, however, points out that there is another kind of magnitude in the physical sciences to which differential calculus (a distinguishing characteristic of Walras’ approach) is also, in principle, applicable. We will call this non-cardinal, or ordinal magnitude. In non-cardinal or ordinal measurement questions about how much or how many times greater are meaningless. Poincaré uses temperature as a clear example of an ordinal magnitude. Here the scales used are limited to determining the order in a continuous increasing/decreasing way, but they do not enable one to say how much or how many times greater. There is nothing in the process of constructing a temperature scale which justifies making such a claim. For instance if the temperature of liquid A measures 122˚F and the temperature of liquid B measures 61˚F these transfer into 50˚C and 16.1˚C in the Celsius Scale. Whereas 122 is twice 61, clearly 50 is greater than twice times 16.1. In this case both scales are ordinal telling us that liquid A is hotter than liquid B but not that it is so many times hotter.7
Clearly, Poincaré is suggesting that Walras’ analogy to the measurement of the mass of body is misleading: the measurement of mass is cardinal, whereas the measurement of satisfaction – where satisfaction is linked by Poincaré to an individual’s preferences – is ordinal. Satisfaction, thus understood in terms of preferences, can be defined by any monotonically increasing function, which in principle opens the door to Walras’ exploitation of differential calculus. As Jaffé perceptively notes, with Poincaré’s very brief analysis ‘we seem to be brought to the threshold of the modern ordered preference analysis of consumers’ behaviour’ (Jaffé 1977b: 304).
As we will see later, despite Poincaré’s critique, Walras continued to use his analogy to the measurement of the mass of a body to justify his mathematical approach to utility. In his correspondence with other interested parties, Walras frequently quotes Poincaré as a supporter of his thesis of the measurability of satisfaction. In December 1906 he obtained permission from Poincaré to quote whole or part of Poincaré’s correspondence. Walras eventually published the second letter in full in an appendix to his piece ‘Economics and Mechanics’ (1909).
To recap, Walras must have been rather pleased with Poincaré’s response thus far. The leading French mathematician and one of the most prestigious European mathematicians at the time has no objection in principle to the use of mathematics in theoretical economics (subject to certain constraints). Moreover he legitimates the (ordinal) measurement of utility, thereby opening up theoretical economics to the rich repertoire of tools available in differential calculus. Poincaré now begins to change direction: reservations and possible pitfalls are signalled.
The first reservation concerns the use of what Poincaré calls ‘arbitrary functions’ in Walras’ Elements (Jaffé 1965, 3: 163, letter 1496). Poincaré agrees that Walras has every right to draw out the consequences of these by the proper use of mathematics. The pitfall, according to Poincaré, lies in the status of the conclusions thus obtained: ‘if the arbitrary functions still appear in the conclusions, the conclusions are not false, but they are totally without interest’ (Jaffé 1965, 3: 163, letter 1496). Consequently one must do one’s utmost to eliminate these arbitrary functions and ‘that is what you are doing’ (ibid.). In his letter, Poincaré unfortunately does not define what he means by arbitrary functions and neither does he expand on how the conclusions are totally without interest. Vis-à-vis the latter, i.e. totally without interest, conclusions impregnated by Walras’ arbitrary functions are presumably without interest to the theoretical economist: since they are correctly derived they would be of interest to the pure mathematician whose primary concerns are consistency and logical validity. Perhaps Poincaré is here suggesting an important research project for the Walrasian theoretical economist. This project would entail (a) identifying the arbitrary functions in Walras’ theoretical economics, (b) ascertaining whether or not the arbitrary dimension is contained in the conclusion, (c) eliminate, if possible, this arbitrary element. If this programme is not successfully completed, i.e. if the arbitrary is not eliminated, the theory is economically speaking without interest. In this connection Jaffé argues that the cardinal measurability of utility is an arbitrary presence as demonstrated by Poincaré and that it remains present in Walras’ ‘proof of the fundamental proportionality of marginal utilities to parametric prices as a condition of equilibrium’ (Jaffé 1977b: 304). Poincaré gives a different example.
I can tell whether the satisfaction experienced by the same individual is greater under one set of circumstances than under another set of circumstances; but I have no way of comparing the satisfactions experienced by two individuals. This increases the number of arbitrary functions to be eliminated.
(Jaffé 1965, 3: 163, letter 1496)
Thus Poincaré is challenging Walras to systematically and comprehensively engage the issue of the interpersonal comparison of utility. Poincaré is clearly cognisant of the individual and subjective characterisation of Walras’ rareté and a priori he can’t see how the raretés of different individuals can be either compared or aggregated. Until this issue is successfully addressed, Walras’ theory is potentially arbitrary and thus possibly without interest to theoretical economists.
Vis-à-vis the issue of what Poincaré means by the expression ‘arbitrary function/hypothesis’, it is useful to consult his Science and Hypothesis. In that work he repeatedly contrasts the successful or fruitful hypotheses of the mathematical sciences with arbitrary hypotheses. All of the hypotheses of applied mathematics discussed go beyond what is given in experience or experimentally verified. These hypotheses have proved to be very fruitful, whereas an arbitrary hypothesis is ‘not fertile’ (Poincaré 1952: xxiii). Moreover, an arbitrary hypothesis ‘is the child of our caprice’ (Poincaré 1952: 136). Furthermore, hypotheses which go beyond the boundaries of experience would be arbitrary ‘if we lost sight of the experiments which led the founders of the science to adopt them and which, imperfect as they were, were sufficient to justify their adoption’ (Poincaré 1952: 110). In short a hypothesis is arbitrary if it is a capricious construct which lacks roots in experience or which fails to approximate to some observable situation. The pitfall of the failure to eliminate the arbitrary from the conclusions of an applied mathematical science has to be balanced with Poincaré’s appreciation of what he calls ‘daring hypotheses’ (Poincaré 1952: 239). These are hypotheses which initially have no experimental support but after a number of years are experimentally vindicated.8 Our suggestion in the previous section, viz. that Poincaré read Walras’ Elements as a pioneering piece of research in its early phase of development was motivated by Poincaré’s appreciation of the pitfall of arbitrary hypotheses on the one hand and his appreciation of daring hypotheses on the other. For instance his challenge to Walrasian theoreticians to systematically and comprehensively engage the issue of the intersubjective comparison of utility could possibly turn this prima facie arbitrary hypothesis into a daring one. Only future research can decide this issue: one has no crystal ball to foretell how this will pan out. However, if the arbitrary still persists, then Walrasian theory will lack interest to economic theoreticians.
