Mathemagics (a tribute to L. Euler and R. Feynman) (Q1595473)

The paper contains a lot of good examples when extension of mathematical techniques and formulas beyond their original scope leads to profound and interesting results. The author starts by showing how the properties of exponentiation \(a^b\) -- which are easy to understand when \(b\) is an integer -- lead to a natural definition of \(a^b\) for non-integer values \(b\). He then shows that Taylor expansion and natural properties of the resulting exponential function \(e^x\) can be naturally extended to complex numbers, to matrices, and to operators (including operators from quantum field theory). From there, he turns to Bernoulli numbers defined by the formula \(B^n=(1+B)^n\) (in which we perform the multiplication and then replace \(B^0=1\) and \(B^k\), \(k\geq 1\), with \(B_k\)), Bernoulli and Hermite polynomials, Heaviside's operational calculus, etc. The set of examples is impressive, and the exposition is clear and profound. What is somewhat questionable is the methodological conclusion that the author deduces from these examples. What he seems to advocate is: if necessary, use the formulas beyond their usual definitions, and do not worry about proving that this application indeed makes sense. This approach was widely spread before Cauchy's ``revolution of rigor'', and it was abandoned because it led to paradoxes. For example, the author cites Euler's derivation of the formula for the infinite sum \(S=1-1+1-1+\ldots\) from reasonable transformations. Specifically, Euler added, element-wise, this series and the same series \(0+1-1+1-1\ldots\) shifted by one element (which should have the same sum \(S\)) and ended up with \(S+S=1+0+0+\ldots=1\) hence \(S=1/2\). However, what the author does not mention is that other similarly reasonable transformations -- e.g., grouping elements together -- lead to different answers \(S=(1-1)+(1-1)+\ldots=0\), \(S=1+(1-1)+(1-1)+\ldots=1\), etc. Such counterexamples were well known to Euler and his contemporaries and the resulting -- often fruitless -- discussions were resolved only when rigor was later enforced. In short, the author's main point (and I think no one disagrees with him) is that it is often fruitful to try and apply techniques beyond their usual definitions. The author is probably right that this idea has been actively used in the past but it may be under-used now. However, the experience of mathematics shows that if we simply make such an application without any further attempt to justify it, we may sometimes end up with paradoxes and wrong answers; so, contrary to the author's implicit claim, proofs are needed.