The Theory of Stretchy Thingys

[jehovajah]

Commentary

I have to say , research shows me why Grassmann was not distinguished as today. The ideas were not as sharply defined as they are today, wth respect to Cartesian coordinates!

I have remarked before that what we call Cartesian have their fixed form by Wallis!  Descartes in fact presented a generalised coordinate system! We can pass through De Fermat and DesCartes to Bombelli and others who acknowledged the Coordinate systems of the Greeks, particularly Archimedes and Apollonius.

Even in DesCartes time DesArgues continued the Pythagorean and Greek systems in his projective geometry of the Conics. DesCartes, somewhat archly corresponded with DesArgues chiding him for retaining this " old fashioned" system!

In any case Hermanns innovation must have seemed just another fancy notational sstem( which it was) doing the same thing as Cartesian coordinates( which it was) . However, it was more flexible and more applicable to more general situations than Cartrsian coordinates. Also it was more general than DesArgues system which was developed for Conics. In that regard it was more general than Apollonius who developed coordinate systems for each shape. Hermanns one system could cope with all rectilinear shapes and coordinate systems.

He did remark that he had more work to do to apply it to circular geometry, but he had started in that area with his work on the ebb and flow of tides. He just needed to clarify his Förderung in that arena.

The principal of lineal spaces and planar spaces, linked to either one of his combined labels or the perceived sum of 2 of his combined labels is the subtle advance, for it sets out the basis of a space! Following through we would look for 3 of his combined labels to describe volume space, and 4 for some other type of space etc.

However, his contact with the Barycentric calculus of Möbius redefined his concepts in terms of the Schwerpunkt or weight point, or if you like point mass of Newton.

The story of his developing ideas continues.

[jehovajah]

String theoretical 2d model

<a href="http://www.youtube.com/v/dv3MzIawR1w&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dv3MzIawR1w&rel=1&fs=1&hd=1</a>

[jehovajah]

Ausdehnungslehre

Therefore In relation to which I was proceeding  now thereby (my lifes work), to continue producing the so found results co dependently and as from the start until now. So that I myself also thought on not one thing in  relation to a single one of them( the results) , to convene branches of mathematics to vet established Record!

and so it revealed itself, that this method of analysis discovered by me was not as it first seemed to me restricted to the fields of geometry,but I confirmed also quickly, that I was measuring the new field of a body of expertise , from which Geometry appeared as a special application.

Already for a considerable ( time) it was enlightening me that in no way could Geometry in the sense  like arithmetic, or combination theory be considered as. simply a branch of mathematics, rather I was Much more already relating itself as given on a thing in Nature( specifically the space) , and therefore it should give rise to a branch of Mathemtics that was pure and abstract in the way related laws could originate from itself, and how these laws appeared to be bound in the geometry of space!

Through this analytical method the possibility had been given to build out of it a pure abstract branch of Mathematics! Yes this method of Analysis was the body of expertise itself, as soon as it was being developed, without some copy to publish already elsewhere in advance  an evidently tested reocord, and it was restricting itself to the purely abstract.

The  essential fundamental advantage, which through this apprehension would be outwardly achieved, was the Form of it, that now all the fundamental statements by which the spatial manifestation or view of space was expressed, were completely pushed out of the way, and so an unhindered , straightforward beginning would be established, just like in Arithmetic, rather the contents of it , so the restriction of 3 dimensions could be put out of the way!
First in this line would be the rules treated as " the light" , in their unrestricted and most general form, setting out there their essentially natural co dependency, with many a principle, which either still are not available in 3 dimensions or only available in a masked way , in this fully general way will unfold in full clarity!

[jehovajah]

Commentary
This incredible lecture is worth repeated watching!

<a href="http://www.youtube.com/v/dIEYch4yI08&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dIEYch4yI08&rel=1&fs=1&hd=1</a>

Difficulty level about 10 out of 10, but breath taking none the less!

[jehovajah]

Commentary

Already it is becoming apparent that Hermann was losing touch with any potential audience. Time and agin he says his general approach and method would make everything clear: clear as light, clear as starlight, sparklingly clear!

When the text was presented, none of his peers could see the clarity he was describing!

This testifies to the revelatory and personal nature of his work, and why I use expertise rather than science in translation, quite apart from science being a word whose connotation has changed since Grassmanns time.

Hermann spent some 17 years composing this work. For 17 years he thought his ideas were completely translucent! He endured a further 17 years finding out they were not, at least to his pears!

In those 17 years he reworked and developed certain ideas, but it was Robert his brother who drove him to redact and republish the work.in his print business, which was quite successful.