Poincaré opens his final paragraph by pointing out that all of the above has nothing to do with his requirement of staying within proper limits stipulated in his first letter. This would suggest that the reservations outlined above do not imply that Walras’ use of arbitrary hypotheses, like the hypothesis of the interpersonal comparison of utility, puts Walras’ project outside the domain of what is legitimate theoretical research in applied mathematics. Future research will ultimately decide whether this range of hypotheses is devoid of theoretical interest. To explain what Poincaré had in mind by his requirement to stay within proper limits, he starts with the truism that all applied mathematical research at the theoretical level uses hypotheses. He immediately adds that if that theoretical research ‘is to be fruitful, it is necessary (as in applications to physics) that one be aware of these hypotheses. If one forgets this condition one oversteps proper limits’ (Jaffé 1965, 3: 163, letter 1496). Poincaré, rather politely, proceeds to provide clear examples of how Walras forgot this elementary truth and thereby overstepped the proper limits. His examples are Walras’ hypotheses that agents are infinitely self-seeking and infinitely clairvoyant. These arbitrary hypotheses, especially the second one, prima facie at least, place Walras’ theoretical work beyond the pale of applied mathematical science. Such arbitrary hypotheses, unlike the arbitrary hypothesis of the intersubjective comparison of utility, go beyond the acceptable limits for a scientific theory.
Why? The reason is to be found in Poincaré’s thesis that ‘isolated theory is empty and experience is blind; and both are useless and of no interest alone. To neglect one for the other would be folly’ (Poincaré 1952: 275). Walras’ hypothesis of infinite clairvoyance (what economists call perfect foresight) falls foul of Poincare’s basic requirement that theoretical hypotheses cannot be taken in total isolation from experience: Walras’ hypothesis falls foul of the vast repertoire of knowledge about humans and their actions accumulated by historians and other social scientists.9 Thus it exceeds proper limits. It is so arbitrary it falls outside the pale of an empirical mathematical science.
Moreover, Poincaré insists that these Walrasian hypotheses cannot be read as approximations. Again he illustrates his point by means of an example. The abstraction from friction in mathematical physics – the principles of mechanics assume that there is no friction – is a good example of an acceptable approximation. The principles of mechanics hold approximately for various surfaces, even though they are based on the assumption of infinite smoothness. Thus one can use these principles to predict how far a train, with smooth wheels on a smooth track, will travel before it halts when a specific force is applied to the brakes. The prediction will be approximately correct. Poincaré’s point is that the Walrasian hypotheses of infinite self-interest and infinite clairvoyance are not like the hypothesis of infinite smoothness in mechanics. These Walrasian hypotheses do not approximate to human reality the way the mechanical hypothesis approximates to imperfectly smooth surfaces. Here is how Poincaré, concisely and subtly, puts the above point.
For example, in mechanics one often neglects friction and assumes that bodies are infinitely smooth. You on your side regard men as infinitely self-seeking (égoistes) and infinitely clairvoyant. The first hypothesis can be admitted as a first approximation, but the second hypothesis calls, perhaps, for some reservations.
(Jaffé 1965, 3: 163, letter 1496)
In this connection, it is useful to recall that when Poincaré read Walras’ Elements, he read it in a very short period of time. Nonetheless Poincaré was gifted with an exceptionally quick mind, capable of grasping the core of a case. One such core claim is stated in Lesson 3 of Walras’ Elements. There he maintains
Everyone who has studied any geometry at all knows perfectly well that only in an abstract ideal circumference are the radii all equal to each other and that only in an abstract, ideal triangle is the sum of the angles equal to the sum of two right angles. Reality confirms these definitions and demonstrations only approximately and yet reality admits of a very wide and fruitful application of these propositions.
(Walras 2003: 71, italics added)
He continues by maintaining that this is also true of his theoretical economics.10 In his letter Poincaré is signalling that he rejects this basic claim. For instance, Walras’ hypothesis of infinite clairvoyance is not like the hypothesis of friction in mechanics.
To sum up, Poincaré’s response to Walras is very concise, rather sophisticated and polite.11 A priori, he is not opposed to the use of mathematics, including differential calculus, in theoretical economics, provided one remains within proper limits. Vis-à-vis Walras’ own programme of the mathematisation of theoretical economics, Poincaré has a hierarchy of reservations, some concern hypotheses which arise within the sphere of proper limits for theoretical economics and other hypotheses which exceed proper limits to theoretical economics. Clearly the latter must be abandoned. Poincaré’s first reservation concerns Walras’ cardinal measure of utility. Poincaré, by linking Walrasian utility to individual preferences, justifies the measurement of utility on ordinal grounds: any reference to cardinal utility in Walras’ conclusions must be eliminated. This reservation does not mean that Walras has exceeded the proper limits imposed on mathematico-theoretical research in economics. Rather Poincaré is outlining a challenge for future research. For instance one could argue that the subsequent development of ordinal utility goes some way in meeting Poincaré’s challenge.
Poincaré’s next reservation concerns another hypothetical assumption on the part of Walras, namely the interpersonal comparison of utility. Poincaré correctly notes Walras’ emphasis on the subjective and individual nature of utility. In Poincaré’s view human beings, the subject matter of social sciences and history, are shown to be ‘too dissimilar, too variable, too capricious’ in comparison with the entities studied in physics and the biological sciences (Poincaré 1956: 19, italics added). Given this conception of human beings, it is not at all evident how one could compare or aggregate the individual, subjective utilities of millions of economic agents. Once again Poincaré is not dogmatic – future research may succeed. However any satisfactory completion of the Walrasian programme has to engage this challenge. The extent that these issues are not satisfactorily resolved, the theory lacks economic interest. Poincaré’s reservation at the pinnacle of this hierarchy, however, is damning. Poincaré is implying that Walras at this stage has gone beyond acceptable limits in his theorising. This reservation concerns Walras’ hypothesis, that economic agents are infinitely clairvoyant. This hypothesis is so far removed from what is shown to be the case by history and the other social sciences, its inclusion in any piece of mathematico-economic research means that that research does not merit consideration as a science.
