Agnès Festré
email: agnes.festre@gredeg.cnrs.fr

Struan Jacobs

Nicolas Brisset

Geoffrey Hodgson

Michael Polanyi's Economics: A Strange Rapprochement *

Keywords: Michael Polanyi, Hayek, Keynes, spontaneous order, State intervention, liberalism, evolutionism JEL codes: B25, B31, B41

In this paper we provide an analysis of Michael Polanyi's contribution to economics. We stress two major influences on Polanyi's economics: first, Hayek's opposition to central planning and his defence of self-organization as a superior mechanism for coordinating individual plans; second, Keynes' macroeconomic disequilibrium approach to the adjustment of savings and investment that called for government intervention. Polanyi blent these two influences and provided an idiosyncratic synthesis, which is an unfortunately neglected contribution in the field of economics. Moreover, we show that this synthesis is consistent with two important features of Polanyi's intellectual background: on the one hand, liberalism, which entailed two overlapping notions of freedom (traditional private freedom together with public liberty); on the other hand, his humanistic end-oriented type of evolutionism tainted with Catholic ideas. The paper highlights that these two features go hand in hand with Polanyi's view of tacit knowledge, which shapes his original and rich reflection about institutions.

Introduction

Michael Polanyi, a very impressive and interesting scientist and philosopher, was born in Budapest in 1891 and died in 1976 in Northampton, UK. He completed a medical degree in 1913 and a PhD in physical chemistry 3 in 1917, both at the University of Budapest. In 1919, he converted from Judaism to Catholicism 4 . He moved to Karlsruhe where he continued to study physical chemistry; then, after a short stay in Budapest, moved to Berlin where he obtained a position at the Kaiser Wilhelm Institute for Fiber Chemistry 5 . He moved to Great Britain in 1933 to take up a position at the University of Manchester after the Nazi regime's imposition of a prohibition on Jews in civil-servant positions. He continued to work in chemistry at the University of Manchester until 1948Manchester until , but in 1937 his interest gradually switched to popular education in economics. As lamented by Melvin Calvin, one of Polanyi's Post-Doc student in 1935 and[START_REF] Keynes | The General Theory of Employment, Interest and Money[END_REF] who later won the Nobel Prize, 'it was hard to interest Polanyi in chemistry subjects anymore' [START_REF] Nye | Michael Polanyi[END_REF], p. 125 referring to Calvin 1991-2). He had many discussions about economics with his brother, Karl Polanyi (1886[START_REF] Polanyi | The Emergence of Man[END_REF], who was living in London. Like Karl, Michael was very concerned, for both professional and family reasons (his mother was from Vilnjus), about the economic and political situation in the Soviet Union, which he visited for the first time in 1928 (Scott and Moleski, 2005, p. 154). In 1936, Michael Polanyi met Bukharin (Polanyi was invited to give a series of lectures to the Ministry of Health Industry in the USSR) who stressed that in socialist societies all scientific research is to be connected to the needs of the ongoing Five Year Plan. In this regard, Michael

Polanyi noted developments in the study of genetics after Lysenko's doctrine received the support of the Soviet State. Similarly, in the same period in Great Britain, the Marxist John Desmond Bernal, was one of the defenders of planned scientific research. Increasingly concerned with the danger of a generalized tendency to planning in science, Polanyi founded in 1940 the Society for Freedom in Science in cooperation with the biologist John R Baker and the ecologist A. G. Tansley. The Society attracted various intellectuals and high profiles scholars working in very different domains. This meeting of minds contributed to making Polanyi acquainted with a variety of disciplines, including economics. Furthermore, it strengthened his inclination towards active engagement in political life. As a scholar untrained 3 Michael Polanyi's son, John Polanyi, won the Nobel Prize for Chemistry in 1986. 4 'Michael Polanyi was baptized into the Catholic Church on October 18, 1919 (well into the era of the White Terror), but it is unclear whether this was an act of faith or a practical step to facilitate his employment in Karlsruhe, Germany, where he was to emigrate shortly' (Frank, 2012, p. 209). 5 For a precise biography of Michael Polanyi see [START_REF] Nye | Michael Polanyi and His Generation: Origins of the Social Construction of Science[END_REF] and [START_REF] Scott | Michael Polanyi, Scientist and Philosopher[END_REF].

in economics he was probably more aware of the layman's exposure to fallacies and misrepresentations and of how they can be manipulated by political leaders. He was particularly concerned about the diffusion of Keynes' ideas in order to cure unemployment and used both moving picture technology and chemical apparatus as pedagogical tools aiming at describing the principles underlying the economic life in a way accessible to the general public 6 .

As well documented by the secondary literature, Polanyi's view of knowledge has some commonalities with Hayek's. Although there is a debate on when and to what extent Hayek incorporated the cognitive psychological content of Sensory Order (Hayek, 1952a) (see Caldwell 2004) and on how Hayek's notion of tacit knowledge differs from Polanyi's one (see [START_REF] Oguz | Hayek on Tacit Knowledge[END_REF], both authors provided some epistemological grounds to the critique of economic planning and promoted self-organization as a more efficient economic coordination mechanism 7 .

At first sight, there are objective reasons why providing a consistent synthesis of Hayek and Keynes' economic ideas can be seen as an unachievable task; it is well known that the two economists disagreed heartily about the role of money in the economy and economic policy.

Although Keynes was not a defender of central planning, he advocated use of government to help regulate the economy in the context of unemployment. Hayek believed that economic freedom was tied to free markets and the less government intervention, the better for the economy. However, they were not completely opposed. Hicks (1967) already highlighted proximities between Hayek and Keynes 8 . More recently, Boehm (1989) and Butos and Koppl (2014) emphasized that, in particular, they shared a subjectivist approach regarding expectations.

Our aim in this paper is to show that there is coherence in Polanyi's economics demonstrated by the rapprochement he proposes between Hayek and Keynes. To explain this coherence the aim is not trying to find proximities between Hayek and Keynes, but to show that from Polanyi's point of view it is possible to defend both conceptions. Our main argument is that 6 We refer here specifically to the two films, the 'economic machines' he designed in his chemical laboratory at the University of Manchester in the 1930s and the visual representations (in particular the 'money belt') he used in Full Employement and Free Trade (1945). See Biró (2017), Beira (2014Beira ( -2015) ) and Moodey (2014Moodey ( -2015)). 7 It is beyond the scope of this paper to analyse the proximity between Hayek and Polanyi visions of tacit knowlege. As well documented by the secondary literature, Gestalt psychology provided a common philosophic background. See [START_REF] Mullins | Michael Polanyi's Use of Gestalt Psychology[END_REF] for Polanyi's use of Gestalt psychology [START_REF] De | The Place of Gestalt Psychology in the Making of Hayek's Thought[END_REF] for the place of Gestalt psychology in the making of Hayek's thought. 8 See also O'Driscoll, 1977), and more recently, Arnon (2014).

Polanyi's and Hayek's conceptions of liberalism and evolution are different, which made it possible for Polanyi to agree that the State should intervene to reduce unemployment while also upholding the concept of a self-organized economy and society.

A secondary objective of the paper is to shed light on Polanyi's approach to institutions.

The paper is organized as follows. In Section 2 we present Polanyi's mixed conception of economics. In Section 3 we present the conceptions of liberalism and evolutionism of Hayek and Polanyi respectively, arguing that these differences help to understand the synthesis provided by Polanyi in the field of economics. Section 4 concludes the paper.

Polanyi's mixed conception of economics

Polanyi's contribution to economics is found in a few articles [START_REF] Polanyi | Economics -Fundamental Data, System, and Spirit[END_REF][START_REF] Polanyi | The "Settling Down" of Capital and the Trade Cycle[END_REF](Polanyi , 1940a) ) and in a book entitled Free Employment and Free Trade (1945). His writings are divided between attacks on the Soviet system and lucid commentaries on the work of Lord Keynes.

Between 1935 and 1950, Polanyi corresponded with a number of important economists including Friedrich A. Hayek and John Maynard Keynes. His position as a scientist and a social thinker led him to become involved with a group of intellectuals in England concerned with social problems. This group, the Moot, was convened by Joseph H. Oldham, editor of the Christian Newsletter (Cash, 1996, p. 7).

In his writings in economics Polanyi is close to Hayek. He was however not convinced by Hayek's conception of economics and especially his defence of laissez faire in the field of economic policy. 9 When he read Keynes's The General Theory of Employment, Interest and Money he was impressed by its economic policy implications 10 .

In fact, Polanyi defends both a Hayekian conception of society and a Keynesian macroeconomic approach. The first is more in line with his conception of knowledge and emergence of novelty, which he describes also as a polycentric process, and the second is more in line with his idea that it is possible to solve economic inefficiencies using what Popper calls 'piecemeal technology' 11 . We examine these two influences, which differ completely in the ways the arguments are organized: In his Keynes-inspired approach, Polanyi does not base his discourse on the idea of emergence but he is deliberately situated at the macroeconomic level. By contrast, in his Hayek-inspired approach, Polanyi's interest is focused on how an order emerges from the individuals' actions and interactions.

