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The 'notational isotopy’ concerns the level of expression, the ’economic isotopy' and the 'mathematical isotopy' concern the level of content. The 'mechanical isotopy' concerns both the level of expression and the level of content. It is defined indeed both by the occurrence of expressions in a particular extract of the text, and also by the membership of the corpus of mechanics, which gives a specific content to these expressions. It can be considered that this isotopy is a non-vernacular extra-textual semiotic function^\ All isotopies are explicit isotopies. The identification of the first three isotopies requires no additional ability than knowledge of vernacular language. That of the 'mathematical isotopy' however, requires specialised mathematical knowledge.

The mathematical isotopy is directly connected to all other isotopies. Its connection with the •mechanical isotopy' is part and parcel of the semiotic function defining this isotopy in other words, it is, for Jevons himself, a legacy from the history of p h y s i c s .i*

Jevons defines the connection between the mathematical and the economic isotopy by using figures and symbolism to represent his arguments. Avowedly, the connection between these arguments and geometrical figures is partial, whereas it seems that he considers that their connection with symbols is comprehensive. As far as the connection between the 'notational' and the mathematical isotopy is concerned, we showed that it is to Jevons a

Emergence of Mathematical

Economics: The Case of Jevons

legacy of the history of mathematics (cf. Chapter 4, and Chapter 5, Section 301).

The connection between the 'mechanical isotopy' and the 'economic isotopy' is achieved at the level of expression with the use of vernacular expressions; /equilibrium/ and /body/ and also with the use of one notational expression in particular: / — = _z_ = 4. / in the two. dy y , .

<f>y dx X

isotopies. We shall now analyse the.role of this latter. At the mechanical isotopy levels,

d>y is

defined as the ratio of two finite forces applied at the ends of a lever (1879, pp.113-114). According to the diagram used by Jevons (1879, p.114), he considers that these forces are functions of the position of the

dy

is a ratio of finite or infinitely small fulcrum.

d.x

displacements AA' and BB' (1879, pp.111-113) IS a ratio defining the position of the fulcrum (1879, p . 113). In / — /, [ = ] is the lav/ of energy (1879, p.111)._dx In / _{dx X}— / [“ ] is the law of virtual velocity (1879, pp. 112-113) , that is, the proportionality of the

finite or infinitely small arcs (Arc%a) of a circle to the radius (R) (1879, pp.112-113). In modern trigonometric terminology, [=] is the equation ArcR=aR.

cD;c"’

is the ratio In the 'economic isotopy'.

Emergence of Mathematical

Economics: The Case of Jevons

the goods exchanged (1879, p . 113). i-S explicitly a ratio of exclusively infinitesimal quantities of the

r

commodities exchanged (1879, pp.113-114). — is the ratio of the final and finite quantities exchanged (1879, p . 114). ■ In / - ^ = — /, [ = ] is the "general form" of the

Ojc dx

definition of economic equilibrium (1879, p . 104 and p . 113). In / — = — /, [ = ] is the Law of Indifference

dx X

(1879, p. 103 and p.114). This law may be called the 'unique price hypothesis'. It states that on a market with perfect knowledge, no parts of given quantities of homogenous commodities can be exchanged in a ratio different from the ratio of the given quantities. Hence Jevons defines not only term by term correspondences between the equation of the notational isotopy on the one hand, and the mechanical and economic isotopies on the other hand; but he also defines correspondences between [ = ] at the notational level (i.e. the rule of substitution of symbols) and scientific laws. These correspondences imply a third implicit correspondence between the mechanical and the economic isotopies: (a) The idea of a 'force' and that of the 'utility' of a good (or 'value') play similar roles. ’ - This is a point Mirowski (1990) has justly noticed about 19th c. neo­ classical analysis in general. (b) Similarly, may be considered at the characteristic feature of the phenomenon (machine, exchange) which is analysed. (c) As

Emergence of Mathematical

^

Economics: The Case of Jevons

Tdy-x

far as — is concerned, the correspondence is less clear, which is confirmed by Jevo ns’ subsequent alteration of the first edition and we shall go back to it l a t e r . ( d ) The trigonometric law used in mechanics corresponds to the Law of Indifference. However, in the economic isotopy, this law, as Schabas (1990, pp.39-40} notices, is specifically an economic law, whereas in the

'mechanical isotopy', it is a mathematical law as well.2° (0) It could be argued that the correspondence between the economic law

o f equilibrium and the 'law of energy' l i m i t e d At the mechanical isotopy level, there is a slight terminological shift in the application of this law to finite quantities (law of energy) and in the application of this law to infinite quantities (law of virtual velocity). This shift does not occur in the economic isotopy.

Consequently, the correspondence between the expressions O t dy y

of the equation / / at the economic and at

0 ,)^ dx X

the mechanical level is not a perfect substitution process. We shall now analyse this limitation in the above case (c) in particular.

Jevons definitely maintains that is a ratio of infinitesimal distances. He does not identify

dv

-7- with a differential coefficient, nor with a virtual t/.rj

Emergence of Mathematical

Economics: The Case of Jevons

not an "abstract number" (1879, p . 90). However from a contemporary standpoint, the physical and trigonometric laws he uses involve such abstract numbers.21 At the economic isotopy level on the contrary, even though Jevons (1871b, p . 93) claims that "the ratio of exchange

dy']

-_dx7- is really a differential coefficient", he expounds _J his economic theory in such a way that at first glance, it does not seem to use differentials nor infinitesimals. For example, he confirms criticisms addressed to him that he does not use the integration process associated with differentials (1879, p.110). He also heavily stresses both implicitly and explicitly that the use of infinitesimals can be avoided (1879, respectively p . 114 and pp.106-108). In addition, he uses finite increments of quantities of goods exchanged when he first expounds the symbolic treatment of the theory of exchange (1879, pp.106-107) in such a way that the introduction in the subsequent passage (1879, pp.107-108) of /dx/ and /dy/ is not convincing at first. It seems that /Ax/ and /Ay/, which stand for finite quantities throughout the text, could be used just as well. No new element is added to

the equilibrium principle used in the passage where /Ax/

and /Ay/ are used, when it is applied in the passage where /dx/ and /dy/ are used. Consequently, it can be

dy considered that

Oy dx is the generalisation of the

Emergence of Mathematical