Despite giving the name vector to the mathematical world, Hamilton descibed his theory as conjugate functions of pure time, and sets of ordered moments in time, deriving their order from the "flow" of time.
For me the motion is the thing, and this is indissoluble from sequence, thus i appreciate a dynamic, sequential "data set" as encapsulating the motion sequence.
And it tis this encapsulation whicj i feel needs to inform our thinking about these ordered sets. These sets capture dynamic and geometric information using what i am calling scalars. Therefore from the otset these ordered sets are information encoders,containers of coded information requiring decoding. Both the sequence and the scalars carry he information.
When we look at a scalar we have a unit of information: the scalar is some ratio of unity, a multiple of unity. This unity defines the focuss of attention, the bit of information we are attending to. Thus if it stands alone it is a single measurement or count of these units. They may be units on a fractal scale of length, or on a relative sizw of density,
However, whatever motions we mase in obtaining this measure or count are not to be discounted. Therefore in numbering we name visible or tactile forms . The forms themselves form the measure or count, wwe merely name them. Thus to be clear numbers are not what we are about, but rather ynamic sequences of geometrical forms and arrangement apprehended by us through maany sequential motions,.
For each motion or from each motion a mark may be drawn, and the tracing of that mark would replay the motion involved in recording the sequece in focus.
We may in fact record these motions by film, and have the minutest detailed rehearsal of all sequences of motion involved in transforming the geometry into information or data, the basic unit being first chosen and then scalars derived by it.
I come to a sequence of scalars as a sequential record of derived measures by a unit or set of units.
The order of the scalars reflects the motions and decisions i made in the application of sme measuring tool carrying or making available the units . So therefore my measuring tool establishes a code decode relationship with the data, and this "codec" enable the fist stage in the processing of the data for whatever purpose.
The codec then is another name for my tool by which i might measure utilising the units in or on or about my tool and through which i might derive scalars in sequence.
The geometry of my codec may also determine the geometry of my sequential data set. I therefore might have a single scalar, a row of scalarrs, a table of scalars and a block of scalars,all based on orthogonal arrangements within my codec, or i may have some other geometrical arrangement.
Now this set of scalars hardly seems a decent conversation, and yet it is a detailed and repetitive conversation about some detailed and repetitive sequential measuring that i undertook. As such its value is not to be judged in terms of entertainment. I may for tha purpose of entertainment, tell you some endearing stories of what happened to me while measuring, and in so doing reveal that my sequential record has not told the whole story!
We have to allow some custom and practice, and some subjective refocusing, with of course some seeking of assurances that the data set is complete and in order!
Hamilton begins to look at the relations between a single scalar and another, andbuilds from there. ““““`it is to e noted that at every stage he endeavours to carry his readers along with him, and thus to show that a necessary convolution exists between notation and information, between the marks on the page and the minds that use the codec to decipher those marks on the page.
hamilton discusses rules for the ordered sets as well as dules for the ordering of mind so as to access those sets information.
Now i do admit to being overly harsh i my last post, because i am a visual lead person, and so i could not see what the Blind geometer might need to comprehend, or even apprehnd a geometrical situaoin. Group theory is such a tool, but if tou are visiual you will find Euclid's books more accessible.
If i discount the drawing of the line. thus the drawing of and the line have a combined significance: they represent thedynamic motion, and he ordered set can be arranged to contain exactly that infomation.
A vector and a matrix therefore carry information about motions, size and direction