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 Author Topic: The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877  (Read 3870 times) Description: Hopefully a steady translation of the whole 1844 version over time. 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #30 on: April 18, 2015, 07:57:59 PM »

Commentary on §15

I must direct the reader to the induction as indeed Hermann constantly does. The labels are not what they might seem to anyone partially versed in the Mathematical arts. In particular the diagrams reinforce the direct line interpretation, but the treatment is equally applicable to circular arc segments and the endless-like circle to which they relate.

In particular do not regard the = sign as equating, but rather as likening. Many of the labels are not only assigned as representations, according to a general principle of representation, or representational theory, but also have a symbolic status as a likeness..

Hermann draws a distinction between this constrained useage here of the labels to the freer use we may adopt after §99. Hamiltonnalo in his theory of couples took a logically sustainable path to support the way we use algebraic symbols andcarithmetic ones, that is quite freely and often very loosely specified.  This freedom is not safe if it's basis is not understood or comprehended. In addition the free use of analystical knitting will lead to negative assignments per se and that has to be justified in the context of line segments and elements of line segments.

Try to establish that [$\alpha\beta$] = (–[$\beta\alpha$]) to grasp the procedural or logical difficulty that we step over without thought.

The difficulty is not in space or our apprehension of a real extensive magnitude, but in our rules of label manipulation.

When the problem or construction task is "visible" we can move spatial objects relative to each other deftly and correctly, but when it is "invisible ", and not possible for us to reorient the construction we can imagine what the result would be. However, sometimes , even quite often, we reverse the spatial orientations of the result, because we overlooked the effect of changing relative positions, or worse still changing labelling. Here both Hermann andvHamilton draw attention to the constraints an "Expert" must consider to get the correct result.

We are taught to be very blasé about such concordances, but when you build a house or send a manned spaceship to the moon and back, reversing the resultant of a calculation or specific solution has disastrous consequences.
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jehovajah
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 « Reply #31 on: April 19, 2015, 07:53:48 AM »

It just dawned on me, after considering Hermanns use of and reference of the figures , that the difference in sense between " ist" and " sei" is the difference between objective being and existence and subjective being and existence . Thus "sei" : let it be, may be , imagine it being , contrasts with " ist" : it is, palpably is, right in front of you being!

Thus (–$\alpha\beta$) is imagined where $\beta\alpha$  is not, as long as everything is running in reverse. Thus $\beta$ becomes the beginning element and $\alpha$  the end element, and we mentally have changed our cognisance of the one objective entity.

That one objective entity is thereby in Superpostion!

Superposition is thus a cognisance we have of an objective entity in which we imagine more than one state/ status exists, but which we can only glimpse unpredictably.

Thus all static situations are in superposition. Dynamic forces, motions, deformations may well be occurring beyond our ken within that object, but in such a way and at such a rate of change that all we perceive is the equilibrium state.

Positing such a state of affairs is in keeping with Hermanns and Justus religious convictions that Geist or God's spirit or mind is active in material reality in a way we must study to perceive, or isolate. The use of labels and symbols expertly and consistently can help the thinker, the meditator, the purchaser of Nature as profound objects of worth( diamonds, crystals, rocks) to keep track of where their imaginations are roaming.

Beyond that intensely personal use, symbols and labels serve to obscure and confuse. Thus to advance that Mathematics is the language of God or Nature is perverse. Such intense runs of symbols are nothing more than a ball ache!  What is useful is when an interpreter comes and explains not the symbols only but the imagination and adventure they help to facilitate.

In a very real sense, a fractal image, sculpture or Chladni resonance pattern are interpreted by symbols and vice versa, if the encoding is expertly done and the automatic machine can decode it correctly .
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 « Reply #32 on: April 20, 2015, 01:11:31 AM »

Commentary on §16

One must not overlook how Hermann initially constructs his elemental structures, rooted on the prior exegesis in the induction , the beginning element and the end element.

He does so by interrupting a line segment creating element. This gives us a general sectioning of a line segment by a interruption anywhere within a perceived line segment.

Having now the end element he can extend the line segment endlessly by additional end elements. Thus he establishes the 3 element structure in which addition occurs as pairs, recursively.

Later this structure or format is repeated in the plane, and again in three dimensions. All extensions are continuous. In higher stages they may be generally contiguous. Arc segments may also be described in the same way.

Normans treatment of affine and projective points is an instance of this kind of structuring.
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 « Reply #33 on: April 21, 2015, 03:47:49 AM »

Tautology is fundamental to our thinking and evolution of thought.

It is self confirming, and rather than begin logically inept( Aristotelian school of thought) it is in fact very confirming and convicting. Those languages that retain the reflexive participles of verbs are well aware of this self affirming nature of the tautological statement. The word itself comes from the greek ,tautes, taute,, which is/was a particularly troublesome version of the word auto ,aute, autes meaning "self". Thus self referencing was given its own participle in agreed. In English there was no equivalent, butbinmaddition Aristotles autistic brain could not handle it!

