Contributed by
Evelyn Leeper
As Asimov notes in his afterword to it (in THE EARLY ASIMOV), it is mostly about the idea of applying mathematical formulae to psychology, which he later did with his psychohistory in the "Foundation" series. A scientist develops formulae that use imaginary numbers but they're okay because the imaginaries all "square out" and just result in a sign change, but then someone develops a formula where the imaginary numbers remain, and the application of it allows an energy field to develop that could engulf the universe.

I would like to point out that this story is not nearly as ridiculous as it may at first appear to a naive reader. In fact, the concept of ``number'' is constantly expanding. In particular, let me begin by noting that only positive whole numbers were considered to be ``numbers'' originally, and that the eventual inclusions of zero, negative numbers, fractions and especially irrational numbers each met with skepticism and resistance at first by people who argued that those were not really numbers. That this was overcome is indicated by the fact that the collection of all of these together are now referred to as ``the real numbers''. Imaginary numbers, those involving the square root of negative one, on the other hand have a name that clearly indicates their present lack of acceptance. They are undeniably useful in mathematics (where they provide important unifying theories behind algebra, geometry, trigonometry and number theory) and even in physics (where quantum fields or wave functions are regularly written in terms of imaginary numbers). Though some regard these as ``tricks'', those who work with them regularly soon begin to view i as being as ``real'' a number as any other and I would not be surprised to see this eventually become common.
Moreover, let me remark that the history of mathematics is full of examples of seemingly abstract concepts eventually being observed in reality. For example, Dirac's original prediction of antimatter was entirely mathematical and abstract. Like imaginary numbers, it was done simply because it worked mathematically and made computations easier, but we now use positrons (the antimatter particle to the electron) in PET scans at hospitals around the world. NonEuclidean geometries were abstract games that seemed to stretch math beyond the real world, until confirmation of general relativity demonstrated that we need to use these ideas to understand the curvature of our own space. And these are just two of many, many examples where our understanding of reality eventually catches up with mathematics research rather than the other way around.
Yes, the idea that imaginary numbers are required to understand squid psychology may seem farfetched, as does the idea that this could be dangerous. But, as the examples above demonstrate, sometimes science discovers that even the seemingly farfetched theories are right.
First appeared in the November 1942 issue of Super Science Stories and reprinted in the 1972 collection The Early Asimov. 