Ratios and proportions as modelling clay.
I could kick myself!
I already wrote about this once. The formal world we live in mentally is made of some gold standards. The right triangle I the most famous, and the sphere the most mysterious. The Arithmoi are the most obscured
At one time, when I was young I believed in fairy tales. Now I am older, I create them . Well we believe our teachers don't we? The things around us when we are born and which we grow up with we just accept. It took me a long slog to realise the fundamental role of the trig ratios. When I I'd I lived them but for a moment and then went after other structures, sequences , series etc. but I can still remember looking down onto a swirling shimmering lake of proportion underneath the crusty notion of number.
Remember creating a new table of proportions for spirals called sint. That was for Theodorus spiral. And I wrote about using trig ratios to model any curve and setting up standard tables. Then I realised that essentially the idea was not new, but common practice. .
But it was not!
I remember meditating on the Hyprbolic plane within the spherical volume, the exponential logarithms and surfaces all linked by the unit sphere. Everybody was investigating these things so I thought they kne. I thought it was obvious.
But it is not. Reading Euclid and Grassmann has convinced me that most do not know the simplifying underlying harmony of the unit sphere and how the trig ratios enable us to make formal models of every curve or surface or form, by simply measuring and making the sequence measures into a table and then calling it a function!
Still I really cannot blame the mathematical tendency to obfuscate, it is more due to my present state of health and it's effect on memory. When I look at Grassmanns words I see what he is getting at before I translate accurately. If I relied on the accurate translation I would miss so so much! Using the ability to project Strecken he can, I can and we cn tabulate any curve. I frequently do, but this is what I mean, it is a tool that perhaps few others know. It constitutes a construction method and it is precisely what Grassmann is exploring: processes of synthesis and how to write them in mathematical terminology!
As a mathematician I understand the excitement in doing this. But I have come to realise that in constraining oneself to this form of term making that much of the applicability of what he was doing, I am doing is lost or obscured. Synthesis is not just a mathematical action! It is the activity of artisans, and Grassmann came to realise this the more he developed and researched. We only have this acute mathematical form because of Robert, his bother, der Mathematiker! The lineal algebra is Hermanns creation, but it's redaction is Roberts Handiwork.he needed to have hermanns work in a particular format to support his own theoretical revision of philosophy called Formenlehre..
In the end Keine Abweichung guided every aspect of their quest, that is " Invariance" laws of conservation ideal immutable forms and relationships. This was not their goal only, but the Platonic dream, the attempt to find those unchangeable forms that were believed to be the superstructure of the invisible spiritual reality, which nevertheless manifested itself in 3 d geometrical forms!
This Socratic platonic belief was lightly imposed by humour, questioning playfulness. But those to whom it became a real driving force it indicated the touch of the Muses . It also forced a choice: either the muses and their world was real or our senses and our world was the reality and the ontology of these forms arose from abstraction.
You could of course believe both! Nevertheless Plato's Theory or philosophy of Ideas / Forms drove the Grassmanns on. But this was not just their motive, the whole of Prussia was in turmoil and change. The Humboldt reform of the education system drove the clergy, those alone were licensed to teach, into extensive philosophical enquiry. Kant was perhaps the most celebrated figure, but the Prussian Renaissance effected every licensed cleric in Prussia. It was their mission to deliver a superior education system.
Justus Grassmann chose to sort out the difficulties in geometrical education as his contribution. Why? Because everyone knew that Euclid improved the thinking capabilities of students of him. What they did not know was that Euclid was a philosophical course in reasoning! They, like nearly everyone else it seems believed it was a text book in Geometry! Consequently they thought it needed straightening out!
Geometry has an interesting history. It derived from Astrology, and the numerology or calculus involved in that topic. This calculus was applied to the measuring of all distant things and to mapping. It was consequently called geometry by the Tekne who were responsible for these mundane calculations. It was even called Gematria by some.mthe technical use of this calculus was also passed on to other technicians: architects, engineers, artisans, military mechanics. It is this mechanical geometry that survived down into the academies of Europe and Greece.
Astrology has always been an elevated topic, the calculus involved refpgarded as divination, and the status of being a qualified astrologer hard to achieve. In the Pythagorean school in Italy that status was called Mathematikos! Thus we have this split in the social standing given to those who essentially applied the same calculus.
By the time of the renaissance the west had access only to the mechanical version of this calculus, and it was called geometry. It received scant academic recognition, because of class distinction. The lower and working class may need to know it, the upper educated class only needed to know of it. The classics and the philosophers were there remit.
This all changed in Europe and England about the time of Eallis and eventually Netom I Saab Barrow returned ith an extensive knowledge of manuscripts in the centres of learning and some copies, among them was Euclids Stoikeioon. Which Barrow translated into Latin!. Academic chairs for geometry were a new development, but academic chairs for Mathematics were non existent. So barrow was able to pass on his knowledge of Euclid as geometry, a minor classics course, but ebpventually he obtained a chair for mathematics a more reputable classics course. Of course he took his geometrical lectures across into that Chair and established a classical link between the Stoikeioon and Mathematics. He may have been the first, but certainly mathematics as an academic ubject was clamouring for an academic chair all across Europe. Geometry had gained a chair first in Europe, but it was a mechanical architectural technical chair in some of the larger art schools in Europe. No one had a Greek version of the Stoikeioon, all worked from Arabic translations of various ancient texts on mechanics.
The rise of Mathematics meant that the schools of geometry either moved into architecture or were subsumed by a mathematical chair. This poor relation status for mechanical geometry persists even today. The mathematical arts were briefly given great prominence by Newton's seminal works the Principia , in which the synthetic geometry reigns supreme. But gradually algebraic geometry and the Cartesian Coordinate system undermined synthetic geometry., and thus with it Euclid's Stoikeioon.
The impact of Newton on Prussian Philosophers was immense!. The realities were that the holy Roman empire had to buy in Genius!. This was unacceptable in the modern industrial world soon coming. Prussian intelligentsia therefore made it a matter of duty and survival to turn their entire education system round and to produce home grown, self actualising genius. The Humboldt educational reforms, lead by philosophers with a clear goal but no mandatory course of action initiated a revolution in discourse between institutes of higher learning and primary educators. Everybodtpy was tasked to deliver the reforms to the mperor!
Notably gauss kept out of this reform movement in mathematics, but he did try to keep abreast of developments in mathematics cross Euope. Such was his anxiety to not fall behind the rapid pace of change, he ignored the work of the Grassmanns as being of interest to primary education only. However Hermann had snt him his work on Ausdehnngs leather. His reply was his usual offhandedly comment, complaining about how busy he was, but he clearly had read it!
The events that transpire between him and Riemann and the Grassmanns I have written on before, suffice it to say, that no one ith any mathematical nous reading Hermanns 1844 work could fail to be astonished by it. Möbius felt it was so revolutionary that it was beyond him. Grassmann thought that it set the Benchmark. So gauss's response is suspicious to say the least!
In any case, nearer to home, Robert needed his brothers work to up port his own theory and together they redacted it and published it to some indigenous success. Hermanns 1844 publication had won him a small international audience, but no response at home! Roberts redaction and publishing skills seemed to make a difference.
Heartened by the uccess Hermann republished his own text with copious notations. This was because Robert had taken uch a controlling hand that Hermann could not recognise his own ideas!
Still the 2 publications in his name have gradually won him a worldwide udirnce. Robert published more on his version of the Ausdehnungslehre after his brothers death, but few have responded to his version as they have to Hermmann's.
This was a family effort, started by Justus, corrected innovatively by Hermmann and finished off by Robert. It has gone on o show that primary educators can rock the world of higher Academia!