Justus Grassmann versus sir William Rowan Hamilton.
I will compare and contrast these two contributors to the early 19th century mathematical landscape. First I will clarify comparison and contrasting, particularly in light of Justus Dialectic approach.
Comparing is an act of examination. This is usually effected by bringing ideas or objects close together(com) . The placing them side by side (pare) is a distinct action involved in the examination, but this is strictly only a part (pare) of the examination, which may go on to involve placing the things in other relative positions to gain further insight. Paro means to equal by placing at the side, thus similarities are looked for first, then distinctions become apparent!
Normally the examination reveals similarities, congruencies and distinctions and differences. Thus an examination results in more ideas and descriptions of the objects relative to each other, or if the examination is of processes or texts more words and processes than were initially examined. The process results in a multiple and combinatorial form of examined things and only leads to simplification if one or more thing is congruent to the others, otherwise it leads to increased complexity.
Because of the increase complexity it may become necessary to contrast the various parts. This is again a comparison, but this time of the newly generated complexities. . The additional generated information are the differences(contra), the notions that are strongly or intensely against(contra) the similarities or congruencies. The idea and procedure is to bring these things to stand(stare) in opposition(contra) to each other, to deliberately point up and mark out the differences, to excite a debate in the mind of the reader or listener by which they may distinguish sharply these differences, usually all agree on these clear distinctions, but it is the subtle ones where debate may ensue and controversy swiftly follow.
Comparing and contrasting belongs to a philosophical method called dialectic. In this method or process any collection of things may be examined and discourse ensued. The result of a dialectic will be philosophically satisfying because from meagre resources it may generate a rich vein of Sophia or wisdom and understanding.
While a dialectic discourse may seem to some just an endless multiplying of words, a shifting around of positions and premises, and sometimes a damn good argument, to a philosopher and his debater it may represent significant shifts in psychological adjustments to events and relations, as well as technological insight into how things work or relate and synthesise. Dialectics therefore are a philosophical praxis for creative thinking which includes analysis nd synthesis.
By engaging in dialectics everyday as a praxis a significant distillation of the philosophers stone may be achieved, the elixir of Wisdom, by which the heart and mind and body of the philosopher may be developed. Extending this praxis to natural dialectics, in which empirical elements, plants and animals are also subjected to the dialectic process in various convoluted stages gives rise to that Sophia called natural philosophy by some and science by all.
If the dialectic is applied to the written or oral words of wisdom then philosophy is generated before it may be distiller into universally agreed Sophia. Ones philosophy can be verified by pragmatics, the way it interacts with the processes of everyday living.mshould it prove useful, it will also gain the epithet of wisdom, but should it prove to be a hindrance, then it may then become known as "folly"!
Now should the dialectic be applied to ultimate questions then it often generates the mythologies nd the fervours of religion. Religion is subject to pragmatics, but such as would deceive continually fulminate against the conclusion of pragmatics, and it is this continual onslaught on pragmatics that reveals the fundamental process of reality for an individual. Reality is created as soon as that individual accepts a pragmatic position.
However, any person who does not exercise this ability to accept what they will, is like a ship on the ocean, tossed this way and that by wind and waves. Such people are in a hypnagogic state, highly vulnerable to suggestion, and capable of being easily manipulated by those who wish to exercise power over others.
The majority of us pass hrough this status of mind in developmental stages. Depending very much on our environment and parenting we develop a stable adjustment into a culture, but it is only by philosophy that we gain our own minds and opinions. Some who seem to be born with their own minds, nevertheless without philosophy reveal themselves to be vulnerable also to those who have wisdom in these things through philosophical praxis..
Depending on what you have been taught is philosophy is the personal subjective grounding you are able to achieve through philosophical praxis, but it is the praxis that enlivens the wisdom, not wisdom itself. There are many things called wisdom, riddles, aphorisms, proverbs, but it is the dialectic praxis that opens these wisdoms to the individual as a personal journey of growth and Sophia acquisition.