Walras published his short piece ‘Economics and Mechanics’ in 1909. This ‘in his opinion provided an outline of the irrefutable basis of his theories’ (Ingrao and Israel 1990: 90). Consistent with his attitude of ignoring non-mathematical economists, Walras’ target audience was mathematicians. We know from his correspondence with Aupetit that his intention was ‘to disseminate’ his method among mathematicians (Jaffé 1965, 3: 339, letter 1666). His core thesis is that ‘his procedure is rigorously identical to that of two of the most advanced and uncontested mathematical sciences, rational mechanics and celestial mechanics’ (Mirowski and Cook, 1990: 208).12 Clearly, this short piece is both rhetorical and methodological. Walras is engaged in Aristotelian rhetoric: he, being an expert on mathematical economics and, after serious reflection on his metier, takes his mathematical audience into account in presenting his core thesis. The piece is also methodological: his mathematico-economic methods and those of both rational and celestial mechanics are identical.13
Hence it is crucial to ascertain how Walras understood rational and celestial mechanics, which for short we call mathematical physics. Fundamentally mathematical physics ‘is the most advanced and most uncontested’ of the sciences (Mirowski and Cook 1990: 208). This view was shared by many of his contemporaries at the close of the nineteenth century, including Poincaré. If the principles of mathematical physics are nothing but contingent experimental laws then they are provisory or, in Walras’ terminology, contestable. These principles, however, have proven to be, in our terminology, much more robust. This robustness is emphasised by those who view mathematical physics as a deductive, a priori science. As we will see Walras is sympathetic to this view and he gives his own unique justification for the robustness of the principles of mathematical physics. In this connection we argue there are four intertwining cords to Walras’ philosophy of mathematical physics which he projects onto mathematical economics. These are (i) Platonic realism, (ii) essentialist scientific realism, (iii) a (Cartesian) deductive method, (iv) causal scientific realism. In short Walras reads mathematical physics, and ipso facto theoretical economics, in a metaphysical way, while Poincaré reads mathematical physics in an empirical-conventionalist way.14 In this section we focus on Walras’ Platonic realism and his essentialist scientific realism. We will discuss his deductivism and causal scientific realism in the next section.
In Lesson 2 of his Elements, Walras introduces his overall conceptions of Science, Arts and Ethics. He starts his brief, skeleton-like, sketch of his general philosophy of a mathematical science with Platonic Philosophy (Walras 2003: 61). In Platonic philosophy ‘science does not study corporeal entities but universals of which these entities are manifestations’ (Walras 2003: 61). This thesis is not a piece of speculative philosophy. On the contrary, according to Walras, this truth was ‘demonstrated’ a long time ago by Platonic philosophy (Walras 2003: 61). In line with this Platonic commitment, the subject matter of all scientific inquiry is ‘universals, their relations and their laws’ (Walras 2003: 61). There is no further elaboration of this very brief sketch. In our opinion it is crucial to appreciate how Walras has synergised Platonic realist metaphysics with modern, i.e. post-seventeenth-century, scientific revolutionary science, especially mathematical physics. More precisely Platonic realism is inextricably conjoined to an essentialist scientific realist reading of mathematical physics, i.e. the realist claim that mathematical physics reveals the essential characteristics realisable in the observable world.
In its conventional presentation in elementary textbooks, Platonic realism does prioritise what Walras calls universals, or what some historians of philosophy call Platonic Forms, over the entities in the observable world. The observable world is, in Plato’s metaphor, but a shadow of the real world of Forms. In Walras’ terminology ‘corporeal entities come and go but universals remain for ever’ (Walras 2003: 61). Moreover, according to Platonic realists, the universals express essential, as distinct from accidental, characteristics. To take Walras’ own example, the universal ‘circle’ specifies the essential characteristics of the Form ‘circle’. A circle drawn on paper is not a true circle. The drawn figure is circular only to the extent that it approximates to, or manifests, or exemplifies the essential characteristics of the Form ‘circle’. Essentialist scientific realism is inextricably intertwined with this Platonic realism by Walras’ additional thesis, namely the uncontested mathematical sciences such as geometry and mathematical physics reveal these essences. Once again Walras starts with geometry – for instance the essence of the universal circle is exposed in the science of geometry – and extends the claim to mathematical physics. In this Walrasian synthesis of Platonic realism and essentialist scientific realism of mathematical physics, scientific explanation is not, as for Popperians and many others, hypothetico-deductive. Explanation for Walras is certainly deductive – there is extensive consensus among scholars on that – but it is not hypothetical. For instance, take a geometrical figure drawn on paper. One asks what is it? One looks and sees it is a circle. The ultimate explanation of this fact is that the relevant empirical characteristics of the observed figure approximate to, or manifest, the essential characteristics of a circle, defined by geometry. Similarly the empirical natures of the entities studied in mathematical physics are ultimately explained by reference to their essential natures revealed by mathematical physics.
Walras puts a little flesh on this skeleton sketch in the concluding section of Lesson 3 ‘Social Wealth and Value and Exchange’. He briefly outlines how mathematical physics arrives at these universals/essences. Firstly, these sciences ‘draw their type concepts from experience’ (Walras 2003: 71). Next, ‘from real-type concepts, these sciences abstract ideal-type concepts which they define and then on the basis of these definitions, they construct a priori the whole framework of their theorems and proofs’ (Walras 2003: 71, italics added for ‘abstract’). In taking these steps mathematical physics goes beyond experience. Poincaré would concur with this but would not accept that the concepts of mathematical physics are derived by abstraction.15 Consider, for instance, the concept of the mathematical continuum which is basic to geometry. Poincaré’s analysis of this mathematical continuum also starts with experience, namely ‘the physical continuum drawn from the rough data of the senses’ (Poincaré 1952: 27). The concept of the mathematical continuum goes beyond this. So far this is in agreement with Walras’ overall position. The concept of the mathematical continuum, however, is not the result of abstracting from accidental characteristics of the physical continuum. Rather it is ‘a creation of the mind’ of the mathematician suggested by his/her experiences (Poincaré 1952: 28). The notion of abstraction fails to do justice to this creative power of the mathematical mind.16 By emphasising mental creativity in the construction of the notion of the mathematical continuum – where, as it were, the physical continuum is simply the catalyst for the creative mathematical mind – Poincaré is signalling that the derivation of various concepts of mathematical physics is much more nuanced and complex than that outlined by Walras.
At the next phase, Walras’ methodology is totally at variance with the methodology of straightforward empirically minded mathematical physicists. After constructing and developing a mathematical model, straightforward empirically minded scientists will test the model by recourse to the bar of experience. If its predictions correspond to the experimental/observable facts then the model is well confirmed. If not, then the model will be challenged. According to Walras this attitude is mistaken vis-à-vis mathematical physics: if this straightforward empirical attitude did apply to the principles of mathematical physics, these principles would be contestable but, as we have already seen, these principles are ‘uncontested’ (Mirowski and Cook 1990: 208). Walras was not unique in holding this position vis-à-vis the principles of mathematical physics. A similar thesis is advocated by Poincaré vis-à-vis the principles of mathematical physics. Indeed at times Poincaré puts the matter more forcefully than Walras.
