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The introduction of the term "factor ", during the multiplication discussion carries into Hermanns Methid some undefined conceptions especially regarding Pre and post subjugation. However what has just dawned on me is that Hermann has designed the product rule in the guise of a factorisation rule!

Factorisation derives from division. In that case one divides a larger whole by a smaller whole, counting as you go. Eventually as you divide off the larger into these smaller sized chunks the chunks acquire the name factor, because the whole is factored or broken down into smaller parts( fractli)  but these parts are made( factum) by the process.  Factorisation therefore is a division process. But now we write the factor pair as a product or multiplication, that is representing the result of a second step knitting.

Where does the factor pair come from? By counting we record or name the number of facta , that is forms made by splitting off pieces the size of the divisor. That count is called the dividend . The divisor and the dividend form the factor pair.

It turns out that if the divisor goes into the whole an integer number of times, then one can find another divisor that cord into the whole an integer number of times . In fact several differing whole number arrangements producing the whole are feasible, thus giving several factor pairs based on that original divisor.

But a special case exists: that is where there are no other arrangements based on that divisor, and you get only one dividend. Such a case is called first Arithmos. Proto Arithmos.  They received this name because in the logos Analogos Method these Arithmoi take the first ot protos position . Thus the logos consists of first and second position: the dividend and divisor take those positions. The analogos consists in the third and fourth position , a factor pir are placed in these positions Fir comparison and contrast.

This process is explored in great length in books 5 and 6 of the Stoikeia. The resultant output of the process is like or differing. If Differring which is the smaller of the 2.
This is the process of proportioning, that is Reasonng or employing rational thought patterns. Books 5 and 6 are one foundation of the Pythagorean dialectic that trains one to think rationally . This was markedly different from Aristotelian logic based on linguistic and grammar analysis.

So what is presented as multiplication is clearly factor pairs from a division process.

Can the factor pairs be switched round?

In practice that is asking is there a divisor that is inter communicant with the count result in such a way that the dividend obtained using it is related to the previous divisor by the same count name? In reality it is a question about co commensurability.  This of course is commutativity , but in general the dividend and the divisor cannot be interchanged. When they are interchanged it physically means we form the whole by multiple parts of the " dividend" being knitted together. This is a different concept to product design by subjugation..

Subjugation restricts thought patterns. Pre and post subjugation issues exist in the division algorithm . Which divisor one uses effects the output result so interchange of divisor is not generally invariant. But if we rewrite it in the product format, because the whole is invariant then divisor and dividend can be interchanged in this product format..

So we have an inconsistency hidden by the label design for the product of factors. And the backwards looking nature of the design, looking back to reconnect to division. The factories are actually not just forelimb and hindlimb but divisor and dividend, and thus by that nature they correspond not to a synthetic knitting but to an analytic one. Thusly the order of writing becomes very important because generally as in all analytical knittings the resultant is one-assigned, that is a unique value to each unique ordered arrangement in the paired format. The analytic knitting thus permits two inter communicant but differing searches with unique outputs. However in the case of divisor and dividend if we fix the object being divided, initial whole, then we can actually vary the divisor, which thusly varies the dividend, and this is how we form our Factorisation Tables( rather than multiplication tables)!

But then through proportion of these divisor variations we can alter the size of the resultant product, that is the initial whole , no longer assumed as fixed, can now increase or decrease in quantified magnitude proportionally. And this relates to commenurability in that we can also allow the tally count to increase in this proportion or maintain the tally count name by increasing the quantified magnitude of the divisor in this proportion.

From this we can take to our sides ( draw out an interesting phenomenon) that the resultant output of the analytical knitting of division may in facy be MULTI-assigned., and so we have to additionally constrain it( by equivalence lasses) into its standard one-assigned format.

By choosing to write the product design as forelimb and hindlimb written side by side without any visible sign we obscure the fact that these are divisor and dividend written together, and there is a deeper connection between them based on factorisation. Calling them factors was meant to recall this deeper relationship, but in fact convention and general useage has altogether subverted this meaning in the context of " Multiplication".

Factors as currently understood are indeed interchangeable, but that is not a consequence of multiplication, rather it is a consequence of division due to divisor magnitude change, and the representation of this magnitude by a commensurable divisor or unit, that is Monas by name.

Monas is the same as Einheit, and it is that indivisible whole used to count everything else in a given system. It is the standard Metron for a system. But it itself is free to vary whimsically! Monads are thus set by assenting to a convention agreed among those who seek to utilise it. This is why every sovereign nation has a functionary responsible for weights measures and time, all metrical standards set by a sovereign nation, and often imposed by conquest of or adoption by another national state.

Lately the international System of units has bloodlessly been adopted by scientists around the world, but you cn still find national measures reasserting their sovereign patina as emblems of national pride and identity, such as the £,$, °F etc.

