Tomorrow is Sunday, and as I prepare to mount my plastic pulpit I will take as my text the introduction to Chapter 5 (Complex Numbers) of A Survey of Modern Algebra by Birkhoff&Mac Lane.  This is a classic and accessible work, first published in 1941, which brought to the American-speaking world what was previously locked up in Van der Waerden’s Moderne Algebra (1931).  The chapter opens with the definition of a complex number and the field C [1], and then continues:
We owe the above definition not to divine revelation, but to simple algebraic experimentation.  First, it was observed that the equation x2 = –1 had no real root (x2 being never negative).  This suggested inventing an imaginary number “i”, satisfying i2 = –1, and otherwise satisfying the ordinary laws of algebra.
Recently, I mentioned how the spin-doctor Alastair Campbell manipulated Tony Blair, saying  “We don’t do God”.  When mathematicians or scientists come out with statements like the one quoted, my reaction is “We don’t do NoGod, either”.  It gives off a rank odour of hubris, as if saying “We, the algebraists, are the Great Ones, and the complex numbers are our creation.”  Not only have I found in practice that this makes for a bad learning experience, but to a complex number jihadi like myself, this is foul heresy.  I would like to explain why, but to lay the ground I will start by showing that their historical analysis is utter baloney, of the type I refer to as

### Piltdown History

I came of age during that cultural K-T boundary layer known in Britain as the “Swinging Sixties”, but starting assimilating scientific knowledge in the 50s, starting with a Pictorial Encyclopaedia which still treated Piltdown Man as real, rather than a forgery.  As the Wikipedia article explains, the Piltdown “skull” fitted many prejudices of the time; putting the man’s cranium onto the apes’ jaw parts (rather than a human-like skull with a relatively small braincase) fitted the idea that it was somehow our mighty brain that drove forward our evolution.  Perhaps as part of the prevailing world-view, the then current version of the history of algebra, especially as expounded by Van der Waerden, was likewise inverted.

In fact, the emergence of complex numbers did not arise from simple quadratic equations, but through the efforts of Italian mathematicians wrestling with cubics, culminating with Rafael Bombelli’s Algebra published in 1572.  Thinking à la positiviste, the “divine revelation” would correspond to Bombelli’s “wild thought” which allowed him to solve an “irreducible” cubic equation with three real roots, but which required the determination of a complex cube root to do so. [2]

Something of the same bad attitude is found in the off-quoted statement of http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kronecker.html Leopold Kronecker
God created the integers, all else is the work of man.
(One of my students quoted that in relation to complex numbers at the head of an essay.  Don’t worry, I didn’t reach for my scimitar, nor even mark him down over it.)

Kronecker’s relations with his colleagues were certainly “chronic”.  Lindemann proved that π is transcendental in 1882, thereby squashing forever any idea of squaring the circle by a Greek-style ruler and compasses construction.  In a lecture given four years later, Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist. He opposed the theory of irrational numbers (now standard) brought by Dedekind, and was against any kind of ‘infinity’ (even infinite series!), making life difficult for Cantor, who gave us all aleph-null und so weiter, und so weiter ...

The master rigorist Edmund Landau thought that analysis and algebra were the only “true” mathematics, and dismissed not only applications, but even trigonometry and geometry as

This means “grease” or “lubricating oil”.  A charitable interpretation might be that it wasn’t a total dismissal, but allowed them the role of “oiling” the mind for what he regarded as real maths, though the structure of his master textbook suggests otherwise.

However, Kronecker’s statement about the integers does not even deserve such a complimentary term.  The creation, even in the least theistic interpretation of that word, is based on complex numbers.  Schrödinger’s wave equation and the Pauli exclusion principle, both of which show how the atoms of your body don’t fall in on themselves, both operate over the complex domain.  Kronecker’s statement is not only plain wrong, it deserves an even stronger epithet, sounding a little bit like Schmieröl, aber aus dem deutschen Skatologie.

[1] The names for the systems of natural numbers (N), all integers (Z), rational numbers (Q), real numbers (R), and complex numbers (C) are easily retained through the mnemonic Nine Zulu Queens Ruled China!

[2] in increasing order of mathematical heaviness, for this I recommend
Imagining Numbers: (particularly the square root of minus fifteen) by Barry Mazur (ISBN 0312421877)
Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire (ISBN 0452288533)
An Imaginary Tale: The Story of "i" [the square root of minus one] by Paul J. Nahin (ISBN 0691127980)