These 4 videos by Norman serve as a basic introduction to Ausdehnungs Groessen. The general approach is where Grassmann ended up after years of trial and development, which is almost always the case in mathematics. We start from the concrete and move to the abstract, se start with the specific and move to the general. Grassmanns argument was precisely this, but also the general should feedback to an understanding of the specific that frees up the subject barriers as well as the mind.
This fifth video shows but one of the applications of these Ausdehnungs Groesse.
Whilst the cyclic groups are based on integers, they do not have to be they can represent any dynamic magnitude with cyclical behaviour. Thus the real valued sin and cos functions can be used and other oscillating functions.
The unit segment in German is called the Eigen segment and hence Eugene values eigen vectors etc …
Grassmann's Ausdehnungs Groesse.
There are 2 forms of expression/ equation: explicit and recursive. Grassmann makes use of both but heavily relies on the recursive form. This is why Grassmann is so useful to computer programmers, not only for implementing software solutions for mathematical functions or procedures, but also for artificial heuristics/intelligence.
The notion of an explicit expression or equation has its exemplars in arithmetic. It arises as a strict procedural process with a strict resultant. It therefore serves many purposes, so that the term expression is a lot misleading!
The use of such descriptors like equation or expression are an indication of a more generalist thought process, tying symbols and combinations of symbols to languaged communication and language useages. The notion derives very straightforwardly from Rhetoric as the overall or overarching academic discipline of an educated person.
An educated person had to be articulate, and therefore had to communicate their knowledge to others according to the rhetorical standard of kairos. Thus no truly educated person would leave their audience confused! They would find a proportionate mean to convey their meaning to whatever the educational level of their audience. This was essentially the discipline of Rhetoric. Thus, as my mother would say, there was more than one way to skin a cat.
The labour of disciplining oneself to say the same thing in 5 or 6 different ways clearly came to be despised, and particularly by so called mathematicians today, who generally frown on so called verbosity. The ability to use an extended range of antonyms and synonyms came to be ridiculed as pomposity, particularly as those who became educated tended to exhibit intellectual snobbery. Rather than accepting the rhetorical duty of Kairos, they ignore it and used their education to distinguish themselves, socially and class wise!
In rhetoric, therefore any sentence was an expression of the intentional meaning. Thus there may be several possible sentences to express the same intention. A sentence then is merely a form of expression indicating a choice of words intended to convey intention. The intention could be a meaning, or an instruction or an observation etc.. The rhetorician only knows whether the intention was received by feedback. Based on feedback the rhetorician would modify the expression of the same intention until the rhetorician received the desired feedback.
So the use of expression has a chequered rhetorical history, bu essentially any collection of sounds or written symbols can be called an expression . The point is what is the intention of said expression?
The explicit equation/ expression is intended to command a process. The process may be obvious or buried within the whole expression. The process has an action or set of actions that is applied to a set of " things" to give a defined result.
Now a question is made from an expression by leaving out essential parts of an expression that has been taught. The implied response is to provide the missing " things". Of course these questions make no sense to someone who has not been taught the standard expression. Even up plying uh a person with the answer does not enable them to make sense, because the expression is explicit, and so they must learn it as explicit!
Now a recursive expression sounds a bit more involved, but it's intention is to provide the receiver with a repeating process. The receiver therefore has to be told the elements of the process and how the process acts on them in order to repeat the process. This expression is structured and self referential. Many languages, particularly Greek are self reflexive. This means that many thinkers think tautologically. This is usually frowned on as circular reasoning, but in fact it is a natural way of thinking for many language groups. The claimed " confusion" is not due to tautology, it is due to the rhetorician not appreciating the reflexive forms in certain languages or modes of thinking. If these are appreciated they can be utilised to give recursive expressions that allow the receiver to understand a sequence.
Everything in my experience is sequential, but some sequences have internal relationships which cannot be succinctly expressed explicitly, but can be recursively. By finding such relationships a process provides a receiver with an unending supply of resultants self generated!
The sum of the first n numbers can be found by an explicit formulation relying on an understanding of sequence and succession, but the generation of the list of Fibonacci numbers requires a recursive process, it is tautological and self referential, but as one carries out the instruction the results lead to a change.
This recursive instruction has been used to give a " proof" by induction. This is not a proof, as nothing can be proven in this modern sense. It is a demonstration which provides the receiver with instruction in what to do so they can verify for themselves. In a similar way the" proof " by contradiction only demonstrates inconsistency. It is up to the individual whether they want to be inconsistent or not!
Grassmanns Ausdehnungs Groesse consist in expressions many of which are recursive, by labelling a combinatorial form by some symbol and then recursively referring to that label as a new magnitude, a new type he could construct more complex types by a recursive process. Once he had reached sufficient complexity he could define resultant expressions at that type level. The consequence of this was that a solution at a higher type level cascaded down to the lowest level of type!
In the other hand a specific solution at the lowest level of type did not indicate how to generalise to more complex levels.