The use of commenurability to define factors numerical names or numerals has also been a big factor in obscuring the role of divisor and dividend. Thus by habit we have called multiplication an inverse process to division, but payed scant attention to the true nature of that reciprocity. In practice we have ignored minor incoherences in order to build a " logical" structure that is linguistically systematic. However we should have built a structure that is dialectically systematic built on geometrical or spaciometric behaviours, not grammatical or linguistic ones.

Justus Grassmann came up against this issue precisely in attempting to define the logical equivalent of multiplication. He could not. He simply had to exhibit it geometrically. Hermann on the other hand was able to derive a systematic design for multiplication based on a dialectic approach, as we have seen. The design of a product was thus revealed to be a design issue with many constraints and behaviours to account for.

But now deeper than this we find that the notions of divisor dividend and commenurability by a common monad having been worked out by the Pythagorean school have through slovenly practices been obscured, and easy attributes like commutativity have been incorrectly employed without proper design demonstration from first principles. In fact, the whole basis of the design was shifted onto some linguistic markers called numeral, representing some notion of accounting for real object , things and magnitudes, but without due care and attention.

In general then multiplication appears commutative, but in real physical Geometries this is not the case because of division, the Ultimate meaning of subjugation! Multiplication in general physical situations is of 2 types, Pre-subjugate or post-subjugate, a fact that surprises Hermann and shakes him to his very psychological core!

Commentary on §11

As discussed in the previous post, the general rule for the dividend does not account for Pre and post subjugate multiplication, and that 2 artforms or skill sets are therefore going to be be essay to do division in Hermanns method, for Hermann does distinguish between multiplication as a a subjugate forelimb or as a subjugate hindlimb! Take very careful note of the footnote. The • signifies which limb the dividend actually represents, even though the divisor cis precisely the same !  Because of commutativity we completely miss this distinction, which arises out of the product design requirement for subjugate systems.

So when thinking of the dividend format, otherwise called the Quotient format, do not go to the " answer" but rather linger at the factorisation stage. Look out for the divisor and the dividend in their guise as factors, and consider which is subjugate to whom?

Hermanns examples are given for the forelimb case, that is the divisor is post subjugating the forelimb, thst is yo be clear, the forelimb is subjugate to the divisor in the system within system set up.

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§12. The in the forgoing paragraphs presented rules  express the general relating of the Multiplication and Division to the Addition and subtraction.

Afaroffagainst, the rules of Multiplication aside themselves do Not go henceforward out of this general relating! how the arithmetic  sets it  out, and which rules declare the everyway toutable for exchange property, and the everyway unionable property of the factors

And are therehere also Not evoked through the general  label of the multiplication .

Much more  we come to learn to know artforms of multiplication in our expertise ,
by considering which artforms:

 the everyway toutable for exchange property  of the factors finds not a place by the least entity;

therefore by which, (By considering which artforms):

  all up to here set out proposition still have their full application .

Also we have thuswith formally evoked the general Label of this  multiplication.

A real label must inter communicate to  this  formal label, if the Nature of the knitted magnitudes is given,  which label declares the creation whole manner of the product, everyway mediated by the Factors.

The relating to the real addition delivers to us a general concording  to this creating whole manner,

 specifically, one of the (real) factors becomes apprehended   As sum of its parts ( concording to §8),

thusly must one place  the  Sum ( factor) of the " product representation forming ", " creating whole" manner concording to the general set Relatings ,  in order to subjugate ( it)  

the ( real) parts of the same manner are able  to subjugate,

and the thusly formed representational Products are able to  add;
 
that brands: there these products once again as "in like sense  created":

 they as parts  are able to knit together to a whole;

That brands: the multiplicative  creating whole manner must be from the artform, that the parts of the factors  go into it on the same manner,

thusly specifically , that if a part of the one is multiplicatively  knitting  together with a part of the other,  
(if a part is creating whole any random magnitude),

Then also, (by considering the multiplicative knitting together of the wholes),  each part of the first factor   with each part of the other factor creates whole such a magnitude, and  indeed the same magnitude,  if these parts are like to the initially taken to oursides for scrutiny parts.

And  it enlightens thuslike:  
that  if
the creating whole manner  has the given to us property ,
also
the multiplicative relating has  its inter-communicant  "knitting together manner" relating to the addition of the like artformed entity ,
and  thuswith  

 all rules of this Relating empower for it.

Also We name therehere already such a Knitting together manner, then,

only if firstly its multiplicative relating to addition of the like artformed entity is concordingly demonstrated,'
or in other words,
only if the "like ingoing" of all the parts of the knitting together limbs in the knitting   is firmly placed in the above given to us sense,

a Multiplication.


The untilhere Presented general knitting Rules satisfy in the essential entity for the  presenting of our expertise, and therehere we go over to this .

The labels of " everyway tout ability for exchange" and " everyway union ability" are labelling concepts that as we have seen, derive from per mutability within sequences ans every way combining within sequences. Thus the foundation behaviours of Hermanns methods are the processing sequences of any kind dealing with any labelled object or idea , particularly involved in analysis and synthesis.

