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Cauchy defined infinitesimal as a variable or a function tending to zero, or as a null sequence.

While I found the definition is not so popular and nearly discarded in math according to the following statement.

(1). Infinitesimal entry in Wikipedia:

Some older textbooks use the term "infinitesimal" to refer to a variable or a function tending to zero

Why textbooks involved with the definition is said to be old ?

(2). Robert Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, P15 (His = Cauchy's) enter image description here Why says 'Even'?

(3). Abraham Robinson, Non-standard analysis, P276 enter image description here why Cauchy's definition of infinitesimal, along with his 'basic approach' was superseded?

Besides, I found most of the Real analysis or Calculus textbooks, such as Principles of mathematical analysis(Rudin) and Introduction to Calculus and Analysis(Richard Courant , Fritz John), don't introduce Cauchy's definition of infinitesimal, Why ? Why Cauchy's definition of infinitesimal was unpopular and not widely used, and nearly discarded?

P.S. I refered some papers still cannot find the answer.

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  • $\begingroup$ Because the "equivalent" epsilontic definition was considered, since the beginning of 19th Century, more rigorous. $\endgroup$ – Mauro ALLEGRANZA Mar 10 '17 at 10:21
  • $\begingroup$ Is Cauchy really giving a definition of an infintesimal there, or just the idea of one. If two infintesimals tend to zero, in what sense are they different or the same? Is one bigger than the other in any sense? $\endgroup$ – Paul Mar 10 '17 at 10:25
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    $\begingroup$ For a modern reformulation, see e.g. Vladimir Zorich, Mathematical Analysis I, Springer (2nd ed, 2016), page 137, for the definition : "function $f$ is said to be infinitesimal compared with the function $g$" and page 177 for the def of "a function $f : E → \mathbb R$ defined on a set $E ⊂ \mathbb R$ is differentiable at a point $x ∈ E$". $\endgroup$ – Mauro ALLEGRANZA Mar 10 '17 at 10:32
  • $\begingroup$ @Paul Some books and papers says Cauchy did give the definition . Does different infintesimals matter ? $\endgroup$ – iMath Mar 10 '17 at 10:36
  • $\begingroup$ @MauroALLEGRANZA Thanks! such infinitesimal is just an abbreviation for variables having limit to zero, does it make more contributions to mathematical analysis than the idea of limit ? $\endgroup$ – iMath Mar 15 '17 at 8:03
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You ask "why Cauchy's definition of infinitesimal, along with his 'basic approach' was superseded?"

The answer is that Cantor, Dedekind, Weierstrass and others developed a foundation for analysis to deal with certain difficulties related to Fourier series, uniform continuity, and uniform convergence. This development resulted in a formalisation that was a decisive moment in the history of analysis and was a great accomplishment. This is all well-known and rivers of ink have been spilt on the subject.

Yet the great accomplishment masked a significant failure that is less frequently spoken about. Namely, these 19th century giants failed to formalize an aspect of the procedures of calculus and analysis that was ubiquitous until and including Cauchy, namely the notion of infinitesimal. Instead, they provided infinitesimal-free paraphrases for the traditional definitions. For example, Cauchy's lucid definition of continuity of $y=f(x)$ ("infinitesimal change in $x$ always leads to infinitesimal change in $y$") got replaced by the familiar jargon ("for every epsilon there exists a delta such that, if $|x-c|$ is less then delta, then $|f(x)-f(c)|$, etc.").

Not only did they fail to formalize it but, unable to do so, some of them became convinced that there was something wrong with the notion of infinitesimal itself, and from this jumped to the conclusion that infinitesimals must be inconsistent or self-contradictory. Cantor went as far as publishing an article claiming to "prove" that infinitesimals were inconsistent. In correspondence Cantor referred to infinitesimals as "paper numbers", "cholera bacillus of mathematics", and even "abomination"; the details can be found in

Dauben, Joseph Warren. Georg Cantor. His mathematics and philosophy of the infinite. Princeton University Press, Princeton, NJ, 1990

and

Ehrlich, Philip. The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60 (2006), no. 1, 1–121.

A solid set-theoretic formalisation for infinitesimals did not emerge until around 1960 and by then Weierstrassian paraphrases were solidly in place, making it difficult to overcome institutional inertia.

Cauchy's idea of representing an infinitesimal by a sequence tending to zero is basically valid, but needs some polishing. Cauchy's infinitesimal specifically is dealt with in a number of articles that you can find here.

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  • $\begingroup$ Haha this is great, do you have sources for where Cantor referred to infinitesimals as those things? $\endgroup$ – enthdegree Mar 12 '17 at 9:04
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    $\begingroup$ Dauben's book on Cantor as well as P. Ehrlich's article from 2006. @enthdegree $\endgroup$ – Mikhail Katz Mar 12 '17 at 9:21
  • $\begingroup$ Thanks very much ! Is the foundation for analysis they developed the completeness of the real numbers ? $\endgroup$ – iMath Mar 15 '17 at 10:32
  • $\begingroup$ The completeness of the real numbers takes a more intuitive form in the hyperreals. Namely, it is equivalent to the existence of the standard part function on the set of finite hyperreals. Actually re-reading your comment I am not sure what you are asking. Can you elaborate? $\endgroup$ – Mikhail Katz Mar 15 '17 at 12:18
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    $\begingroup$ @iMath: those foundations were more about the $\epsilon, \delta$ definitions. Completeness has a different role to play in the traditional presentation of analysis (mainly it guarantees the existence of various limits and is the key to all significant theorems). But as Mikhail Katz mentions here, in non-standard analysis completeness is the key ingredient which allows us to get back to the reals after we have done some gymnastics with infinitesimals via the standard part function. $\endgroup$ – Paramanand Singh Feb 10 '18 at 2:46
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Because Cauchy's formulation doesn't quite correspond to what we want it to be. For example, suppose $y$ is a function of $x$. Per Cauchy, we take $dx$ to be some decreasing value. What does that mean mathematically? Decreasing with respect to what? Evidently, $dx$ is to be a function of some other variable, which I'll call $t$. $dx = dx(t)$. Calling it a sequence as one of your quotes does just restricts the domain of $t$ to the natural numbers. The requirement on the function is that some appropriate limit with respect to $t$ of $dx(t)$ is $0$.

