And a slideshow
Just to get a feel for Lagranges influence on Hamilton and Grassmann. Lagrange was all about finding the minimalist solution that does the job! This is a calculus notion: find the Maxima and minima of some curve equation generalised to differential space in general. Justus Grassmann believed in the "differential space", that is ultimately everything extensive has to be described in terms of differentials. The notion of extensive space is Cartesian, and differentials are the main way that variation is introduced into an extensive space.
Extension by the way is a very very subtle idea, so do;t be fooled into thinking it is just about length!. Cartesian extension was challenged principally by Spinoza, Leibniz only modified it and Newton Essentially Quantifies it in what many see as the correct way. The notion of an empty space, a creation ex nihilo is an absurdity, but the only logical (illogical) challenge to extension.
This is to show how obscure the board of mathematicians have become! What on earth are tey describing, depicting or labelling in this article!?
In case you wondered, this is meant to be an explanation of the previous topic!
One wonders if mathematicians really do talk to each other? They certainly do not speak to the general public!
I have illustrated, i hope, the complexity of notation that arises by mathematical instinct! One might on first reading Grassmann be tempted to think he is doing the same. However, Grassmann is starting at the other end of the scale.
The world and reality we experience is complex or symplect(greek) Deal with it>
HOW DO WE DEAL WITH IT? with the right notation is Grassmann's reply and a fearles heart! Du not accept the platitiudes of dealing with a simple case and then expanding to the more complex or more general. This sounds nice but is in fact a recipe for misunderstanding! Start with the complex, differential case and integrate toward the simpler notation! This is an anlogy of how our brains work!
When i identified conjugacy, it was to approach the fundamental analytical notion of my experience of being Human and interacting with space. It was to somehow link the inner and outer experiences of my experiential continuum so that it supported and explained the consistent and persistent "mathematical" Form" of notation, the notion of conjunction the notion of aggregation.
Why beyond the training and inculcation did this "form" make such philosophical and linguistic sense? Why ultimately does every philosopher try to reduce there concepts to Pseudo mathematical terminology? It is because the fundamental underpinning action of the brain is to conjugate, that is to factorise or tabularise the experiential continuum, and that leaves didtinguished aspects as adjugates to that conjugation_ pieces that do add up to make a sensible whole for a conjugate region!
So why do we end up with all these seemingly unrelated rules, conventions, subject boundaries etc? The answer is congruency. We are individual and incongruous even in a shared experience! thus the work of the board of Mathematicians, the governance of any society , tribe or group, is to promote and maintain a congruent set of notions so we might communicate efficiently if not effectively with one anaother. Deeper study and discipleship is the way for an individual to become in tune with another individual, and a willingness to overcome ones innate perspectives , to share mind space with a revered individual is a life long act of obeisance and apprenticeship. The mark of a true master is the ability to recognise when the acolyte is no longer receiving knowledge and wisdom through them, but directly from the same source as the master. And in such knowledge a true master releases the student from the apprenticeship to become a fellow traveler along the streams from the same source! Then, indeed the master has a companionfrom whom and to whom insight is freely received, shaed and given.
So Hamilton starts with the full experience. This i have calles the proporionscape, but in fact we may call it the differential landscape! For it is the proportions of these diferentials that i measure and distinguish, that is differentiate by labeling or languaging or terminologising. In this one act of naming i reduce a complexity to a simple sound pattern, or mark.
The measurement of proportions is achieved by the assent to and application of any metron. In assenting to a metron immediately compare and contrast all other adjugates and conjugates to that metron, Thus the metron immediately factorises space or Shunya into multiples and adjugate fractions of those multiples. It is these adjugate fractions i term as differentials, fractals , particles, and any other host of other names my languages will supply. Each name preserves a significance of the context in which it was first applied. This means the conjugate of the named object is as significant as the object itself. The conjugacy relationship is necessary to recover the whole.
Grassmann's analytical method, therefore duals the interminable verbiage of mathematical specificity with the simplicity of the labels. The trick is to keep track of the labels and only open them up to their full verbal diarrhea when absolutely necessary, and hopefully at the end of the process of solution, discussion or proportioning.
What i have illustrated briefly is examples, bad ones where this rule has not been easily applied. Thus to show off mathematical "rigour" forms that do not communicate, terms hat alienate have been chosen. Sometimes this has been mere hubris, on other occasions it has arisen because to distinguish in the mind of learned mathematicians the precise taxonomy of what is being discussed, the indexing has had to be complete.
When a taxonomy serves to obscure and alienate, it is time to change the taxonomy. The technical term for this taxonomy is the Ontology of the topic, For those to whom taxonomy is a foreign word think of a tabulated index which refers one to each topic or page in the book of mathematicalknowledge, or the librarians system of categorising books. Useful for librarians, but impenetrable for most of us. Nevertheless we can navigat it through its appeal to sensible headings and topics!
This in a nutshell is Grasmann's argument: sensibly name what yo are discussing, and everyone can follow along, even if they skip the necessary detailed bits!
How is it i can come i from the great wide world and find a particular book in a library to satisfy my interest?
Th answer to that question is the answer that Grassmann potentially provides in is Ausdehnungslehre 1844. The 1862 version is corrupted by interaction with Gauss and Riemann!