An experimental law is always subject to revision; we may always expect to see it replaced by some other more exact law. But no one seriously thinks that the law of which we speak (the principles of inertia) will ever be abandoned or amended.
(Poincaré 1952: 95)
This robustness enjoyed by the principles of mathematical physics renders them apart from other experimental laws.
The crucial issue here is raised by Poincaré’s next question, why? Why are the principles of mathematical physics so different from other experimental laws or hypotheses? Poincaré’s explanation of the robustness of the principles of mechanics is very different to Walras’ explanation. According to Walras the principles of mathematical physics are robust precisely because they reveal the Platonic essences. Having grasped the true essences, there is no need for any subsequent empirical testing. Thus for Walras, after the deductive development of mathematical physics, mathematical physicists ‘go back to experience not to confirm but to apply their conclusions’. (Walras 2003: 71, italics added). Having abstracted the real essences, the only avenue open is to see how to apply these to the observable world. The issue of further testing does not arise.
Poincaré refused to read mathematical physics in a metaphysical way. The mathematical physicist starts with ‘a law (which) has received sufficient confirmation from experiment’ (Poincaré 1958: 124). Mathematical physicists now have a choice: they ‘may leave this law in the fray; it will then remain subjected to an incessant revision, which without any doubt will end up demonstrating that it is only approximate’ (Poincaré 1958: 124). This is the normal fate of experimental laws: they remain in the fray of experimental testing. There is, however, an alternative. The mathematical physicist ‘may elevate it (the law) into a principle by adopting conventions …’ (Poincaré 1958: 124).17 By elevating the law into a conventional principle, the mathematical physicist is taking the conventional principle out of the fray of the normal testing of empirical laws. According to Poincaré ‘great advantages have often been found in proceeding in that way’ (Poincaré 1958: 125). Among these advantages are abbreviation and simplification in the enunciation of empirical laws. These conventional principles also supply scientists with additional linguistic resources, used in the articulation of scientific facts (Poincaré 1952: 91–92). For Poincaré this conventionalist approach is subject to two indispensable caveats. First ‘it is clear that if all laws had been transformed into (conventional) principles nothing would be left of science’ (Poincaré 1958: 125). The conventionalist strategy is to be justified by its fruitfulness. Secondly, while the purpose of the conventionalist strategy is to ensure that the principle is elevated above the fray of normal testing, ‘if a principle ceases to be fecund, experiment without contradicting it directly will nevertheless have condemned it’ (Poincaré 1958: 110, italics ours). In other words ‘experiment without directly contradicting a new extension of the principle will nevertheless have condemned it’ (Poincaré 1952: 167). Clearly for Poincaré the principles of mathematical physics are unique. They are much more robust than other experimental laws. Their robustness, however, is conventional. Moreover, this conventionalist strategy of taking its principles out of the fray of normal testing must be fruitful. If the strategy ceases to be fruitful, the mathematical physicists will reject the conventional principles. This possibility is excluded by Walras’ Platonic realist reading of these principles.
According to Walras the methodology of theoretical economics is the same as that of mathematical physics. Theoretical economics obtains its type concepts such as those of exchange, supply, demand, capital and so on from experience. Next the theoretical economists ‘abstracts and defines ideal-type concepts in terms of which it carries out its reasoning’ (Walras 2003: 71). Finally ‘the return to reality should not take place until the science is completed and then only with a view to practical applications’ (Walras 2003: 71).18 As Walras expressed it earlier ‘corporeal entities come and go but universals remain for ever’ (Walras 2003: 61). So too with the universals of theoretical economics. For Poincaré, the universals of mathematical physics, though more robust than empirical universals, do not remain forever. Their life span depends on how fruitful they will prove in the future developments of that science.
In the previous section we identified four themes central to Walras’ understanding of mathematical physics. Thus far we have dwelled on two, viz. his Platonic realism and his essentialist scientific realism. We now turn to the other two, namely the deductive method and causal scientific realism. Jaffé (1977a) and others have extensively discussed Walras’ deductive method and, in particular, the influence of Descartes. It is clear from the previous section that Walras wished to develop economic theory along the Cartesian lines of ‘more geometrico’. Time and again he cites geometry to legitimate his methodology. This deductive method is inextricably linked to his Platonic realist reading of the principles of mathematical physics. In a Cartesian fashion this Platonic realism, combined with essentialist scientific realism, provides Walras with a solid starting point for his deductive inferences, namely Platonic universal truths. Descartes’ principle that clear and distinct ideas are true is replaced by the methodological principle that mathematical physics identifies true essences. Moreover, mathematics furnishes the theoretician with a logically impeccable deductive method. With these pillars, Walras’ methodology of mathematical physics, and ipso facto of mathematical economics, is based on solid foundations. Walras, however, does not stop here. He buttresses this methodological edifice with another corner stone, namely causal scientific realism.
In his ‘Economics and Mechanics’ he used Poincaré to introduce his causal scientific realism. In Science and Hypothesis, Poincaré rejects the thesis that mathematical physics is a deductive, a priori science, with the emphasis on a priori. Walras sums up Poincaré’s position as follows.
After quoting and then criticising the attempts at definition of mass by Newton, Thompson and Tait and of force by Lagrange and Kirkhoff, one of the masters of modern science (Poincaré) concluded that: masses are co-efficients which it is found convenient to introduce into calculations.
(Mirowski and Cook 1990: 213)
We have altered the translation by substituting ‘attempts at’ for ‘essais on’. When stated so concisely Poincaré’s position is, to say the least, enigmatic. Walras is clearly aware of this. After quoting Poincaré’s position in italics (as it was in the original), Walras immediately remarks ‘A la bonne heure!’, which Mirowski and Cook correctly translate as ‘Fine!’ Collins French Dictionary draws our attention to the irony of this phrase in French and translates it as ‘that’s a fine idea!’ (Collins Robert French Dictionary 2006: 479). If Poincaré’s conclusion accurately sums up all that can be legitimately said about the notion of mass in the differential equations enunciating the fundamental principles of mathematical physics, then mathematical physics has nothing of significance to say to anybody interested in the real physical world. Thus, according to Walras, Poincaré’s conclusion is not satisfactory. Indeed according to Poincaré himself it is ‘a confession of failure’ of the a priori deductive interpretation of mathematical physics (Poincaré 1952: 103, italics ours). Thus according to both Walras and Poincaré an alternative explanation to that of the a priori deductive one is required. Unlike Poincaré, the alternative espoused by Walras is to combine the deductive character of mathematical physics with causal scientific realism. For the causal scientific realist ‘forces would be the causes of the traverse of space, masses the causes of the elapsed time’.19 Similarly ‘utilities and raretés would be the causes of supply and demand …’20 In short, according to Walras’ causal realism, the principles of mathematical physics, enunciated as differential equations, are causal, i.e. they correctly identify masses and forces as ontologically fundamental causal factors operating in the real physical world. Similarly the principles of mathematical economics are causal differential equations, which correctly identify utility and rareté as the fundamental causes of supply and demand and thus the fundamental source of (exchange) value.
