Euler and infinity
What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can anyone elaborate?
Edit: If I remember correctly Euler's original solution to the Basel problem is as follows.
Using Taylor series for $\sin (s)/s$ we write $$\sin (s)/s = 1 - {s^2}/3! + {s^4}/5! - \cdots $$ but $\sin (s)/s$ vanishes at $\pm \pi$, $\pm 2\pi$, etc. hence $$\frac{{\sin s}}{s} = {\left( {1 - \frac{s}{\pi }} \right)}{\left( {1 + \frac{s}{\pi }} \right)}{\left( {1 - \frac{s}{{2\pi }}} \right)}{\left( {1 + \frac{s}{{2\pi }}} \right)}{\left( {1 - \frac{s}{{3\pi }}} \right)}{\left( {1 + \frac{s}{{3\pi }}} \right)} \cdots$$ or $$\frac{{\sin s}}{s} = {\left( {1 - \frac{{{s^2}}}{{{1^2}\pi^2}}} \right)}{\left( {1 - \frac{{{s^2}}}{{{2^2}{\pi ^2}}}} \right)}{\left( {1 - \frac{{{s^2}}}{{{3^2}{\pi ^2}}}} \right)} \cdots$$ which is $$\frac{{\sin s}}{s} = 1 - \frac{{{s^2}}}{{{\pi ^2}}}{\left( {\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \cdots } \right)} + \cdots.$$ Equating coefficients yields $$\zeta (2) = \frac{{{\pi ^2}}}{6}.$$
But $\pm \pi$, $\pm 2\pi$, etc. are also roots of ${e^s}\sin (s)/s$, correct? So equating coefficients does not give ${\pi ^2}/6$.
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$\begingroup$There is another way Euler "lacked rigour" in nowadays terms.
He used the idea of "something infinitesimally small" in his Introductio in analysin infinitorum (chapter 7, §115). He just gave this meaning to a variable and identified the term with its limit. So he would have said "$\frac{1}{\delta}=0$ for $\delta$ infinitely small". (This is something people use to do nowadays - at least when they aren't mathematicians.)
Clearly Euler didn't have the notions of mathematics from Cauchy, Weierstrass and so on. So it's kind of mean to say he lacked rigour. (By the way: I recommend reading (or at least browsing) the Introductio once - it is quite interesting to see how he develops all these equalities between trigonomic, rational and exponential functions.)
$\endgroup$ 5 $\begingroup$Basically they just mean that many of his arguments involving, for example, infinite products and sums are not rigorous by modern standards. Sometimes, for instance, he manipulated them in ways that make sense for finite products and sums but that we now know don’t always make sense for infinite products and sums. Fortunately, he was an extraordinarily good mathematician and had an excellent sense of when these manipulations would actually work.
In particular, Euler predates rigorous notions of convergence, so his proofs ignore convergence issues. An example can be seen in this sketch of his proof of the product formula for the zeta function: he simply carried out the infinite manipulations, but by modern standards of rigor they require some justification.
$\endgroup$ 1 $\begingroup$Euler worked before calculus was placed on rigorous foundations by Cauchy, Riemann and Weierstrass. One of his favorite techniques was to exploit analogies between polynomials and power series, viewing power series as polynomials of infinite degree. His keen intuition allowed him to avoid pitfalls, often obtaining results that could be later translated into rigorous proofs. Below is a prototypical example, excerpted from historian Judith V. Grabiner's Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus.
To respond to your question concerning Euler's treatment of infinity, note that Euler did indeed "treat infinity differently" from the way it is viewed today. Our conceptual framework today is dominated by the work of Cantor, Dedekind, and Weierstrass, who sought to eliminate infinitesimals and replace them by epsilon, delta procedures within the context of an Archimedean continuum devoid of infinitesimals. Euler, on the other hand, worked with infinitesimals galore, and used infinite numbers freely. Thus he viewed an infinite series as a polynomial of infinite order. In the terminology of the historian Detlef Laugwitz, his arguments contained some "hidden lemmas" that require further justification, which can indeed be provided in light of modern theories.
Other than that, Euler's techniques and procedural moves are closely mirrored by techniques and principles developed in the context of a hyperreal extension $\mathbb{R}\subset{}^{\ast}\mathbb{R}$, and his "infinite numbers" admit of proxies in the hyperreal approach, namely hyperreal integers in $^\ast\mathbb{N}\setminus\mathbb{N}$. Thus, an infinite series is approximated (up to infinitesimal error) by a polynomial of infinite hyperfinite degree. These can be manipulated like ordinary polynomials by the transfer principle.
Euler obviously did not have the semantic foundational frameworks developed from 1870 onward such as ZFC, but his syntactic procedures are successfully and faithfully mirrored in the hyperreal approach. Historians critical of Euler's techniques are generally ignorant of hyperreal techniques and therefore hostile toward them.
A number of articles in the literature successfully interpret Euler's procedures in terms of modern infinitesimals (with the syntactic/semantic proviso stated above), including the work of Kanovei, Laugwitz, McKinzie, Tuckey, and others.
Note that Euler did not jump from the equality of zeros to the equality of sine to the infinite product. He gave an elaborate, and essentially correct, argument in favor of such equality. More specifically, Euler provided an elaborate 7-step argument in favor of the decomposition. In fact his argument can be formalized, step-by-rigorous-step, in the framework of modern theories, as discussed in this recent article.
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