To these process sequences Hermann adds certain constraints as implicitly or explicitly suggested by the nature of the objects or the goal of the process. The overall goal, as you may recall from the induction, is to establish rules that make sense of the processes around us, and divert us from unbearable confusion. Thus we deliberately remove ourselves from a extremely wide range of confusing processes and behaviours in order to construct a rule governed model.

The rules we create and the constraints we set are of our own devising, no matter how closely they produce results as found in the natural behaviours around us. The resultant outputs rather than telling us more about " infinity" tell us more about ourselves and how we work. Thus we can learn to know many different products of our own design that we may also call multiplication!

Normans preamble.

<a href="http://www.youtube.com/v/t5gbivTuk6Q&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/t5gbivTuk6Q&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/t5gbivTuk6Q&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/t5gbivTuk6Q&rel=1&fs=1&hd=1</a>

I apprehend now the role of 3 elements in the overall design of a system in which synthesis and analysis must take place or have an expression, and in which also a subjugating synthesis must take place and be of a next higher " rank" or type, and which synthesis must support a next higher rsnk of analysis in an inter-communicating way.

These next higher knitting designs are called multiplication and division in the system if they satisfy or please certain constraints on " formal multiplication " in its relating to addition or subtraction.

I have also looked at the signature or the centripetally acting quality , or the centrally active property of a systems designed product, –1, 0 , 2 (or as some prefer 1).

When set out in a rank array or Cayley table these form the central diagonal evaluations/ property / quality. To tie them hard and fast to a numerical meaning is misleading, although of course in a metrical system this is a useful and powerful simplification, and allows algebraic manipulations or processes to be done, and like and differing outputs to be crisply distinguished.

Because Hermann has given a foundational explanatory role to synthesis and analysis, he also has to evaluate the property or quality of the sum of these centrally acting products to complete the product design. This constrains the behaviour of the sum of the next lower level elements without actually needing to specify precisely how , because that is imposed from " above", and worked out concordingly in the full specification of the designed system, by working through a few special and useful generalities.

In looking at this formal product design process it struck me that a product design relevant to a fractal generator is the iterated " function" design at the core of the iterative process. A little thought should make clear how z = z + c is a fundamental synthesis , but which is clearly recursive or iterative ;
and  z =z² + c is a product design which is again recursive with constraints
z.1 = z.z + c.1 to conform to the general product pattern c(a + b) = ca + cb at least by analogy.


The use of analogy, you may recall is a basic skill in the toolset of this Expertise.

I have consistently translated Begriff as label, but the better apprehension is a handle attached to some entity  so that it uniquely identifies that entity ( as a label) but also enables us to manipulate the entity according to its nature, or notion . As we are sometimes not clear about a notion, and indeed may label the precisely identical notion severally, it is important to be rigorously strict in our interpretation of the labe, often having to demonstrate that one label is identical to a number of other seemingly distinct labels, when and if they are.

So the signs and labels and symbols form a formal model of real dynamics and behaviours and kinematics, conforming to " common" mathematical liturgies, but rigorously defined at the design stage by the designer and therefore pertaining only to that interpretation. No matter how familiar a string of symbols look, using the method of Grassmann constrains the interpretation of and the application of the labels.

Commentary on page 375 of the annals of 1877

Hermann here critiques Dillners presentation of the Quaternions.

Firstly he criticises the Prussian predilection for preferring foreign ideas and presentations to the less "grande" Or less "international" homegrown variety. Second he is not impressed by the Unsociable or unfamiliar words used in place of more familiar and simpler ones. Thirdly , Dillner does not even dream of mentioning the more general and foundational propositions of his own 2 works.

This then is a nationalist appeal to his own countrymen to support and promote homegrown talent .

It is to be noted that Hermann does not claim he discovered Quaternions at the same time, at least in writing, as Hamilton did. No, rather he says this work of Hamilton ought to be deriveable from his more general theory in a more straight forward manner, and that should be plain from his previously published results on the different product design types of multiplication, as well as the one Scruple of the Doctrine: the design of all magnitudes and any magnitude!.

The interesting take is on the numeral design of these magnitudes ( Arithmoi). This design is drawn from an account tally of all the differing magnitudes, very much like a till receipt from supermarket..

Hermann now ties these various product design concepts ( multiplication genres) to 3 groups of equations or better likening methods.

If we relate them to a Csyley-Grassmann table we see they are characteristics of the diagonal and the off diagonal entries or elements.

Quatenionic present themselves when the account tally for differing monads is 3. And the tying up of them occurs then, but the product design is represented by a group of constraining likenings( equations) . Consequently the product is a "Middle- man" or intermediary one!

Consider that Hermann dedignsted the " aussere" , "innere" and now the "mittlere" products and you hopefully get the picture that the mittlere product design is some mediated combination of the other 2. Thus to mediate the product one must perform a quite complex task. The product design is thereby no simple arithmetic average or even an algebraic average. It is a mediated combination of 2 well defined products and the essence is in the detail of that mediation process!