Walras’ starting point in deriving his causal realism, namely his brief summary of Poincaré’s critique of the a priori deductive approach to mathematical physics, totally ignores the context of Poincaré’s specific analysis. Poincaré begins his discussions of the principles of mathematical physics by noting two dominant views of mathematical physics. According to one view it is an experimental science. According to the other view mathematical physics is an a priori deductive science. In Poincaré’s opinion neither view ‘distinguishes between what is experiment, what is mathematical reasoning, what is convention and what is hypothesis’ (Poincaré 1952: 89). The identification of the diverse contributions of these to mathematical physics is essential to the project of ascertaining the correct understanding of the principles of mathematical physics. Indeed the task of achieving the correct understanding of the fundamental principles of mathematical physics is more complicated than that. The principles of mathematical physics are not presuppositionless: to comprehensively understand these principles, one must unearth their presuppositions. Among these presuppositions Poincaré identifies (1) Newtonian absolute space, which does not exist, (2) absolute time which erroneously assumes that the notion of the simultaneity of two events occurring in two different places to be unproblematical and (3) Euclidean geometry, the choice of which in Newtonian absolute space is conventional.21 Poincaré sums up his earlier chapters which explored in detail these presuppositions as follows. These ‘no more existed before mechanics than the French language can be logically said to have existed before the truths which are expressed in French’ (Poincaré 1952: 90). Poincaré is forewarning the reader that the correct understanding of the principles of mathematical physics is much more complex than that suggested by the direct ontological reading of causal realists. The causal realist ignores (a) the diverse presuppositions of the principles of mathematical physics, and (b) the divergent roles of experiment, mathematical reasoning, convention and hypothesis in that unique, distinctive science. Poincaré’s empirical-conventional account of the principles of mathematical physics, outlined in the previous section, fulfil these conditions.
Clearly we need to examine this empirical-conventionalist view of the principles of mathematical physics to see why, in Poincaré’s eyes, they cannot be read in the causal realist way suggested by Walras. Prima facie, Walras’ causal realist interpretation is plausible. Numerous mathematical physicists, including Poincaré, maintain that the laws of mathematical physics are expressed by differential equations. Hence when these laws are combined with appropriate initial conditions they furnish causal explanations. The principles of mathematical physics are also expressed by differential equations. Hence according to realists these principles are also integral to causal explanation. Poincaré, however, rejects this latter causal realist claim. The fundamental principles of mathematical physics are not ontologically causal: they are not fundamental causal truths. To appreciate Poincaré’s rejection of the causal explanatory role attributed by realists to the principles of mathematical physics we need to reflect on (a) what exactly is a scientific causal explanation: or, if one prefers, the domain of scientific causality; (b) the construction of the linguistic-conceptual scheme of science; (c) the relationship between an experimental law and a fundamental principle; (d) the implications of the history of science for scientific conceptual schemes.
Vis-à-vis the domain of scientific causality, Poincaré is not exceptional in limiting it to the domain of observation/experimentation. In physics ‘experiment is the sole source of truth. It alone can teach us something new’ (Poincaré 1952: 140). If an explanation, be it causal or otherwise, is not open to experimental investigation it is not a scientific explanation. In the physical sciences the only way of discovering and investigating causal relationships is by observation/experimentation. If, for some reason, one identifies something which is not open to empirical investigation as a true cause then one has stepped beyond the bounds of science. Some varieties of metaphysics engage in this non-scientific kind of explanation. For instance, Thomistic metaphysicians identify God as the first cause of the universe. This causal explanation is metaphysical because it is not open to empirical/experimental investigation. For Poincaré the domain of metaphysics and the domain of science do not overlap. Causes in science, unlike metaphysics, must be subject to empirical investigation.
Moreover experimental sciences are not reducible to crude untrained observation. As distinct from crude observation, scientific observation has two indispensable components: the human contribution, namely the language in which scientific facts and laws are articulated, and, secondly, the contribution of the world gained by trained observation and experimentation. Poincaré illustrates the importance of both by means of the following example. An untrained visitor is in the laboratory with Poincaré while he is conducting an experiment in electricity. Both observe the galvanometer and Poincaré asks is the current passing? The untrained visitor ‘looks at the wire to try to see something pass; but if I put the same question to my assistant who understands my language he will know what I mean’ (Poincaré 1958: 119, italics ours). The progress of science occurs on two distinct fronts: the construction of more and more sophisticated experimental equipment to gain further access to the world and the construction of a more sophisticated conceptual-linguistic scheme in which scientific facts and laws can as accurately as possible be articulated.
We now turn to Poincaré’s crucial distinction between a principle of mathematical physics and a law. These principles are not scientific laws. A scientific law is open to experimental scrutiny and thus open to revision. Indeed quite often this empirical scrutiny ends up by demonstrating the law as an approximation. As we saw in the previous section a principle is much more robust or durable. In Poincaré’s terminology they are ‘elevated above the fray of experimental enquiry’ (Poincaré 1958: 124). Crucially for Poincaré the robustness of a principle is not due to anything in the real world which can be discovered by observation or experimentation. Rather the robustness is due to scientists’ conventional decision to place the principle, for a time, beyond the bar of experience. However, by so removing them, principles are denied scientific causality. As we have just seen in our summary of the domain of causality, anything not open to empirical investigation cannot be deemed to be scientific causes, and principles, by virtue of the decision of scientists, are explicitly excluded from empirical investigation.
This becomes clearer when we see how Poincaré analyses the relationship between a principle and a law. This relationship is a genetic one: a principle is not a law, but it originated from a law. When a law has received sufficient confirmation from experiment, theoreticians have a choice: they may leave the law in the fray of experimental enquiry or else they ‘may elevate it into a principle’ (Poincaré 1958: 124). Schematically the procedure of elevating a law into a principle is as follows. The experimental law expressed an approximately true relationship between terms A and B. The mathematical physicist now introduces an intermediary term C and ‘C is by definition that which has with A exactly the relation expressed by the law’ (Poincaré 1952: 139). Thus, the law is transmogrified into a principle, expressing an absolute, rigorous, robust, universal relation between A and C. But this is only one part of what the mathematical physicist is doing. In addition to decomposing the original, contingent law relating A to B into a rigorous robust, universal principle relating A to C, mathematical physicists complete the process by adding to that principle another revisable experimental law expressing the relation of C to B. Thus the original law is ‘broken up’ into a principle and another law, ‘thereby it is very clear however far this partition be pushed, there will always remain laws’ (Poincaré 1958: 125). If the new law expressing a relation between C and B were not added the whole of mathematical physics would become devoid of scientific interest. Poincaré offers us the following example of this genetic process. Take the claim the stars obey Newton’s law. This may be broken up into (a) gravitation obeys Newton’s Law and (b) gravitation is the only force acting on the stars. (a) is a principle which ‘is no longer subject to the test of experiment’ (Poincaré 1958: 124). It is neither true nor false. (b), however, is subject to experimental testing. If (b) were not added, the whole process would be devoid of scientific interest.