Polanyi's Hayek-inspired approach to inter-individual economic coordination 12

What Polanyi takes from Hayek is the idea that freedom is an essential element of coordination 13 . In The Republic of Science (1962), Polanyi compares the market mechanism à la Hayek and the functioning of science 14 . He advocated against the idea that pure science 9 'In the controversy between Laissez Faire and Planning my outlook leans distinctly towards the former… Yet I have proposed some measures, which many people may call "planning". If we understand by "planning" the setting aside -in whatever instance -of an alleged automatism, which economic science used to consider as inescapable, and its replacement by government action, then I have certainly suggested definite measures of "planning'" (Polanyi, 1945, p. 149). 'Besides, if I recalled the memorable campaign through which he [Friedrich Hayek] defended for a time some indispensable truths against the turbulent hopes of contemporary opinion, I should also have to criticize to a certain extent the position he took up in this struggle. He addressed an age obsessed by the fear of mass unemployment while turning an indifferent eye on this problem. This surely was a mistake' (Polanyi, 1949, p. 267). 10 'But since the early 1930s a new and more enlightened opinion has begun to dawn upon economic science, and through the publication in 1936 of The General Theory of Employment, Interest and Money, by J.M. Keynes, the light finally broke through' (Polanyi, 1945, p. 2). 11 By analogy with the central role of piecemeal experiments in the sciences, Popper argues that piecemeal social engineering is the only type of reform (in contrast to economic or social planning) that can be rationally justified, namely, a conception of public intervention that is small-scale, incremental, and continuously amended in the light of experience. Note that Polanyi never refers explicitly to the notion of piecemeal technology. 12 For a comparison between Hayek and Polanyi, see Mirowski (1998Mirowski ( -1999)). 13 'The essay I am offering here for the celebration of my honoured friend Fritz Hayek is on a subject close to his major interest" (Polanyi, 1997, p.183). This essay is part of a book published in 1969: Roads to Freedom: Essays in Honour of Friedrich A. von Hayek. This Festschrift of 'honorary essays' presented to Hayek on his 70th birthday includes essays by Erich Streissler, Jacques Rueff, Peter T. Bauer, James M. Buchanan, Gottfried Haberler, George N. Halm, Ludwig M. Lachmann, Friedrich A. Lutz, Fritz Machlup, Frank W. Paish, Michael Polanyi, Karl R. Popper, Günter Schmölders, and Gordon Tulloch. 14 For the debate over who, Hayek or Polanyi, is the father of the notion of spontaneous order, see [START_REF] Jacobs | Michael Polanyi's Theory of Spontaneous Orders[END_REF] and the pro-Hayek response from [START_REF] Bladel | Against Polanyi-Centrism: Hayek and the Re-Emergence of "Spontaneous Order[END_REF]. Moleski would seem right in writing (personal communication): 'I am not sure it is fair to say that Polanyi "took" this theme from Hayek. I think it developed quite naturally from his instinctive dislike of Soviet economics and a philosophy of science that made it the handmaid of totalitarian ideals'. could be based on social demand 15 . In The Republic of Science he begins by analysing the functioning of a free market à la Hayek, and stresses the idea that the scientific community should be subject to the same rules16 : What I have said here about the highest possible co-ordination of individual scientific efforts by a process of self-co-ordination may recall the self-co-ordination achieved by producers and consumers operating in a market. It was, indeed, with this in mind that I spoke of "the invisible hand" guiding the co-ordination of independent initiatives to a maximum advancement of science, just as Adam Smith invoked "the invisible hand" to describe the achievement of greatest joint material satisfaction when independent producers and consumers are guided by the prices of goods in a market (Polanyi, 1962, p. 56).

He called this process of coordination polycentricity: 'I think I have proved in earlier writings that the production and distribution of modern technological products can be conducted only polycentrically, that is, by essentially independent productive centers distributing their products through a market ' (ibid, 183).

The idea of polycentricity is another avatar of Polanyi's seminal distinction between two kind of orders which one finds in Polanyi's 1941 essay "The Growth of Thought in Society", where he developed his vision of the organization of science17 . The first one 'consists in limiting the freedom of things and men to stay or move about at their pleasure, by assigning to each a specific position in a pre-arranged plan' (Polanyi, 1941, 431). By contrast, in the other order, which characterizes both natural and human settings, 'no constraint is applied specifically to individual particles (…). The particles are thus free to obey the internal forces 15 'I appreciate the generous sentiments which actuate the aspiration of guiding the progress of science into socially beneficent channels, but I hold its aim to be impossible and indeed nonsensical. An example will show what I mean by this impossibility. In January 1945, Lord Russell and I were together on the BBC Brains Trust. We were asked about the possible technical uses of Einstein's theory of relativity, and neither of us could think of any. This was forty-years after the publication of the theory and fifty years after the inception by Einstein of the work which led to its discovery. It was fifty-eight years after the Michelson Morley experiment. But, actually, the technical application of relativity, which neither Russell nor I could think of, was to be revealed within a few months by the explosion of the first atomic bomb. For the energy of the explosion was released at the expense of mass in accordance with the relativistic equation e = mc2 an equation which was soon to be found splashed over the cover of Time magazine, as a token of its supreme practical importance' (Polanyi, 1962, p. 62).

acting between them, and the resultant order represents the equilibrium between all the internal and external forces' (ibid.).

In Polanyi (1948, 248-249) we find a demonstration, through the use of graph representations, of the impossibility of economic planning through the comparison between two opposed coordination devices that characterize the two kind of orders, namely, corporate authority and spontaneous order (see Figures 12345in the Appendix).

What Polanyi wants to demonstrate is the superiority -in terms of efficiency -of spontaneous order over corporate authority when the number of interacting individuals becomes significant. In other words, 'the impossibility of central economic direction lies in the much shorter span of control [the number of adjustable relations] of a corporate authority as compared with a self-adjusted system' (Polanyi, 1948, p. 256).

This approach based on the importance of the span of control in the definition of the principles of management was first proposed by Graimucas (1933), then revived by [START_REF] Meier | Ode to Luther Gulick, Span of Control and Organizational Performance[END_REF] and developed by [START_REF] Theobalt | The Many Faces of Span of Control, Organizational Structure Across Multiple Goals[END_REF] 18 .

According to Meier and Bolt (2000, p. 120), any quadratic formula relating production (Y) to span of control (X) within a given set of organizations can be defined by three key parameters, as denoted in the following equation:

where is the highest value that Y can attain (i.e., maximum output),  is the value of X (span) where output is maximized, and is a constant that adjusts the curve to be either narrower or wider. Furthermore, they find a relation between organizational performance and span of control similar to Polanyi: 18 The notion of 'span of control' has been first introduced in Management by [START_REF] Graicunas | Relationship In Organization[END_REF][START_REF] Graicunas | Relationship In Organization[END_REF]. This notion came from the psychological notion of 'span of attention'. The formula known as the Graicunas formula can be written as follows:

2 2 1

where r is the number of relationship and n the number of subordinates. Accordingly, the 'span of control' is defined by n, the number of cross relationships equals n(n-1) and the direct group relationships n(2 n /2 -1). See Gulick and Urwick (1937), Urwick (1956[START_REF] Urwick | V. A. Graicunas and the Span of Control[END_REF], Nickols (2011). Polanyi completes his analysis of market efficiency (compared to corporate authority) with an analysis of the incentives needed for economic actions when their performance is not necessarily in the actors' interests. Accordingly, he lists the 'five determinants of societal action, which, for the sake of further discussion', he prefers 'to write in the sequence: Powers, Tasks, Tests, Rewards, Accession' (Polanyi, 1997, p. 186).

"Power" defines people's possibility to act. It is complete in a polycentric system, and limited inside a corporation. "Tasks" are the set of actions assigned to individuals. They are determined by senior individuals within a corporation and rationally defined by the individuals in a polycentric system. "Tests" are the set of tools used to evaluate individual performance. They are based on a contract in a polycentric system, and on a fair appraisal in corporations. "Accession" is linked to the procedure of "promotion or demotion". It is defined by a contract in the case of a business corporation 19 .

Polanyi then analyses the institutions that, in a polycentric system, 'provide the persons making decisions with the five factors of responsible societal action' (Polanyi, 1997, p. 189).

