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I was reading the preface of Marx's Mathematical Manuscripts. They explain the situation of calculus in the time of Marx, it seems that at the time analysis as we know today was still being forged by Cantor, Cauchy, Dedekind, Weierstrass. Also, it seems that there was some resistance to the appearance of calculus as we know today and Marx was reading some old books:

This more precise work was unknown in the English universities at that time. Not without reason did the well-known English mathematician Hardy comment in his Course of Pure Mathematics, written significantly later (1917): ‘It [this book] was written when analysis was neglected in Cambridge, and with an emphasis and enthusiasm which seem rather ridiculous now. If I were to rewrite it now I should not write (to use Prof. Littlewood’s simile) like a “missionary talking to cannibals”,’ (preface to the 1937 edition). Hardy had to note as a special achievement the fact that in monographs in analysis ‘even in England there is now [i.e., in 1937] no lack’.

Marx studied textbooks of differential calculus. He oriented himself with books used at courses in Cambridge University, where in the 17th century Newton held a chair of higher mathematics, the traditions of which were kept by the English up to Marx’s day. Indeed, there was a sharp struggle in the 20s and 30s of the last century between young English scholars, grouped about the ‘Analytical Society’ of mathematicians, and the opposing established and obsolete traditions, converted into untouchable ‘clerical’ dogma, represented by Newton. The latter applied the synthetic methods of his Principia with the stipulation that each problem had to be solved from the beginning without converting it into a more general problem which could then be solved with the apparatus of calculus.

In this regard, the facts are sufficiently clear that Marx began studying differential calculus with the work of the French abbot Sauri, Cours complet de mathématiques (1778), based on the methods of Leibnitz and written in his notation, and that he turned next to the De analyse per aequationes numero terminorum infinitas of. Newton (cf.ms.2763). Marx was so taken with Sauri’s use of the Leibnitzian algorithmic methods of differentiation that he sent an explanation of it (with application to the problem of the tangent to the parabola) in a special appendix to one of his letters to Engels.

Marx, however, did not limit himself to Sauri’s Cours. The next text to which he turned was the English, translation of a modern (1827) French textbook, J.-L. Boucharlat’s Eléments de calcul différentiel et du calcul intégral. Written in an ecletic spirit, in combined the ideas of d’Alembert and Lagrange. It went through eight editions in France alone and was translated into foreign languages (including Russian); the textbook, however, did not satisfy Marx, and he next turned to a series of monographs and survey-course books. Besides the classic works of Euler and MacLaurin (who popularised Newton) there were the university textbooks of Lacroix, Hind, Hemming, and others. Marx made scattered outlines and notations from all these books.

It seems that at the time, people were doing "weird" things in calculus:

Differential calculus is characterised by its symbols and terminology, such notions as ‘differential’ and ‘infinitely small’ of different orders, such symbols as dx, dy, d²y, d³y ... dy/dx³, d²y/dx², d³y/dx³ and others. In the middle of the last century many of the instructional books used by Marx associated these concepts and symbols with special methods of constructing quantities different from the usual mathematical numbers and functions. Indeed, mathematical analysis was obliged to operate with these special quantities. This is not true at the present time: there are no special symbols in contemporary analysis; yet the symbols and terminology have been preserved, and even appear to be quite suitable. How? How can this happen, if the corresponding concepts have no meaning? The mathematical manuscripts of Karl Marx provide the best answer to this question. Indeed, such an answer which permits the understanding of the essence of all symbolic calculus, whose general theory was only recently constructed in contemporary mathematical logic.

But to me, it's not clear at all what are these weird things. I tried to take a look at Sauri's book but it not only seems that the are constructing different things but the typography is a bit weird. I'd like to know more what were these weird things people did back in the time.

In particular, what opposition people had to writing "dx/dy=0/0" as is pointed out in a letter sent to Marx by Engels:

Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. And yet it is clear that dy/dx can only be the pure expression of a completed process if the last trace of the quanta x and y has disappeared, leaving the expression of the preceding process of their change without any quantity.

I know what is the modern opposition to this, but I don't know if the opposition was the same back in the day.

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    $\begingroup$ I assume they are referring to infinitesimals, or "the ghosts of departed quantities" as Berkeley called them in his attack on Newton. $\endgroup$
    – nwr
    Oct 18, 2023 at 4:42
  • $\begingroup$ @nwr Yeah, I figured out. But it seems that not only the infinitesimals were weird but also the constructions they made by "adding" more infinitesimals (assuming I understood what I read from Sauri's book). $\endgroup$
    – Red Banana
    Oct 18, 2023 at 4:48
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    $\begingroup$ I think the issue brought up is the use of limits in calculus, which wasn't in widespread use until roughly the last third of the 1800s. I've read many prefaces 1800s calculus texts in which the author specifically calls attention to using "the method of limits". $\endgroup$ Oct 18, 2023 at 16:05

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Besides infintesimals, there are other ideas and concepts that mathematicians used to work with up until roughly 1940, that have not been properly formalized in modern mainstream mathematics, like the ideas of variable quantities, constants and surrounding concepts like function of something and change/difference of a variable ($\Delta x$, or in infinitesimal form $dx$). Although people still use these words and ideas nowadays, they either use them informally without proper definition, or the modern definition doesn't match the idea of the past. The most notorious example is the word function which got a new definition between 1900 and 1940. See Who first considered the f in f(x) as an object in itself, and who decided to call it a function?

Further SE questions that might help you get a feel for these issues are:

Formalizations of the idea that something is a function of something else

Variable-centric logical foundation of calculus

If d/dx is an operator, on what does it operate?

What did Alan Turing mean when he said he didn't fully understand dy/dx?

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