How Philosophy Enters Mathematical Reasoning
Marx, Peano, and Differentials
Providence, RI 02918
One of the important philosophical steps in the history of the calculus was the replacement of the differential by the derivative as the fundamental concept of infinitesimal analysis. This process was carried out by Lagrange and Cauchy, but had its beginnings with Euler. Behind it was the foundational problem posed by differentials, for which there were self‑contradictory claims. Before this replacement was made, the foundational problem could hardly have been solved; after it was made, the problem of the interpretation of the differential still did not have a satisfactory solution until near the end of the 19th century, when Karl Marx, working independently in Londonand without knowledge of the foundational work that had been done by Cauchy and later mathematicians on the Continentarrived in 1881 at the concept of the differential as an operational symbol for taking derivatives.
This concept could not have been achieved in the time of Leibniz. As Henk Bos [p. 4] has pointed out: "There are three processes in the history of analysis in the 17th and 18th centuries which are of crucial importance for the history of the concept of the differential. The first is the introduction, in the 1680's and 1690's of the Leibnizian infinitesimal analysis within the body of the Cartesian analysis, which at that time may be characterized as the study of curves by means of algebraic techniques". The second, according to Bos, was the separation of analysis from geometry, which took place in the first half of the 18th century. The third, just mentioned, was the replacement of the differential by the derivative as the fundamental concept of infinitesimal analysis. He then shows that, in the Leibnizian calculus, the derivative would have had to be interpreted as a ratio that was correlated to a variable having the dimension of length. This implies that the operation of derivation cannot be repeated in a natural way because it is not clear what sort of quantity it would correlate with a ratio. "Thus the derivative could not occur in the geometrical phase of the infinitesimal calculus" [Bos, p.8].
Although Marx was not aware of contemporary developments in the foundations of calculusindeed he began his study with an 18th century text (of Abbé Sauri)the basic concept for himwas the derivative and he said of the differential: "dy = f'(x)dx appears to us as another form of dy/dx = f'(x) and is always replaceable by the latter" [Marx, p.62]. What. then, do the symbols dy and dx represent? Marx answered this question by means of a dialectical analysis of what happens in mathematics in the crossing over from algebra to a differential calculus. (On Marx's approach to this question, cf. Kennedy .) This aspect of Marx's study was already brought out by V. I. Glivenko in 1934. one year after the first publication of a part of Marx's mathematical manuscripts [cf. Kennedy 1978]. Glivenko [p. 85] concludes: "As a result of his investigations also appears the concept of the differential calculus as its own kind of algebra, constructed over the usual algebra and containing, besides numbers, differential symbols" and (referring to the opening pages of Hadamard's Cours d'Analyse of 1927) finds confirmation that "mathematicians, too, are beginning to arrive at such a concept of the general character of the differential calculus".
The philosophical question. however. remains: What is it that is reflected by the symbols dy and dx? One answer is that the differential is the principal linear part of an increment. Thus. if y = f(x) and ∆y is the increment brought about in y by an increment ∆x of x, then the principal linear part of ∆y is dy = f'(x), ∆x. (In this context, the increment of x is necessarily linear, so that dx = ∆x.) This idea goes back at least to Euler and, according to S. A. Yanovskaya, editor of Marx's Mathematical Manuscripts. Marx was aware of it, and of course it is known to all later mathematicians. But this interpretation is valid only for first order differentials and only for functions of a single independent variable. The difficulty shows up in the case of functions of two variables, each of which is a function of another variablea case studied by Marxand it shows up even more strikingly in the attempt to define second order differentials, as Glivenko pointed out.
Thus, according to this interpretation, if y = f(x), then dy = f’(x) ∆x and d²y = d[f’(x)∆x]. Following the usual rule for differentiating products, since ∆x is independent of x, the derivative of f'(x) ∆x is f'(x)∙0 + f''(x) ∙∆x, so that, since dx = ∆x,
(1) d²y = f''(x)dx².
But if x is a function of t, then by the chain rule for differentiating composite functions, we have dy/dt = f'(x)(dx/dt), and a second differentiation leads to
(2) d²y/dt² = f'(x)(d²x/dt²) + f’’(x)(dx/dt) ².
Multiplying through by dt² yields
(3) d²y = f'(x)d²x + f''(x)dx², which does not agree with (1).
The dangers arising from such difficulties are beautifully illustrated in a story told by Henri Poincaré in 1899. He says he was present at an examination at which the candidate explained the theory of the speed of sound as follows: "We have to integrate the equation d²z/dt² = a²(d²z/dx²). I divide by d²z and multiply by dx². I then have (dx/dt)² = a² from which dx/dt = ± a, which proves that sound may be propagated in both directions with speed a." According to Poincaré, the examiner, an excellent physicist whom he does not name, replied: "That's remarkable; your proof is much simpler than all those I know," and he gave him a mark of 19 of a possible 20 [Poincaré 1899; quoted in Peano 1957, p. 384].
