Physical intuition in the imaginary world

Paging through a wonderful book “An imaginary tale: The story of \(\sqrt{-1}\)” by Paul J. Nahin (strongly recommended!), I discovered this episode of history.

On 18 October 1740 Euler wrote to John Bernoulli that the solution to differential equation of a harmonic oscillator

\(y”+y=0\),  \(y(0)=2\), \(y'(0)=0\)

can be written in two ways:

\(y(x) = 2 \cos x\)

and

\(y(x) = e^{ix} + e^{-ix}.\)

He concluded from that

\(2\cos x =e^{ix} + e^{-ix}.\)

which was first step to his famous formula.

Obviously, Euler was using the uniqueness of a solution with given initial values. I bet his belief in the uniqueness was rooted in physical intuition. For him, expansion of mathematical language did not change his vision of the world.

Perhaps, 2oth century physicists weret thinking that an “imaginary” solution corresponds to something in the real world, something that was not discovered yet.