When the children of Shunya first appeared, they were misunderstood. Brahma had said something to his astrologer Brahmagupta, and Brahmagupta advised his colleagues, astrologers all, not to be confused by the Hellenistic ideas they were absorbing, for Brahma had given to his Brahmin a truthful tradition that all things come from and return to Shunya.
The contemplation of Shunya must continue, The contemplation of Ho Monas was given to the Greeks.
At this stage I may opine that they are one and the same contemplation, but culturally clearly they are not. Because of the cultural mythologies they had engendered the contemplation of one or the other was a demand that culturalists could make. This demand divides man. Culturalists divide man.
Surprisingly it is empires and conquests that unite man, but on the basis of superseding a cultural paradigm, of finding a way to symbiotically link cultures into a middle,higher ground, drawing all devotees onwards and upwards out of their cultural ghettos into a new and sunlit plane.
This does not happen overnight, and it is true that it is always darkest before the dawn! So Brahmagupta was dividing off his Brahmin from the hellenistic influences, but he was unable to then contribute to the Hellenistic roller coaster that took his ideas and spread them around the world. So his ideas by slips of the tongue, mistranslations became misrepresented and the mighty Shunya was reduced to an empty voided shell!
Everything comes from Shunya. Everything returns to Shunya.
Brahmagupta advised that Shunya provides both good fortune and misfortune, and that Brahmin mus not think of one without the other. Nobody wanted to think about misfortune! But Brahmagupta demonstrated that even in the figures, the Vedic numerals it could be seen for 1-1 returns us to Shunya. This was his first and main advice on misfortune, that it is everywhere combined with fortune, that they are conjugate. In expressing this notion from Brahms he advised that Brahmins could think of misfortune as being cut away from Shunya, just as fortune could be thought of as adding to Shunya. Of course he did not really say this, it is a misinterpretation foisted on us. What he said was to Shunya things are added, and from Shunya things are taken away, meaning that Shunya is the stage, the backdrop against which the Brahmin must view the fortunes and misfortunes of their lives. Within their contemplative consciousness they can perceive fortune and they can perceive misfortune as standing out against the backdrop of Shunya and misfortune takes away from the Brahmin while fortune aggregates to the Brahmin. There are many other analogous descriptions of the interaction psychologically ,mentally between the Brahmin and these particular children of Shunya!
The application of this posited mental state of mind was far reaching, especially in understanding how these children of Shunya fortune and misfortune worked out in every aspect of ones life. Of course to the merchants,seeking good omens from the Astrologer in residence, this advice was of psrticular relevance, for it meant they could keep track of misfortune and make plans and efforts to avoid it. Thus fiscal management is one of the most direct and long established applications of Brahmaguptas advice.
It is also the means by which his Avicenna was spread from nation to nation by traders ho noted the unusual facility and success of the Brahmins . But when they sought to understand the " secret" they totally misunderstood, and cobbled together their own Greek version.. Nothing comes from nothing was the wisdom around the world, even brahmagupta would have laughed and laughed! Shunya means everything! How could they get it so wrong? Black really became white and white black in this case or red in the Chinese case!
There are cultural reasons, agents of misinformation involved in the transmission of this idea, which is hy it is so misrepresented, and it is time to clear away the misinformation and look clearly on the children of Shunya.
The very next opportunity for correction occured through Rafael Bombelli.
These were the times of the Spanish Inquisition! The Arabs grip on southern Europe was waning and Catholic monarchs were re asserting there power with vigour. The Arabic empire had in the main been a force for good, for learning and culture. The Roman See during this time was particularly weak and pragmatically worked cooperatively with the Arab leaders.mthus the benefits of Arabic scholarship made their way piecemeal into Italy and southern Spain and Portugal. But by the time of Bombelli, the Catholic monarchs were massing against the Arabs, seeking to regain control of their native lands. Some were rich enough to raise equip and train armies, others were more brutish and mercenary, raising mobs to war with a potent combination of bribery ,terror,religious supremacy and demonising of the enemy. Some sided with the Popes stance, others were virulently against it and sort to depose the pope along with the Arabs.
Within each country, then, the reaction to Arab culture was markedly different.,some welcomed it others took what benefited them, while some of the poorer less refined brutally destroyed any and everything to do with the hated " Moors". By the time of Bombelli the Arab empire was waning and the seat of learning established in Spain, was being forcefully expunged. Only a few of the Arab books of learning were available for study, and then only those approved by the monarchs. Private collections and collectors had to resort to smuggling books across Europe!
