I found this question:

Are Leibnizian infinitesimals thought to be logical fictions by Leibniz scholars?

The winning answers had this to say:

"This is the thickness of lines in his diagrams. Summing an infinity of such lines required to fill the figures gave the areas of the figures. But, according to Beeley, in his later communications to Leibniz, he writes they -- the lines -- have literally null thickness, he has decided. Leibniz disagrees. Leibniz's own idea in modern notation is closer to Wallis's first statement, and more precisely the above conception."

I haven't read Leibniz directly, but I read "metaphysical principals of the infinitesimal calculus" (1946), and while I am not a historian or mathematician, in my opinion it's a scholastic work reviewing Leibniz's various writings and attempting a broad philosophic understanding.

In this book, Guenon says something like this: points, lines, surfaces, and volumes are all of different species... you cannot integrate or differentiate up and down a dimension.

However elsewhere in the book I found that when you differentiate you can turn a volume into a surface, a surface into a line, and a line into a point, but the reverse (integration) is not possible. I may be misinterpreting, but this seemed bizarre.

What are the different views on this? Is my source disreputable or am I misinterpreting?

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    $\begingroup$ Please, note that René Guénon is not an historian, but a metaphysician "having written on topics ranging from metaphysics, "sacred science" and traditional studies to symbolism and initiation". Thus, you can hardly see it referenced into current works by history of mathematics' scholars ... $\endgroup$ Feb 10, 2015 at 9:50
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    $\begingroup$ You can see at least Carl Boyer, The History of the Calculus and Its Conceptual Development (1949) : the "metaphysical" discussion around the "foundations" of the calculus was exactly about those topics : if e.g. a line has $0$ thickness, how is possible, "adding" lines together (also an infinite number of them), to produce a shape with a thickness ? You will not find a satisfactory "metaphysical" answer; nevertheless, during the 19th century, the development of the theory found a rigorous mathematical foundation. $\endgroup$ Feb 10, 2015 at 9:56
  • $\begingroup$ Other references : C.H. Edwards Jr, The historical development of the calculus (1979) and Margaret Baron, The Origins of Infinitesimal Calculus (1969). $\endgroup$ Feb 10, 2015 at 10:00
  • $\begingroup$ It's not clear from your text if "you cannot integrate or differentiate up and down a dimension" is what Guenon says or your own conclusion. Either way it does not follow from "points, lines, surfaces, and volumes are all of different species". The latter is a classical position of ancient geometers still shared by many in the 17th century, but they had no problem synthesizing surfaces from lines or volumes from areas. Cavalieri's method uses it to compute volumes and areas, and Leibniz was most certainly aware of it. $\endgroup$
    – Conifold
    Feb 10, 2015 at 21:00
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    $\begingroup$ Perhaps "reverse is not possible" refers to the fact that (indefinite) integration unlike differentiation can not be done uniquely. $\endgroup$
    – Conifold
    Feb 10, 2015 at 21:03

1 Answer 1


You can see at least the recent overview by John L. Bell in SEP :

according to the intuitive view of the continuum

to be continuous is to constitute an unbroken or uninterrupted whole, like the ocean or the sky. A continuous entity — a continuum — has no “gaps”. [...] While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit. This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom — that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible.

Closely associated with the concept of a continuum is that of infinitesimal. An infinitesimal magnitude has been somewhat hazily conceived as a continuum “viewed in the small,” an “ultimate part” of a continuum. In something like the same sense as a discrete entity is made up of its individual units, its “indivisibles”, so, it was maintained, a continuum is “composed” of infinitesimal magnitudes, its ultimate parts. (It is in this sense, for example, that mathematicians of the 17th century held that continuous curves are “composed” of infinitesimal straight lines.) Now the “coherence” of a continuum entails that each of its (connected) parts is also a continuum, and, accordingly, divisible. Since points are indivisible, it follows that no point can be part of a continuum. Infinitesimal magnitudes, as parts of continua, cannot, of necessity, be points: they are, in a word, nonpunctiform.

