Interpreting the Infinitesimal Mathematics of Leibniz and Euler

Abstract

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

Appendix: Analysis of Euler’s Proof

In Sect. 3.5 we summarized Euler’s derivation of the product decomposition for sine. The derivation of infinite product decompositions (9) and (10) as found in (Euler 1748, §156) can be broken up into seven steps as follows. Recall that Euler’s i is an infinite integer.

Step 1 Euler observes that

$$\begin{aligned} 2\sinh x= & {} e^x-e^{-x} = \left( 1+\frac{x}{i}\right) ^{i}-\left( 1-\frac{x}{i}\right) ^{i}, \end{aligned}$$

(23)

where i is an infinitely large natural number. To motivate the next step, note that the expression $x^i-1=(x-1)(1+x+x^2+\cdots +x^{i-1})$ can be factored further as a product $\prod _{k=0}^{i-1}(x-\zeta ^k)$, where $\zeta =e^{2\pi \sqrt{-1}/i}$; conjugate factors can then be combined to yield a decomposition into real quadratic terms.

Step 2 Euler uses the fact that $a^{i}-b^{i}$ is the product of the factors

$$\begin{aligned} a^2+b^2-2ab\cos \frac{2k\pi }{i},\quad \text {where}\quad 1\le k<\frac{i}{2}, \end{aligned}$$

(24)

together with the factor $a-b$ and, if i is an even number, the factor $a+b$, as well.

Step 3 Setting $a=1+\frac{x}{i}$ and $b=1-\frac{x}{i}$ in (23), Euler transforms expression (24) into the form

$$\begin{aligned} 2+2\frac{x^2}{{i}^2}-2\bigg (1-\frac{x^2}{{i}^2}\bigg ) \cos \frac{2k\pi }{i}\,. \end{aligned}$$

(25)

Step 4 Euler then replaces (25) by the expression

$$\begin{aligned} \frac{4k^2\pi ^2}{{i}^2} \bigg (1+\frac{x^2}{k^2\pi ^2}-\frac{x^2}{{i}^2}\bigg ), \end{aligned}$$

(26)

justifying this step by means of the formula

$$\begin{aligned} \cos \frac{2k\pi }{i} = 1-\frac{2k^2\pi ^2}{{i}^2}. \end{aligned}$$

(27)

Step 5 Next, Euler argues that the difference $e^x-e^{-x}$ is divisible by the expression

$$\begin{aligned} 1+\frac{x^2}{k^2\pi ^2}-\frac{x^2}{i^2} \end{aligned}$$

(28)

from (26), where “we omit the term $\frac{x^2}{{i}^2}$ since even when multiplied by i, it remains infinitely small” Euler (1988).

Step 6 As there is still a factor of $a-b=2x/{i}$, Euler obtains the final equality (9), arguing that then “the resulting first term will be x” (in order to conform to the Maclaurin series for $\sinh x$).

Step 7 Finally, formula (10) is obtained from (9) by means of the substitution $x\mapsto \sqrt{-1}\,x$.

Euler’s argument in favor of (9) and (10) was formalized in terms of a proof in Robinson’s framework in Luxemburg (1973). However, Luxemburg’s formalisation deviates from Euler’s argument beginning with steps 3 and 4, and thus circumvents the most problematic steps 5 and 6. A proof in Robinson’s framework, formalizing Euler’s argument step-by-step throughout, appeared in the article Kanovei (1988); see also McKinzie and Tuckey (1997) as well as the monograph (Kanovei and Reeken 2004, section 2.4a). This formalisation interprets problematic details of Euler’s argument on the basis of general principles in Robinson’s framework, as well as general analytic facts that were known in Euler’s time. Such principles and facts behind some early proofs exploiting infinitesimals are sometimes referred to as hidden lemmas in this context; see Laugwitz (1987a, 1989), McKinzie and Tuckey (1997).