Most of us understand the role of emphasis and it's importance in convicting and developing confidence, but for Aristotle the truth is not made truer by emphasis! For most others of us emphasis and reemphasis does confirm, reaffirm and convict again the truth of an issue.

I use truth as an undefined notion here, as I do not actually usebthebconcept myself. I prefer the complartive " true". Such a comparison admits andcevenndemandsvtautology, for one thing may be truer to a standard than another, yet both are true to the standard!

$\alpha\beta$ = $\alpha\beta$ + 0 = $\alpha\beta$ + ($\beta\alpha$ -$\beta\alpha$)

Now $\beta\alpha$ we know Hermannmhas not yet deconstructed into its elemental structures. But if we look at his format we would have to describe it as "forward running. ". The issue then is the confusion about the term" forward". However if we say that $\beta\alpha$ is "running into against" then we have a sensible Definition.

This definition means we can say
$\beta\alpha$  = -$\alpha\beta$.

When we substitute that into the above likening string we get

$\alpha\beta$ =$\alpha\beta$-$\alpha\beta$ -$\beta\alpha$=$\alpha\alpha$ - $\beta\alpha$ = 0-$\beta\alpha$ =-$\beta\alpha$

Thus a tautology confirms the substitution we have made, the decision regarding the definition of $\beta\alpha$ as being consistent with the procedures we are establishing for the labels.
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jehovajah
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 « Reply #34 on: May 09, 2015, 03:58:20 PM »

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction  of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§16.  I now take to my side to scrutinise, in order to reach around to the knittings of differing artformed line segments  , to nearby  me, 2 differing artformed fundamental varyings
And let  the first fundamental varying ( or the running into against set entity of it ) whimsically continue forth an element,
And then the thus varied Element in the second manner of varying likecasely  whimsically stride forth
Thusly I come to be capable therethrough to create whole out of an Element an unending like crowd of new elements
And the totality of the thusly creatable whole Elements I name a System of second Step.

Further I take a third fundamental varying to my side for scrutiny,
which varying does not  once again guide out Beginning elements   from that second step System to an element of this System of second step
And which varying I besign of its behalf  as independent from both those first ones, and let one whimsical element of that system of second step this third varying( or the running in against set entity of it)  whimsically  continue forth,

thusly comes to be the totality of thusly createable elements in order to build a representation of a system of third step
And there is no restriction set to this manner of creating whole concording to the label
thusly I  come to be capable upon this method to stride forth to Systems of whimsically higher steps .
Hereby is it important to hold firmly
That all upon this manner created whole elements, are not permitting to become apprehended  as by othermeans already given entities  , rather as  "springing out of a source like" created  whole entities
And that they therehere all,
sofar as they are created whole "springing out of a source like" through differing varyings ,
also seem outwardly as differing, their label concording  .

Thereagainst, is once again clear
That
Concording to the entity the elements one time are created,
they appear off from there  as givens,
and their differing quality or  identical quality can not by another  come to be  to decide upon, as if one goes back  onto  the original creating whole event.

Before I now go over to our personal task, specifically to the knitting of differing varying manners, want I the manifesting  through the Geometrical trackings to come to aid  .
It is specifically clear
That the system of second step to the plane inter-communicates,
and the plane therethrough created comes to be thought of
That all points of a direct line concording to a new,  not enholden in it  direction ( or concording to its running in against entity)  themselves kinematically move forth , whereby then plainly the totality of the thusly creatable points builds a representation of unending- like plane .

It appears thuslywith the plane as a totality from parallel lines which all a given Direct entity intersect

And it is outwardly seeable
That, there these parallel lines do not cut themselves,
and also the "springing out of a source like"   direct entities not yet a second time are hit, all upon that manner created points from one another are Differing ,  and thuslywith the analogy a completed entity is.

Thusly so one reaches to the  complete un end-like space, as to the System of third step, if one the points of the plane concording to a new, not in the plane lying  direction ( or its running into against set entity) kinematically move forth

And further can the geometry not stride forth,
while the abstract expertise knows no boundary!

Footnote
• how sometimes in the doctrine of space, all points are already originally given through the prior set out Space .
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jehovajah
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 « Reply #35 on: May 10, 2015, 01:47:49 AM »

http://philsci-archive.pitt.edu/2155/1/Method.pdf

http://www.math.rutgers.edu/~cherlin/History/Papers1999/kijewski.html
Commentary on §17
I was going to start by referring to Newtons Dream of a dynamical point, but I cannot find the reference I have so clear in my minds eye. It is likely that recent scholarship has made that particular presentation inaccurate .

