Although I have not put Euler in the title I think I want to introduce this topic by the Euler Equivalence classes on the Integers, that is the cyclic groups on Z , the modulo n groups.
Now it occurs , right at the outset, that the conjugacy of the equivalence classes, which are used constructively to derive more abelian or swappable groups, is used to define the independence of each term, element or basis term of the constructed group. In so constructing a derived group we are careful to enshrine the independence of each part, by exemplifying how the process of identifying each element of the new group is performed; by specifying how this structure is maintained through addition of group elements, and by mapping labels across in homeomorphic moronic or isomorphic function transformation or or mapping relationship.
Thus, we do not have a lineal independence or even a linear independence Peres at the foundation of our group structure, but a conjugate independence!
Associated within each conjugate are concomitant adjugates, but these adjugates are not independent of the associated conjugate. In fact we use these adjugates to comprise or define the structure of the conjugate. The conjugate is always paired with its conjugate! That is it paired factor. This is fundamentally what conjugation means. Shunya is factored into 2 complementary conjugates. These conjugates have a fundamental, but mysterious "scalar" relationship such that as one is "scaled up" the other is reciprocally" scaled down", and contrariwise, as one is "scaled down" the other is reciprocally "scaled up. "
But also, on the level of Shunya the conjugates are adjugates of Shunya, so they are not reciprocal in the sense of inverses of each other. This type of reciprocity is called factorisation. They are reciprocal in this factorisation sense. Again this reciprocity s also called scale variance, and the arithmetic example of such is the scalar reciprocity. Thus if one conjugate trebles in size, the other conjugate is trebled also in size, making the new size 6 that of the original, or inversely, the original 1/6 th of the new size. This s map scaling whereas the previous one is quantity of metron scaling.
These kinds of quantizations only have adjectival or adverbial force, as no metron has been rigorously applied. To do so requires a finite form, and a formulaic process of application and interpretation, in the course of which black may become white and white black!
Conjugacy and adjugacy are indeed the very limit of our ability to apprehend.
So the independence we structure into our descriptions of magnitudes in space are independence of conjugates, and therefore independence of associated adjugates Between the two conjugates, whereas the adjugates within a conjugate are not independent because their associated conjugate is comprised of them by simple aggregation.
Suppose I focus on a slotted spoon. This structure is immediately conjugate to the rest of Shunya! But compared to the rest of Shunya the slotted spoon is very simple: a handle, a single shaped lump of metal with holes in it. The holes make no difference in 3d, but may in a 2d representation. So now the adjugates of the slotted spoon are the handle and the teal, and yet the sdjugates of the rest of Shumya defy adequate description! The spoon also is relatively simple to describe, while the rest of Shunya is of immense complexity. And yet I naturally think that by understanding the spoon I might better understand the rest of Shunya. This kind of likeness or mapping s called analogy or metaphor, and it is one of our profound processes for grappling with the complexity of Shunya. But to work the spoon and it's adjugates are mentally deemed independent of the rest of Shunya and it's adjugates. This is how we roll mentally!
Conjugacy, factorisation and scale are how we define independence. It is therefore not down to linearity or even planarity, but it is due to conjugacy , a mental processing of spatial disposition in relationships and relativity. Because it is a mental process, every aspect of it is in fact a formal terminology, a set of labels and rules applied to our experience of space, a set of formal definitions, marks, symbols and forms applied by a rule governed set of processes as a map or model or mesh network of our spatial experience.
Grassmann divided this into formal knowledge production, and real knowledge production, into 2 concomitant experiential continuum or Sein, one formal and mental and subjective, the other real and physical and, presumably although this too is a mental assumption, the other is "objective". What he does recognise is how inseparable they are, using the phrase body and soul, Lebe und Seele, to describe our geometrical intuitions of der Raum..
Extension I have described as much more subtle and complex a notion tha just a line stretching away or toward you! Consequently to define independence in terms of linearity is totally inadequate. Independence, I claime is defined in terms of conjugacy..
Now we may see the Euclidean/Platonic conjugacy into real and unreal forms, the real forms contrariwise being the Ideal, the formal ones, as far as Plato and Socrates would argue, the Eideai or ideas being more real forms than there concrete expressions in everyday experience. We also see the Grassmannian conjugacy, which is a reworking of the platonic in which the concrete forms are real and the ideas are formal, immaterial abstractions from these real forms. Thus we have approaches from either side of the conjugate divide, both of which are fundamental to our apprehension schemes of space, how we interact with space and/or our experience of it.
