Wigner's effective mathematics and contradiction

TL;DR

This paper examines an inconsistency in complex phase angles within Kirchhoff's scalar diffraction theory to explore implications for Wigner's view on the effectiveness of mathematics in physics.

Contribution

It introduces and analyzes an inconsistency in complex phase angles and studies its impact on the physical predictions of Kirchhoff's diffraction theory.

Findings

Excluding the contradictory phase angle allows a nonzero diffraction amplitude.

Including the contradictory phase angle results in a zero diffraction amplitude.

The experiment can distinguish between the inclusion or exclusion of the phase angle.

Abstract

Complex numbers are basic. An inconsistency would question Wigner's unreasonable effectiveness of mathematics. A vehicle to study this question is Kirchoff's scalar diffraction theory. In the paper, an inconsistency in complex phase angle is presented. When this inconsistency is introduced in Kirchoff's theory we can study its influence on the experimental success of this theory. There are no \emph{a priori} reasons to include or exclude phase angles. Referring to Wigner, an experiment can provide more insight. In the experiment a weak intensity, small wavelength source can be employed. When the contradictory phase angle is excluded, a nonzero diffraction amplitude appears physically possible. If it is included, this amplitude vanishes.