He uses the figure below (slightly modified to clarify):

19 Note that Polanyi does not analyse Accession in the case of a polycentric system. The double-headed arrows connecting the symbols expresses the following two way processes:

The W's allocate themselves to jobs offered by M (managers) while M's give jobs to W's. Similarly the L's assign sites to one of the plants controlled by an M, while an M decides to puts his plant on a site placed at his disposal by an L. The I's allocate capital between different M's, while the M's allocate their ideas to one of the I's. On the right of the M's we see them distributing their products between the C's (consumers) while the C's choose between different M's in allocating their custom. (Polanyi, 1997, p. 190) In the above quotation, the W's refers to the workers, the L's to landowners, while the I's stand for the investors. According to Polanyi the polycentric system permits the five determinants to function efficiently if, (i) there is a system of contracts that bilaterally links the different parties (W and M, L and M, I and M and M and C); (ii) this system of contracts being 'enforceable by a legal order of private law ' (ibid., p. 190) and (iii) a price system exists that allows the different parties rationally to allocate their resources.

What distinguishes capitalism and socialism is mainly the contractual relationships between the I's and the M's. First, under socialism the I's 'are appointed by public authorities' (ibid, p. 193) and people cannot directly finance the M's and reap no profits from this: 'All risks of enterprise are pooled in the public treasury. The State bears all the losses and takes all the profits ' (ibid., p. 194). Second, in the competitive order I ↔ M, we still have the managers boosting their projects in competition with each other and trying to attract capital in the direction of greater profitability; but the I's being all officials of the same government cannot possibly pursue as many different and conflicting opinions as private individuals could' (ibid, p. 194).

Judging by what precedes, Polanyi's conception of the market economy sounds very close to Hayek's. More specifically, Polanyi argues that self-organization is a more efficient coordination device than planned organization because (i) from a threshold number of people, the span of control does not permit the hierarchy to organize the flows of information between the different levels of the organization efficiently; and (ii) to use modern parlance, the system of incentives is poor within an organization whereas it is strong in the market.

Nevertheless, Polanyi was conscious that the State had a role in making the market and the entire economy function efficiently20 . As he wrote: 'The State supervises commerce by controlling the standard forms of contracts through which it operates and by supporting the organization of markets which offer scope for public competition' (Polanyi, 1997, p.135). This is a traditional function of the State, namely, to define and enforce the rules of the economic game, in particular, contract law and to guarantee property rights.

However, according to Polanyi, roads, town halls and armaments should be funded by the public authorities: the modern State 'considers, in particular, that the care for children, for the sick, the old, and the unemployed is a public concern, and it provides services for these from public funds'. But 'the great majority of human satisfactions are, however, of distinctly individual character, and are parceled out through the market to individual consumers on commercial basis' (ibid). Thus, the State is obliged not only to defend freedom in economics, science and society in general but also to finance public goods and services and to avoid poverty and inequalities 21 . From this perspective it is interesting that Polanyi considers that the best way to reduce inequalities is to suppress inheritance. On the topic of speculation, Polanyi considers that 'limitations on certain speculative opportunities and on the movement of "hot money" may be imposed permanently. However, he adds that the public will hardly approve of the permanent establishment of a broad battlefront between an army of officials and the private users of money' (Polanyi, 1945, p. 87).

To sum up, according to Polanyi, the State has to ensure that the conditions for polycentricity (or self-organization) are warranted. In particular, property rights and contract law must be defined and enforced. Moreover, the State must first, finance public goods and public services and second, try to reduce poverty and economic inequalities as well as economic dysfunctions such as speculation for instance. The latter two tasks are absolutely not in line with Hayek's conception of economics.

Polanyi's Keynes-inspired approach to government economic intervention

In a letter to Toni Stolper, written in September 1937, Polanyi states that his beautiful vacation in the Bretagne 'resulted in [the] valuable achievement of understanding Keynes' book General Theory of Unemployment! A grandiose oeuvre that will offer nourishment to social powers for many years' (Beira, 2014(Beira, -2015, p. 9), p. 9).

In his book Full Employment and Free Trade (1945) Polanyi takes as point of departure what he calls the 'money circle' and used the visual representation of a belt (the 'Money Belt') to characterize different states of "equilibrium" regarding the level of employment (see Figures 1 and2 in the annexes) 22 . Thus, trade cycles are possible. When the economy expands, prices increase and 'such a tendency makes prospects of further new business investment appear profitable' (Polanyi, 1945, p. 25). The increase in new business investment creates a monetary expansion and prices rise again: 'A business expansion is therefore a self-accelerating 21 Polanyi also notes that 'if a local deficiency in health or education can be noted, national funds should, without question, be made available to remedy the weakness' (Polanyi, 1945, p. 99). 22 Polanyi's book was reviewed by [START_REF] Arndt | Review of Full Employment and Free Trade by Michael Polanyi[END_REF], [START_REF] Gilbert | Professor Polanyi's Full Employment and Free Trade[END_REF], [START_REF] Lindblom | Review of Full Employment and Free Trade by Michael Polanyi[END_REF], [START_REF] Lutz | Review of Full Employment and Free Trade by Michael Polanyi[END_REF][START_REF] Phelps Brown | Review of Full Employment and Free Trade by Michael Polanyi[END_REF], [START_REF] Stead | Review of Full Employment and Free Trade by Michael Polanyi[END_REF], T.M.R. (1945), [START_REF] Harrod | Full employment[END_REF] and [START_REF] Woolbert | Recent Books on International Relations[END_REF].

process' (ibid.). Similar but reverse reasoning can be applied to the process of contraction. However, expansion is not an indefinite process:

Once a retardation has set in a contracting force may gain the upper hand. Increased depreciation allowances allocated with respect to the increment of constructional equipment which was created in the upspring, combined with a comparative dearth of renewals (as to be expected in the immediate wake of a wave of new construction), will tend to produce a down-turn. Such a downward trend will go on self-accelerating for a while (ibid., p.25).

In 1938 Polanyi proposed an explanation for the existence of a cycle of reinvestment:

The cycles of reinvestment will be shown to be accompanied by regular changes in the age distribution, average age, and value of capital during which payments for reinvestment oscillate around the steady rate of amortisation, causing alternative phases of liquidation and re-absorption of capital and ending up eventually with the net liquidation of one half of the originally invested capital. The whole of this process is called 'The Settling Down of Capital' (Polanyi, 1938, pp. 153-154).

In this way Polanyi explains the Trade Cycle as the relationships between amortization payments and reinvestment. The first exceeds the second during a downturn, and the reverse effect is a source of revival following a slump. This phenomenon, according to Polanyi, is a consequence of what he calls the Gap, that is, the difference between savings and investments. This Gap is due to the fact that savings and investments depend on two sets of independent decisions that do not necessarily converge and 'expansions and contractions of monetary circulation arise in consequence' (Polanyi, 1945, p. 26). There is a tendency for gaps to be self-sealing because variations in the national income induce variations in both savings and investments in the same direction: 'This process of self-sealing, however, is achieved at the cost of adjusting Employment and National Income to some level which is not likely to be a desirable one. In fact, advanced industrial communities tend to adjust the level of employment well below full capacity' (ibid., p. 26).

Accordingly, because the automatic forces that fill the gap are not efficient in the sense that full employment is not naturally achieved, the gap has to be filled 'artificially':

The problem of Full Employment can now be seen to consist in the task of filling this gap. In other words: to achieve Full Employment it is necessary to fill the Gap at Full Employment-the gap by which Savings exceed new commercial investment when Full Circulation is maintained (Polanyi, 1945, p. 27, emphasis in original).

But this should not be achieved by any means. Polanyi realized that an insufficiency of demand meant an insufficiency of money. Therefore, the government should finance its deficit by issuing new money.

Interestingly, Polanyi introduces the principle of neutrality, according to which 'the process undertaken in order to create sufficient circulation need involve and must involve no material sacrifice to speak of. It should be, and can be, carried out in a neutral form, i.e., in a way requiring no materially significant economic or social action to accompany it' (ibid, p. 29, emphasis in original).

Clearly, Polanyi defines the 'principle of neutrality' differently from the conventional way monetary economists do. Traditionally, the principle of neutrality expresses the idea that nominal variables have no effect on real ones. This is the basis of the quantitative theory of money: If the quantity of money supplied increases then prices increase without any effect on GDP real value.

To a certain extent, the neutrality principle can be interpreted not only as an analytical tool but also as a pedagogical device aimed at making a clear distinction between what must be the province of markets (to provide for the (including monetary) needs of the public and what are the prerogatives of the State (i.e., the traditional regalian functions, but also providing for public goods, limiting inequalities, and importantly, acting as the guardian of the monetary circulation). The neutrality principle implies that the issuing of money during periods of depression should not constitute a financial burden for the public (condition 1), nor that it should produce mal-or overinvestment through inadequate tax (condition 2).

For Polanyi, an expansionary monetary policy based on fiscal deficit, and whose amount is limited to the usual government expenditures will meet the first condition because, in this case, the issuing of new money cannot be considered as borrowing:

If, employment being depressed for lack of circulation, the Government covers some of its expenditure by the issue of new money in order to supplement circulation and to restore employment, this must not be looked as a process of borrowing: but that the operation must, on the contrary, be regarded as a definitive financial act by which the Government discharges for the time being its obligation, as guardian of the level of monetary circulation (ibid, pp. 36-37).