But what of Hamilton?
Hamilton did not arrive at this fundamental level until he discoverd by rude and brute force and determination, the quaternion reference frame. Prior to this he had conjugated space in an Aristotelian way, somewhat similar, but still at odds with Plato and Euclid. We may distinguish the difference as being that of the Pythagorean school of reasoning, which Plato adopted, even if he did not fully understandi, but crucial parts of which Aristotle rejected as illogical.
Thus Hamilton worked at the level of Newton, Bombelli, Cotes and De Moivre who pioneered the constant quantity Euler later referred to as i. This constant magnitude is the ratio of the semi periphery to the diameter or the quarter periphery of the circle to the radius, that is pi/2. This ratio was missed, obscured, misunderstood, denigrated, until wallis took hold of it. And while he could not definitely clear the matter up, he was sufficiently clear for Newton, De Moivre and Cotes to make substantial headway decades before Euler. However Euler added a modern legitimacy to the notion through his extensive work on infinite series, no less I might add, and no more than the extensive work in the same field of sir Isaac and De Moivre and Wallis themselves. The rewriting in terms of the Euler constant e was a fashionable thing to do, but no more significant than Cotes
Nevertheless the European juggernaut of mathematical research and expertise had finally arrived at the level of understanding of Newton, Cotes, and De moivre and delivered to Gauss the material which he then shaped into its enduring form, right at the time when Grassmann and Hamilton were about to rock the boat!
Hamilton followed the British and Irish mathematical tradition, and spent years trying to demonstrate the reality of the imaginaries. This was not so much as a physupical reality, but as a spiritual reality, an intellectual grasp of the reality of god!
These notions were not fanciful, for spiritual reality was taken for granted. Where the objection lay was in the affront to the sensibilities of the time. Even Lewis Carroll could not stand it, but he got an enduring best seller out of wittily mocking the mathesis of the imaginaries!
Despite Gauss overbearing handling of European mathematics, Hamilton was able to demonstrate the fundamental importance of conjugate relations, conjugate functions in the differential partitioning of time(not space, but immediately applicable to space or any dynamic regressive extensive form). And, I have to admit, by pure sophistry demonstrating that the imaginaries are a fundamental solution to a coupled pair of conjugate differential equations. Now what the hell difference that makes I really do not know, but underneath that, independent of spatial coordinates, he demonstrates an intricate relation between pairs of moment pairs! This is of course on the face of it all lineal, and so apparently within the normal real measurement paradigms, and yet these imaginary solutions must result! Really Hamilton took the then philosophers stone of calculus and showed it supported the imaginaries. Those in higher mathematics would have to concede to the reality of these quantities or disown calculus!
Well on the continent some were quite willing to disown calculus! But Dedekind, again by a platonic dialectic invested Wallis's measuring line concept with the force of the "number" magnitude and quantity properties, ideas that up until then had not needed to be defined. Now they were defined as somehow being "real " numbers. Having settled the minds of those who were skittish, differentials could now be discussed as real quantities, and distinguished from imaginary or vanishing ones. This made Hamilton's analysis of couples even more significant, for if Dedekind and his kind of calculus was accepted, then there was no escaping the reality of the imaginary quantities, carefully distinguishable from these an isin quantities as they had been.
Thus Dedekind enabled the construction of different types of " numbers" , and as shocking as it was, complex numbers were a theoretically sound part of such constructions. Not everybody was happy, but most people could not escape the logic, and so they shut up controversing, mumbling into their beers! Only Lewis Carroll seemed to turn his mumblings into a profitable venture!
Hamilton thus recognised the fundamental distinction, but he was not on the same conjugate page as Plato/Euclid or grassmann, and he recognised that on reading Grassmann's 1844 ausdehnungslehre. Only when he had his insight on quaternions was he in a mental position to start conjugating reality differently, and this too was a visionary experience that connected him to Cotes and De Moivre in ways he spent the rest of his life exploring. It is not an accident that he called his last work on quaternions the Elements, for he had at last arrived at a revisioning of the then held Euclidean view of the conjugation of Shunya.