The concept of the differential as an operational symbol in the sense of Marx‑Hadamard can be extended to second order differentials and Hadamard proposed to write (3) as designating that and only that which holds in (2), whatever the functional dependence of the variables x and y on the parameter t. Thus, as Glivenko [p. 84] remarks: "The concept of the differential as the principal linear part of an increment turns out to be an interpretation useful only in certain special cases. . . . The result at which we have arrived may explain just this, that precisely the operational concept of the differential calculus correctly and completely reflects reality," and he adds in a footnote: "Even if only because in reality there are no absolutely independent variables".
The anecdote by Poincaré was quoted by Giuseppe Peano  in an article proposing a radical solution of the problem of the concept of the differential. Peano simply identified the differential with the derivative: "Modern texts of infinitesimal analysis usually define the derivative of a function is the limit of an incremental ratio. They then define the differential of a function as the product of its derivative and the differential of the independent variable. This latter is defined as an arbitrary quantity, constant or variable, or as an increment of a variable. finite or infinitesimal: and the infinitesimal is variously treated. Some authors, such as Todhunter, Veblen, consider dy'dx as a symbol to indicate the derivative, indecomposable into the elements dy and dx. The affair becomes much simpler if differential is defined as synonymous with derivative. The identity between differential and derivative will be explained here with logical and historical arguments. The very simple logical argument i's that wherever differential is written, one may read derivative, and the truth of the proposition remains" [Peano 1957. p. 369].
Thus far, Peano would seem to be in agreement with the operational view just described. at least to the extent of saying that differential formulas have just the same content as the corresponding derivative formulas. But I think he goes too far in suggesting that Leibniz, for example, thought in derivatives and not differentials, thus attributing to Leibniz the sophisticated thought processes of Poincaré : "As for myself, I ordinarily use the differential notation, first because it is the language most of my contemporaries speak, and then for the small practical reasons just mentioned. But if I write in differentials, most often I think in derivatives" [quoted in Peano 1957, p. 383].
Though this sophistication was possible for Poincaré, the historical reasoning given by Bos makes it seem hardly possible for Leibniz to have thought in the same terms. But I suggest that Peano's error was due less to any lack of concern for historical accuracy than to a lack of consideration of the philosophical question concerning the meaning of dy and dx. Peano made no pretence of being a philosopher and. indeed, denied competence in this field. Fearing, perhaps. the excesses of the 'schools' then current in the philosophy of mathematics (formalism, logicism, intuitionism), he drew back from a philosophical discussion even of the concept of numbereven though he is best known for his Postulates for the Natural Numbers.
I am suggesting that it was Peano's failure to consider philosophical questions that allowed him to fall into the historical error regarding Leibniz. This is the other, and necessary, side of the touchstone: “Objects are best understood in terms of their historical development'' [Adler, p. 59]. An understanding of the historical context helps us appreciate the philosophical questions, a concern for philosophical problems alerts us to historical possibilities. Peano's viewpoint further ignores the fact that philosophical questions of mathematics are also relevant to an understanding of more general philosophical problems. As the author of the article on "Mathematics" in Rozental’s Philosophical Dictionary, [p. 230] wrote: "The philosophical questions of mathematics . . . have always appeared in the arena of the struggle between materialism and idealism."
Adler, I. 1980 "Basic concepts of dialectical materialism," Science and Nature No. 3: 58‑59
Bos, H. J. M. 1974 "Differentials. higher order differentials and the derivative in the Leibnizian calculus," Arch. Hist. Exact Sci. 14: 1‑90.
Glivenko, V. I. 1934 "Ponyatie Diferentsiala u Marksa i Adamara," Pod Znamenem Marksizina Nr. 5: 79‑85.
Kennedy, H. C. 1977 "Karl Marx and the foundations of differential calculus." Historia Mathematica 4: 303‑318,
Kennedy, H. C. 1978 "Marx's mathematical manuscripts," Science and Nature No. 1: 59‑62.
Marx, K. 1968 Matematicheskie rukopist (Moscow, Nauk).
Peano, G. 1912 "Derivata e differenziale," Atti Accad. sci. Torino 48: 47‑69
Peano, G. 1957 Opere Seelte Vol. 1.
Poincaré, H. 1899 "La notation differentielle et l'enseignement," Enseignement mathématique 1 : 106‑110.
Rozental, M. M. ed. 1975 Filosofskii Slovar. 3rd ed. (Moscow).
SOURCE: Kennedy, Hubert. "Marx, Peano, and Differentials," Science & Nature, no. 5 (1982), pp. 39-42.
Science and Nature, Table of Contents, issues #1-10 (1978-1989)
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