As for the extent of the monetary policy and its potential redistribution effects, Polanyi writes that Governments must use existing channels of public expenditure for issuing new money and do not undertake new public enterprises or deviate in any other way from the otherwise desirable course of economic policies, merely for the purpose of bringing money into circulation (Polanyi, 1945, p. 147).

Note that the two conditions for long-run efficient of monetary policy aimed at curing unemployement stand in sharp contrast with the dominant Keynesian position of the time.

Referring to [START_REF] Meade | Consumers' Credit and Unemployment[END_REF], Polanyi writes that:

He expounded this method [the Keynesian theory] here with unsurpassed clarity, but failed to round off the picture by adding to it the principle of neutrality. His suggestion to distribute money as a supplement to social services in amounts compensating for circulatory deficiencies is in sharp conflict with this principle' (Polanyi, 1945, p. 124).

By the same token, Polanyi criticizes President Roosevelt's 1937 decision to impose a special tax on undistributed profits because such a policy induces a 'misdirection of human efforts, or human relations' as well as employment policies as well as Beveridge's employment policies based on additional taxation (Polanyi, 1945, p. 29 et p. 57).

Polanyi is also strongly opposed to Keynesian massive fiscal policies that imply that the Government consolidates a floating debt by placing government bonds with the public. In these cases, government intervention is at odds with the principle of neutrality. Moreover, it has some counterparts in terms of seigneuriage tax, which translates in a reduction of money circulation (see Polanyi 1945, p. 18).

In other words, Polanyi stresses the idea that the State, in order to solve unemployment due to the Gap, has to inject new money into the economy without compensation on the part of the public so long as this injection does not produce distortions in the real economy.

To the extent that the neutrality principle guaranties that State interventions do not distort individuals' incentives and resource allocation, they do not infringe the freedom of people to behave and coordinate. Therefore, they do not prevent spontaneous order from emerging.

As suggested by Manucci (2005, p. 156), the neutrality principle is a safeguard of the functional separation between economics and politics.

To sum up, Polanyi's half-forgotten contribution to macroeconomics is more important than acknowledged.

First, Polanyi provided an idiosyncratic coherent synthesis of macroeconomic Keynesian and microeconomic Hayekian ideas, which offers a room for a fiscal policy based on money issue as a means to cure unemployment23 . A related interpretation provided by Roberts24 and van Cott is that Polanyi synthesized Keynesian economics with the monetary school of economics later associated with Milton Friedman25 .

It is true that contrary to Keynes who damned monetary policy, Polanyi considered money was all what mattered. On the other hand, he was not an expert of monetary theory. In particular, he did not consider, as Hayek did extensively, the role of banking credit in the cycle and its deleterious -in terms in over or malinvestment -interaction with government manipulation of money. He nevertheless warned against the risk of the misuse of the newly issued money in particular when it takes the form of the undertaking of new public enterprises.

Second, he was a visionary man correctly predicting that many of Keynesian policies were pointless, because they were mainly aimed at increasing the sphere of influence of government without solving the main problem according to Polanyi, namely, the needs of the public for cash balances.

As will be developed in the next section, Polanyi's economics must be put in perspective with his overall achievement, and in particular with his visions of liberalism and evolutionnism.

Polanyi and Hayek's conception of liberalism and evolution

In the former section, we have provided a historical reconstruction of Polanyi's economic ideas. We have reached the conclusion that although Hayek and Keynes exerted a strong influence on his vision of economics, Polanyi nevertheless developed a consistent idiosyncratic synthesis. First, Polanyi stressed the idea that polycentricity (self-organization) is efficient compared to planning. Second, he supported State intervention -provided certain constraints are fulfilled -to help reduce unemployment. According to us, this synthesis can be better understood if put in perspective with Polanyi's views of liberalism and evolutionism.

As a liberal, Polanyi introduces an important difference between private freedom and public liberty. As an evolutionist, Polanyi disagrees with Hayek's non-teleological interpretation.

Polanyi's liberalism

As highlighted by [START_REF] Mullins | Michael Polanyii's Early Liberal Vision: Society as a Network of Dynamic Orders Reliant on Public Liberty[END_REF] and [START_REF] Biró | Changing knowledge in the early economic thought of Michael Polanyi[END_REF], Polanyi envisions society as a network of overlapping layers of orders (a system of dynamic orders) that are governed by two kinds of coordination mechanisms (spontaneous market-like vs. planned organization-like). Despite differences among the various orders in society, the system as a whole cannot be maintained without what Polanyi calls 'public power'. Unlike the invisible hand, public power 'shelters and controls' the 'economic institutions of society'. In other terms, by making co-operation the coefficient of the economic system, Polanyi appears to be rejecting competition in favor of co-operation. But competition is one of the constituent elements of the laws of the market, for which the 'invisible hand' is a metaphor (Moodey, 2014(Moodey, -2015, p. 32, p. 32).

Polanyi's distinction between two notions of freedom, i.e., 'private freedom' and 'public liberty' reflects this dual view of coordination. Private freedom as 'individualistic' and 'selfassertive' (defended by utilitarians, romantics and nihilists) permits people to do what they please, 'whereas judges, scientists, theologians, and members of other dynamic orders have their freedom from public purpose' (Jacobs and Mullins, 2008, p. 127). This second kind of freedom relates to the notion of 'public liberty':

The freedom with which we are concerned here is not for the sake of the individual at all, but for the benefit of the community in which dynamic systems of order are to be maintained. It is freedom with a responsible purpose; a privilege combined with duties, as exacting as any that are shouldered by man. It may be called, therefore, Public Liberty -as opposed to Private Freedom (Polanyi, 1941, p. 438).

According to Polanyi, 'responsible public liberty sets a limit to irresponsible private freedom; but we must not think of the two as enemies. On the contrary, they merge into one another and mutually stimulate each other' (Polanyi, 1941, p. 440). In The Foundations of Academic Freedom (1947), Polanyi shows that public liberty supposes that scientists and scholars more generally respect standards that are based on tradition and discipline: 'These principles can be readily generalized for scholarship in general. Academic freedom can claim to be an efficient form of organization for discovery in all fields of systematic study controlled by a tradition of intellectual discipline' (Polanyi, 1947, p. 584).

For Polanyi, liberalism is grounded in a combination of the two kinds of freedom: private and public. When Polanyi takes the examples of the scientist and the judge, he stresses the idea that both have to respect standards inherited from tradition and, that it is on that basis that they can express their originality, that they can discover new results or propose new sentences. Scientists as well as judges act on 'spiritual grounds'. Both professions are committed to their job according to socially accepted rules 26 . By contrast, Hayek's liberalism relates to a notion of spontaneous order usually seen as a single entity, which is equivalent to free society as a whole (see [START_REF] Jacobs | Spontaneous order: Michael Polanyi and Friedrich Hayek[END_REF].

Polanyi's liberalism is not in line with Hayek's in particular because Hayek endorsed Popper's theory of open society and, as well known, Polanyi was explicitly critical of Popper (cf. [START_REF] Jacobs | Relations between Karl Popper and Michael Polanyi[END_REF].

But despite some important differences, both Hayek and Polanyi were looking for ways to defeat the scientific socialism which was dominant during WW2. In a letter from July 1 st , 1941, Hayek explained that he attached 'very importance to the pseudo-scientific arguments on social organization being effectively met and I am getting more and more alarmed by the effects of the propaganda' of the left scientists which 'discredit the reputation of science by such escapades' 27 .

Polanyi is on the same line when he is concerned about the difficulty to diffuse liberal ideas that rely on propositions which are too abstract and not enough scientifically grounded, such as the notion of invisible hand. The problem with the invisible hand was precisely its inaccessibility, which frustrated the agent's economic activity from its larger social and moral sense, a void which central plannism fulfilled.

After the publication in Nature of Hayek's 1941 article on planning, science and freedom, Polanyi wrote him to express his commitment to their joint enterprise, stating that "the only real aim in my view is the starting of the literary and philosophical movement of our own for the renaissance of Liberalism' 28 .

From this perspective, Hayek and Polanyi can be viewed as two pivotal figures of early neoliberalism. They shared the idea that the epistemological ground of neoliberalism (against planning) lies in a theory of knowledge that focuses on the tacit nature of knowledge and its related implications for science and society.

They however departed from one another concerning the implications in terms of organization of science. Polanyi focused on the notion of public liberty whereas, according to some commentators, Hayek's emphasis is on individual freedom 29 . 27 Hayek to Polanyi, 1 July 1941, Box 4, Folder 7, Polanyi Papers, University of Chicago Library. 28 Polanyi to Hayek, 18 November 1941, Box 78, Folder 35, Hayek Papers, Hoover Archives. 29 See [START_REF] Allen | Beyond Liberalism: The Political Thought of F[END_REF] and [START_REF] Jacobs | Friedrich Hayek and Michael Polanyi in Correspondence[END_REF].

