In his essay on Couples or Conjugate functions which form the second part of his Science of Pure Time, Hamilton sets out to establish the imaginaries as part of his developmental sequence for the various magnitudes. In so doing he introduces the complex pair of ratioed moments as functions of conscious apprehension. That is to say, in line with his development thus far he sets up a natural relationship between pairs of moment pairs! This is of course a fractal pattern, but what Hamilton was doing was laying a complex structure of relationships for subjective experiences of time, starting with the fundamental experience of a moment, and then the comparison of pairs of moments, from which he elegantly derives the real magnitudinal stucture as alimit process of the rationals magnitudinal structure he establishes on the ordinal relationships embedded in the flow of time.
At one level this is pure sophistry and at another it is pure genius, it is certainly magical, and involves a certain amount of mystification and misdirection for a good cause! It provides Charles Lewis Dodgson with all he needed to write his sarcastic critique Alice in Wonderland, in which all the characters perforce represented leading mathematicians of the day including Hamilton, possibly as the Mad Hatter.
Lewis Carrol, his nomme de plume, introduced the cheshire cat into his story, an ebdearing creature who disappeared leving only its smile. This was another characterisation of the behviour of the imaginary numbers, real, but are they really there? All we ever really see are their smile!
Hamilton pulls off another clever trick by looking at pairs of pairs of moments. We could consider the pairs to be pairs of epochs, eventful periods of time, Each Epoch is of course independent of any other, yet obey the "laws of time", especially now Hamilton has demonstrated them equivalent to the real numbers. to compare these independent epochs Hamilton used the newish idea of a Function. With this idea he was able to establish a functional relationship bbetween the epochs so that a differential equation expressed that relationship. One has to ask Why go this way? The answer is simple, he caan pull off a trick which hides the underlying construction of the x, y coordinate plane. However, despite this trickery, he in fact draws upon the vector algebra he has been unconsciously building from the start, and reveals the one to one correspondence between the coordinate pairs and the properties of the algebraic or rather combinatorial functions. He is able to establish a relationship between the coordinate pairs 1,0 and 0,1 that reflects the vector relationship between a, *1 ans a*√-1. This is a pure analogy whuch he can then found by a definition, making the imaginaries a complex functional anlogy of the real s, and dependent on the one flowing magnitude of time. As i said, it is very clever stuff, but it is sophistry, and Carrol sensed this, hence his book.
Schroedinger took the cat and used it to explain complex magnitudes rasther than tp mock them! And in addition he sets out to Kill the Cheshire Cat!
It is Minkowski's decision to adopt complex magnitudes which is the reason why Schroedinger comes upon his great Equation, but for Dodgeson it was a step too far, and he attacks what he sees as the root of the problem in his second book. Alice through the Looking Glass. He sees the problem arising because mathematicians allowed Negative magnitudes into the fold!
We have to return to Brahmagupta to unserstand the intoduction of negative ciphers, or numeral signs, and we have to divest the thin covering og "mathematics", to reveal its core stem which is astrology. And in astrological terms the introduction of "negatice" marks makes no sense unless you understand them as marks of good and bad fortune. While good and bad as adjectival terms are broadly antithetical, they are by no means "polar- opposites" or even poles on the same spectrum. This conventional fallacy underpins the insecurity in applying such a model analogically, and especially to magnitudes. Nevertheless this is precisely what we have done over the millenia.
Having done this, how do we make sense of it? The aphorisms that Brahmagupta discusses are for a wholly different purpose than creating a "number "system. His aim was to redress an imbalance in Philosophical interpretation, which he saaw as becoming too hellenistic, and not Brahman. Nevertheless his comments found an anlogous use in the financial dealings of humans, where "money" was already an unreal concept of fortune. It was not the concept that was being used, but the method of application. By this method, another individual could be assigned misfortune, which could only be reassigned by an accumalation of fortune. From the outset then, these astrological notions were not welcomed!
