Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

I do believe that whatever has been of interest and of great importance in Manipume¨ has come from the study of and spaciometric analysis of motion in space, whether that motion be little or large or in equilibrium.

[jehovajah]

There is a taxonomy of plane curves in my mind and an analogous one for curved surfaces.

The plane curve is distinguished by a tool which is a unit circle with a unit tangent. Any curve is measured by this tool and thus classified by whatever modofications are needed to describe it with this tool.

So for example an angle is a curve distinguished from the tangent and lying between it and the circle or intersecting the circle in some chord. Then by degrees i can describe all the polygonal curves until i eventually come to the continuous curves including those that sweep back and intersect the circle like pedal curves etc.. The logarithmic and exponential curves fit nicely in between these dimensions, which if we parametrise will even give a coedinate reference to the curves. Spirals will also fit in nicely

The analogy in 3d a sphere with a tangential plane, allows a classification of the curved surfaces, and of particular interest will be the conic surface and other vorticular surfaces.

I do not know if it can be extended to a classification of volumes, but it would need to include surface distinctions if it can.

Curves that lie entirely within the circle or sphere would form a special group of interest.

[jehovajah]

What is the fuss about negative numbers and √-1? The greeks based their maths on geometry, and the negative numbers make no difference there.

So negative numbers are a different arithmetic object, that had no geoemetrical meaning until Wallis.Even Descartes coordinates did not deal with negative numbers until Wallis.

Bombellis' secret tool was his set square and his neusis, but he tried many methods including 3d models for measurement.

Geometrically -1  is  a square like anyother.



But Bombelli had a way of representing "negative" squares geometrically by placing it under an arm as if in a balance .

 Then usisng his set squares he arranges a system which produces a line or square above the line. This i believe is a positive and negative value system that is geometeical. As you can imagine it required fiddling about a bit until it was just right- this is precisely what neusis mrans!

Now as usual in maths you get to a point where your symbols totally confuse you! What the hell does this mesn you say to yourself and you become lost and confused, but however you plod on and somehow get the right answer!


"In modern notation, Cardano's multiplication was (5−√−15)(5+√−15), and applying the rule for brackets this becomes 25−−15=40."

Bombelli simply observed that the rule of signs still applied to the surd bracket.

Now here comes the red herring! Who knows what a surd bracket means?  Bombeelli did not neither did Cardano or Tagliatelli, and nither for that matter do we.



What the hell does "radice" mean!? By that i mean, how do you "radice"? Let me simplify the question to get to the point of it--How do you square root?

The "operation" of square rooting, the "algorithm" for finding a root is not taught. We usually skip that and learn them by rote or use tables.

There are geometrical and numerical operations to approximate to a square root. Theodorus spiral is a brilliant example! It exemplifies that geometrically Square rooting is rotating a vector around the circle in a certain way!

Numerical equivalent is to iterate between two values successively "adjusting " the starting point of the calculation and "measuring" how close the "operation" brings you to the desired result.

Whichever way it is looked at the "process"  of radice is a fiddling about operation well described by neusis.

The red herring is : it is not -1 that is the "imaginary" number generator it is the surd operation itself. In short the issue has never really been about number, but about "algorismo" how to proceed in calculating. What Bombelli observed is that it does not matter how you calculate the surd the result still obeys the sign rules if it comes back signed.

Negative numbers were and are the issues in general and square rooting them just highlighted how much western mathematicians with their greek traditions hated them a snot being "arithmoi" that is geometrical objects used as magnitudes for arithmetic.

Bombelli showed that geometrically we can distinguish them by drawing them either side of a line. Wallis later refined this into the number line concept and applied it to an extension of cartesian coordinates to graph the same thing.

How do we square root, cube root nth root in general? we iterate a neusis.

It is the "operation" of neusis that makes the square root of -1 into an operator that rotates the plane! Bombelli did nor have a cartesian plane he had a set square instead, and he rotated it. This has come down to us as rotating the plane, and gone on to develop the notion of relativity through other affine transforms.

Theodorus did the same thing earlier, but o course his ideas were buried by Pythagoras or by his adherents. Archimedes was brave enough to ressurect some of them, and perhaps the most useful of them, and that is why Archimedes is such a master of engineering: he recognised the importance and value of neusis to engineers, despite what the academics were pontificating.

Without neusis it is hard to think how one might deal with many geometrical issues and construction issues. Neusis is after all the greek way of saying "trial and error" through iteration till the error becomes indistinguishable.