In short what is unique or distinctive about mathematical physics in comparison with other sciences is the creative way in which mathematical physicists can transmogrify a well-confirmed experimental law into a rigorous, conventional robust principle and a new experimental law. In the case where the original scientific law was correctly used in a causal explanation, this causal explanatory role is transferred to the new experimental law, but not to the principle. Since the principle is placed beyond the bar of experience it cannot be used referentially in a causal explanation.22 As we have already seen, any explanation, if it is scientific, rather than metaphysical, must be open to experimental investigation.
Does this mean that for Poincaré the principles of mathematical physics are metaphysical truths? Not at all. The domains of science and of metaphysics are distinct and separate. They do not overlap. Principles, by virtue of the manner in which they are elevated above the bar of experience, are not located in metaphysics. Rather they are new additions to the evolving conceptual-linguistic scheme indispensable to scientific progress. In this sense they are fundamental. Contrary to Walras, being fundamental to the current scientific conceptual-linguistic scheme does not imply that the principles are causally fundamental. Because an experimental law and a principle share the mathematical structure of a differential equation, scientific causal realists like Walras incorrectly assume that both are ontologically indispensable to causal explanation in science. Also, because they are robust, realists incorrectly assume principles are causally more fundamental than laws. Science, however, unlike metaphysics, is limited to observation and experiment in exploring the causal structure of the observable world and thus, though principles are fundamental to its conceptual scheme, they are incapable of revealing fundamental causes. Hence after identifying the appropriate principle anyone claiming, as Walras does, that ‘force is the cause of motion is talking metaphysics’ which is ‘absolutely fruitless’ to the scientist (Poincaré 1952: 98).
We now turn to the implications of the history of science for the scientific conceptual-linguistic scheme. By reviewing in broad sweep the history of science, Poincaré sees no reason as to why one should ontologically or causally privilege the principles of mathematical physics fundamental to the highly fruitful conceptual scheme used in the 1890s. Poincaré sums up his position as follows. ‘Descartes used to commiserate with the Ionians. Descartes in his turn makes us smile, and no doubt some day our children will laugh at us’ (Poincaré 1952: 141). The scientific conceptual scheme used by Poincaré and his contemporaries, in which the principles of mathematical physics are indispensable, has served contemporary science very well. According to Poincaré, however, its continued success is not guaranteed. It all depends on its fruitfulness going into the future. Future research may force scientists to abandon it, including the presently robust principles of mathematical physics. We know from the history of science that successful scientific conceptual schemes of past eras have not lasted the test of time. The same fate awaits any dominant scientific conceptual scheme. Though no one can foretell how it will happen, the successful principles of mathematical physics will in the future be replaced when a more fruitful scientific conceptual-linguistic scheme is constructed to accommodate unforeseen empirical discoveries. Thus these principles have no ontological standing: they do not enunciate the fundamental causes operational in the observable world. Their standing is gauged by their pragmatic usefulness or fruitfulness in supplying the linguistic resources required for the articulation of new facts and new laws. Moreover, their future demise will not entail the demise of current empirical causal knowledge. That true empirical knowledge will be re-articulated and indeed added to in a better conceptual scheme. In short Walras’ scientific realist analysis of the principles of mechanics fails to recognise the sophisticated and complex character of the mathematical science of mechanics and its non-causal, robust, conventional principles.
It is evident that Walras read Poincaré’s Science and Hypothesis. Yet in his final defence given in ‘Economics and Mechanics’, where he quotes Poincaré and uses him as a launching pad for his causal scientific realism, Walras makes no reference to Poincaré’s rejection of this causal scientific realism outlined above. In this connection one may be tempted to extend Jaffé’s thesis concerning Walras’ attitude to historical forerunners to his contemporaries. According to Jaffé, Walras’
only interest was either to bolster his own theoretical contributions by invoking the posthumous support of respected forerunners, or else to berate as fatal flaws anything he found in the writings of others that did not accord with his own ideas.
(Jaffé 1977a: 26–27)
We suggest that this explanation of Walras’ failure to even mention Poincairé’s alternative analysis of the principles of mathematical physics may be complemented by the following hypothesis. Walras was so deeply committed to his metaphysical reading of these principles that any reference to Poincaré’s alternative would lead to a pointless polemic, distracting from his central claim that mathematical economics reveals rareté, not labour as claimed by classical economists, as the fundamental source of value. Be that as it may, Walras had a deep-seated commitment to a metaphysical interpretation of the principles of mathematical physics, whereas Poincaré had a different deep-seated commitment which unequivocally excluded any metaphysical essentialist or fundamental causal interpretation. It is this metaphysical interpretation which Walras transfers to his mathematical economics.23 As we have already seen, Walras sees no need to test his principles of mathematical economics by recourse to the bar of experience. His principles are applied to but not tested by the economic world. The basic reason for this is his metaphysical reading of economic principles, by analogy to his metaphysical reading of the principles of mathematical physics. If one does not subscribe to this metaphysical reading, which Poincaré does not, then mathematical economics loses its ontological privilege. In short, even if Walras’ principles of economics were analogous to the principles of mechanics which, as we have seen they are not, for Poincaré Walras’ principles would be incapable of revealing the causal structure of economic systems.
It is evident from Walras’ extensive correspondence that he was a passionate mathematical economist with a clear mission. This mission was to persuade open-minded, reasonable moral scientists and political economists, that, despite appearances to the contrary, theoretical economics is an applied mathematical science which, when pursued along the lines of his Elements, would attain the same success as that achieved in the unique and distinctive science of mathematical physics/rational mechanics. Thus the paradigm for theoretical economics is not experimental physics. Rather it is the distinctive science of mathematical physics. In particular, Walras’ mission includes a number of central theses. Firstly, contrary to the received view, theoretical economics shares a specific deductive method with mathematical physics, namely the mathematical resources of differential calculus. Walras’ mission was to convince open-minded moral scientists and political scientists of this unprecedented methodological discovery. To this end he frequently uses Poincaré’s correspondence to argue from authority for this methodological innovation – the most famous French and perhaps European mathematician and mathematical physicist of that time supports his innovation. Moreover, there is little doubt that with this methodological innovation, theoretical economics would undergo a revolutionary change: by recourse to differential calculus, a novel research programme, undreamt of in classical economics, is opened up to theoretical economists.