Now $dy$ is also some function of $t$, determined by the relationship between $y$ and $x$. In particular, $dy(t) = y(x + dx(t)) - y(x)$. Well and good, but by this definition, $\frac{dy}{dx}$ is a function of both $x$ and $t$. That is not what we want. We want $\frac{dy}{dx}$ to be the derivative, which is a function of $x$ alone. By the Cauchy definition, $$y'(x) = \lim_t \frac{dy}{dx} \ne \frac{dy}{dx}$$ Since the latter depends on $t$.

So Cauchy's idea came closer to putting the idea on a solid foundation, but it still needed refining. Once suitable refinments were established, it has fallen out of favor.

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  • $\begingroup$ Paul, I am not sure your answer is historically correct. Cauchy clearly understood that when passing to the limit one needs to discard a higher order term, so this was not a shortcoming of his definition of infinitesimal. $\endgroup$ – Mikhail Katz Mar 12 '17 at 10:11
  • $\begingroup$ @MikhailKatz - Thank you, but I never even hinted that Cauchy failed to understand this. After all, he is the one who invented what you call the "familiar jargon". At best, I may have been unclear in saying "By the Cauchy definition" when what I really meant was "Using Cauchy's concept of differentials, as I have interpreted it here". I was not addressing Cauchy's understanding of the subject, which of course was impeccable, but rather why i believe the informal description of differentials he provided is no longer used. $\endgroup$ – Paul Sinclair Mar 12 '17 at 17:22
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    $\begingroup$ Paul, there is a common misconception that Cauchy invented the "familiar jargon" of epsilon-delta definitions. This is not the case. For example, in the case of continuity Cauchy gave an infinitesimal definition without mentioning epsilon-delta and without using alternating quantifiers. $\endgroup$ – Mikhail Katz Mar 13 '17 at 8:52
  • $\begingroup$ As far as Cauchy's treatment of differentials, you got it exactly backwards. While in definition of continuity Cauchy used infinitesimals, Cauchy defined the differentials in a recongnizably modern form. Here $dx$ is an arbitrary number (not necessarily infinitesimal) and $dy$ is defined by the relation $dy=f'(x) dx$. $\endgroup$ – Mikhail Katz Mar 13 '17 at 8:54
  • $\begingroup$ @MikhailKatz - I'm sorry, but you are mistaken. Cauchy gave "definitions" which used infinitesimal language, in accordance with the language used by his predecessors. But when actually proving things, he reinterpreted them into the epsilon-delta formulation. When later writers actually wrote the definitions themselves in the epsilon-delta formulations, they were merely following what he used in his proofs. Cauchy originated the concepts, and even the use of $\epsilon$ and $\delta$ in their familiar roles. $\endgroup$ – Paul Sinclair Mar 13 '17 at 16:39
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Robinson's definition of infinitesimal can be seen as a refinement of Cauchy's. In Robinson's definition, a hyperreal number is an equivalence class of sequences of real numbers, and it's a key theorem (for the relationship between standard and nonstandard analysis) that such a hyperreal number is infinitesimal (in the ordering that Robinson defines) if and only if any/all of the convergent sequences in its equivalence class converge to zero. (The equivalence class will also have some divergent sequences, although zero will still be a cluster point of all of them.) If we formalize a "variable" that converges to zero as a sequence that converges to zero, then every infinitesimal in Cauchy's sense gives rise to an infinitesimal in Robinson's sense.

Not everything is the same. I expect that if you presented Cauchy with $x = (-1)^n/n$ as a variable that converges to zero (as $n \to \infty$), then he might accept that as defining a nonzero infinitesimal that is neither positive nor negative. But every nonzero hyperreal number is either positive or negative. The choice of free ultrafilter used in Robinson's construction of hyperreals determines whether the hyperreal associated to this particular sequence is positive or negative (it is definitely nonzero), and that is completely arbitrary. But it must be one or the other.

I suppose that one would argue that $x = (-1)^n/n$ is simply not going to be useful in any argument with infinitesimals, and certainly Cauchy usually worked with infinitesimal variables such as $x = 1/n$ (which is definitely positive). But I don't know that Cauchy would never find it useful or that every argument of Cauchy's can be directly translated into an argument in Robinson's nonstandard calculus.

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  • $\begingroup$ Possibly Cauchy's definition is closer to the work of Schmieden-Laugwitz which exploited the Frechet filter instead. $\endgroup$ – Mikhail Katz Feb 10 '18 at 20:08
  • $\begingroup$ Interesting! I had not known about Schmieden & Laugwitz, but I see that Laugwitz has a chapter on it (modestly attributed entirely to Schmieden in the chapter title) in Reuniting the Antipodes. (I have just read most of this chapter on Google Books.) By the way, Laugwitz states at one point ‘there exist oscillatory infinitesimals’ (which are neither positive, negative, nor zero) in what he calls ‘the Cauchy continuum’. $\endgroup$ – Toby Bartels Feb 12 '18 at 11:41

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