My research has shown that the philosophy of Rational Mechanics had progressed up the academic slope to become considered by Natural Philosophers who were basically theological philosophers. A treatise on rational mechanics was kicking around but it seems not to have been definitive.

Newton read widely influential authors, Huygens, Wren to garner speculUationsand to inspire his own explorations. The chief principles of rational Mechanics were formulated by Aristotle and then Archimedes, followed by Nicolaus and others  this principle was that of continued motion in a straight line.

It was this motion, or dynamic that Newton extended. Into his idea of locii traced in space. But then he used an inductive argument to describe surfaces traced out by motions and finally spaces traced out by moving surfaces.

Whether these dynamic principles are in the rational Mechanics Of his times, or whether he vouchsafed them for Rational mechanics of his time I cannot find out, but it is clear that these notions are key to both Newtons and Hermanns construction of a representation of space.
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 « Reply #36 on: May 10, 2015, 11:24:02 AM »

Randys combinatorics for V9
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 « Reply #37 on: May 10, 2015, 08:43:56 PM »

Randy Powells Topology for his combinatorics
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 « Reply #38 on: May 11, 2015, 02:28:51 AM »

This morning I wrote a meditation on this  § and it was not good enough so Shunya swallowed it back!
Maybe it will be time soon to write it in the mean time bits and pieces are being put forward here.

Combinatorics is not a " mathematical" subject. It is how I, we construct models of space and how space constructs these models in me or us.
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 « Reply #39 on: May 11, 2015, 10:34:45 AM »

This is brilliantly clear but ultimately flawed!

As you know space exists and we exist in and of space, but our metrics are entirely thought patterns . As a consequence we are not restricted by orthogonality or curvature. These restrictions we I pose on our subjective experience of space.

We play a game, we set up the rules, but then we forget it is only a game. However this is brilliantly appropriate to Grassmanns thought patterning and exposition.

We moderns who reject mind and A god as entities have only a fractal pattern to resolve our inconsistencies, but those of us who accept Gods, mind and spirits can describe these instances in these terms  with consistency.

What Newton argued is that we must do whatever we do Empirically, not arbitrarily or whimsically hypothesising occult powers to suit our own purposes.
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 « Reply #40 on: May 11, 2015, 12:31:56 PM »

We have a demonstration of how combinatorics progresses from the arithmoi to the Geometry!

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 « Reply #41 on: May 12, 2015, 05:01:42 AM »

Commentary on §17
The main points to observe is the inductive synthesis of the Systems. Each step is an inductive step in an inductive process.
The main constraint is the varying ( a) . The creative element is undefined but is constrained by a' the system step(s) is the count of the number of varyings thus
s:=#{ a}
s:=#{a,b} =#{a}+#{b} = #{b}+s= #{b,s}

It is this last symbolic arrangement which is inductive.
On the face value it always gives the tally 2 but s can be any valid tally and so s is transformed by the process of tallying.

This is an infinite counting " loop" or an iterative definition.
Consider  z:= z+ c
We know this does not make arithmetic sense , but it makes combinatorial sense! The := sign is one I learned only from computer syntax and it thus was described as a computational operator. Really it is a combinatorial marker, allowing the process on RH side to be subjugate to the label on the LHS. It is o ly inside a for loop that its significance appears clearly.

Since Hermann is describing an inductive process he stops the reader here to point out that each varying creates an element for the next step! Thus the process never has more than 2 " statuses" :the creating element, the created element output as a result of the varying.

However, the " combinatorial tally" of varyings increases at each creation of an element. The" element" become increasingly more " complex" combinatorially with each varying.
We can notate this combinatorial tally using subscripts
sn+1= sn+ 1 for n= 0,1,...  s0=0
The subscripts or superscripts or even index marks do the counting.

It is this use of subscripts or superscripts that hide the combinatorial nature of our labels.
Wallis called them potentials or exponentials in his algebra. Thus they became powers or exponents rather than combinatorial tally markers. They soon became lost in the fog of logarithms and lost their simple combinatorial association.

However the move toward these subscripts super becoming logarithms was accompanied by a fresh insight into the organising power of these combinatorial markers. They took on a role in a topic later called combinatorics , but very much obscured within larger subject bases.

Sequences and series appear as if out of nowhere, their combinatorial origins down played and their combinatorial significance From geometry and later topology , completely adumbrated.

As a child I learned to count, add and multiply, but never to combine the counted objects into combinatorial series. 'I explores sequences without realising the combinatorial structure I was imbibing.  Learned disconnected rules about dimensions and units without realising a combinatorial doctrine was being imposed, or that one even existed.

The work that Justus did not only makes that clear it illuminates a murky basis to all so called "mathematics". It connects simple counting to increasingly mor complex counting driven by the  increasing complexity of varyings in the objects counted .