As suggested by Jacobs and Mullins (2015, p. 16), this is one of the reasons why Polanyi wrote an ambivalent review of Hayek's Individualism and Economic Order.

According to Polanyi, it not clear how to interpret Hayek's reference to ' "Acton and Burke

[who] tell us that tradition is the only true bulwark of liberty" ' and his notion of individualist order which ' "must rest on the enforcement of abstract principles" '. For Polanyi, abstractness is a non-operational criterion for distinguishing between good and bad traditions or convention because we don't know 'how to distinguish between abstract principles on which we should base the order of the economy and speculative ideas, the fascination of which we must firmly resist?' (Polanyi, 1949, p. 267). The interpretation is even more problematic if we agree with Hayek that ' "the individual (…) must be ready and willing to adjust himself to changes and to submit to conventions which are not the result of intelligent design, whose justification in the particular instance may not be recognisable, and which often may appear unintelligible and irrational" ' (Polanyi, 1949, p. 268).

Polanyi's notion of public liberty is also associated with the importance of traditions as guides for individual behaviour. But it implies the idea of deliberate and meaningful commitment. By contrast, if for Hayek social norms and traditions are also important, it is mainly because individuals are essentially and involuntarily rule-followers.

From this perspective, Polanyi's liberalism is close to Keynes's conception of politics. Indeed, according to Keynes (1931, p. 344), the political problem of mankind is to combine three things: Economic Efficiency, Social Justice, and Individual Liberty. The first needs criticism, precaution, and technical knowledge; the second, an unselfish and enthusiastic spirit which loves the ordinary man; the third, tolerance, breadth, appreciation of the excellencies of variety and independence, which prefers, above everything, to give unhindered opportunity to the exceptional and to the aspiring. Moreover, in a series of lectures he gave in Oxford in 1924 and in Berlin in 1926, Keynes argued against the naïve doctrine of laissez-faire, saying that the world is not so governed from above that private and social interest always coincide.

It is not so managed here below that in practice they coincide. It is not a correct deduction from the Principles of Economics that enlightened self-interest always operates in the public interest (Keynes, 1931, p. 312, emphasis in original).

To sum up, the idea of commitment makes Polanyi's conception of liberalism compatible with economic interventionism, under certain limits, as exemplified by his neutrality principle. In the following quotations, Polanyi defends Keynes's economic interventionism:

The orthodox Liberals [rejecting Keynes' theory out of ignorance or misunderstanding] maintain that, if the market is limited by the fixation of some of its elements, then it must cease to function, the implication being that there exists a logical system of complete laissez faire, the only rational alternative to which is collectivism (Polanyi, 1940, p. 56). Also, in a 22 February 1937 lecture to the Manchester Political Society, Polanyi stressed that the work of Keynes has brought an understanding of the trade cycle which seems also to lead up a proper definition of public responsibility in an industrial system. At last we have before us a fundamental criticism of liberal economics which avoids the mistakes of Communism (Polanyi 'On Popular Education in Economics', lecture delivered in February 1937 to the Manchester Political Society, box 25, Folder 9;cited in Jacobs and Mullins, 2015, p. 5, note 25;republished in Tradition & Discovery in Polanyi (2016)).

As will be developed further, the importance Polanyi gives to co-operation as opposed to competition, and to public liberty as opposed to private freedom, is close to his conception of mankind evolution.

Polanyi's evolutionism

Michael Polanyi's evolutionism has not been much discussed in the literature. As it can be easily imagined, it is deeply rooted in his view of knowledge. As recently argued by Biro, Polanyi's economic thought is evolutionist in a sense that "it is based on changing knowledge in cognitive, behavioural, social and technical domains" (Biro, 2018, p. 207). Biro further draws a parallel with Dopfer and Potts' social evolutionism conceived as a multi level organized society that articulates -from both a bottom-up and top-down perspective -micro, meso and macro segments of society, and that is based on the concept of rules, which are 'units of knowledge and therefore the building blocks of wealth and the locus of evolutionary change': 'all economic knowledge and the growth of knowledge can be analysed as a process of coordination and change in rules' (Dopfer and Potts, 2008, Preface).

There are textual evidences of the fact that Polanyi has always been an evolutionist, referring as early as in his 1941 essay to Köhler's notion of "spontaneous equilibration". But he gradually developed his own interpretation inspired by his vision of tacit knowledge and based on the notion of emergence. Paksi (2011Paksi ( -2012) discusses Polanyi's understanding of evolutionary biology, as it can be reconstructed from Part IV of Personal Knowledge "The Logic of Achievement", along these lines. More precisely, in his attempt to situate the human knower in the largest context of biological evolutionary history, Polanyi is indeed an emergentist, not rejecting neo-Darwinism but arguing that it is logically unsatisfactory because it lacks an emergent principle of evolution, which entails finalist causes or end-directed processes 30 . As he states, the theory of natural selection, by subsuming that evolutionary progress under the heading of adaptation as defined by differential reproductive advantage, necessarily overlooks the fact that the consecutive steps of a long-range evolutionary progresslike the rise of human consciousness -cannot be determined merely by their adaptive advantage, since these advantages can form part of progress only in so far as they prove adaptive in a peculiar way, namely on the lines of ascending evolutionary achievement (Polanyi, 1958, p. 385, emphasis in original) 31 .

Polanyi claims, 'Darwinism has diverted attention for a century from the descent of man by investigating the conditions of evolution and overlooking its action. Evolution can be understood only as a feat of emergence' (Polanyi, 1958, p. 390, emphasis in original). 30 Polanyi claims, 'Darwinism has diverted attention for a century from the descent of man by investigating the conditions of evolution and overlooking its action. Evolution can be understood only as a feat of emergence' (Polanyi, 1958, p. 390, emphasis in original). 31 'Polanyi opposes moral relativism and advocates a society in which liberty is defended on the grounds that it facilitates the pursuit of transcendental ideals' (Goodman, 2005, p. 55).

As emphasized by Paksi (2011Paksi ( -2012, p. 51), p. 51), Polanyi describes living beings (as well as artefacts like machines) in terms of 'hierarchical levels which impose or are instantiated within boundary conditions; each higher level configuration operates within boundaries or boundary conditions left open (i.e., possibilities) by the lower level. Each level of harnessing and controlling the lower level(s)'. Processes operating at different levels must be independent and cannot be deduced from one another, otherwise the lower level processes wholly determine the upper ones. From this, Polanyi concludes that 'the logical structure of the hierarchy implies that a higher level can come into existence only through a process not manifest in the lower level, a process which thus qualifies as an emergence' (Polanyi, 1958, p. 45).

The parallel between Polanyi's view of tacit knowledge and his interpretation of evolution is striking. In describing the concepts of (focal vs. subsidiary) awareness, he refers to stereoscopy 32 and the notion of 'clues': 'we don't look at these two [the stereoscopic pictures] in themselves, but see them as clues to their joint appearance in the stereo-image' and 'the relation of clues to that which they indicate is a logical relation [italics due to M.P.] similar to that which a premise has to inferences drawn from it" (Polanyi, 1965, p. 799).

The articulation between the two types of awareness also implies an emergence. Subsidiary awareness of a whole cannot be deduced from focal awareness of its constituting parts; it is strongly linked to imagination and invention or new knowledge: 'the clues of a problem anticipate aspects of a future discovery and guide the questing mind to make the discovery" (Polanyi, 1967, p. 188).

As we have hinted at in the introduction, Hayek's view of the tacit dimension of knowledge differs from Polanyi's ones. A closer look at Hayek's notion of tacitness permits to contrast his evolutionism to the one of Polanyi.

As well known, Hayek's notion of tacitness is related to the human mind's classification, i.e., to the idea that perception is the result of this classification and not the other way around.

Moreover, the process of knowledge creation is partly unconscious and not entirely 32 Stereoscopy refers to the techniques used by 3D imaging for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. Most stereoscopic methods present two offset images separately (about four inches from one another) a to the left and right eye of the viewer (a metaphor for subsidiary awareness). These two-dimensional are then combined in the brain to give the perception of 3D depth (a metaphor for focal awareness). accessible: 'it is impossible that our brain should ever be able to produce a complete explanation (as distinguished from a mere explanation of the principle) of the particular ways in which it itself classifies external stimuli'. In other terms, 'to "explain" our own knowledge would require that we should know more than we actually do' (Hayek 1952b, p. 47). This explains, why for Hayek, evolution is a blind process, independent of individuals' intentions:

though by moral conduct an individual may increase his opportunities, the resulting evolution will not gratify all his moral desires. Evolution cannot be just. Indeed, to insist that all future change be just would be to demand that evolution come to a halt' (Hayek, 1990, p. 74, emphasis in original).

