Euler and infinity

Question

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 ${\pi ^2}/6$.

Answer 1

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.

Answer 2

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.

Answer 3

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.)