This is the fundamental notion of Empiriscism and the methodology of science, and the aim of reductionism;with the meta purpose being to synthesise all the perfect parts into the perfect whole.

[jehovajah]

Euclids proposition fourteen I believe is Bombelli's basic neusis as well as his basic "radice" or square rooting algorithm.

I can see Bombelli's set square and i can see the semi circular rotation required in the construction. Thus a rotation by π is needed but the value is read off at π/2 at the specific corner of the rectangle.

This site more than adequately illustrates my meaning of neusis, but more importantly it demonstrates why Geometry was so loved by the Greeks: motion and transformation through motion of specific elements!

When i learned geometry we had a text book, a set square, protractor, and a pair of compass with a marked ruler. We did not have a java applet or even a film to suggest moving the elements of the figures. Those of us who were good at geometry "spotted" the required basic elements of a proof, and naturally moved the book or the diagram around in space or in our imaginations, others who struggled did not move the elements or even the book!
Nobody ever said this is what you were upposed to do, and in fact are rquired to do in some "proofs". Nobody mentioned "neusis"

At primary level or even at secondary level, Nobody mentioned in my day that ruler and compass was in fact frowned on by the "greeks", who desired a "pure reason" for their demonstrations, not an approximation.

Even the Greek academicans were up themselves! Archimedes was an engineer and whatever worked he used! So was Bombelli.

In the development of their abstraction there are many false assumptions that can be pointed out in the greeks "fashionable" derivation of basic theorems, but in this case Neusis is my concern. It was absolutely relied upon until it became so commomplace that it was not referred to, and then it was forgotten and then repudiated as infereior. The fact is that parallel lines are the formalisation of nuesis, along with the circular rotation. Because for everything in Euclids elements one could "replace" neusis with these formal constructions, i think that  is probably why it fell out of favour.

However as the master of spirals observed , not every curve is a circle! Archimedes in fact demonstrated the necessity of retaining neusis in geometry by trisecting the angle easily and elegantly, and then he went on to show how a spiral neusis path could do the same thing.

Sometimes engineers and pragmatists have to rescue mathematics from stuffy academicians, you know. Sometimes you have to give it back to the people to play with! A sentiment that resonates with Bombelli, Duerer and other great communicators and innovators.

It is of inerest to me that in this construction of the square root a type of spiral curve is evident, linking back elegantly to Theodorus. The points of intersection G and E can be used as generators of an Archimedian style spiral.

[jehovajah]

Note to myself:

Pythagoreans constructed the arithmoi from the proto arithmoi. The geometry of space suggested to them that a unity existed, because they could construct, so it seemed any rectilinear figure  and reduce it by neusis to a rectangle an from there to a square.

Transformations of this sort they believed happened all the time and that was the explanation of the construction of the cosmos.

The special transformations were parallelogram transformations and triangle transformations between parallel lines,plus transformations by rotation in a circle. These were fundamental because the area magnitude was not changed, meaning matter was not created or lost and therefore there must be a fundamental unit that is unchangeable. They called it the "atom" and it was a unity that scaled up to every other form of matter and it was constantly moving through space, and changing thereby through interactions with other portions of matter.

The geometry with its portions and proportions was an exact model of these atoms  it was thought  and the proportioning revealed the ratio of the amounts of atoms, the fundamental unity.

By constructing various rectilinear shapes they found they could transform all to rectangular or square forms and one could explain how unity worked to build portions. They found the proto arithmoi as objects that could not be transformed beyond a simple rectangle. They therefore were proto forms that could be developed into other forms, but were irreducible to other forms.

Like one or unity, the unit square they could be scaled to other forms and so with unity formed the prime elements of the cosmos. Finding them became important to fully describe the geometry of the universe.

It soon became apparent that they scaled, but not independently! They crossed each scaled arrangment in a mesh form

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The structure was used to organise the proportions into arithmoi or scalars, which later mathematicians named as integers and fractions and used them as a unified measuring scale. Geometrical figures or arithmoi had become numbers, and numbers went on to lose their scalar significance, until the development of vectors.

Therefore there exists a fundamental group of geometrical figures for every abstract ring, group, field,equivalence class, or number set.

Descartes demonstrated that the basic operations of number had geometric counterparts, not realising that for the west arithmoi are the root source of the number concept and operations, that is greek geometrical thinking produced the scalars and their operations and Eudoxus organised their application in proportioning arguments , based on the neusis and transformations of geometrical forms.

