DESCRIPCIÓN DE LAS CATEGORÍAS Y SUB CATEGORÍAS DE MI PRÁCTICA RECONSTRUIDA:

(j)jf Ay

equation — = — (which Jevons states in words only) to <t>y Ax

infinite quantities, on an exclusively symbolic basts.

Following this interpretation, it can be argued that the formalism of differential calculus is redundant and that it only aims at making the law of exchange literally look like the law of mechanics. To summarize this argument, it could be pointed out that Jevons does not consider

dy

that / -7- / in an indivisible expression or a singular dx

concept, but that it is the compound of /dx/ and /dy/. In this connection, it can be argued that there is no

dy proper economic concept corresponding to that of

dx in

the mechanical isotopy (i.e. that of velocity).

However, there is also evidence that Jevons (1879, pp.106-108) used two mathematical properties of a "utility function" which first are not consistent with the definition he gives of utility, and secondly, are two mathematical features defining the idea of the differential of a function according to today's standards. On these grounds, it can be argued that the mathematical entity occurring in this extract is the differential of the utility function. These properties are (a) that it is linear with respèct to the second variable; (b) that according to Jevons, the (total) utility U of a good is not separable from the exchange of a finite quantity x of this good. Consequently, in the following quotation, the two occurrences of /utility/ do not refer to the same function.

Emergence of Mathematical

Economics: The Case of Jevons

y

"Now the increment of beef, Ay, is — times

X

as great as the increment of corn, Ax, so that, in order that their utilities shall be equal,

y

the degree of utility of beef must be ^ times as great as the degree of utility of corn."

(Jevons, 1879, p . 107).

If it were the same function, in order to be correct, Jevons’ reasoning would imply that the utility function is linear with respect to scalar multiplication. It would also imply that this function is not dependent on the fixed quantity exchanged. /U(Ax)/ would indeed refer both to the utility of an increment Ax in the exchange of X and also the utility of the exchange of Ax) .f- Similarly, we consider that (a-x) . dy = ViY* dx is an equality between differentials: given the initial endowments and zero of the individual 1 in the two good exchanged, given his utility functions Qj and for each good, the equilibrium of exchange occurs for the quantities x and y of these goods such that daQ^ (x) = dgvy % (y) where duf(t) is the value at point t of the

differential of f at u.^^

Emergence of Mathematical

Economics: The Case of Jevons

NOTES TO CHAPTER 5

^ However, Zylberberg(1990) and Zouboulakis (1993) suggest that as far as France is concerned the case did not differ from the British. 2 According to R.D. Collison-Black (1970), Jevons received Walras' Principes d'une Théorie Mathématique de l'Échange in 1874. The latter helped the former with the compilation of the bibliography on mathematical economics. (Jevons, 1879, Preface to the second

edition). ,

2 R. KôneKamp (1972) reports that Jevons wrote a 'notice of a general mathematical theory of political economy'in 1862. She probably means the drafts of Jevons' (1866) article. R. Kdnekamp adds that in 1857, Jevons was to write on 'Formal Economics', a study of which there is no trace.

^ He mentions that in the literature, utility either means

numerical ratio, mental state or mass of commodities. Value either means value in exchange or in use.

^ This view is identical to G. Boole's belief that the Algebraic Calculus can be applied to disentangling theological controversies. He applied it indeed to the analysis of Clarke and Spinoza's work,

(cf. An Investigation into the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. Dover

Publication, 1854).

® In this view, Universal Arithmetic does not recognise essentially negative quantities. This explains why in the second edition of the Theory of Political Economy, he adds comments on the appearance of negative values in Economic Calculus.

These formula are commonly used in contemporary binary logic and also in mathematics as in integration theory and probability theory,

'a' and 'b ' stand for propositions. '1-a' is the negation of the proposition 'a'; it is also written '-a' in contemporary logic. The negation operator '-' is defined by the truth-table:

a 0 1 “ia 1 0

'a + b ' stands for proposition ' a or b ' (inclusive); written 'a v b' in contemporary formal logic. The '+' defined by the truth-table:

it is also operator is b'

Consequently the two formula do not have the same truth-table. In the first formula, 'a and b' can not be true at the same time, whereas it is possible in the second formula.

® This machine was constructed in 1869 and in 1962 it was displayed at the History of Science Museum in Oxford according to W. Mays

(Mays, 1962).

^ Cf. Jevons' own words (Jevons, 1879) and J. Passemore (1966, p.130).

Emergence of Mathematical

Economics: The Case of Jevons

^ ^ It seems that one of the main subject matter discussed by philosophers in the last two decades of the 19thC. in connection to Economic science was precisely this discrepancy (cf. The Monlst, vol.l to 6 and also chapter 8, section 1§4).

Some details on these authors can be found in Theocharis(1993) and in Theocharis. 1988. 'C. Courtois: an early contributor to cost-benefit analysis', History of Political Economy, 20, pp.265- 273.

However, Von Thünen's writings as well as their translation into French occur in the bibliography on mathematical economics which Jevons (1879, Appendix I, pp.301-310) attached to the second edition of the Theory of Political Economy.

For a. retrospective mathematical analysis of this aspect of Jevons' thought see Reid (1972).

^ ^ The criticism could be addressed to us that the 'economic isotopy' ought to depend on 19th c. vernacular English categories instead of 20th c. categories. However, the linguistic analysis which this definition would imply is beyond the scope of this thesis. This limitation is not as great a lacuna as it may seem at first sight. The first reason is that, the economic expressions we are concerned with belong both to the 19th c. and 20th c. economic lexica since in particular, the corpus of Neo-classical economics to which this text belongs, is a general feature of 20th c. economics (cf. Chapter 2, Section 1). The second reason is that even though we do not study 19th c. economic lexica, we replace 19th c. economics into 19th c. context (cf. Chapter 3, Chapter 4).

^ ^ The criticism could also be addressed to us that 19 th c. mathematical categories ought to be used to define the 'mathematical isotopy ' . The same answers hold here as that provided about the 'economic isotopy' (cf. previous note). First, the use of 19th c. categories is out of the scope of this thesis; and secondly 20th c. mathematical categories are grounded on 19th c. mathematics (cf. Joseph et al (1994)). In addition, we believe that, generally speaking, the reading of mathematical texts is de facto anachronistic since it relies significantly on the knowledge of contemporary mathematics.

Contrary to what is the case in the other text we analyse (cf. Chapter 6) it seems impossible to define a 'theoretical isotopy' that differs from the general semiotic function of the text.

18 Furthermore, it is considered in the History of Science that the bridging-in of connections between Geometry in particular and Physics is a defining feature of classical mechanics. This connection consisted both in considering physical space as a Euclidean space and in considering constructions and mechanical devices as providing solutions to geometrical problems (cf. Blanche, 1969; Grosholz, 1991; Holland, 1991). In addition, in Antiquity, according to Lefebre (1995) the analysis of the motions of the planets was often carried out on the pattern of the analysis of

CHAPTER 6

THE CONSOLIDATION OF MATHEMATICAL