Bombelli as an engineer travelled widely and one of his passions was old documents. Thus he gradually built up a collection of literature which contained Arabic material. Fortunately in Italy having such a collection was not such a dangerous past time, and consequently Italy became a land for "educational" tourism. Many artisans would travel to places where books might be had and studied.Dürer for example skirted the hordes of Italy picking up literature where he could, while enjoying the delights of some less salubrious venues.
So Bombelli was witness to the most extraordinary sight of intellectual rivalry between Cardano and Tartaglia. Both excepted the challenge to do better than the Arab/Persian who worked out from Euclid's gnomon, how to find any square from any combined shape. To do so required a style of rhetoric called by Al Khwarzim. " Al Jibr" and subhead lined as the Indian way of balancing.
This was lost in translation. Al Jibr means agonisingly twisted! Thus this Indian way of " reasoning" that is by Euclidean duals was the most convoluted method ever come across! But what Al Khwarzim was referring to, was the particular manipulations that Brahmagupta had done with misfortune and fortune to show how these things are everywhere within our experience.mthe careful and subtle changes that have to be made to go from -5 *-5 to + 25 are pure algebraic manipulation!
This, plus the Indian method or algorithm for finding approximate or accurate squares, for expanding things iteratively as repeating fractions, we're all called "Al Jibr" and Cardano and Tartaglia excelled and revelled in it as being " the hardest " thing you could possibly tackle!
From this contest and it's successes and failures Bombelli resolved to write a book on this method of reasoning, making it more accessible, particularly to engineers. Thus he brought forward Brahmaguptas rhetoric on fortune and misfortune and codified it into the sign rules! He went a step further and set out rules for Cardano's oddities the squares of meno or nrgative numbers. Now I mention the squares rather than the square roots, because the object of square rooting is to construct a square! So the issue never was really the square root of a negative number, but how do you contruct a negative square!
Meanwhile Bombelli set rules down on how to calculate with such surds which had negative squares within this bracket. He mused and then called them piu di meno that is pluses and minuses of these negative squares. As such they obeyed Brahmaguptas rules. Then he adjugated them to numbers merchants dealt with every day to form what he called adjugate numbers- and what we now call lineal combinations.
Bombellis own research into methods of finding squares had taken him to documents containing Euclid's circle construction of a square from a rectangle. This involved him in using a carpenters square, and measuring off a value accurately. While using this method it occurred to him thst if Euclid used a segmented line to represent a rectangle then naturally if the line pointed one way it could be said to be a positive rectangle, and if the other way negative rectangle. Finding the square was the same but done in reverse! Therefore he felt emboldened to consider his adjugate numbers as somehow a little mor or a little less than ordinary numbers depending on whethrer the we're piu di meno or meno di meno.
Because of his construction he also noted tha his adjugate numbers always come in pairs. He called adjugate pairs Conjugate. Thus Bombelli revealed and named 2 more Children from Shunya .
Ptolemy had spent his life analysing and tabulating astronomical data. Not only had he recorded his own, he also had access to Babylonian and Egyptian records. There was some considerable skill and mental sagacity required to collate all these data "files", because the Babylonians used a modulo 60 number system, and the Egyptians used tables developed on their syllabic systems which ran any where between 52 and 100+ syllables. But both the related civilisations had one common idea: the sphere and it's cognate in the flat papyrus or the wet clay the circle. Ancient records of star "charts" show considerable sophistication and artistry, but it is only in the early Egyptian,Harrapan-Dravidian and Babylonian cultures that we have evidence of the sphere as a significant astrological tool.
Maybe to someone used to using syllabaries and alphabets to enumerate, the Egyptian system was not too much of a stretch for Ptolemy. We forget that numbers are a modern invention by Dedekind et al. We forget about the connection between proportions and magnitudes, and the simple elegance of that mindset.
Why do we prefer number to magnitude? The history is involved, but it has damaged our understanding and apprehension of space, and cut us off from earlier non numerate societies!
So to move to a modulo 60 system seems hard in terms of numbers, but really in terms of magnitudes it is the ratio of the sun to the moon! For the moon and the sun move in this 60 ratio with respect to the seasons and the Tymes of a year.
