Tension between Intuitive Infinitesimals and Formal Mathematical Analysis

Anticipating the problem as early as 1908, Felix Klein reflected upon the success of a calculus textbook dealing in “mystical schemes”, namely the textbook by Lübsen […] which appeared first in 1855 and which had for a long time an extraordinary influence among a large part of the public […] Lübsen defined the differential quotient first by means of the limit notion; but along side of this he placed […] what he considered to be the true infinitesimal calculus-a mystical scheme of operating with infinitely small quantities […] And then follows an English quotation: “An infinitesimal is the spirit of a departed quantity” [17, p. 216-217].

In his visionary way, Klein adds:

The reason why such reflections could so long hold their place [alongside] the mathematically rigorous method of limits, must be sought probably in the widely felt need of penetrating beyond the abstract logical formulation of the method of limits to the intrinsic nature of continuous magnitudes, and of forming more definite images of them than were supplied by emphasis solely upon the psychological moment which determined the concept of limit [17, p. 217].

In the closing months of World War II, the teenage Peter Roquette’s calculus teacher at Königsberg was an old lady trained in the old school, the regular teacher having been drafted into action. Roquette reminisces in the following terms:

I still remember the sight of her standing in front of the blackboard w[h]ere she had drawn a wonderfully smooth parabola, inserting a secant and telling us that / y x ∆ ∆ is its slope, until finally she convinced us that the slope of the tangent is dy/dx where dx is infinitesimally small and dy accordingly [26, p. 186].

This, I admit, impressed me deeply. Until then our school Math had consisted largely of Euclidean geometry, with so many problems of constructing triangles from some given data. This was o.k. but in the long run that stuff did not strike me as more than boring exercises. But now, with those infinitesimals, Math seemed to have more interesting things in stock than I had met so far [26, p. 186].

But then at the university a few years later, we were told to my disappointment that my Math teacher had not been up to date after all. We were warned to beware of infinitesimals since they do not exist, and in any case they lead to contradictions. Instead, although one writes dy/dx […], this does not really mean a quotient of two entities, but it should be interpreted as a symbolic notation only, namely the limit of the quotient / y x ∆ ∆ . I survived this disappointment too [26, p. 186-187].

Then, some decades later, the old lady turned out not to have been so far off the mark:

when I learned about Robinson’s infinitesimals [24], my early school day experiences came to my mind again and I wondered whether that lady teacher had not been so wrong after all. The discussion with Abraham Robinson kindled my interest and I wished to know more about it. Some time later there arose the opportunity to invite him to visit us in Germany where he gave lectures on his ideas, first in Tübingen and later in Heidelberg, after I had moved there [26, p. 187].

The results of the ensuing collaboration were reported in [25] and [27].

Roquette mentions an infinitesimal calculus textbook published as late as 1912, the year of the last edition of L. Kiepert [16]. He speculates [26, p. 192] that his old lady teacher may have been trained using Kiepert’s textbook.

Kiepert and other infinitesimal textbooks seem to have been edged out of the market by Courant’s textbook [6]. Courant set the tone for the attitude prevailing at the time, when he described infinitesimals as “devoid of any clear meaning” and “naive befogging” [6, p. 81], as well as “incompatible with the clarity of ideas demanded in mathematics”, “entirely meaningless”, “fog which hung round the foundations”, and a “hazy idea” [6, p. 101], while acknowledging Leibniz’s masterly use of them:

In the early days of the differential calculus even Leibnitz1 himself was capable of combining these vague mystical ideas with a thoroughly clear understanding of the limiting process. It is true that this fog which hung round the foundations of the new science did not prevent Leibnitz or his great successors from finding the right path [6, p. 101].

How is it that they were in a position to find the right path? The Russian mathematician and historian Medvedev asks the million dollar question:

If infinitely small and infinitely large magnitudes are regarded as inconsistent notions, how could they serve as a basis for the construction

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