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 ${\pi ^2}/6$ and there is no justification for writing $$\frac{{\sin s}}{s} = {\left( {1 - \frac{s}{\pi }} \right)}{\left( {1 + \frac{s}{\pi }} \right)}{\left( {1 - \frac{s}{{2\pi }}} \right)}{\left( {1 + \frac{s}{{2\pi }}} \right)}{\left( {1 - \frac{s}{{3\pi }}} \right)}{\left( {1 + \frac{s}{{3\pi }}} \right)} \cdots$$ but we can using Weierstrass factorization theorem.

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Dear glebovg, Euler was surely aware that $(e^s \sin s)/s$ had the same zeroes of $(\sin s)/s$, but presumably also understood that the former had a much faster rate of growth than the latter, and so didn't have the same "polynomial-like" behaviour. Regards, –  Matt E Oct 22 '12 at 3:04
But we can still factor ${e^s}\sin (s)/s$ just like we factored $\sin (s)/s$. –  glebovg Oct 22 '12 at 3:16
Dear glebovg, I'm not sure what you mean, but let me be more precise: for two polynomials $f$ and $g$, we know that if $f$ and $g$ have the same zeroes, and the ratio $f(x)/g(x)$ goes to $1$ as $x \to \infty$, then $f = g$. (Checking the limit at infinity removes the ambiguity scaling.) Similary, Euler's infinite product for $(\sin s)/s$, and the function $(\sin s)/s$ itself, have the same zeroes, and their ratio is well-behaved as $s \to \infty$. This is not true if we divide $e^s (\sin s)/s$ by the infinite product: while we get a zero-free function, it blows up at infinity. Regards, –  Matt E Oct 22 '12 at 3:36
P.S. I should say that of course, ultimately we know that the ratio of $(\sin s)/s$ and the infinite product is equal to $1$, but I am claiming we can prove this by making an analysis of the growth of the ratio, and my guess is that Euler would have been sensitive to this sort of phenomenon and possibility: he was, after all, one of the very greatest mathematicians of all time. Regards, –  Matt E Oct 22 '12 at 3:39

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.

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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.)

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So, in some sense, he disregarded the Archimedean property of the real numbers. –  glebovg Oct 18 '12 at 16:13
I don't think he went that far. It was just his way of explaining this intuition. He didn't thinkt about its (algebraic-)theoretical implications. Besides I'm not sure if this principle was known in his time. –  AndreasS Oct 18 '12 at 16:33
I think he must have realized that if $\delta$ is infinitesimal then multiplying it by $1/\delta$ gives 1. Perhaps he believed in the existence of infinitesimals because it makes sense in philosophy. –  glebovg Oct 18 '12 at 16:39
I believe George Berkeley wrote a book about infinitesimals and Newton's reasoning, claiming that it was nonsense. –  glebovg Oct 18 '12 at 16:46
Newton's reasoning weren't the most concise - as far as I know. George Berkley argued against infinitesimals but he argued against mathematics as a whole, as well. I'd say Euler just used this way as a "façon de parler" but could not make satisfactory sense of it - as nobody could at that time. Besides, the Introducio was a textbook. Maybe Euler simplified his reasoning there a little (although he certainly didn't think in "$\epsilon$-terms") –  AndreasS Oct 18 '12 at 17:25

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.

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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.