Gradually over time the pythagorean ideal was eroded to allow for the real and hypereal nymbers, but the fact that they were still scalars was being lost at the same time as vectors were being found as fundamental elements of description.

There is no fundamental basis for unity, only a fundamental scalar role, and the mysterious atom has been fractured enough times to support the notion that infinitesimally small divisions may be possible forever.

Using the Planck constants as unity only re emphasise the scalar nature of measurement systems and the vector basis of notFS,that is the motion field.

Now magnitude is a visual and kinaesthetic sensation, allied to direction and called a vector. However we have other sensations which are allied to direction and should also be understood as vectors" taste, smell and hearing. As these are more directly linked to the brain through specific mesh sensors located regionally in the head and magnitude does not seem appropriate measure, but Intensity does.

Therefore a vector is an entity that has intensity and direction and rotation,or magnitude and direction and rotation.

[jehovajah]

So the research comes full circle.

Bombelli's vector model when generalised to 3d produces a vector shape like Hamilton's Quaternions.

So in fashioning a measure i took design cues from a sextant, a theodolite, and a sundial. I could not get the measure to measure within the body of the unit sphere, so i went for a half sphere. Looking at the yin yang symbol i realised i could fix the measure orthogonal/tangential to a sphere shaped round a yin yang symbol. Then i realised that Bombelli had stumbled onto an n dimensional space!

Firstly because he was using a set square he was  utilising the fundamental metric of space and carrying with it trigonometrical relations. Therefore he had everything he needed of the cartesian coordinate system without needing the coordinates. Thus the model vector in its entirety is needed to visualise a vector in 2 space.

In 3 space the model vector is able to rotate freely about a point on one of its corners. The resolution of the model vector is always the hypotenuse.

Now for generalised coordinate vectors the set square is not square, but a general triangle with resolution dependent on the sides of the triangle and the angles between them.

Concentrating on the right triangle it suddenly appears that for each resolution i can choose an orthogonal vector to form with it a resolution2 . Then repeat the process with resolution2 to produce resolution3 and so on. This set of orthogonal vectors are a tangential set of vectors to the curve traced by the resolution vectors. Thus the resolution vectors , the curve and the tangential vectors form an n-dimensional space, in which the tangent n is orthogonal to the space of resolutions n-1 .

Now this system of tangents and resolutions is similar to Theodorus's spiral, and therfore a n dimensional vector space has a tangential envelope which is a spiral form.

By choosing the curves appropriately i could combine n dimension vector spaces to delineate a surface, but i will need to understand better about bundling, nearness of curves, any cross relationships and system constraints or conditions such as Lagrangian or Laplacians. I would need to study Hilbert spaces to see if this has already been done, but nevertheless i have a correspondence between vortices and spirals and n dimensional orthogonal vector spaces.

My spider friends have been building n dimensional vector spaces for a while it seems and the cone spider shows that they can be used to cover surfaces and volumes.

[jehovajah]

Why mathematics should be a sub branch of computier science.

I think so because maths is about space and motion in space and relative equilibrium on space,and we no longer have to specify that the objects we play with stay still or follow simple curved paths. In fact we can understand dynamic systems better by watching animations of first analyses. What is always forgotten is that the mathematical geniuses of the past and even today were often prodigious calculators,and could quickly compute variations to formulae or propositions to get numerical confirmation of suspicions. Euler for example loved infinite series!

For those of us not so gifted a calculator or better still a computer puts the same intuitive power in our hands. Therefore Runiter, spacetime, mathmatica, mathmaxima are essential ools worth learning.

If one can learn to programme that is good, but the applications most relevant require a concerted effort from many to be recognised as useful.

[jehovajah]

Maths online giving a historical perspective.We see the growing diffusion of greek scientific methods and ideology, and a move away from arabic rhetoric to symbolic notation.

I find also the theory and definition of "number" arithmoi clearly understood as geometrical objects and being able to measure and scale and be compounds of units.
The essential and mysterious notion of unity is expressed here as existing, existence, being,occupying space and moving.

An account of greek Geometry which makes the mistake of separating mathematics out and so failing to perceive that the whole was the greek science.

We find a link back to an older science of the Sumerian, Egyptians and Dravidians.