The concept of an indivisible is closely allied to, but to be distinguished from, that of an infinitesimal. An indivisible is, by definition, something that cannot be divided, which is usually understood to mean that it has no proper parts. Now a partless, or indivisible entity does not necessarily have to be infinitesimal: souls, individual consciousnesses, and Leibnizian monads all supposedly lack parts but are surely not infinitesimal. But these have in common the feature of being unextended; extended entities such as lines, surfaces, and volumes prove a much richer source of “indivisibles”. Indeed, if the process of dividing such entities were to terminate [...] it would necessarily issue in indivisibles of a qualitatively different nature. In the case of a straight line, such indivisibles would, plausibly, be points; in the case of a circle, straight lines; and in the case of a cylinder divided by sections parallel to its base, circles. In each case the indivisible in question is infinitesimal in the sense of possessing one fewer dimension than the figure from which it is generated. In the 16th and 17th centuries indivisibles in this sense were used in the calculation of areas and volumes of curvilinear figures, a surface or volume being thought of as a collection, or sum, of linear, or planar indivisibles respectively.

Newton's during the year 1665–66 issued the invention of what he called the “Calculus of Fluxions”, the principles and methods of which were presented in three tracts published many years after they were written : De analysi per aequationes numero terminorum infinitas; Methodus fluxionum et serierum infinitarum; and De quadratura curvarum. Newton's approach to the calculus rests [...] on the conception of continua as being generated by motion. [...] In the Methodus fluxionum Newton makes explicit his conception of variable quantities as generated by motions, and introduces his characteristic notation.

Newton later became discontented with the undeniable presence of infinitesimals in his calculus, and dissatisfied with the dubious procedure of “neglecting” them. In the preface to the De quadratura curvarum he remarks that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities [...].

Newton developed three approaches for his calculus, all of which he regarded as leading to equivalent results, but which varied in their degree of rigour. The first employed infinitesimal quantities which, while not finite, are at the same time not exactly zero. Finding that these eluded precise formulation, Newton focussed instead on their ratio, which is in general a finite number. If this ratio is known, the infinitesimal quantities forming it may be replaced by any suitable finite magnitudes—such as velocities or fluxions—having the same ratio. This is the method of fluxions. Recognizing that this method itself required a foundation, Newton supplied it with one in the form of the doctrine of prime and ultimate ratios, a kinematic form of the theory of limits.

Leibniz's essays [devoted to the development of the infinitesimal calculus] Nova Methodus of 1684 and De Geometri Recondita of 1686 may be said to represent the official births of the differential and integral calculi, respectively. His approach to the calculus, in which the use of infinitesimals, plays a central role, has combinatorial roots, traceable to his early work on derived sequences of numbers. Given a curve determined by correlated variables $x, y$, he wrote $dx$ and $dy$ for infinitesimal differences, or differentials, between the values $x$ and $y$: and $dy/dx$ for the ratio of the two, which he then took to represent the slope of the curve at the corresponding point. This suggestive, if highly formal procedure led Leibniz to evolve rules for calculating with differentials, which was achieved by appropriate modification of the rules of calculation for ordinary numbers.

Although the use of infinitesimals was instrumental in Leibniz's approach to the calculus, in 1684 he introduced the concept of differential without mentioning infinitely small quantities, almost certainly in order to avoid foundational difficulties.

Leibniz's attitude toward infinitesimals and differentials seems to have been that they furnished the elements from which to fashion a formal grammar, an algebra, of the continuous. Since he regarded continua as purely ideal entities, it was then perfectly consistent for him to maintain, as he did, that infinitesimal quantities themselves are no less ideal—simply useful fictions, introduced to shorten arguments and aid insight.

Although Leibniz himself did not credit the infinitesimal or the (mathematical) infinite with objective existence, a number of his followers did not hesitate to do so. Among the most prominent of these was Johann Bernoulli [...].

  • $\begingroup$ "Finding that these eluded precise formulation, Newton focussed instead on their ratio, which is in general a finite number." I thought the leibniz notation was misleading since it implied dy/dx was a fraction but it really wasn't $\endgroup$ Feb 11, 2015 at 6:27
  • $\begingroup$ "[leibniz] regarded continua as purely ideal entities" I would argue the opposite, numbers are ideal, yet continuity is premised on measurement, which is the real world. But then maybe it's incompatibility between measurement and numbering. measuring (say length for example) has the limitation of the instrument, and even given an indefinitely precise methods of measurement it can potentially be irrational. And irrational are sort of incompatible with number, at least early definitions of numbers. Even non-reciprocal fractions are questionable for guenon, which explains Egyptian unit fractions $\endgroup$ Feb 11, 2015 at 7:01

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