For instance, a hidden lemma behind Step 4 asserts, on the basis of the evaluation of the remainder R of the Taylor expansion

$$\begin{aligned} \cos \frac{2k\pi }{i}=1-\frac{2k^2\pi ^2}{i^2}+R\;, \end{aligned}$$

that the quadratic polynomial $T_k(x)=2+2\frac{x^2}{{i}^2}-2\big (1-\frac{x^2}{{i}^2}\big ) \cos \frac{2k\pi }{i}$ as in (25) admits the representation

$$\begin{aligned} T_k(x)= C_k\,\big (U_k(x)+p_k\cdot x^2\big ), \end{aligned}$$

where $C_k$ and $p_k$ do not depend on x while

$$\begin{aligned} U_k(x)= 1+\frac{x^2}{k^2\pi ^2}-\frac{x^2}{i^2}, \end{aligned}$$

and for any standard real x and any finite or infinitely large integer $k\le \frac{i}{2}\,$ the following holds:

• (1) if k is finite then $p_k$ is infinitesimal, and

• (2) there is a real $\gamma$ such that $|p_k|<\gamma \cdot k^{-2}$ for any infinitely large $k\le \frac{i}{2}\,$.

This allows one to infer that the effect of the transformation of step 4 on the product of factors (25) is infinitesimal. See (Kanovei 1988, §4) as well as equation (11) on page 75 in Kanovei and Reeken (2004) for additional details.

Some hidden lemmas of a different kind, related to basic principles of nonstandard analysis, are discussed in McKinzie and Tuckey (1997, 43ff; see below).

What clearly stands out of Euler’s argument is his explicit use of infinitesimal quantities such as (25) and (26), as well as the approximate formula (27) which holds “up to” an infinitesimal of higher order. Thus, Euler exploited bona fide infinitesimals, rather than merely ratios thereof, in a routine fashion in some of his best work.

We now provide further technical details on a hyperreal interpretation of Euler’s proof of the product formula for the sine function. Our goal here is to indicate how Euler’s inferential moves find modern proxies in a hyperreal framework.

We discuss the hidden lemmas related to basic principles of nonstandard analysis following McKinzie and Tuckey (1997, 43ff), where it is argued that the Euler sine factorisation and similar constructions are best understood in the context of the following hidden definition in terms of modern nonstandard analysis. The following definition is borrowed from McKinzie and Tuckey (1997, 44).

Definition. A sum $a_1+ a_2 + a_3 + \cdots$ is Euler-convergent (E-convergent) if and only if

• (i) $a_k$ is defined by an elementary function,¹³

• (ii) for all infinite¹⁴ J, the sum $a_1+ a_2 + \cdots +a_J$ is finite, and

• (iii) for all infinite pairs $J<K$, the sum $a_J+ a_{J+1} + \ldots +a_K$ is infinitesimal.

Similarly, a product $(1+b_l)(1+b_2)(1+b_3)\ldots$ is Euler-convergent if and only if (i) $b_k$ is defined by an elementary function, (ii) for all infinite J, the product $(1+b_1)(1+b_2)\ldots (1 + b_J)$ is finite, and (iii) for all infinitely large $J<K$, the product $(1 + b_J)(1 + b_{J+1})\ldots (1 + b_K)$ differs infinitesimally from 1.

Next, McKinzie and Tuckey present a series of hidden lemmas implicit in Euler’s argument. The first such hidden lemma asserts that if the sums $a_1+ a_2 + \cdots$ and $b_1+ b_2 + \cdots$ are E-convergent and $a_k\simeq b_k$ (meaning that $a_k-b_k$ is infinitesimal) for all finite k, then

$$\begin{aligned} a_1+ a_2 + \cdots +a_N\simeq b_1+ b_2 + \cdots +b_N \end{aligned}$$

for all N finite and infinite. To prove this lemma, it suffices to note that if $a_k\simeq b_k$ holds for all finite k, then, by Robinson’s lemma (see e. g. Theorem 2.2.12, in Kanovei and Reeken 2004, 62), there is an infinite K such that $a_1+\cdots +a_k\simeq b_1+\cdots +b_k$ holds for all $k\le K$.

The second hidden lemma asserts a similar property for products. The third hidden lemma asserts that if, for all finite x, the sums

$$\begin{aligned} f(x)=a_0+a_1x+a_2x^2+\cdots \quad \text {and}\quad g(x)=b_0+b_1x+b_2x^2+\cdots \end{aligned}$$

are E-convergent and we have [$f(x)\simeq g(x)$]. This means that $a_0+a_1x+a_2x^2+\cdots +a_Jx^J\simeq b_0+b_1x+b_2x^2+\cdots +b_Kx^K$ for all infinite J, K. Note that the choice of J, K is immaterial by (ii) and (iii) of the definition of E-convergence. Then $a_n\simeq b_n$ for all n finite and infinite. A detailed analysis in McKinzie and Tuckey (1997) shows that these three lemmas, together with an additional sublemma, suffice to formalize Euler’s derivations step-by-step in a hyperreal framework.