Justus grassmann was born in 1779 died 1852 while sir William Rowan Hamilton was born in 1805 died 1865. When Justus was in higher education therefore he was under the influence of scholars who had barely got to grips with Newtons Principia and his method of philosophy. Thus Kant was his contemporary and the first to publicly favour Newton over other European philosophers and Mechanists, including Leibniz. Despites Leibniz Monadic views , developed in contrast to Descartes dualism, there were many important and innovative ideas he propounded but which we're ignored to certain extent in the light of European events, and intense competition with the Bernoullis and Euler. So Leibniz programme slipped quietly into the undergrowth while rising stars like Kant and Hegel and Schliermacher took the stage.
The Humboldt reforms gave Justus the opportunity and the responsibility to prepare the next generation for the new order that was in the wings. His teachers therefore were very traditional in outlook and backward looking. To create something new Justus had to innovate as only young men and women can, and this involved deep philosophical research into the nature of things or Die Sache.
His starting point was the received wisdom that mathematics consisted of 3 subjects: geometry, Algebra. And Arithmetic; and these all derived from Euclid. He did not know this was a false premise, and that in fact the curriculum he was introduced to derived from the Aristotelian school of pedagogy!
Nevertheless he took the presentation of geometry and mathematics to task for its illogicality and it confusing arrangement. But he went further. He was tasked to teach juniors. He needed to have an accessible curriculum and an engaging for young minds. This meant he had to completely rethink geometry from the ground up. Those who were also engaged in doing this were Kant and Schelling. Kant wanted to model mathematics on Neton's Principia, Schelling wanted to remodel mathematics from the ground up. He recognised that geometry in particular knot just constructive, but also discursive. Thus he did not want geometry assigned to a lower unimportant level in the reformation. He wanted it to take its rightful place as the lingua Franca, the common language of Mathemtics . He felt that intuitively geometry made sense of the structures in the world, therefore the axioms of it must enable and enhance this. The constructive nature of geometry was the constructive basis of philosophy, and the discursive nature of Philosphy was the discursive nature of geometry. To divide them as sharply as Kant seemed to imply would damage philosophy and mathematics. Thus Schelling was a champion for the axiomatic approach to both mathematics and philosophy. It was Fichte who demonstrated that mathematics could be logically constructed on axioms, thus showing that ynthesis was philosophical and therefore discoursive as per Kant
However Justus felt that geometry was not truely constructive! He set out to show that it was in fact a mixed up mess not clearly distinguished as Kant made out. Consequently he felt mathematics was not truely constructive. He set out to demonstrate his contention nd to provide a truely constructive geometry and mathematics.
His false premise was that ther is a clearly defined notion called mathematics, and mathematical, and this was to be found in arithmetic. Thus arithmetic was his measure by which he determined constructive principles and the constructive nature of the other branches of mathematics.
However he could not justify choosing arithmetic without some general principles and axioms to justify this, and he obtains thes by a dialectic analysis of the fundamentals of mathematics: Magnitude– continuous and discrete, and equivalence– equal or not equal. These principles he felt were the foundation of mathematics. The role and methods of mathematics were to reveal relationships between magnitudes on the basis that thy were equivalent or not.. This analysis of mathematics was Grassmann's own and represented a radical departure from mathematics as an intellectual adornment. Mathematics had a job to do in constructing the new Prussia. Mathematics and mathematicians had to be able to build a technological infrastructure to bring Prussia into the modern dynamic age!
Compare this with Hamilton, who seeing logical inconsistencies in modern mathematics caused by the confusion over number and geometric length, constructs the whole of mathematics through an algebraic construction process!
This process not only constructs all mathematics but every root and branch and includes the construction of the real numbers and goes on to construct the imaginary magnitudes. At the foundation of this construction was the human apprehension of relations in time! Hamilton climed that a similar construction had been done in terms of space, and by this he meant the work of Euclid! His view of Euclid's influence was consistent with the old notions of mathematics deriving from Euclid., but Hamilton showed there was no inconsistency in Euclid's approach. Thus it revealed that the difference was fur to the interpreter of Euclid, or the material that was referred to. Hamilton read the original in Greek and found no contradictions. Therefore Justus was working from a different version of what Euclid wrote, or the version that Hamilton read.