Of course nothing is as precise as 60, but when you try to organise thes movements around a circular disk, this is the best fit! This best fit comes from not one observation but from centuries of observations. It took the Sumerians centuries to move to this circular system, and as they did so, their script and their language and their accounting and their syllabaries were all gradually brought into line. This is the cultural power of astrology and astrologers. We are indeed children of the stars!
So Ptolemy may have not had it as bad as some modern mathematicians who are bound by numbers! Nevertheless Ptolemy improved on the system by using some of Euclid and Eudoxus spherical geometrical relations, and he particularly used a ratio called the periphery to the circumference. This ratio has been the most troublesome to uncover because it got 2 names or more, despite being a constant!
The first name was Pi. The second name was i.
In the limited set of circle to diameter ratios pi and I are conjugates of each other. They are not adjugates except in the special construct we cal complex numbers, they are logically linked. i is –pi that is not pi. So i is everything else to do with the rati relationship between the semicircle and the diameter, it is the algebraic label for that ratio, and the Mamet we give that ratio. We also refer to it by its numerical label pi
We form these ratio relations in a dynami proportion landscape, and in such a proportionscape constant proportions stand out. We do not always give names to these rationic relations, but even when we do we do not agree on thst name. However we all agree on the proportion the label refers to.
Sometimes the ratios cannot be described, but become apparent through algebraic notation. Thus the ratio
x2 –y2 to 2xy has no other description, but when it is iterated through all x and y, the portion between x = -2 and x =1 has a particular form in the x,y display plane. But it has to be differential lay coloured to visualise it.this form is called the Mandelbrot set.mbut as you can see it is only found by displaying, searching and distinguishing in special and intricate ways. Similarly if we display , search and distinguish the ratio of semicircular arc to diameters we find a constancy which we recognise and call i
Again, if we approach it in terms of Euclid's Formal relations we find that to demonstrate that all circles are similar figures, and in al such circles the ratio of the semicircular arc to the diameter is the same requires a special framework of definitions and presuppositions in prior theorems to establish collinearity, from which we can deduce by duality that a radial arc trisects a circular arc into 6 equal sector arcs irrespective of which radius chosen! Thus the rati of arc to radius is a constant no matter what circle. This makes the ratio of semi circular arc to diameter a constant. We do not name this constant anything other than i the ratio of sector arc to diameter is also not named. But once we establish a metron, we can enumerate or count the ratio. This is not the same as experiencing the ratio as a magnitudinal constant! That is what i is , and why it is different to, but related conjugately to pi.
One can enumerate this ratio in many ways, but non of them come to pi because that ratio is a special kind of enumeration.
Thus we have 6 arcs to one diameter or 2 radii, or 6 radii, or 6 sectors or 6 chords. Or 6 arcs to one periphery. All of these different ratios enumerate, but all differently, and pi comes fro an alternative scheme with a common metron. This metron is called length, but in reality it is a flexible stretch resistant string.
If we take a flexible, stretch resistant string as a metron, we can evaluate a ratio that is pi!. It is constant because i is a constant, but it has a special relationship: it cannot be precisely written in numerals! And yet all other forms of the ratio can, so what is the advantage to us of this indefinite ratio?
One important thing it demonstrates: numerals are not fundamental. The fundamentals are magnitudes and algebraic ratios and metrons. From these we construct all our languages and proportionscapes, and our numerate descriptions are always conjugate to our algebraic intuitive descriptions.
Why would we adjugate them?
Fundamentally we live in an adjugate experience.
We apprehend it in many ways: dual; subjective- objective;inner/outer;bounded/unbounded; infinite/finite.
The opening words of Grassmann's first chapter in Ausdehnungslehre 1844 ring true
"The topmost partition of the Knowledge producers is into the real and the formal…"
In the formal experience our reality is rule and definition bound like Euclid's Stoikeioon, but in the real experience our our reality is defined by relativities which are constant whichever orientation they are interacted with. In reality real things refuse to be trodden underfoot, in formality real things refused to remain covered over! Thus i refuses to go away because it is a real relationship of magnitudes, similarly pi refuses to be covered over because it is a real process result for those defined magnitudes.mto get the full picture about the ratio of semicircle to diameter we have to take I and pi as conjugates and then adjugate them!: i + pi
This is an adjugate on of a real experience with a formal one and yet for millennia i has been considered imaginary! In contra distinction, he subjective labels called numbers have come to be considered as real! We even call them the Real numbers!
The next person in this revelation of the children of Shunya is De Moivre.