So far, my hero Hamilton has been amazingly astute, but I have heard of the blarney stone and recognise a schmooze!
I have to remind myself how extraordinary his conception is and how fine his thinking was and is so as not to be too in contemporaneous . Hamilton sounds and seems so modern exactly because he ushers in the modern way of rigorous analysis and axiomatic construction as it relates to algebra.
I have not had opportunity to read Al Kwarzihm nor yet Brahmagupta , and I have merely glimpsed at Newton,Euler,Spinoza,DesCartes, Bombelli so I cannot speak authoritatively but my impression and reading indicates that no one quite put it together like Hamilton.
Nevertheless, as I say he tends to hypnotise one into accepting what ought to be questioned earnestly. So it is that I question his assertion that arithmetic is a prior art so simple and fundamental as to precede his analysis and to in fact encompass and ground his assertion of the relations in the science of pure time.
To me this is a cop out. His analysis is of course based on a deep antipathy with the ordinal nature of arithmetic, and a sympathy with it's structural progression. And he therefore acknowledges that as. A source of inspiration. But no one else to my knowledge has constructed the ordinals from the relations of order or succession or flow or pure time as clearly and as rigorously as he.
Even if , as is evident he owes his training to Euclid and Eudoxus, neither of these had his purpose in mind nor his clarity of thought. I therefore think that Hamilton establishes the logical foundation of the ordinals on the related notions of succession and pure time. He also, by his method establishes the foundation of the rationale,irrationals and reals as a set of ordinal "measures". And finally it is his aim in which he already succeeds to establish the complex relations as a set of relations based on the ordinal real relations, which by a trick of notation we may write and express algebraically as we do arithmetically.
Now ,with every precise meaning exposed and made clear, particularly as to he to use and interpret the symbols and notation and conciseness of language, all may see the sensible applicability of these relations to ordinal analysis in the real world.
By these means also he educates the reader in the rigours of the ways of algebra, so as to avoid slipshod expression,reasoning and interpretation,through which come confusion, misunderstanding and contempt.
Unless there is any other it would appear that Hamilton lifted the subject out of the quagmire of being termed " imaginary" and rebooted and resulted it as " complex" . It is also clear that he related it purely to the ordinal, successive property of the so called natural numbers, and thus not to numbers at all but to relations,as he says in time.therefore it is his audience which has foisted the notions of number onto the subject where clearly it created and creates today a problem in understanding.
Wessel prior to Hamilton had dealt with the issue of imaginary numbers by attributing direction to the numbers. Though this helped immeasurably, it did not clear the confusion or promote acceptance of negative or imaginary numbers among the general mathematical society. Academics and geniuses to one side, negative and imaginary had a poor reputation.
However , Hamilton provides a tour de force grounding of the relationships involved which are indisputable as to reality and applicability, but because the necessary ground is in the logic of order, it appears as magical that c
Negative and imaginary " numbers" arise out of that. This is why, I feel Hamilton cops out, and in effect leaves it ambiguously stated that arithmetic supports or gives rise to the same set of relations.
Hamilton would not be the last nor is he the first to misconstrue what he had discovered: a priority relations underpin all our languaging and numbers are adjectives in our language not a separate language.
We attach, by custom an extraordinary significance to number which has it's roots in commercialism not in so called mysticism Pythagorean or otherwise. If anything the mysticism so called was a empirical approach to harmonise and categorise relations perceivable in the world ie science by another name.
It is quite extraordinary exactly how much relativistic language Hamilton has to use because he is dealing with relations , which is taken up and used by the scientific community in all sorts of fields, Einstein 's theory of relativity, quantum mechanics Maxwell's electromagnetic theory etc. At one time Hamilton was the business! But he eventually fell out of favour, but his influence and legacy live on.
I have reached the point where Hamilton introduces the notion of a complex sign denoting combination to transition to a specified moment. Here as all the way through Hamilton is at pains not to ignore the observers contribution to the "meaning" of what is being described or enacted. This is precisely where one has to keep alert because Hamilton has chosen such a suggestive notation that one naturally goes in the direction he is leading, and thereby failing to apprehend the subtlety of his pleading. In another context this would easily be considered "mis-leading" but Hamilton is at pains to point out the subtleties. Nevertheless this is precisely where the author himself can make unwarranted assumptions.
I believe Maxwell turned on Hamilton, after first being an ardent advocate, because it is so easy to be misled by the subtlety of the reasoning. I have not read his work on quaternions to this depth, but I see Hamilton's style as being easily twisted by unscrupulous opponents to make him say the opposite of what is intended and indeed clearly expressed once viewed calmly free from polemics.
Hamilton ,in his treatment of succession suggests that "numbers" first arise as ordinals and then as quotative cardinals, that is names of aggregations, and this is a suggestive thought to be pursued, but it is clear that number is not apriori as comparison supersedes it in that position. It is my contention that measurement is fundamental to all language development in specific terms The Logos Summetria Response.
Since i have closely followed Hamilton in developing my exploration of the motion field as the source for all things conscious and computable i have here to disagree with the suggestion that order is is a continuous flow, and i have outlined my take on Hamilton's sources. That order is included in the mix of a motion field rather than utter incoherency is an axiomatic given based mainly on the observation of rotational order.
Rotational sequent motion to me is everywhere apprehensible, plus it has the added quality of being capable of producing exacct and close approximations to every known motion, including so called "random" motions. These motion curves are studied under the name roulette but i prefer trochoids, Indeed trochoidal surfaces are being produced by Lazarus Plath.
So, Yes i agree on the primacy of order, but not the type. And this is where Hamilton differs from me interms of, not only time but also succession: for he perceives it linearly whereas i perceive it rotationally and thus fractally,scaled and more often convoluted.
An analysis such as Hamilton's may be applied to a rotational Order and i will look to see if it has been done, if not by Hamilton in his discussion on Quaternions, then elsewhere by another author. Certainly the mathematician who studied the subject of Clock arithmetics, with Gauss may have some thing to say about ordinality that is fundamental.
In the meantime Like hamilton explore spaciometry and create your own algebra. You may be missing out on a lot of fun!:jester:
As a post script i find this type of analytical/analogical thinking as being best exemplified in a group of: Hamilton and Boole and Cantor, and absolutely foundational to modern computer science. This kind of relational thinking therefore has had a wide influence and of course traces back to the Greeks particularly Eudoxus.