Some 800 years later Bmbrlli updated these financial notions and practices in order to apply them to engineering. Tataglia. Cardano and others had derived, using them some pretty "funky" functions to solve for roots. They did not have todays notation, nor todays conventionalisms, and they adtually spoke to each other rhetorically! Algebra, if nohing else in mathematics was a technical conversation, not a slippery glossy symbolic "trip"
Thus we find Bombelli, like Vieta and all around him, introducing innovative notations and marks to identify the referent of their rhetoric. Bombelli, who as an engineer used and constructed engineering drawings, and he used a set square. He also studied what little information of the methods of "the Fathers" that he had access to, by which he meant the roman-Greek heritage he had. In this study and construction, he became more and more convinced that Neusis of the set square solved the major parts of the difficulties Cardano and Tartaglia encounterd. This enabled him to posit a meaning to parts of the function they used to solve degree 3 and some degree 4 polynomials for roots, which made no sense as magnitudes actually made sense functionally, as neusis or construction instructions. In his book La Algebra, he explains all this with copious examples, referring to what we now know are the Euclidean constructions based on the Gnomon. Where Neusisi came into it was that for certain polynomials the solution was not constructible in the normal way, and not even in a mirror fashion as was the case in some instances, but required a circle construction in which a rotation by π/2 was executed. Not only did this give the correct result, it gave 2 results. Thes 2 results were called conjugate, from conjugare meaning loosely 2 related results.
Within the funky funcyions therefore, the √ of "numbers" was involved, and this was an operation that produced a "number value". Of course this was not the case. Magnitudes and quantities were what were involved, and a magnitude result was what was expected. Thus a negative magnitude made no sense, let alone a square root of a negative magnitude. Bombelli, by constructing these magnitudes and their gnomonic relationships, was convinced that negative magnitudes in this rhetoric were only "apparent", and in fact relative. His best analogy was a mirror line. One side represented the magnitude apprehensibly the other side representedthe magnitude as a mirror reflection! Similarly he was able to work out that the square root represented either a double reflection or a rotation .
Bombelli is famous now for his piu di meno and meno di meno poem, which gave simple action rules to cope with the funkiest parts of the functions the apparent roots of negative magnitudes, which were in addition to Brahmaguptas "rules" on "negative" ciphers. Suddenly Brahmagupta is rehabilitated into the more adonine "negativ"e or "meno" magnitude, breaking the astrological link. But it did no good, people still hated "negative" magnitudes.
Bombelli not only invented conjugate roots, but also adjugate magnitudes, that is linear combinations of magnitude and these wierd funky procedures.
We have to mention Descartes, who surveyed Bombelli's work as well as tartaglias and Cardanos, and called these PROCEDURES imaginary numbers. This is where we entered into a great difficulty with number and magnitude. Suddenly number could include not solely magnitude bu also procedures to obtain magnitudes. Gradually as number became the preferred quantitative parlance, magnitude slipped into obscure uses, and the distinctions coverd by the one term became entirely confusing.
Gauss and Grassmann and Hamilton, Wessel, Argand and Cauchy all worked on the problem, but failed to distinguish the magnitude from the procedure conceptually, as Bombelli and the early geometers, astrologers had done. In addition Gauss was given credit for the linear combination form that Bombelli had innovated.
So Hamiltons important paper established the imaginaries as a "proper" number system, and thus confusingly as a proper magnitude system in the mind of physisicista after Newton. es[ecially Minkowaki, Maxwell etc. This was because Physicists were taught by Newton to consider only magnitudes and quantities, not number! Thus they imported, on the sayso of "mathematicians" at the time the "complex number" systems as magnitudes applicable to 2 dimensions, and then 4 as quaternions.
So how do we understand this curious creatue consisting of a magnitude and a procedure? Well apparently cats are very useful creatures to explain them!
When the work on Probability theory was sufficiently mature it was possible to describe it in terms of these "complex numbers", thus the background was set for Schroedingers famous thought experiment, an attempt to explain the odd looking schroedinger equation based on probability theory and "complex numbers" confused with magnitudes.
Considering Lewis Carol's, no holds barred attack on Hamilton et al. it is no wonder Schroedinger wanted o kill the cat! But Charles Dodgeson is having the last laugh i think, as complex numbers are unraveling before my eyes in this modern computer science programming language world. We can at lat properly deal with this odd mixture of magnitude and procedure properly. And the exciting thing is where that leads us to next in terms of modeling real biological systems.
Unfortunately for physics, especially theoretical physics, they are going to have to set their house in order, as magnitude and process combined in this way is not physics per se.
Grassmann's Strecken notions in his Ausdehnungslehre, and his Ausdehnungsgroesse wait in the wings for their day in the sun.