Algebra has had a long history. It picked up its Arabic name from a book by Al Khwarzimi on the Indian method of " reconciliation" or al jibr. In the vernacular the Arabic means " mind fuck!" but literally it refers to tautological, self referencing, convoluted and twisted reasoning!
Al Khwarzimi implies that the Indian Astrologers were extremely adept in its subtleties and derived many surprisingly true statements by its methods. The methods as eposited by Khwarzimi were largely rhetorical. This rhetorical style was largely the reason why it was so subtle and so self referential. The changes and manipultions described in the rhetoric, especially in writing, we're do mind numbing, or even hypnagogic that students were turned away from the subject in terror! It required a strict and interesting teacher to capture and engage the mi d of those who would learn Al Jibr!
Indian gurus are renowned for enforcing this strict discipline on their disciples. The end result was a ki d of epiphany or a brainwashing in which the turntable suddenly becomes able to use apply and understnd the rhetorical gobbledygook
However, we can trace the Indian astrologers influence back to the early Hellenistic influences of Greek philosophical thought as expounded by the Pythagorean Scholars, and of course prior to them the Egyptian, Harappan/Dravidian and Babylonian scribal traditions. However, it was the Pythagorean School which made public these otherwise arcane methods, practices, rituals in a rhetorical form. The Indian astrologers elaborated. Skilfully on these Pythagorean principles.
I perhaps should then mention the influence of Brahmagupta in this area, and several of his students and rivals at the time, but the net effect was to provide a body of knowledge for the Arabic empire to harvest, redact and translate into Arabic wisdom, after purification. It is the Arabic purified form that seeps slowly into the renaissance movement, but surprisingly it is Bombelli who is the first to publish a major work on Algebra.
Up until Bombelli most books only had a chapter on it because arithmetic reigned supreme. That part of algebra which could be clearly made analogous to arithmetic was all most authors were interested in. However, Bombelli had seen the power of Algebra in the contest between Cardano and Tartaglia.
In his work Bombelli draws on his own creative notation, but he has influences from Viete and Herriot as well as Cardano and Tartaglia evident in his work.
As Mersenne would lament mathematicians were a secretive lot! They would publish results in arcane ways and big their prowess up as much as possible. There is an economic reason for this. Most mathematicians were hobbyists, but professionals required a patron, and that was not easy to achieve without self promotion, and protecting saleable skills or methods. Thus Algebra remained the personal and private thinking of many mathematicians and scientists, including Newton.
It was really the few who published text books for students, and it was Wallis who made the next significant contribution. To Algebra.
Whoops I left out Descatres and De Fermat!, but they would insist on calling Algebra geometry!, which marks the division of Algebra between Arithmetic and Geometry, a division only healed by Grassmann and Hamilton.
Geometric Algebra and Arithmetical algebra are 2 completely different rhetorics, joined only by their common use of mnemonic symbology.. In one the symbols good for a growing conception of some notion called number, while in the other the symbols were labels for various definite magnitudes or quantities, found in space not just in the feverish minds of so called mathematicians.
Descartes boast was that his geometry was better than the ancients because it combined the best of geometry(?) with the best of Algebrs(?). Desargues at the time disagreed and continued to explore the Spaciometry of magnitudes free from the arithmetic implied in Descartes Algebra. He worked with the magnitudes directly immersing himself in the interaction. This contact with Spaciometry is such an important difference to arithmetical geometry where the magnitudes are suddenly replaced by a numeral which numeral is a thing in and of itself, and brings with it the whole machinery of arithmetic, distorting pure space , wrenching it from intuitive extensions and replacing its fellowship with the rote mechanisms of tabulated forms.
The geometrical algebra of Desargue was full of light and shadow, projection and extension dynamic slipping and sliding of forms. And in this dynamic he looked for what was invariant. His symbols were designed to capture invariant relationships, so their very position in the symbolic chin was significant. Changing the symbols about was changing the fundamental Spaciometry. The slackness of commutativity in the tables was not allowed in this notation.
Descartes did not understand why Desargues did not stick to arithmetical terminology as he did. He thought it was a mistake to invent new terms for what he clearly thought of as Arithmetic. This notion of Algrbra as Arithmetic persisted until the 1800's when group theory and ring theory were beginning to take shape. Among the early pioneers after Galois were the Grassmanns and Hamilton.
Hamilton called his effort the Mathesis or doctrine of the imaginaries, not just because he wanted to settle the carping about the imaginary quantities expressed by Euler and others, but also because he claimed it was a science of Pure Time, a knowledge that was derivable from unquantifiable magnitudes, perceived in the memories and minds of mn, and yet as real as length, or length was as imaginary as any 2 moments in time!
Hermann Grassmann, following his father simply observed that the foundation of any mathematical branch was due to human thought about 2 sorts real things or formal ideas! Real yhings could not be stepped over if they were true, and similarly formal ideas could not be forgotten if they were true. However defined forms as ideas remained invariant just as real things always dig you in the ribs if you say they do not exist!.
Both these men were heavily influenced by Lagrangian philosophy, promulgated via the French Ecole, but Lagrange himself was a died in the wool student of Newtonian philosophy especially that in the Principia Mathemtica, the principles of Astrology!
Newyon's stance was that we pay homage to the wisdom of the past, and carry on the traditions of empirical enquiry, not beholden to one method or one teacher but to derive from all insight into the majesty of the enterprise.
