What does this Jacques Hadamard quote mean?
The shortest path between two truths in the real domain passes through the complex domain.
Is this a philosophical statement? what is its mathematical background?
It's actually misquoted. From: http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html
A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".
Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)
(TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.
So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.
An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]
it seems very likely this quote means something in the spirit of:
Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.
The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!
And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.
Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.
I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.
Just to illustrate the quotation from Hadamard: The exponential function exp(z) satisfies for complex arguments z, w the addition theorem
exp(z+w) = exp z * exp w
The theorem follows from the definition of the exponential function as a power series and the binomial formula. To derive an addition theorem for trigonometric functions like sinus and cosinus one uses the Euler formula
exp(iz) = cos z + i * sin z
Here „i“ denotes the imaginary unit satisfying
i**2 = -1
The addition theorem
exp(z+w) = exp z *exp w
cos(z+w) + i * sin(z+w) =
cos z * cos w - cos z * cos w + i * (sin z * cos w + cos w * sin z)
Specialize to real arguments x=z, y=w and compare real- and imaginary part. You get the well-known additions theorems of real trigonometric functions:
cos(x+y) = cos x * cos y – sin x * sin y
sin(x+y) = sin x * cos y + cos x * sin y
This computation illustrates Hadamard's statement
„the shortest and best way between two truths of the real domain often passes through the imaginary one.“