[jehovajah]

Commentary
Hermann's vision of an abstract ,pure mathematics, unmcumbered by constraints of 3 dimension is largely the current agenda of mathematics. But in that regard he was ahead of his time, because mathematics was derived from mechanics through geometry. Most intuition is therefore rooted in 3 dimensionl experience.

This is how we embed ourselves in reality. But Hermann believed there was another level, a spiritual one that did not mix with reality. This was the experience he was discovering and bringing his method into being for!

To him this was clarity, this was god peering into the darkness and saying  "Licht an!"

Of course, the religious metaphor was completely in keeping with his cultural and social mileu. But many arguments about the nature of this reality existed at the time , a huge European debate about this philosophical question existed, a huge and bitter controversy. Kant was responsible for brokering a kind of peace in his time, but it was a fractured one.

Hermanns method was a solution ahead of its time, and not sought for in his time until Riemann's Habilitation speech!

This video touches on this theme but without recognising Grassmann's contribution.
<a href="http://www.youtube.com/v/kkKTW-Q0qkc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/kkKTW-Q0qkc&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

Grassmann states that the usual distinctions fall by the wayside in his method. Thus whatever reference frame we use in 3 dimensions will make no difference in his analytical method. This means we have to put these distinctions in at the synthesis stage!

In this regard, the point becomes the only concept left to cling onto, that is to label distinctly. Evenso it is not the usual idea of a point, whatever that might be. The Schwerpunkt therefore is not a simple point , but an idea of a space measure!

Within the Schwerpunkt all space is enfolded. It is truly an equivalent idea to Shunya, but not as a single poit in space, or in time. It is a mental experiential point in our psyche, or rather distinct such " points". How we synthesise such points I have yet to read, but where they come from is the method of Analydis so far alluded to.

One clue on synthesis is given by the product rules for 2,3,4 points and the summation rules for multiple points, the Barycentric calculus a la Grassmann, or Grassmann style.

Let us see how Grassmann teases us further in what is only an introduction after all.

[kram1032]

Why didn't you add that video to the thread on Homotopy Type Theory?

[jehovajah]

Commentary

Those that engage space, design and model it have a different view to those who revere number. For too long, the Mechanical basis of geometry and therefore mathematics so called has been treated as second class. And yet Newton highly revered mechanics!

This video provides some back ground or motivation to terms scattered throughout mathematics.
<a href="http://www.youtube.com/v/JuNDS4R-OwI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/JuNDS4R-OwI&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

This is not restricted to geometry, Hermann said!

While this video restricts it to vectors , makes the point that it is a sum of products!

Hermann realised those products could be anything! Specifically any 2 things could be produced in the sum, including geometric point notation, A, B, C!

<a href="http://www.youtube.com/v/A3PvLqLzdcw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/A3PvLqLzdcw&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

Hermann and most modern mathematicians like to disarmingly say:

"Well now I can throw away the picture and just do this Algebraiclly! I could just define this product or sum algebraically ".

Well of course we could, but what happens then is one psychologically starts to look for "meaning" in all this " algebra". One tries to " interpret" what one is doing!

Algebra in the vernacular Arabic means " mind fluff! ", or less venally "mind contortions!". It has taken me a while to see the joke Al Khwarzimi has played on us, in succeeding generations! The Indian system of numerical notation is based on cycles and Gunas and Ganitas. The modulo or clock arithmetics form the fundamentals of their system position and order carry significance in terms of size, thus the decimal system of the Hindis reduced calculation, aggregation and dis aggregation to a mechanical whirring and rotation and twisting of the clock or dial indicators!

This subtle replacement of one to one counters, beads, pebbles etc by a mechanical clock system where Shunya meaning " full" tipped the mechanism into the next cycle or circuit for accounting, is what we call the decimal placeholder system. In this system Shunya takes on a dual role" full/empty".

Shunya can never mean empty or nothing, it always means full or everything. But in this system, once a pot is full ( Shunya) it is moved into a different cycle and sn empty pot replaces it. The empty pot has no name! Shunya is not its name. That is the name of the completion of the filling process.

Given this mechanization of calculation by the Hindis, Brahmans Harappians and other Dravidian or Indus Valley peoples, the Arabs found this constant cycling very disconcerting. Thus the term Al Jibr was used, which, as I say has a vernacular meaning that is apt!

Thus we deal not with mythical Algrbra but actually with Greek geometrical mosaics. These space filling forms were counted, identified and related meaningfully to space and spatio temporal events.

The greatest mystery was our interaction with space . Psychologically we needed to have things or experiences called seemeioon ( singular) or seemeia( plural). These terms are usually translated as point or points, but they are more related to signal or signals. Thus space attracts our attention or focus in some way that directs us to investigate, to understand, to find meaning. Seemeia are all those signals that define a region of interest.