Vis-à-vis mathematical physics, a small number of differential equations constitute the fundamental principles of that science. Moreover, as is evident in the writings of Poincaré, these principles were perceived by numerous scientists as privileged, being distinct from the experimental laws of the other sciences. This unique, distinctive feature is evident in the fact, acknowledged by Poincaré and others, that these principles are not tested in the manner in which the laws of experimental physics are tested. Walras’ mission was to convince his readers that mathematical economics had the same kind of core principles. In this connection, Poincaré was not convinced. The principles of mathematical physics are path-dependent: they are the outcome of a long historical process of trial and error, grounded in experimentation. The principles of Walras’ Elements lack this tried and tested historical trajectory. His Elements, at best, mark the birth of an innovative research programme which lacked the validation of extensive empirical research. Hence it is premature to identify its core hypotheses as principles, à la mathematical physics.
Thirdly, like Poincaré, Walras holds that the distinctive robustness of the principles of mathematical physics requires explanation. He proceeds to explain this robustness by recourse to the realist thesis that these principles reveal the essential, as opposed to the accidental, characteristics of their domain of application. Fundamentally, they reveal the basic causes of the motion of bodies in the universe. Similarly, the principles of mathematical economics reveal that rareté is the fundamental source of exchange value. Poincaré, however, offers an alternative, non-realist, explanation of the robustness of the principles of the exceptionally successful discipline of mathematical physics. One might say that he skilfully uses Ockham’s razor to shave off the excessive explanatory beard of scientific realism. Poincaré starts with the methodological truism that in any scientific discipline, causality is explored by recourse to observation and experimentation. If any scientific hypothesis is not open to rigorous experimental investigation, it cannot be used as a causal explanation. Ipso facto this applies to the principles of mathematical physics: these principles cannot function as scientific causal explanations precisely because their privileged status within mathematical physics rules out any immediate effort at probing them by experimentation. Secondly, the privileged status of the principles of mathematical physics can be explained internally, i.e. within the boundaries of science understood historically, without recourse to essentialist scientific realism. According to Poincaré mathematical scientists operate under two indispensable constraints: on the one hand the extension of the boundaries of the observable by the invention of sophisticated experimental equipment, and on the other the extension of their conceptual-linguistic schema necessary for the precise articulation of scientific facts and laws. These two constraints give rise to a choice situation in mathematical physics when, after years of testing, a law-like generalisation continues to be well confirmed. The physicist may leave the law as it is, i.e. subject to more experimental testing, or transmogrify it into a principle, which is now privileged, and another experimental law, which is not privileged. By opting for the second alternative, the principle is integrated into the conceptual linguistic scheme while the new law is experimentally probed. Scientific realists, à la Walras, fail to appreciate (i) the two constraints noted above on mathematical scientists, (ii) the historical trajectory of a principle, (iii) the only route to causality in science is observation/experimentation, (iv) current conceptual schemes, like those of the past, may be shown to be inadequate by future experimental research. Thus Poincaré sums up as follows:
the object of mathematical theories is not to reveal to us the real nature of things; that would be an unreasonable claim. Their only object is to co-ordinate the physical laws with which physical experiment makes us acquainted, the enunciation of which, without the aid of mathematics, we should be unable to effect.
(Poincaré 1952: 211)
Contrary to Poincaré, according to Walras, both mathematical physics and mathematical economics reveal the real natures of things. The equations of mathematical physics reveal the essential, as opposed to the accidental, causes of motion, while the fundamental equations of mathematical economics reveal the essential cause of exchange value. For Poincaré this kind of claim is scientifically ‘unreasonable’ (Poincaré 1952: 211) whereas for Walras it is based on ‘a truth long ago demonstrated by the Platonic philosophy’ (Walras 2003: 61). In short, Poincaré’s deep-seated commitment is to a non-realist reading of the principles of mathematical physics, whereas Walras’ deep-seated commitment is to a realist reading of these principles. Clearly the philosophy of applied mathematics one adopts is crucial to the account one gives of the privileging of the principles of mathematical physics and to the methodological evaluation of Walras’ equation of the principles of mathematical economics with those of mathematical physics.
Moreover, Walras’ synthesis of essentialist realism with his causal realist reading of his principles of mathematical economics necessarily entails that mathematical economists are locked into these principles. This lock-in is not like the historical, path-dependent lock-in, highlighted by Brian Arthur and others in their economic analyses of technological innovation (Arthur 1994: 13–33). Rather it is an ontologically based, not a contingent path-dependent, lock-in. Like mathematical physicists, mathematical economists are blessed with being in possession of essential truths conveyed by their principles. Being essential truths, these principles are invariable. Future empirical research cannot upend them. Since mathematical economics has correctly identified essential causes, aside from discovering other essential causes, the only scope for further empirical research is in the domain of accidental characteristics. Thus there is no inconsistency in Walras’ central thesis that economists do not test their fundamental equations, rather they merely apply them to the actual economic world. This ontologically based lock-in guarantees that future observational research will never succeed in showing these principles to be in need of radical restatement.
As we have already seen, Walras’ case for this ontological privileging of his principles of theoretical economics rests on (a) his equation of these principles with the principles of mathematical physics and on (b) his essentialist causal realist reading of these principles. As we have outlined in the previous sections, Poincaré rejects both of these claims. Walras’ mathematical economics is in its infancy whereas mathematical physics is a mature science. Furthermore some of Walras’ principles, e.g. perfect foresight, are not analogous to the principles of mathematical physics which evolved out of well-confirmed laws. Others, e.g. interpersonal comparison of utility, require further investigation and thus the jury is out on their scientific status. Also Walras’ essentialist causal realist reading of the principles of both sciences is open to question. In short, if one is sympathetic to a Poincaré-type empirical analysis of Walras’ Elements, then at best one can say is that his Elements has achieved interesting results. Ironically, in a Poincaré-type empirical approach, the Achilles’ heel of Walras’ ontological lock-in to the fundamental equations of Walrasian economics lies in the manner in which he equated mathematical economics with mathematical physics. For any empirical minded economist, the ontological lock-in to the principles of neo-classical economics would require an altogether different legitimation to that supplied by Walras.