"   Around the year 390, Plato visited Sicily, where he came under the influence of Archytas of Tarentum, a follower of the Pythagoreans. Archytas studied, among other mathematical topics, the theory of those means that are associated with Greek mathematics: the arithmetic, geometric and harmonic means. Plato returned to Athens in 388, and in the next twenty years, his Academy came into existence. The purpose of the Academy was to train young people in the sciences (mathematics, music and astronomy) before they undertook careers as legislators and administrators. The two main interests of the Academy were mathematics and dialectic (the Socratic examination of the assumptions made in reasoning). While Plato regarded the study of mathematics as preparatory to the study of dialectic, he nonetheless believed that the study of arithmetic and plane geometry, as well as the geometry of solids, must form the basis of an education leading to knowledge, as opposed to opinion. Plato’s teaching at the Academy was assisted by Theaetetus, whom we have mentioned above. Eudoxus of Cnidus, a pupil of Archytas and an important contributor to the emerging Greek theory of magnitude and number, also taught from time to time at the Academy. Plato’s role in the teaching at the Academy was probably that of an organizer and systematizer, and he may have left the specialist teaching to others. The Academy may be seen as a place where selected sciences were taught and their foundations examined as a mental discipline, the goal being practical wisdom and legislative skill. Clearly, this has relevance to the nature of university learning nowadays, especially as it relates to the conflict between a liberal education, as espoused by Plato, and vocational education with some special aim or skill in mind.

Plato’s enthusiasm for mathematics is described by Eudemus, writing some time after the death of Plato:

• Plato . . .caused the other branches of knowledge to make a very great advance through his earnest zeal about them, and especially geometry: it is very remarkable how he crams his essays throughout with mathematical terms and illustrations, and everywhere tries to arouse an admiration for them in those who embrace the study

of philosophy.

Aristotle (384-322 BCE), the famous philosopher and logician, came to Athens in 367 and became a member of Plato’s Academy. He remained there for twenty years, until Plato’s death in 347. As we noted above, in Plato’s time, dialectic was of primary importance at the Academy, with mathematics an important prerequisite. Aristotle held that the mathematical method then being developed was to be a model for any properly organized science. Greek mathematics at the time was distinguished by its axiomatic method, and sequence of reasoning, from which irrefutable theorems are derived. Aristotle required that any science should proceed as mathematics does, and the mathematical method should be applied to all sciences.

Aristotle is important for laying down the working method for each demonstrative science. Writing in his Posterior Analytics, he says:

• By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first terms and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight line is, or what a triangle is must be assumed, but the rest must be proved. Now of the premises used in demonstrative sciences some are peculiar to each science and others are common to all . . .Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is meant by ‘odd’ or ‘even’, ‘a square’ or ‘a cube’, and geometry to the question what is meant by ‘the irrational’ or ‘deflection’ or the so-called ‘verging’ to a point.

Aristotle notes that every demonstrative science must proceed from indemonstrable principles; otherwise, the steps of demonstration would be endless. This is especially apparent in mathematics. He discusses the nature of what is an axiom, a definition, a postulate and a hypothesis. It is quite difficult to distinguish between a postulate and a hypothesis. All these terms play a leading role in Euclid’s Elements.

Aristotle’s influence on later European thought was immense. For many centuries,     "

Although the word mathematics is used here liberally, it means science not mathematics. This science as Plato enthused was an analytical, dialectical, deductive mathesis, that is, doctrine of praxis. Of course it was applied to space in motion, and motion in space. It was truely a spaciometry, not just a geometry.

[jehovajah]

The most detailed account of foundational indian science i have read so far.

The information highlights the rhetoric of early thinkers, which put another way is using words to stand for things, relations, operations and aspects and not the customary referent. The other way of thinking of rhetoric is analogous thinking, or metaphor. In this sense there is a connection with Chinese I ching and Nine Chapter formulations .

It is clear that Indian science is different to greek science, preferring to use all aspects and attributes of form in their rhetoric. and enjoying the relationships and poetry of forms in their exposition or exegesis. Thus the rhythm and metre, the arrangement and juxtaposition of rhetoric conveys a major part of the sense of the "advice". Like a song the advice relates analogous things regardless of context, and so meaning of the "advice" can be found in all sorts of contexts. This is the true heart of Algebra, a generality of applicability, an essence of relation.

The rhythmical and metrical nature of indian science refers directly to the process of iteration. Of all the fundamental things we know of being human, and existing in a world outside of human subjectivity, iteration has got to be the one fundamental common action.