The Pythagoreans over time aggregated these geometrical regions into standard measuring patterns or mosaics called Arithmoi. It is these Arithmoi that we inhabit and name when we count.

Lest we forget the Indian contribution to this process I coined the term Shunyasutras.. These are the same Arithmoi but without the rectilinear bias one picks up from being taught legendary geometry books 1 to 7.

Of course Euclids Stoikeia is all supposed to be introductory and in that sense elementary, but after book 7 Arithmoi theory predominates, untill the treatment of solids  and spheres is introduced. By this time a solid reference frame experience applied flexibly to measurement and proportion should have been ncountered by the student.

If the whole course was taught now as Euclid laid it out, I doubt if it could be done in less than 2 years!

This was and is sn undergraduate course in Pythagorean philosophy! That it has been mistaken for a mathematical text book is due to a misuse of the degree status "Mathmatikos!"  Mathematikos is a qualification in Astrology. One would need to study more thn the Stoikeia to obtain it.

Both Plato and Aristotle were studying for this qualification. I do not know if Plato ever achieved it, but Aristotle certainly never did. In fact he rejected the qualification on several key points to do with Arithmoi!

Dissent was not uncommon, so it was an unfortunate political situation that prevented Arstole from perhaps resolving his difficulties with some Pythagoren concepts through further study in the Academy. Certainly both Eudoxus and Euclid obtained the qualification, and it would seem that both Archimedes and Apollonius obtained it also.

Aristotles Lyceum granted a rival qualification, for a while, until geopolitics again disrupted his school of thought, transferring it into the hands of Islamic scholars, who were much persuaded by it!

At the end of history, the clear, spatial mosaics of the Pythagoreans were obscured by the later interpretations of succeeding generations who did not always get it, and often covered true descriptions by mistaken ones.

Whatever we claim to be doing Algebraically is in fact being done arithmetically, by the methods derived for the Arithmoi. Symbolic arithmetic is a very powerful and cogent interaction ith space.

In this regard all Grassmann did was to not leave out orientation, but further to make it explicit and in combination with the quantity of the magnitude.

We do not use magnitude in its Grrek sense much, nowadays, so the distinction between quantum and magnitude is not clear. A quantum is a heap or a lump of some magnitude. A magnitude is an extensive experience.of something.

Thus the magnitude of space is an experience of its extensivenesses. But a quantum of space is a defined regional and bounded amount of space. These bounds do not necessarily have to be physical, as one can mentally bound or envisage a boundary to a region.

So
When Rⁿ is used to specify a set of ordered real numbers, it's use is I'll defined at the level of "number". Confusingly it is used to specify points in a reference frame And vectors in that reference frame. The reference frame is not defined.
Grassmann's method starts with the primitives that define a reference frame, and these primitives are lines!

Usually hey are termed unit lines, and for that reason the concept of a unit has to be added to the fundamental definition.. Because these primitives are lines we can in fact model the primitive reference frame by the unit sphere in hich any radial represents a primitive line of unit length.

To define a unit length one radial is distinguished as the fundamental Scalar of the whole system or reference frame. Which one it is is a matter of choice, but some conventions exist around the use of this choice.

Thus we can hold a primitive sphere in our minds as modelling not only the primitive reference frame but any reference frame. We can then constrain the elements according to our design purpose.

Once you have a model of a primitive reference frame, one can understnd definitions of points as vertices of parallelapioeds or spherical pyramids  with apexes at the centre.

In this regard, the dot product can distinguish angles between radials in the same great circle.

<a href="http://www.youtube.com/v/k5hqdAkPYzg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/k5hqdAkPYzg&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary
This is a fun representation of a reference frame. This machine projects spherical surface points into space describing planar geometric forms. The whole experience is the model of the reference frame primitive. What is missing is the unit scale reference. But suppose you set that as a ray length to a specific point on a wall, then relative to that point you will see forms scaled accordingly!


<a href="http://www.youtube.com/v/udP6ak5StKY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/udP6ak5StKY&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

We can begin to see how the product sums become layered and constructed, renamed and reinterpreted by different constraints and different mentalities.

At the primitive level it is Grassmanns analytical method tht underpins.
The vector cross product over powere the simpler so called wedge product. This innovation derives from Gibbs reinterpretation of Grassmanns method and his algebras. It is retained because engineers and others like to know where a normal is pointing! However that information is contained within the surface of an object. So in terms of manipulation it is not needed and is a hindrance. However, at the end it can be put in to indicate the normal direction to those who like to process surfaces in this way.