As the quotation clearly states, this does not imply that Hayek do not consider the personal and moral dimension of human beings. Social norms are indeed very important in his theory of evolution. In particular, they play a very important role at the individual (natural) level of selection of rules of conduct. By contrast, the social or cultural level of selection of rules (i.e., "the social order of actions", see Hayek, 1967, p. 67) that take place at the group level is detached from individuals' intentions (see [START_REF] Festré | The Economic Analysis of Social Norms: A Reappraisal of Hayek's legacy[END_REF]. As well known, Hayek's conception of evolution is Lamarkian (because of the inheritance of acquired characteristics) and, for him, fitness is due to the capability of groups to resist competition 33 .

Competition then is the principle of social and economic selection and the rules of conduct are selected in so far as they permit group's population to grow 34 . This is the bedrock of Hayek's argument of group selection 35 . By contrast, for Polanyi, the personal and moral dimension of human beings is at the core of the higher level processes of biological evolution: I have not hesitated to value the more comprehensive levels of life as the higher form of existence, for the absence of value judgments in science is but a pretence, which, if followed strictly, would render biology blind not only to evolution, but to life itself.

For the value of life comes into existence with life itself (Polanyi, 1964b, p. 24). 33 See Hayek, 1990. 34 Concerning the debate on Hayek's conception of fitness, see [START_REF] Garrouste | Le problème de la cohérence de l'évolutionnisme hayékien[END_REF] and [START_REF] Hodgson | Hayek's Theory of Cultural Evolution: an Evaluation in the Light of Vanberg's Critique[END_REF]. 35 There is a sharp debate concerning the coherence of Hayek's evolutionism. See [START_REF] Festré | The Economic Analysis of Social Norms: A Reappraisal of Hayek's legacy[END_REF]. Now, at the social level, evolution proceeds by new forms of knowledge that are embedded in persons and that can be disseminated in the society thanks to heroic engaged scientists: 'the scientist's decision depends on the strength of the beliefs in the light of which he interprets his observations, and we approve of this decision if we share these beliefs' (Polanyi, 1950, p. 30).

According to Polanyi, "Man" becomes better through evolution 36 . Polanyi's vision of Man is such that he considers that 'man's responsibility to standards of truth and rightness establishes him as a rational person' (Polanyi, 1959, p. 90). This is why evolution is teleological -in the sense of non-purely materialist but not implying vitalism -for Polanyi.

Conclusion

Polanyi's life followed a very fascinating trajectory. His initial training was in natural sciences as a physical chemist. After moving to England, he repeatedly changed his mind and participated in societal debates on planning in economics as well as science. He defended the free market in line with Hayek's conception of economics and society, and proposed a Keynesian approach to full employment. We have shown that Polanyi offered a consistent idiosyncratic synthesis of Hayekian and Keynes' insights. Our interpretation is that this synthesis provided by Polanyi goes beyond the opposition between Hayek and Keynes. First, Polanyi, like Keynes and Hayek did not defend socialism and was a promoter of a free-market economy. Second, the neutrality principle can be interpreted as a means to reconcile State intervention and polycentricity. We have also emphasized that Polanyi's conception of liberalism as well as his approach to evolution was key to understand his synthetic approach in economics.

From the perspective of institutional economics, Polanyi's achievement illustrates his synthetic method by mixing different orders (spontaneous and deliberatey organized) and is in sharp contrast with Hayek binary Manichean approach. As put forward by [START_REF] Nye | Michael Polanyi and His Generation: Origins of the Social Construction of Science[END_REF], 36 'This is the point at which the theory of evolution finally bursts through the bounds of natural science and becomes entirely an affirmation of man's ultimate aims' (Polanyi, 1958, p. 404). Suppose now that the nodes are directly connected to each other as in Figure 3, which represents a spontaneous order situation.

Figure 3. A system of spontaneous order of 9 individuals (source: Polanyi, 1948, p. 249) In the above figure, the number j of connections between the nodes equals (9.8) = 72, that is, for l nodes l(l-1), and for every new node the increase in j equals l-1.

Let k be the difference in the connections by nodes between the two organizational devices (spontaneous order and corporate authority). Accordingly:

1 -3 2 ⁄ 1 1 3 ⁄ As shown in Figure 4, k begins to decrease and then increases from l = 1.5 onwards.

As far as the slope of k is concerned, it increases up to 1 and then remains constant (see Figure 5). (source Polanyi, 1945, p.7)

Figure 6 .

 6 Figure 6. Span of Control and Outputs: The Theoretical Relationship

Figure 7 .

 7 Figure 7. Resources allocation in an economic system

Figure 2 .

 2 Figure 2. The curve of the function i = (3/2)(1-(1/3) l-1 with 1

Figure 4 .

 4 Figure 4. The curve of the function k = (l -1) -(3/2)(1 -(1/3) l-1 ) with 1

Figure 5 .

 5 Figure 5. The curve of the function dk/dl = 1-(9/2)(ln3)(1/3) l , with 1

For a critique of this conception see[START_REF] Knight | Virtue and Knowledge: The View of Professor Polanyi[END_REF],[START_REF] Buchanan | Politics and Science: Reflections on Knight's critique of Polanyi[END_REF] and the response of[START_REF] Roberts | Politics and Science: A Critique of Buchanan's Assessment of Polanyi[END_REF].

Note that the concept of polycentricity diffused to governance studies, thanks to Vincent and Elinor Ostrom and the Bloomington School of institutional analysis (see[START_REF] Aligica | Challenging Institutional Analysis and Development: The Bloomington School[END_REF].

We do not develop this aspect of Polanyi's conception here; however, in our view, those elements are due to his strong humanism. SeeKelleher (2008Kelleher ( -2009) ) and below.

See Spector and van Cott (2007) for a critique of macroeconomics textbooks presentations of pure fiscal policy that systematically neglect its underlying monetary mechanisms.

Paul Craig Roberts is an economist who was one of Polanyi's last graduate student.

Roberts and Van Cott make such claim: 'Being untrained as an economist allowed Polanyi to avoid pitfalls that confused economists. It also left him unaware of the magnitude of his achievement. He saw himself as a Keynesian, but in fact he achieved, in the early years of Keynesianism, before the monetarist critique, an integration of the two approaches that economists did not reach until the 1970s'.

The role of commitment is essential in Polanyi's conception of science. See[START_REF] Hall | The Role of Commitment in Scientific Inquiry: Polanyi or Popper?[END_REF].

In his 1948 article Polanyi apparently makes an error when he evaluates m as r l-1 . However, it would seem to be a typographical error because the value of i is calculated using the value of m = 3 l-1 .

Michael Polanyi's original account of institutions is to be found in his work on the social dimension of science and provides a noteworthy connection with his brother Karl.

Furthermore, as we have emphasized, Polanyi considers that seeing is believing. Therefore, perceptions can be intentionally manipulated. This explains why he devoted some time to imagine new pedagogical devices based on visualisation that could serve the population to better integrate knowledge and fight against prejudice ideas: 'Modern education breaks down our natural predilections which favor the magical outlook and inculcates in us certain conceptions which primitive people do not possess' (Polanyi, 1950, p. 32).

Appendix

Following Polanyi, let us suppose a leader and three subordinates who each have three subordinates, who, in turn, three subordinates each (see Figure 1). Figure 1. A pyramid of authority of 4 levels (source Polanyi, 1948, p. 248) The total number r of connections between the nodes is given by:

If one generalizes the three-level graph to l levels, the total number of connections is given by:

The number m 37 of ultimate subordinates equals 3 l-1 and i = r/m is the number of relations per person governed by the corporate authority (see Figure 2). In other terms, i can be interpreted as the number of connections by node and is given by the following geometric series:


E Gouillart

N Kuncio

O Dauchot

B Dubrulle

S Roux

J.-L Thiffeault

Walls inhibit chaotic mixing

We report on experiments of chaotic mixing in a closed vessel, in which a highly viscous fluid is stirred by a moving rod. We analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and observe a slow algebraic decay of the inhomogeneity, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel boundary, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1-D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results.

Low-Reynolds-number fluid mixing has a variety of applications ranging from geophysics to industrial mixing devices. While turbulent flows lead to highly efficient mixing, simple laminar flows with chaotic Lagrangian dynamics also promote rapid homogenization [1]. Dynamical systems approaches based on flow kinematics have provided a first insight into chaotic mixing [2,[START_REF] Jana | [END_REF]. A deeper understanding of homogenization processes is gained by examining the interplay between chaotic stirring and diffusion. Several experimental [4,5] and numerical [6,7] studies obtained an exponential decay for the variance of a diffusive scalar concentration field in a chaotic mixer. This behavior is attributed to an asymptotic spatial structure of the scalar dubbed strange eigenmode [8], that results in a global exponential decay of the spatial contrast. However, these theories focus on ideal mixing systems, e.g. with periodic or slip boundary conditions, far from the reality of industrial mixing devices with solid no-slip walls. It has been suggested [9,10] that mixing might be slower in bounded flows, but experimental evidence is still lacking.