Once you really get that you will understand the post I am writing for V9 about the origins of the Arithmoi in the Stoikeia. The Arithmos is a combinatorial mosaic. The Greek ' ar" means connection in a rhythmical pattern. This pattern is everywhere and it is fractal. Even in our disease arthritis it appears referring to the connecting combination of bones we call joints.

Combinatorics sits at the very roots of our rational systems : of thought, mathematics,architectural design,physical chemical and biological expertises, and technological innovations from mechanics to engineering to computer manufacture and design,to software design and implementation. The list goes on.
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 « Reply #42 on: May 12, 2015, 05:27:25 AM »

We cannot really progress much further without acknowledging the simple crowd doctrine Hermann and adjusts use. There are end-like crowds and unend-like crowds, thus we have bounded and unbounded, or unbound able crowds.

These crowds are crowds of any object Or entity.

Now let us consider the geometrical instance of these crowds. In that case the creating element may usefully be identified as a creating point, that is, an entity that creates points!
Thus first varying is a whimsically chosen orientation and the " point" moves only in that orientation obeying the variation constraints. Such a combination of orientation and motion is called a directed motion or a Direction. . The creating point under this direction creates a Line or Line segment.

The line segment is an end-like  crowd of points, but it has no Finite tally! Thus a finite crowd is perhaps a misleading descriptor of it.

In addition let us also consider the motion that creates an arc segment or a circle. In this case  the varying constricts the creating point to a fixed measure from some central point . That point is mysterious until we develop the second step system.

In the second step system a second varying causes any created point from the first step that obeys its command to move . This creates in fact an entity that consists of directed line segments or circular arc segments that are parallel ( or concentric ) and non intersecting. Or these lines and arcs may also be unend-like . No matter which, they consist of an un" countable" crowd of created points.

Now it is clear that the running into against set direction or motion constraint on the circular arc will produce similar results.. Thus the system of the second step is a finely ruled surface called a plane, consisting in either parallel lines or concentric circles that intersect the system of the first step.

If the system of the first step was a circular arc, then the pattern becomes more intricate. The parallel lines no longer occur. Directed lines that intersect at 1 pont called the centre radiate through the arc or circle;

or  arcsegments or circles of the same constraint as the first pass through this point called the centre;

or arcs from one central point (a point which is the centre for these arcs) as concentric circles of expanding size intersect this circle in 2 points or just one, of which there are 2 such points called points of tangency;

Or indeed this may be the case for every point created in the original arc or circle.

Such a surface is indeed a " plane" surface and clearly has more complex potential than the first type of surface. But they too are an uncountable crowd of creating elements.

From these types of crowds of created elements consisting of so ordered and arranged point creating elements we may proceed to ever more complex but finely ruled lines and / or arcs in the system of the third step.  Such a system usually called space is thus neither empty nor in fact simple, but rather full of every potentiality of motion.

Now Hermann draws an oft misquoted distinction. " geometry" can go no higher than step 3, but his expertise knows no bounds!

Geometry, my friends is not space!

It is clear from the outset that these systems are interpretable by Geometry, but in his day Geometry was restricted to 3 orthogonal axes. Lagrange in fact felt that this was the nature of space also despite pursuing these notions as generalised coordinates. But then everybody believed space was Euclidean, the new nonEuclidean spaces were fictional, and Newton was a Divinely inspired Messenger from god. His laws were immutable because they were the Laws of God and Nature!

For a theologian this is not too far a conclusion to leap to. However it confuses empirical data with "Rational" opinion. In those days Rational meant inspired by the word(Logos) of God. Today scientists and atheists have commandeered the word to attack any religious or superstitious belief system! What poetic justice is that?! Such systems are Irrational while the Rational system remains curiously undefined! For to say it is the "scientific method " is to fall into the greatest state of gullibility ever created!

Any professional scientist will soon disabuse you of the fairy tale nature of the so called scientific method!

So it is then, that some would have you believe that space is an empty nothing, that it consists of only 3 possible " dimensions" and it is predominantly ruled by straight lines. Finally that all forces or impulses act only in straight lines!

If you have a moment read both Örsted and Ampères accounts of their empirical research findings!
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 « Reply #43 on: May 12, 2015, 11:42:14 AM »

One has to step back for a moment and realise what has just been dome:
We have constructed a representation of space From a combinatorial combination of prior labels!, by means of appropriate constraints.

We will see how Arrays of such label combinations essentially encode Combinatorual arrangements of these creating points in our instanced space.

And we must acknowledge the Moment as a fully realised space in a moment of perception. Such moments in a crowd can encode dynamic variations in space- like systems.
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 « Reply #44 on: May 15, 2015, 02:38:14 PM »

V9 will be so important to the understanding in this thread that I urge you to read that thread alongside this one.
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