<a href="http://www.youtube.com/v/ZpvlrvwcXk4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ZpvlrvwcXk4&rel=1&fs=1&hd=1</a>

One very important point. Do not be misled. There are no defined vector nulls arising from vector products! What  are defined as such must be an identity with a geometric point. Thus since a bivector null, trivector null etc cannot be a geometric point! They cannot be vector nulls.. They are not even the same type of product as a vector null. A vector null is a scalar product.

Here is Peano's take on Grassmanns ideas. His work lead Levi and Ricci to develop Tensors. A bit over complicated maybe, but they work, apparently.

<a href="http://www.youtube.com/v/UdFl_4xWDlk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/UdFl_4xWDlk&rel=1&fs=1&hd=1</a>

[youhn]

Algebra in the vernacular Arabic means " mind fluff! ", or less venally "mind contortions!". It has taken me a while to see the joke Al Khwarzimi has played on us, in succeeding generations!

This indeed sounds very funny. Do you have a reference on that, because I cannot seem to find it. Translations I can find are things like "restoration", "completion", "reunion" or "reunion of broken parts".

Algebra is a branch of mathematics that deals with properties of operations and the structures these operations are defined on. Elementary Algebra that follows the study of arithmetic is mostly occupied with operations on sets of whole and rational numbers and solving first and second order equations. What puts elementary algebra a step ahead of elementary arithmetic is a systematic use of letters to denote generic numbers.

Mastering of elementary algebra which is often hailed as a necessary preparatory step for the study of Calculus, is as often an insurmountable block in many a career. However, the symbolism that is first introduced in elementary algebra permeates all of mathematics. This symbolism is the alphabet of the mathematical language.

The word "algebra" is a shortened misspelled transliteration of an Arabic title al-jebr w'al-muqabalah (circa 825) by the Persian mathematician known as al-Khowarismi [Words, p. 21]. The al-jebr part means "reunion of broken parts", the second part al-muqabalah translates as "to place in front of, to balance, to oppose, to set equal." Together they describe symbol manipulations common in algebra: combining like terms, moving a term to the other side of an equation, etc.

In its English usage in the 14th century, algeber meant "bone-setting," close to its original meaning. By the 16th century, the form algebra appeared in its mathematical meaning. Robert Recorde (c. 1510-1558), the inventor of the symbol "=" of equality, was the first to use the term in this sense. He, however, still spelled it as algeber. The misspellers proved to be more numerous, and the current spelling algebra took roots.

Thus the original meaning of algebra refers to what we today call elementary algebra which is mostly occupied with solving simple equations. More generally, the term algebra encompasses nowadays many other fields of mathematics: geometric algebra, abstract algebra, boolean algebra, s-algebra, to name a few.

[jehovajah]

This is the original reference I stumbled on.

http://www.argyrou.eclipse.co.uk/GreekEtymology.htm

And this was the blog post it inspired!

http://my.opera.com/jehovajah/blog/2012/03/04/algebra-always-has-hurt.

When I read the translations such as you quoted and compare the evidence which is still observable today in students reactions I have to think: is that a good translation? The answer is as usual yes and no! For about 95% of the human race my allusion makes sense! For the other 5% they would object to the vernacular as disreputable! Lol!

However, when I tell mathematicians they are not actually making things clearer by jargon and symbology they do not believe me!

In that regard, Grassmanns assertions about clarity and light are strongly idiosyncratic in my experience, but I do appreciate that some notations are more reader friendly than others, and actually can aid meditation. However I agree with Newton, algebra is too personal to be an aesthetic medium of communication.

The machintions of ones mind and the whirring of the cogs can be too complex to notate clearly, but the tested conclusion often can be presented aesthetically and elegantly. The analysis leads to a profound synthesis that is often beautiful in its appearance. De Analyse was Newtons attempt to place his conclusions in that light.

Grassmnns use of the word Satz I am still mulling over, but it seems to be referring to the idea of " a public record"  or record in the sense of "it is on the record". In that regard he seems to me to be saying he did not think his ideas neede scrutiny because they were confirmed by well known " facts" or by an accepted statement of facts. While this was deeply encouraging, it missed the point about communicating with ones peers!

So that is the point about algebra in a nutshell!

Historically Al Khwarzimi ( his nickname)  was guided by the Greek rhetorical tradition. This may or may not have made his exposition clearer, but the gradual and increasing tendency to symbolisation, snd the attitude that conciseness of symbols is somehow elegant ( calligraphically this is true) misses the aspect of communication! This is not to downplay the necessity for meditation in mathematics: the so called " stare"! However, it is to recognise the difficulty presented to students by mathematicians who cannot see they are not communicating!

I could go on, but I will stop here!