In this Letter, we study experimentally dye homogenization by chaotic mixing in a 2-D closed flow. Precise measurements of the concentration field yield "slow" algebraic decay of an inhomogeneity, at odds with the expected exponential decay. We relate quantitatively this slow mixing to the chaotic nature of trajectories initially close to the no-slip wall, which end up escaping in the bulk and slow down the whole mixing process.

A cylindrical rod periodically driven on a figure-eight path gently stirs viscous sugar syrup inside a closed vessel (Fig. 1 (a)). The stirring scale is comparable to the vessel size, in contrast to other devices such as arrays of magnets [4,5]. This protocol is a good candidate for efficient mixing: we can observe on a Poincaré section (Fig. 1 (a)) -computed numerically for the corresponding Stokes flow -a large chaotic region spanning the entire domain, including the vicinity of the wall. The signature of chaotic advection can also be observed in Fig. 1 (a), where a complex lamellar pattern is created by the stretching and folding of an initial dye blob into exponentially thin filaments. The fluid viscosity ν = 5 × 10 -4 m 2 • s -1 together with rod diameter ℓ = 16 mm and stirring velocity U = 2 cm • s -1 yield a Reynolds number Re = U ℓ/ν ≃ 0.6, consistent with a Stokes flow regime. A spot of low-diffusivity dye (Indian ink diluted in sugar syrup) is injected at the surface of the fluid, and we follow the evolution of the dye concentration field during the mixing process (see Fig. 1). The concentration field is measured through the transparent bottom of the vessel using a 12-bit CCD camera at resolution 2000 × 2000.

Despite the exponential stretching occurring in the bulk, the resulting variance σ 2 (C) of the concentration field (measured in a large central rectangular region) decays surprisingly slowly with time t as a power law t -m with m ≃ 3.2 (Fig. 1 (b)), and not exponentially as expected. This behavior persists until the end of the experiment (35 periods), by which time the variance has decayed by more than three orders of magnitude. Moreover, concentration probability distribution functions (PDFs) shown on Fig. 1 (c) exhibit wide power-law tails on both sides of the most probable value. The probability of "white" (zero) concentration decays very slowly with time, whereas the peak shifts towards lower values.

In order to understand these surprising scalings, we first describe the various mechanisms at play during the mixing process. A blob of dye, initially released close to the vessel center, is transformed into a complicated pattern expanding towards the boundary with time. We distinguish at each instant the growing "mixed region", delimited by the advected blob frontier, and the remaining boundary region where C = 0, in the vicinity of the vessel wall. This distinction is obvious in Fig. 1 (a) where one can observe a central heart-shaped mixed region and an annular unmixed boundary region. As the chaotic region spans the entire flow, fluid particles initially close to the zero-velocity wall eventually escape from the boundary to wander through the whole chaotic region. Trajectories escape along the unstable manifold of parabolic separation points on the boundary [START_REF] Jana | [END_REF]11]. The signature of such an escape path can be visualized on Fig. 1 (a) where unmixed fluid from the boundary is "sucked" inside the mixed region through its white cusp, close to the rod. This results in the periodic injection of broad white strips that can be observed inside the mixed region. The mixing region then grows towards the boundary to make up for this mass injection. Incompressibility combined with zero-velocity condition at the boundary leads [9] to a shrinking distance between the mixed region border and the boundary scaling as d(t) ∝ t -1 . This scaling is verified experimentally. The area of unmixed fluid from the boundary injected at each period inside the mixed region then scales as ḋ(t) ∝ t -2 . As a result, the mean concentration value inside the bulk decreases with time as (1 -d(t)) -1 . Simultaneously, the mixed region is stretched and folded at each half-cycle of the rod movement (see Fig. 2 (a)) in a baker's-map-like fashion [12].

However, it should be noted that the two folded parts are not stacked directly onto each other but separated by the newest injected white strip. Also note that the part "attached" to the rod has experienced more stretching than the one "left behind". Briefly, (i) chaotic stretching imposes that the typical width of a dye filament in the bulk shrinks exponentially down to the diffusion or measurement scale, yet (ii) wide strips of unmixed fluid of width ḋ(t) ∝ 1/t 2 are periodically inserted between these fine structures. In the following we derive how these two effects lead to the observed scalings.

For this purpose, we simplify the 2-D problem by characterizing only 1-D concentration profiles C(x, t) along a secant to the stretching direction -the dashed segments on Fig. 2 (a) -that is, we neglect the variation of the concentration along a dye filament on a scale comparable to the vessel size. We thus call from now on "mixed region" the intersection of the 2-D mixed region with such a segment. The effect of the mixer during a half-period then amounts to the action of a one-dimensional discrete-time map that transforms concentration profiles by inserting an interval of width ḋt of fluid from the boundary between two inhomogeneously compressed images of the mixing region at the previous time (see Fig. 2 (a)). Such a map f , defined on [0, 1] for simplicity, evolves concentration profiles as C(x, t + 1) = C(f -1 (x), t) and meets the following requirements: (i) it is a continuous doublevalued function to account for the stretching/folding process; (ii) x = 0 is a marginally unstable point of f -1 ; the correct dynamics close to the boundary are indeed reproduced by imposing f -1 (x) ≃ x + ax 2 + • • • , a > 0 for small x; (iii) because of mass conservation, at each x, the local slopes of the two branches add up to 1. Other details of f are unessential for our discussion. Diffusion is mimicked by letting the concentration profile diffuse between successive iterations of the map. This model is a modified baker's map [12], with a parabolic point at x = 0, whereas the dynamics are purely hyperbolic in a classical baker's map. Numerical simulations for a specific choice of f lead to results shown in Fig. 1 (b) and Fig. 3: both the power-law evolution of the variance and the different aforementioned features of the experimental PDFs are reproduced by the simulations.

The map transforms an initial blob of dye of width s 0 into an increasing number of strips with widths s 0 γ 1 • • • γ t , resulting from different stretching histories inside the mixed region, where γ t is the compression experienced at time t. White strips also experience this multiplicative stretching from their injection time. As the mixed region grows towards the boundary, different values of stretching will be sampled. It will nevertheless be justified below that the concentration measures only require knowledge of the stretching histories traced back during a finite number of periods. This allows us to define in a "quasi-static" approximation a slowly varying instantaneous "Lyapunov exponent"

t t + 1 2 d(t) (a) w B w px l i = s 0 γ (i) 1 ..γ (i) t C ⇑ C = i Ci = i C0 li wB (b)
(x, t + 1) = C(f -1 (x), t), f (x) : f1(x) = x -ax 2 + (γ -1 + a)x 3 ; f2(x) = 1 -ax 2 + (γ -1 + a)
x 3 with a = 0.9 and γ = 0.55. Upper (resp. lower) inset: light (resp. dark) gray tail, P (C) against |Cmax -C|. We observe the same power-law decay (Cmax -C) -2 on both sides of the maximum as in the experiment (Fig. 1 (c)).

Γ(t) = exp( -log(| ∂f -1 (x) ∂x |) M R )
, that is the geometric mean of the compression taken over the mixed region (MR) at time t. Note that as the white strips are injected close to the center of the domain, the two branches of f have comparable mean slopes, yielding the estimate Γ ∼ 0.5. Without diffusion, dye strips would have a typical width s 0 Γ t at time t. However, the balance between stretching and diffusion imposes that the width of a strip stabilizes at the Batchelor scale w B = κ/(1 -Γ 2 ), where κ is the diffusion coefficient. w B is thus the smallest lengthscale that can be observed in the concentration profile, and different elementary strips may overlap (Fig. 2 (b)). Since the concentration field is probed by averaging it on the pixel size w px , which is smaller than w B , the concentration at a pixel is given by adding the contributions from strips contained in a box of size w B around the pixel. Hence we characterize P (C) by considering the different combinations of strip widths for a zero-diffusivity dye -the concentration profile on Fig. 2 (b) -that one might find in a box of size w B . We will distinguish between three generic cases corresponding to three different regions of the histogram P (C) (see Fig. 1 (c) and Fig. 3): a white (W) peak at C = 0 corresponding to injected white strips still wider than w B , light gray (LG) and dark gray (DG) tails corresponding to respectively smaller and larger concentrations than the peak (mean) concentration. Once we have quantified the proportion of boxes contributing to these different values of C, the variance will be readily obtained as

σ 2 (C) = (C -C ) 2 P (C)dC = σ 2 W + σ 2 LG + σ 2 DG . ( 1
)
Let us start with white (zero) concentration measures that come from the stretched images of white strips injected before t. White strips injected at an early time have been stretched and wiped out by diffusion, that is their width has become smaller than w B . Hence the oldest white strips that can be observed have been injected at the time t i (t) such that ḋti Γ t-ti = w B . Note that from t white defined by ḋt white = w B , the injected white strip is smaller than w B and no white pixels can be observed. Before t white , the number of white pixels is pro-

portional to n W = d ti(t) -d t ∝ (t -t i )(t i t) -1 for large t (using d t ∝ t -1 ). As t -t i ≃ (2 log t + log w B )/ log Γ, n W ≃ (2 log t + log w B )/(log Γ × t 2 ). We deduce σ 2 W ≃ (2 log t + log w B )/(log Γ × t 2 )
for t < t white and σ 2 W = 0 after t white .