His argument was that philosophy foundered because it was of private invention! There were principles of thought which were evident in the past which were as applicable to issues of his day as they are to ours. These principles were Pythagorean and these he drew out afresh in the beginning passages of his Principia.
The use of measure, in particular to model what cannot be determined by senses was a key notion. The use of these models to measure unseen or unfelt things by the variations in these measures; the confusing of these measures and ratios with the actual referred thing or experience, which was always to be avoided, the distinction always held aloof from the measure.
His argument was that if these measures be true, what might we say about what they measure? If by combination of measures another measure be true what might that mean about the underlying referrents?nthus symbol and manipulation only give indication of how we may philosophise about the " realities". They are not the realities themselves.
Hermann Grassmann approaches the Spaciometry of magnitudes from this high philosophical standpoint as addressed by Newton. Hamilton approached the algebra fom the academic physical model point of view. His work in optics had trained him in the academic mathematical approach. His mathesis was highly philosophical, but overtly mathematical and clearly algebraic . But instead of embedding algebra in arithmetic Hamilton subversively embeds arithmetic in Algebra!
While Grassmann deals thoroughly with the logical basis for notation and reference of magnitudes by symbols, and how it serves to denote combinatorial properties of spaciometric magnitudes, Hamilton demonstrates that underpinning every arithmetic is an encompassing "step " algebra. Both derive the group and ring combinatorial rules for their subjective materials, but Grassmann derives them in increasing generality, while Hamilton derives them as he narrows den to a specific arithmetic, that of the so called real numbers, but which Hamilton at the time denotes as continuous magnitudes.
Dedekind would still be promulgating his concept of real numbers at this time, and both Grassmann nd Hamilon show knowledge of this continuous property of numbers, but it is Grassman who looks beyond the continuity of arithmetic to the discreteness of magnitudes nd the combinatorial laws that algebraically govern these discrete magnitudes in combination.
It is Grassmann ho has the vision that Gauss Intenive and Extensive units or quantities were governed by algebraic rules, that combinatorial rules applied to them, that continuous and discrete quantities while different were the same , that is they were treated the sme by algebra. This similitude was t a different level to arithmetic, they were in fact reated at a level we have come to call group theoretical, which Grassmann after his father called Verbindungslehre!
Th implications of this realisation were vast, and frankly overwhelming. He pioneered as far as his time would permit and appealed for fellow travellers, fellow researchers in his life's quest. His appeal fell seemingly on deaf ears!
The radical nature of his work could only be appreciated by a few, Hamilton was one of them. There were a small handfull of other international student who recognised what he had found nd what he was appealing for help with.
It is my contention that Gass recognised Grassmanns genius, but had plans for his own man Riemann. He was either to busy or too manipulative to help this primary school teacher advance his research. In stead he marked his book like a term paper: some good ideas but need to be more clearly expressed!. I think Gauss, who was predominantly arithmetical , failed to rte the algebra Grassmann was using. Like Descartes he wanted arithmetical terminology even in Algrbra.
While gauss wrestled ith Lobachesky and o called non Euclidean Gometries, he failed to see the isr of the ing nd group theorists as anything signiicnt. Consequently he stamped all over Bolyai o stake his clim in non Eucliden geometries, while ignoring the fundamentals of Spaciometry as encoded in algebra since Eudoxus. He felt geometry as packaged by Legendre was Greek geometry, and he felt that the 5 th postulate was an embarrassment to logical , axiomatic purity of thought. Because of it geometry was in criis! Only he and Riemann could save the day, light the path forward to a great mathematical revolution, tied in with Physics.
Actually, he was wrong. It was Hamilton and particularly Grassmann who, respecting the ancients, as Newyon advised, progressed geometry into geometric algebra. But it has taken until now to revert the misleading arithmetical algebra into its proper place.
For this to be clear one needs to understand the notion of number is not necessary, but the notion of Arithmos and counting is. And from these notions we can develop 2 algebras: the intensive nd continuous algebras requiring an inner product ratio, and the extensive and combinatorial algebras requiring an outer product.
It is the exterior algebras we have confused ith arithmetic, a nd the interior algebras we have confused with continuity as opposed to contiguity. But it is the logos analogos of both that is required to describe the calculus of discrete nd the calculus of continuos magnitudes.
And this is only possible through the begränze Linie, the Strecke, that which Gibbs obscured as the vector, a specific term Hmilton invented for an algebraic fom for a line in 3d space .
The concepts are I liar but not idntical. In line ith Euclid a Strecke is a gramme cut out of a greater gramme by 2 cuts. Such cuts may be identified ith seemeioon, but in fact a gramme is defined as made up from a continuum of dual seemeia. The consequence of the cut is that they pick out one seemeioon at each end, an impossible thing to do with any accuracy, but sufficiently clear to begin synthesis.
To this Strecke I as the observer add 3 more properties, orientation relative to me as a third point in space, direction of travel between the 2 cut ends and displacement of extension as a Metron in and of itself.
Thus a Strecke is not a vector. It is a line with these properties. It will be used by Grassmann to ymbolise iscrete quantities in space. Only quantities with the same description are the same. Thus orientation, dinfection, displacement must all match for the notion of duality to hold.
One other misfortune of number. The notion of duality fades because a notion of equality is applied instead. Equality covers a multiple of difference.5 cats do not equal 5 dogs, and yet all will say they are equal in number! Yet say : they are URL in number and experience the hidden difficulty..
Say again: they are dual in count and experience what has been lost to the notion of Equality through number.