Programmers recognised the usefulness for them of this type analysis.
Early computers were unbelievably restrictive in terms of memory and processing ability. Also physical objects had to be used to represent a value by state so the meaning symbol divide human thinking had been trained in ,mthere the symbols were physical material objects, but the meaning was immaterial subjective apprehension did not work! Turing had to explore how the state of a physical object could represent the state of 2 or more other objects! The answer was labelling!
The labels used in computing are called pointers. By these pointers whole systems of physical objects could be referred to, especially if a sequential movement of the pointer could be controlled locally.
Thus a block is designated as a pointer. Pointing information is looped to that block. It therefore only points to those blocks identified by the information . This system forms a" structure" in the programming language C. It has to be declared or initialized before the rest of the programme runs.
The operating system parses and sets up the limited memory objects to fulfill this role. These processes are written in sub routines that the general programmer does not have to bother with, but the programmer that programmes in machine code has to know each string of signals that set up a flow of other signals driven by the clock. By opening and shutting " gates" the flow of signal is directed to particular places in the memory where it is cycled accordingly. The release of this flow is again governed by opening a " gate" on demand and releasing the flow to its purpose.
I am sure this is a quaint way of describing Turings detailed analysis of states and state functions in his machine, but I do not mean here to get too technical. Both Grassmann and Turing laboured hard over the precise nature of these things and it is only pleasant to those who find such precision and stricture pleasant. Justus Grassmann and others laboured with conviction that being meticulous would provide the ultimate solution! This I do not believe. But I do believe that such meticulous distinctions enable a finer appreciation of the fractal nature of our apprehension, and of our experience of consciousness, my experiential continuum.
Why are the combinatorial practices as they are?
The rules of combination are " fit" and "type".
Fit refers to the relationship between 2 pieces. The notion properly derives from segmenting a whole into pieces. The whole is then reinterpreted as being constituted of these pieces. The only thing that has changed is the observers mental apprehension!
Things which are always together, or necessarily together may come to be regarded as fitting each other. Things which are designed to be together are also regarded as fitting each other. By this means you may appreciate that it is by will that elements are said to fit, regardless of their spatial relationships. Two objects always found together mat be deemed to fit by some but not by others. It is therefore necessary to specify and demonstrate what elements you mean to regard as fitting togethery and specifically how.
So as an example Newton defined how the measure called the quantity of matter was to be constructed. The elements of volume and density were to be fit together, and the action of this fitting was that the increase in one or the othe should increase the measure, but the increase in both simultaneously shoul result in the combined increase. These increases were in the form of factors of a multiple form, not an aggregation.
Now should 2 items be distinguishedmp, then they could be define by Ann aggregation of the two using a distinguishing mark+ between them. Now the notion of type comes ino play.
Anything is it own type, self reflexively, but I may give a type to anything I choose as a label. Thus I could type a cat as a kind of dog that growls miaw! I can then count cats as dogs!.
In any aggregation I could combine cats and dogs by type and replace a combination symbol by a multiple symbol.
As you see I am free to type things as I please and so reduce aggregations of symbols to multiples of types. The loss of information is profound, but if every stage of the labelling of type is known this information may be recovered.
Grassmann applied these freedoms in a remarkable way. He used the then study of combinatorial relations to construct a process to establish aggregations that he then labelled or typed
Having done that he realised this was a recursive process and he could take it to any level he wished.
Next he defined how to combine these types as a kind of complex table of products . The important thing to beable to go further was to devise a way of combining all the parts into one form he called an Ausdehnungsgroesse . For it to work the new quantities had to posses closure under the combinatorial operation. This means the operation produces its own kind or type!
He realised that he could not only define the type of anything he could define the process that produced the type from his table combination. When he did that he had consistency, but he did not know if it was all a piece of sleight of hand! Only by testing it out on several important Maths questions did he realise that he had hit upon a viable method!
Today this approach has developed into ring and group theory. Justus , his dad was in fact n early pioneer in the field , in touch with many early theoreticians struggling to make sense of the hodge podge delivered to thm as mathematics. Before Algebra had even come of age group and ring theorists were struggling to understand how Maths worked and how it should work, but Grassmann and Hamilton had more or less laid out the bones of the fledgling subject of algebra with 2 stunning new methods of analysis and calculation.
Today algebra and group and ring theory have fallen pray to subject boundary wars. Grassmann had hoped to melt these boundaries and forge a new subject that subsumed them all, called Formenlehre.
Grassmann's vision, although that of his fathers, nevertheless discerned the difference between him and his father. Robert his brother however cme along and muddied the waters with regard to hermanns distinctness. Frankly he used Herrmanns work to promote his own. There was no ill wished upon his brothers work, which up until Robert had receive little ( bu significantly very influential) attention in Prussia. After he had redacted it with hermanns collaboration under protest, his work became very well known. Herrmann realised he could relaunch his original on the success of the redaction, and it is his original I have referred to .