We now concentrate on the distribution of light gray values corresponding to white strips that have just been compressed below the cut-off scale w B . We propose to approximate the measured value C as the average of the biggest white strip with width λ < w B , and mixed "gray" fluid whose concentration is close to the most probable concentration C g . A box with a white strip of scale λ thus bears a concentration C λ = C g (1 -λ/w B ) and we can relate P (C) to the distribution of widths of the images of the injected white strips Q(λ). A white strip injected at t 0 is transformed into 2 t-t0 images with scales ḋt0 Γ t-t0 . Therefore Q(λ) = (λ/ ḋt0 ) log(2)/ log(Γ) × (1/λ log Γ), resulting in

P (C) = ḋ log 2/ log (Γ -1 ) t w B C g (t) w B (1 - C C g ) log 2 log Γ -1 = g(t) C g -C (log 2/ log Γ)-1 . (2)
P (C) thus has a power-law tail in the light gray levels whose exponent depends on the mean stretching Γ. We observe satisfactory agreement between this prediction and both experimental data and numerical 1-D simulations (see Fig. 1 (c) and Fig. 3) where for this tail P (C) ∝ (C -C g ) -α with α 2, consistent with Γ 0.5, a rather homogeneous stretching. Also note that the amplitude of the light-gray tail decreases as a power law

g(t) ∝ ḋ log 2/ log(1/Γ) t ∝ t -2(log 2/ log(1/Γ))
. We deduce

σ 2 LG = g(t) Cg Cmin (C g -C) 2-α(Γ) dC
where α(Γ) = 1 -log 2/ log Γ, and C min is the smallest concentration observed (C min = 0 for t < t white and C min = C g (1 -ḋt /w B ) for t > t white ). For t < t white the integral is constant and σ 2

LG ∝ g(t) ∝ ḋt ∝ t -2 . On the other hand, for t ≥ t white ,

σ 2 LG = g(t) 2 + α(Γ) [C g -C min ] 3-α(Γ) ∝ t -(6+2 log 2 log Γ ) .
For α(Γ) ∼ 2 as we observed, the exponent in the above power law is about -4.

Finally, the dark gray tail corresponds to boxes containing images of mixed regions from early times -thus with an important percentage of black -that have experienced little stretching. It is therefore not sufficient to consider only the mean stretching Γ as before, since stretching histories far from the mean are involved. Using the large-deviation function S for the finite-time Lyapunov exponents distribution [START_REF] Ott | Chaos in Dynamical Systems[END_REF], we derive

P (C, t) = 2 t-t0 exp[-(t -t 0 )S(-log w B t-t0 + log Γ)] (C -C ) 2 (t -t 0 ) 2 ,
where t 0 (C) is the earlier time at which the mixed region had a mean concentration C, so that

C = C (1-d t0 ) -1 .
At a fixed time the dark gray tail decreases as the dominant contribution (C-C ) -2 , however the probability to observe a fixed concentration value decays exponentially, allowing us to neglect σ 2 DG in (1). Both the (C -C ) -2 shape and the rapid fall-off of the dark-gray tail can be observed on Fig. 1 (c) and Fig. 3.

We now sum these contributions to obtain σ 2 (C). In the experiment, the crossover t white is estimated as 30 periods. However, 3-D effects inside the fluid prevented us from conducting experiments for more than 35 periods. For this early regime, fitting the data with σ 2 W ∝ (2 log t + log w B )/t 2 (black line on Fig. 1 (b)) gives good results, except close to t white where the contribution of the light gray tail starts to dominate. In contrast, in numerical simulations we observe (Fig. 1 (b)) both the (2 log t + log w B )/t 2 behavior (black line) , which can be interpreted as in the experiment, and the t -4 decay after t white (100 periods for the case studied) given by σ 2

LG . Note finally that the algebraic nature of d(t) permits crude "first-order" approximations such as considering only the mean stretching given by the Lyapunov exponent as we did. The injection process dominates other mechanisms put forward to analyze concentration distributions, such as the evolution of the distribution by self-convolution due to the random addition of concentration levels [13,14]. The strange eigenmode formalism also fails to describe this nonasymptotic regime as the spatial mixing pattern is still evolving.

In conclusion, we propose a universal mixing scenario for mixers with fixed walls belonging to a chaotic region. The injection of unmixed fluid from the boundary inside the bulk affects the whole concentration field, no matter how far from the walls. The algebraic scalings for the variance and concentration distributions can be predicted by studying in physical space the filament-like stirring pattern generated by the combination of stretching, folding, and injection of fluid from the boundaries. Our reasoning could be extended to other algebraic expressions of d(t) resulting from different hydrodynamics at the wall. In the present case (no-slip wall), we derive a power-law (asymptotically t -4 ) evolution for the decay of the concentration variance, and we find very good agreement between our analytical analysis, experimental and numerical results. Let us briefly mention that Stokes flow simulations of the well-studied blinking vortex flow [1,[START_REF] Jana | [END_REF] also yield the same algebraic decay for the variance of a coarse-grained concentration, obtained by advecting a large number of points [16]. Future work will address the generalization of our approach to other systems characterized by continuous injection of inhomogeneity, such as open flows.

FIG. 1 :

 1 FIG. 1: (a) Chaotic mixing experiment in a closed vessel: a rod moves periodically on a figure-eight path (see Fig. 2 (a)) and transforms an initial spot of dye into a complicated filamentary pattern. Inset: Poincaré section obtained numerically for the corresponding Stokes flow. Note that the chaotic region spans the entire domain. (b) Evolution of the variance of the concentration field in a fixed central region. : experiment , •: numerical simulation (see Fig. 3 for a description). Solid line fits: contribution of the white pixels σ 2 W ≃ (2 log t + log wB)/(log Γ × t 2 ) derived below. (c) Experimental concentration PDFs in the bulk at time t = 13, 17, 31 periods. Both sides of the peak can be fitted by power laws (Cmax -C) -2 (red and blue plots). Inset: left ("light-gray") tail of the peak, P (C) against |Cmax(t) -C|.

2

 2 FIG. 1: (a) Chaotic mixing experiment in a closed vessel: a rod moves periodically on a figure-eight path (see Fig. 2 (a)) and transforms an initial spot of dye into a complicated filamentary pattern. Inset: Poincaré section obtained numerically for the corresponding Stokes flow. Note that the chaotic region spans the entire domain. (b) Evolution of the variance of the concentration field in a fixed central region. : experiment , •: numerical simulation (see Fig. 3 for a description). Solid line fits: contribution of the white pixels σ 2 W ≃ (2 log t + log wB)/(log Γ × t 2 ) derived below. (c) Experimental concentration PDFs in the bulk at time t = 13, 17, 31 periods. Both sides of the peak can be fitted by power laws (Cmax -C) -2 (red and blue plots). Inset: left ("light-gray") tail of the peak, P (C) against |Cmax(t) -C|.

  FIG. 1: (a) Chaotic mixing experiment in a closed vessel: a rod moves periodically on a figure-eight path (see Fig. 2 (a)) and transforms an initial spot of dye into a complicated filamentary pattern. Inset: Poincaré section obtained numerically for the corresponding Stokes flow. Note that the chaotic region spans the entire domain. (b) Evolution of the variance of the concentration field in a fixed central region. : experiment , •: numerical simulation (see Fig. 3 for a description). Solid line fits: contribution of the white pixels σ 2 W ≃ (2 log t + log wB)/(log Γ × t 2 ) derived below. (c) Experimental concentration PDFs in the bulk at time t = 13, 17, 31 periods. Both sides of the peak can be fitted by power laws (Cmax -C) -2 (red and blue plots). Inset: left ("light-gray") tail of the peak, P (C) against |Cmax(t) -C|.

FIG. 2 :FIG. 3 :

 23 FIG.2:(a) Transport mechanisms: (i) the rod stretches and folds the mixed region and (ii) a white strip of unmixed fluid is injected between the two folded parts. (b) Stretched strips of dye (black) are smeared out by diffusion on a scale wB (gray). The concentration C of a pixel x is then given by adding the concentrations coming from strips inside a box of size wB around x.

The authors thank E. Villermaux and F. Daviaud for fruitful discussions.