We use the line as a most potent symbol, and to this symbol we attach magnitudes of orientation magnitudes of direction(such as velocity, acceleration, force ) and just the magnitude called length. We also attach the subjective notion of connection to this same symbol, as well as the notion of continuity, curvature(including straightness), boundary and process.. All the while it remains what it is a mark or a fine cord like the spiders web. We also analogise the symbol so that what is not a line is given its definitive properties. We cannot therefore expect to be clear as to its meaning in any given application without a substantial context. …
I would like to use Mr. Newton's Method of Fluxions, His phikosophy of Quantity to underpin the ntions of Mr. Grassmann's Ausdehnungsgroesse. In particular Mr . Grassmanns notions of the law of 3 strecken involved with a parallelogram.
In fact i would ike to use Newton's Philosophy of Quantity to underpin the notion of Kombinationslehre.
As i have developed it Kombinationslehre is a development of sequence theory and sequence theory is all about motion, motion sequents and sequences. It is the joining of sequences and the division of sequences that form the basic notions of Kombinationslehre.The content of these sequences is at last revealed as the structure of points as the final indivisible analysis product from which we may synthesise in degrees. Ihave also attributed to the point(seemeioon) the notion of a web of orientation connections and a vorticular envelope of motion sequents as trajectories. Thus the naked point as such is clothed with invisibles, lines and surfaces are selected to make them symbolically or analogically visible and to give referent to any discussion.
We may import the notion of a line as a drawn trace or a marked out set of points into the description of the structures and connections around a distinguished point( and so to all ) by the very process of analysis. That is we may start with a physical line and mentally reduce the finenes of that line until the lements of that line distinguish themselves. We may then precede further until we reach the planck lengths associated ith each of those elements. By this stage, were we to look around we would be in an unfamiliar uncertain landscape. However, holding our nerve we precede on to the point of our own exhaustion. Ehstever stage that is is for us, me relatively the point, the thing i can no longer be bothered to divide furhter!
Now this point, as i say will in no wise appear close to any other point that i identify by the same process from the original line drawn, and indeed i may see other nearby points which i would not have arrived at by this process, except if i continually redefined and redrew the line with the finer and finer elements i discover, uncover on the way to the point. Thus my very notion of a line from points is produced by a fractal iteration process and the notion of connections, if represented by "lines" becomes a web, a mesh of relations,a graph with infinite nodes, and each such line analyses into a collection of points which are in no way contiguous or continuous.
Thus is revealed the fractal attribution of continuity and contiguity of points by the formal notion i call "line", which symbolises my constructive active motion of drawing a connecting line from some imagined material between the connected points.
Eventually when i have come to a stop i may take my indivisible "point", which by no means has to be fixed, stationary or even solid, and which may in fact be quivering before me in mortal terror!
Such a thing i may place in contiguous position to other such points until i reach one of my analysed points, and this arrangement of quivering points i may at last accede to be called a "line" through sheer exhaustion.
Such a line of points underpins any and all notions of line from thenceforward, and i can use this notion of line to trace out the connections and to demonstrate the motion envelope around a given point, but in this case an analysed point. With the analysed points as centres i can utilise the unanalysed points to construct my attribution to the analysed point, but if the points move dynamically how can i realise a pretty picture? My only recourse is to take film of the arrangement and to freeze frame for each particular instance, Thus my attribution to a point is dynamically variable, and the notion itself is a formal analogy drawn from instances that `are freeze frames of the actual dynamical system.
Thus, i say, Mr Newton's Musings are eminently applicable, even if the whole scenario is fraught with tautology and analogy.
Thus rising up from these considerations to the level of sequence, it is the sequence of analysed points, distinguished as postulates, that is then used to determine division and combination of structure as if the structure was in freeze frame. It is also my subjective processing centre that supplies the notion of continuity and contiguity where in the dynamic situation there is none.
Just as we may freeze frame to apply subjective notions, we may also isolate distinct or distinguished actions to support those notions. Such actions may just be one of many unconscious actions that actually perform the task, nd thus such ctions my be regarded s symbolic of some more complex underlying processes.
Such processes it is clear are sequential in nature, but also many processes may be parallel in the sense of synchronicity. and this is where i divine the points attachment to a region of complex data transmission which exhibits sequential and synchronous outputs. These outputs i define as intensities without much prevarication at this stage, so that i may move on to egional intensities associated with an analysed and distinguished point.These regions reflect the unanalysed points in dynamic motion and connection in freeze frame mode.
Rising to the form level i may collect in some sequential way these analysed points into a form which posseses solidity, and the sequential process of doing that is where i wish to begin my kombinationslehre, although of course, i have already begun it.
Before this, however, i return to Euclid to remark that the "points" which i have arrived at by analysis are not shapeless, but the possible shapes of the points are many and varied. Euclid thus under the influence of Pythagoras and Plato distinguishes specific shapes of points a they may be cut from a circle or Sphere. By understanding the properies of these shapes both open and closed he establishes the basis for combining them to form wholes.The idea is to return to the monad.
It may be well said, and i say it, that Euclid took 3 books to develop the notions and properties of a circle from a point, and a further nine books to develop the notions of a point into a sphere. By this i mean that the schema and sterea are an extended meditation on the forms a point may take and the relationships that may exist between points. The instantaneous dynamics of points are also included as froze forms.