While mathematics has lately distinguished itself by complex notation, this has been a move by people who tired of the constant repetition. But the indians enjoyed the rhythm of the repetition and so carried the relationships in hymns and and songs and poems.. This is markedly different to the greek discoursive dialectical style, and is in many ways the source of the clash between greek and indian science.  

[jehovajah]

Technical treatment of number. Dealing with this metaphysical analysis .  Again it is a science not a mathematics that the greeks were exploring, a science of space and motion. Arithmoi

In short arithmoi are scalars of a unity. You may choose the unity, and all its properties are scaled. However each unity consists of space which is part of unity and not a scalar. This part can be proportioned but not represented by arithmoi scalars .

Thus arithmoi are an attribute to space which i as  a animate attribute and having attributed proceed to count, scale and manipulate. However space itself within these confines is also being manipulated. Arithmoi enable me to scale space,so when the arithmoi become a mouthful i can rescale and calculate at that new scale. Thus the greeks had self similaity built into their scalars.

Their analysis of the space within a unit attributed to it divisibility and aggregation,but no mensuration. However by changing scale for unity mensuration could be effected, but the same condition applied to the space within the unity. Therefore not only did they appreciate self similarity they had the apriori of iteration. Thus the greek idea of space was necessarily nested abd fractal.

The Platonic notion of space was that it was reducible to two fundamentals: unity and extensible/elastic magnitude.  Pythagoras had thought that there was a fundamental unity which by definition was fixed and merely scalable upwards and thus was the measure of all things.

[jehovajah]

A nice treatment of indian sensibilities in their science

Brahmagupta the man who invented negative numbers.

"I......n the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt......."

Brahmagupta wroote a hymn to his goddess in which the void was no longer excluded, but reverenced as the opening or unfolding of the universe. His attribution of debt or fortune to numbers is a worshipful insight into the hidden world of his goddess. This kind of attribution was not strange or uncommon in indian sensibilities, as they understood unity to be in and of itself, the qualities and attributes that the space utilised to standardise a measure has. Everything attributed to a unity was a referent to that unity and is scalable with that unity.

So in his consideration of the Astronomy of his time and the origin of the  universe he considered the unfolding from the void as the ultimate source of everything,but true to indian philosophy of astrology it unfolded good fortune and bad fortune first in equal measure. This is the meaning of fortune and debt, and why negative numbers are so hated!

[jehovajah]

Out of respect for Brahmagupta and his goddess it is probably best to explain what he was about.

BG defined unity in the usual customary way in india, replete with all its attributes and potentials. Any advice he would give would have to have applicability in many analogous situations and so it was usual not to constrain a definition unecessarily. The greeks however did not obseve such niceties,attempting instead to grasp the robust core of the essential nature of space. they left a lot of things out.

Having defined unity, in which no measurement could be made because it itself was the thing, the space by which measurement was made, BG observed that unity could be taken from itself to leave the void. No one had done this before because it is understood that unity and all measurement are relative to each other in scalar ratio forms, and all things spatial exist in a form.

That form no matter how small is a unity, thus to have no unity implies that it does not exist. The idea that all is nothingness is neither a greek or indian or ancient idea. In fact although i once thought it was a medieval christian idea it is not a serious idea in any culture, except unphilosophical scientists! They of course blame Newton,but that does not hold up as Newton was a Descartesian.

So for BG unity "offing" itself was not "annihilation" introducing nothingness, but returning to the infinite void from which all things came, and in which is all potentiality in indistinguishable form and activity. The void and therefore 0 was a dynamic state of superpositional potential, anything could happen , and anything could explode out of the void, and frequently did!

So having made what seemed an innocuous observation that unity "subtract" itself returns to the void that is 0 BG advises how to use this entity with regard to our custumary manipulations of unity. The customary manipulations of Arithmoi became called arithmetic, and arithmoi strictly were the integers/scalars  above unity. If our unity was a calculus(stone) then our customary manipulations would be called calculation. in any case ar(ea) were the unity of the greeks and the ar(ithmoi) were geometrical shapes and volumes based on those and arithmetic is the manipulation of the ars(areas,volumes)

So what was this new observation to be called? BG may have given it a name, but it like unity was a separate idea to the scalars.

Today we have a candidate name called nullity and infinity, and some axioms based on it. BG saw nullity as a relationship with unity which defined the meaning of our measure and our operators. When doing everyday calculations our operators have one meaning, but when dealing with nullity and unity our operators take on a subtly different meaning.