1 For a more detailed analysis of Poincaré’s role in the Dreyfus affair, see Rollet (1997).
2 As noted by Bru, for instance, Poincaré was sceptical about the application of probability theory to concrete issues in the moral sciences (Taqqu (interview of Bru) 2001). Poincaré’s reasons for this scepticism are not confined to a Comteian influence. Some of these reasons are outlined in his chapter ‘Chance’ in his Science and Method.
3 Ingrao and Israel (1990), among others, tell us that Antoine Paul Piccard, Professor of Mechanics at the Academy of Lausanne from 1869 to 1881, helped Walras with his mathematics.
4 A terminological confusion may arise here. Walras distinguishes theoretical economics from applied economics. But in so far as mathematical physics or Walras’ economics exploit mathematics, though each is theoretical, each is also a branch of applied mathematics.
5 According to some philosophers ‘Poincaré was categorically opposed to the naissant mathematical logic’ of Russell in England and Frege in Germany (Schmid 1978: 9). In our view this utterly misrepresents Poincaré. For instance he concludes his chapter ‘The New Logic’ of Science and Method as follows: ‘To sum up Mr. Russell and Mr. Hilbert have both made a great effort, and have both of them written a book full of views that are original, profound and often true. These two books furnish us with subject for much thought and there is much that we can learn from them. Not a few of their results are substantial and destined to survive’ (Poincaré 1956: 176, italics ours).
6 As van Daal and Jolink point out, the origin of Walras’ general equilibrium theory has often been attributed to Poinsot’s Élements de Statiques. We are not concerned about the extent of this influence – a point disputed by van Daal and Jolink against Jaffé. Clearly it exerted some influence.
7 For a detailed account of the differences between what we called ordinal and cardinal measurement, the reader may usefully consult chapters six and seven of Carnap (1966).
8 Poincaré uses the phrase ‘daring hypothesis’ vis-à-vis Maxwell’s theory where Maxwell’s ‘conception was only a daring hypothesis which could be supported by no experiment; but after twenty years Maxwell’s ideas received the confirmation of experiment’ (Poincaré 1952: 239). We are not suggesting that Maxwell’s conception was, in Poincaré’s eyes, arbitrary. Rather our suggestion is that there is no evidence to suggest that Poincaré would outlaw the use of a daring hypothesis at the conception/birth of a science.
9 Poincaré held history in very high esteem. Indeed theoretical scientists who forget the histories of their disciplines could easily misunderstand their disciplines.
10 In the next section we will examine in detail how Walras draws this analogy.
11 In order to expose its implications, we have taken the politeness out of Poincaré’s response.
12 Unless otherwise stated we are using Mirowski’s and Cook’s translation.
13 In particular, he argues for a perfect similarity between (a) the formula of minimum satisfaction and the formula of the equilibrium of the Roman balance and (b) the equations of general equilibrium and the equations of universal gravity.
14 In this connection the Bridel and Mornati (2009) article, with the perceptive title ‘De L’Équilibre Général comme “Branche de la Métaphysique”’ is very interesting. While they largely focus on Pareto’s critique of Walras, they do not make our four-fold distinction.
15 Walras does not elaborate on his understanding of abstraction. In philosophy the term, at times, connotes abstracting from accidental characteristics and thereby giving one immediate access to the essential characteristics. For instance, a mathematician sees a specific circular figure. Its circumference is coloured and it has a specific thickness. The mathematician, however, abstracts from those accidental characteristics and thus discovers the essential characteristics.
16 For instance, the mathematical continuum presupposes Dedekind cuts and Poincaré is clearly correct in claiming that the notion of a Dedekind cut is not derived from abstracting away the accidental features of the physical continuum.
17 As Carnap points out ‘Poincaré has been accused of conventionalism in this radical sense (which says that all concepts and even the laws of science are a matter of convention)’ (Carnap 1966: 59). Carnap correctly notes that this accusation is based on a misunderstanding of his writings. ‘Poincaré can be called a conventionalist only if all that is meant is that he was a philosopher who emphasised, more than previous philosophers, the great role of convention. He was not a radical conventionalist’ (Carnap 1966: 59).
18 Numerous commentators, e.g. Jaffé (1977a) note the influence of Natural Law on Walras. Perhaps it is worth noting that much the same kind of reasoning informs Natural law morality: the moral codes/rules/institutions found in different societies are truly moral only to the extent that they approximate to the essence of morality revealed by Natural Law. Thus one does not test Natural Law against the observed moral practices. Rather one applies Natural Law to these practices. In particular, in connection with the influence of Natural Law on Walras, Jaffé concludes that Walras’ ‘latent purpose in contriving his general equilibrium model was not to describe or analyse the working of the economic system as it existed, nor was it primarily to portray the purely economic relationships within a network of markets under the assumption of a theoretically perfect regime of free competition. It was rather to demonstrate the possibility of formulating axiomatically a rationally consistent economic system that would satisfy the demands of social justice without overstepping the bounds imposed by the natural exigencies of the real world’ (Jaffé 1977a: 31). Walras’ Platonist reading of the principles of mathematical economics may help us in appreciating how Walras reconciled his mathematical economics and Natural Law. Both are to be applied to their respective domains because both unearth what is essential in each. And, for a Platonist, the essence of goodness/justice cannot be incompatible with the essence of truth, discovered by mathematical economics.
19 Les forces seraient ainsi des causes d’espace parcouru, les masses des causes de temps employé au parcours desquelles résulterait la vitesse dans le movement, des causes physiques plus constants mais plus caches … (Walras 1909: 325).
20 Les utilitès et les raretés seriaent des causes de demande et d’offre, desquelles resulterait la valeur dans l’eschange, des causes psychiques plus sensibles mais plus variables (Walras 1909: 325).
21 Poincaré is drawing our attention to the manner in which Newtonian mechanics and physics conceptualises space and time. Poincaré, like Lorentz, had done much work akin to Einstein’s special theory of relativity. This is extensively discussed in Greffe, Heinzmann and Lorentz (1996).
22 Like facts, scientific laws have two dimensions: (1) a linguistic component and (2) an empirical component. The contribution of a principle to the enunciation of a fact or a law is limited to (1).
23 According to Mirowski, it is the energy metaphor which is transferred from mathematical physics to mathematical economics (Mirowski 1989). For us it is the Platonist-realist methodology which is transferred by Walras. It should be noted that we do not agree with Mirowski, or indeed anyone else’s reading of scientific models as metaphors. For our position on this general methodological theme see Boylan and O’Gorman (1995). In this work, however, we have chosen to avoid this general methodological issue.