One may reasonably ask if this is all about the pointn then why are additional points and lines included as parts of figures particularly if the point is indivisible. My answer is that it serves to highlight the fundamental tautological nature of our formal knowledge productio, that we think analogically and by bias, and give weight to conclusions in contexts which can only be described as proportionate. W fundamentally ignore or substantially diminish that which we feel or find to be of no importance in terms of direct relativity or relavance . Thus we process in a fracal way all information and output results that are fractal in nature and relationship. We thus always have to take care to distinguish and to compare and comment at the appropriate level. We hus have to observe the tenets of Kairos.
If now, in my meditations i propose that the distinguished point or points have the shape of a larger collection of points, it is only to say that the fractal relationship will serve to magnify what can be iknown about the point without dividing the point any further. It is also to say that the point achieved is one due to exhaustion not to absolute teleos, but we may learn nothing new by proceeding further.
However, as i have described the undistinguished points serve in other ways that symbolise connections and motion tracks around a distinguished point arrived at by analysis. Thus the lines may be understood to be symbolic and the points ever ready to be put to some use in describing motions.
The relativity of a point is a subjective thing, brought about by unconscious processing that fits the output to some goal point. Thus as that goal point changes so does the computed output, and we may experience this happening by turning our head and eyes to quickly to focus on anything. Yet as soon as they rest the output is in place! The gaps in the visual experience are not dark "screens", but persistence of vision and focus adjustment blurring fills the void.
So we may use in analogy the very things we seek to explain providing we admit and recognise the tautology. The reader is then left with choices and some experiencing to do. These as Euclid puts it , are our Items, our postulates that we beg of you. Go experience, go do, go figure so that you may have a referent to what i am talking to you about.
Newton and Steiner are perhaps the greatest synthetic geometers that we know of. But Newtons understanding of Euclid surpasses that of Steiner in its dynamic application. Steiner on the other hand demonstrated a rich understanding of the combinatorial nature of Geometry. Hamilton had a deep understanding of the methods of Euclid and Eudoxus and applied it constantly, but to new , or so he thought, conceptions especially time. Grassmann understood the dynamic nature of Euclidean geometry and sought to bring it "up to date" for modern algebraic use. All Three had to tackle the notion of instantaneous motion, the freeze frame of reality which opened up the dynamic situation to Euclidean methods.
However, it is the philosophy of quantity that Mr. Newton derived after so many years of meditation that has proven to be the mos influential, and without which Grassmann and Hamilton would not be accesible or well founded. Thus both Grassmann and Hamilton symbolised a dynamic quantity by a line. Hamiltom called it a vector,and based upon his work with the imaginaries he defined rules of addition and nultiplication and division. These represented scalar "fields for the essentially unit step in space or time. The step derived its motion from the sequential flowing of time or a river in space.
To say that Grassmann did the same is to misrepresent Grassmann. Grassmann took three punkt, without question and in the normal geometric sense of his fathers geometric lessons, and 3 segments that joined them, and he noticed a notational pattern that implied that if 2 of the segments flowed the 3rd segment was the resultant line. This is th law of 2 Strecken and is fundamentally different to Hamilton who only started with 2 points.
Now both come to the Newtonian example fr fluxions, Grassmann in his study of the area of parallelograms, and Hamilton in his theory of couples to support the mathesis of the imaginary , and more particularly his friend.
Hamilton deals with the imaginary in terms of the derivative of a function of 2 arguments, very similar to Newton's fluxion method. Grassman deals with it in terms of 2 strecken multiplied by a third in his law of 3 strecken , very similar to Newton's moments of a rectangle lemma . Neither acknowledge the similarity to Newton's Methods of fluxions, because neither had probably read it , being taught the "foreign " method of infinitesimals as Newton 's preface puts it. Newton did not use infinitesimals he used vanishing quantities, quantities that arise because we have exhausted our will to divide them any further. Quantities i have defined as points.In addition these pints are not assignable. Thus, due to the number line concept if i assign a value of 1 to anything, one immediately and easily sees the possibility of halving it. These vanishing Quantities do not have that property, they have no part, But they can be gathered together and combined. And this is where Grassmann starts with Schwerpunkt: points that can b added and become "heavier". Hamilton did ot start there, he assumed differentiation as a continuous process, not a quantised one.
Now Grassmann's option means that coincidence actually makes a point heavier, and so does collinearity and coplanarity. We cannot , in Grassmann assume that a point has "swallowed up" any other point, or a line any other line or a plane any other plane in the combined Ausdehnungsgroesse. Neither can you do the same in Newton's Method of fluxions.
Thus the arrangement of these distinguished points, arrived at by analysis in Newton's case is contributory to the quantities in the figure. When Newton allows a side to extend, it takes on extra magnitude, and that magnitude contributes to the area magnitude in a form. Similarly Grasmmann's three Strecken contribute to the forms they are in area magnitudes. As two streckenin procession define a line segment those same 2 strecken but this time in contemporaneous jostling motion define an area segment in the presence of parlle lines. That area segment can be partitioned by a third strecken into 2 related area segments that sum to the first providing the correct signs/directions are observed are observed, again in the presence of parallel lines.Grassmann's Three Sreaks encapsulates the dynamics of change and division with change all in one formulation. These grassmann products as they are called are Netonian rectangle moments summed to first order, for a change in a direction defined by the third Strecken.
The development of the law of 2 strecken i will pass over as it is substantially scalar addition and multiplication.
The development of the law of 3 Strecken i am going to look at as a special cse of Newton's Momment lemma in which one of the sides extends a segment to it in any direction