To start with how can one take unity from itself? you can make a relative motion of unity to another place but unity still exists, just in another place: by this BG advises on relativity.

The only way was if unity was returned to the void by an equal but opposite unity which exists for that purpose: by this BG advises on quantum phenomena.

BG as an example decomposes the void by an astrological measure: fortune and poverty. AS i say this is an example of its applicability that BG advances. He expects fully that others would draw out the other applications, and there are probably others he had in mind, but by aligning his thinking with the astrologers he fulfilled his other hat as a keeper of the astrological knowledge of indian astronomy.

".....Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

Brahmagupta then tried to extend arithmetic to include division by zero:-

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero...."

Actually BG was right, because he did not make the mistake of thinking that unity and nullity were arithmoi to coin a greek phrase.

What they are we have only just begun to realise: they are seeds of all our measurement and manipulation and interpretation of a dynamic space, a motion field if you will.

[jehovajah]

Some fundamental relations:

1-1=0  additive inverse /diffusion
1/1=1  multiplicative inverse/ condensation
cos^2+sin^2=1^2  rotational equilibrium
√-1*√-1=-1  dynamic equilibrium

e^(π*√-1)+1=0 rotational dynamic equilibrium(?) observer
0/0=0
n÷0=n/0  - ratios or fractions of the void /uniqueness of relativity
circumference: radius =2π : 1 radian measure.

ø*√-1=ln(cosø+√-1*sinø) rotational dynamic equilibrium(?)subjective

[jehovajah]

Having no fear of astrological misfortune, and neither impeaching or besmirching any god, i gaze steadily at the void and the decompositon  of it that BG has brought to my attention.

Firstly the attribution of plus and minus to a magnitude, referring to the fact that independent of our involvement, should such attributed magnitudes come into contact they would automatically subtract and leave a balance.By this BG advises us of a ceaseless activity in space that is computational.

However at present there seems to be no automatic emission or decomposition of the void into these components in what BG has formulated, at least by report.

However i do not worry, as this is a new apprehending of his insight and over "time" more will flower from it.

Now we may also look for other attributes like debt and fortune,+ and - which are of this bipolar structure and which give a "zero sum"  in some sense. Dynamic equilibria as well as static equilibria are fruitful candidates.

Concentrating briefly on +1 and -1 as unities, the BG rules imply that all transformations in one apply equally to the other scalar arithmoi, the spaciometry of forms and their manipulation and transformation- in a simplified set of operation equivalent to an arithmetic.

Where we do have to nake alteration and be mindful is whre the two unities mix, and the rule arelaid out clearly there: -+=-;  +-=-.

So bearing this in mind we have no trouble determining√-1.

Firstly √ has been defined as the ± of the root of the magnitude. A tautology is used in this definition which i think is simpler expressed as: the magnitude is without sign and is a scalar of unity. To this scalar we attribute sign as necessary. The rules then are rules of how we attribute sign before and after calculation.

That being said we have been defective in our definition of √. We have defined ± as plus  or minus but have not defines plus AND minus +/-. As you can see i have had to switch them about when ±would more naturally mean plus and minus.

The difference this makes is simple : √-1 can now be defined consistently as plus and minus 1 (±1)

Where √+1 is now defined as +/-1(plus or minus 1).

These 2 definitions represent a dynamic equilibrium and 2 static equilibria.

[jehovajah]

Shunaya simply put means replete/swollen with every conceivable and non conceivable attribute.

So when Bhudda says the world is empty of self, he advises not to pursue self ,or that which does not exist independently. As all things are dependent, he advises to seek all things, a wider apprehension. By these advices he directs any listener to a life of active learning and appreciation of all things as an interdependent connected web.

Brahmagupta therefore invites a meditation on the source of all unities by introducing shunaya into our basic conceptions of unity and their scalar arithmetic, or manipulation,or calculation.

There is one thing Brahmagupta advises that is overlooked, and it ought not to be overlooked, and that is the yoke he so deftly and lightly lays on the shoulders of these attributable unities. As light as it is yet it binds stronger than death quantiies and magnitudes in proportions of exactness, in relations of cunjugacy, adjugate companionships and bilateral even multilateral activity and actions and behaviours. Thus what we see intimated is merely the tip of a far more extensive range of attributable properties and decomposition of a very swollen void.

Without shunaya we would not have the Cartesian Algebra as it is extended to today, neither affine transforms or tensor analysis,nor vectors in any recognisable form.