Kroenecker has been considered as carrying forward the Pythagorean Ideal. But on closer inspection it is The Grassmann's concept of Formenlehre that is closer to the Pythagorean school of thought. Consequently Ring and Group Theorists are Neo Pythagoreans! …
How are sequencing , proportions and mosaics at the base of combinatorics?
How do the rules of combination in Algebra lead to the generation of Ausdehnungs Groesse.?
How do the methods of combinatorics extend to the process of producing Ausdehnungs Groesse?
How did the tables of combinations of elements into sequences of n elements lead to matrices and determinants and the notation for producing Ausdehnungs Groesse?
From the questions you can already see that a wide range of mathematical topics are drawn together into the notion of a Grassmann Algebra. This reflects the fundamental nature of his Analytical method. This approach is now obscured in the axiomatic approach of group and ring theory.
The amazing experience of our subjective processing centre is how it can take a scattering of objects and or points and pick out a sequence or an array or a mosaic. This is so basic or fundamental that we overlook it.
The research into our neural network meshes, starting at our sensors reveals that we can receive multiple signals at activation levels for the sensors, but that a system of node filter this mas of data by only firing when a certain group of sensors are active. Then a sequence of filters each built on top of the previous level of filters reduces the signals to maybe a few complex ignal paths. It is these nervous pathways, both central and peripheral on which our consciousness is founded. First at a deeply unconscious level, but then as the processing sequences become more and more elaborate, but dealing with highly sequenced and massively parallel data streams levels of conscious awareness are evoked.
It is therefore not surprising that at a conscious level we think in highly sequenced patterns of relationships. The visual representation system tends to serve as a focus for the other representational systems sequence structures, thus I am able to conceive of anything by a visual representation of it and particularly by a visually laid out sequence or a sequence of such sequences called loosely an array.
The ancient Greeks called such arrays Arithmoi and I prefer to call them mosaics.
Fundamental to Spaciometry a mosaic has a monad, a unit form, which inspires counting. But mosaics are necessarily dynamic, so often we have to freeze frame an expression in array form to apprehend it..
Arrays themselves become useful as tables to arrange things on, and so the Arithmos becomes an essential organising pattern. Not all forms have a natural array. So dividing the form into its natural monads is an important process. This is called factoring a form into monads, and it is n intensive action. Using a monad to measure an object is an extensive action.
We are used to factories ign a form into a standard pattern called a gnomon array, but a factorisation can take any form, particularly the curved gnomon for curved objects.
Once we have our monads it is natural to use a cultural metronomic sequence called a count, which is arrange so that each element in the sequence succeeds the previous one and precedes the next. This count is actually based on a complex arrangement of Arithmoi, so an Arithmos that has fewer monads comes before one that has more monads. This is the foundation of the notion of quantity.
The construction of these mosaics introduces the notion of combination. When all the monads are the sme, the combinatorial rule is to gregge them into the whole from which thy came. When the monads are different the rule is to aggregate according to type. Thus any sufficiently complex mosaic is made up of a combination of multiples of types.
The notation I use to denote these types and multiples is similarly a combination of symbols alphanumerical in nature, but many other symbols appended.
The rules of combining symbols in this way is defined by choice. Either an individual choice or a cultural one. The cultural choice is not always adequate, and may be misleading.
Utilising the mosaic pattern of an arithmos certain multiple forms can be laid out in its sequences. The Arithmos itself provides a arithmetical relationship which may or may not be helpful in expressing the desired thought either about Spaciometry or quantity.
For example I can use a 10 by 4 gnomon array( mosaic or Arithmos) to count in 4's or tens up to 40" thus this array is the model of the factorisation of 40 by 10 and by 4.
This same mosaic could be used to write don the multiples of 10 up to 400.
The same array / mosaic could record the different proportion s of a combining of 2 wholes, one that can be divided into 4 and another into 10. What different sequences of proportions can be chosen from these 2 constituents without repetition. The arrangement of the constituents is not distinguished.
Now let me go to any monad in this mosaic, any " cell" in this array, any row column in this table, and look at what I have witten first let me extract an attribute called the order of writing, then let me distinguish each symbol in the sequence even if is from the same constituent. Now let me count each such distinguished sequence element
To find out how many different orders of writing I could have used I start by using a tree diagram. At each branch I place the options of succession in the sequences and branch again until I have used all my options. Counting the branches enables me to name what I can visually inspec, number and different arrangements of these written order sequences.
Now by inspection I find that for each distinguished element I could create a mosaic square and place all the 2 node sequences. I could create a mosaic cube and place all the 3 node sequences. I could then collect al the cubes into a super rectangular block and do the same. I could collect the super rectangular blocks to form a super square, then I could form a super cube etc. thus I can associate more and more sequence nodes to more and more complex mosaics. The thing about these is they are self similar to scale.
They do not even have to be blocks, they can be any form. The fractal nature of associating sequences to Arithmoi, that is mosaics becomes clear.
Bearing all this in mind it is clear that Grassmann saw this fractal disposition and realised he could explain spaciometric disposition and omens ions by a sequence of symbols. The sequences were naturally combined by combinatorial processes, and each sequence of distinguished or non distinguished constituents traced out a kind of axial reference frame that was totally unique and independent. Using these as axes he could span any space. Representing each of these sequences as a line of an axis was a conceptual simplification. The way to write these symbolic sequences had to be decided. Initially Grassmann just wrote them as a sequence of points. Later he realised he could distinguish aggregates using the + sign as a combinatorial marker. His Brother Robert introduced this kind of notation which seemed to work , much to Grassmann's surprise! In the start of defining abstract Ausdehnungs froess of this form in the 1862 version he comments " the form of Ausdehnungs Groesse seem to be this general lineal combination of multiples!"
Well we have seen that they are even more general than that.
Notation is crucial to what you want the Ausdehnungs Groesse to do. So rewriting them as vectors fits a specific purpose, rewiing them again as lineal combination serves the purpose of findings solutions to systems of linear equations. Rewriting them agin as Barycentric coordinates serves another purpose? They can also be rewritten as polynomials. The transformations seem to be fluid and endless!
Although Grassmann did not invent the notation of matrices which was only invented in the 1850's, he laid the groundwork for its use in fundamental algebras not just for solving systems of equations. The determinant has a long history, going back to Chinese mathematicians in the 2nd century BC, and certainly known to Gauss, Cauchy and others, Grassmann used it and defined the determinant for a system of n linear equations before Cayley in 1877!.
It took a while for people to take the Arithmos array and treat it as a whole thing called a Tableu and later a matrix. It only really took off when the determinant was clearly linked to matrices and matrix division, something Grassmann had defined in his lineal algebra of 1844 and 1862.
This next video introduces Jakob Steiner. The reason is because of the heuristic approach to the solution to a problem he posed. The Grassman algebra is only part of Grassmann's method of analysis and synthesis. Each Ausdehnungs Groesse has to be tailored to the problem that needs a solution. The Asdehnungslehre 1844 goes through the stages to developing an extensive magnitude for a specified problem. As you will see sometimes the Ausdehnungs Groesse is written explicitly, sometimes recursively.
This following set of scanned papers shows how the Arithmoi have been misunderstood.