What is algebra?
It is the combinatorics of arithmetic. In combinatorics we pay attention to the sequence and arrangement of elements. In arithmetic these elements tend to be objects that we call numbers and symbols that we call operations. In fact in the written form arithmetics is totally symbolic. By this I mean that the marks that we call numerals are symbols, and the symbols that we use for operations are symbols, and any other signs marks or elements or brackets that we use in arithmetic are symbols.
What distinguishes algebra from arithmetic, is that we tend to use general symbols for everything. if we do not have a typeface for a particular operation or expression of something that we are trying to perform or process that we are trying to depict then we can create a new symbol and give it a precise definition.
Historically algebra was developed by an Islamic scholar called Al Khwarzim. and the symbolic methods or processes that he wrote down our more correctly called algorithms. Algebra was devised from a common term in Arabic Al Jibr which referred probably to the swinging of a balance. The balance would have two sides in which different objects replaced and the object would swing until it achieved a balance. however there was a more vernacular meaning to the word al Jibr and from there we get the idea that algebra was a “mindfuck” involving twisting and contorting the mind or brain in order to obtain some meaningless (apparently ) solution. Also from this connection we derived the word gibberish.https://en.m.wikipedia.org/wiki/Gibberish
Underlying algebra therefore is the fundamental nature of combinatorics, in which elements which might be objects or items or ideas are placed together in some kind of sequence or pattern. These sequences of patterns might be visual objects or they may be auditory objects or Tones ,musical notes; they may also be patterns of movement or patterns of flavours and tastes. This combinatorics is not limited to visual objects but can include every aspect of our interaction with our experience( experiential continuum).
How do we combine things?
The methods and systems and objects and elements that we used to combine help us to define the boundaries of certain subjects. So if we use mostly symbols then we may define our subject as logical or mathematical. If we use mostly the elements of nature and the environment that we may define our subject as chemical and if we use mainly the elements of the zoetopia then we might define our subject as zoology.
The common Underlying idea is our language the language of our mind and the ideas of our mind. The symbolic representation of our language may then take on the position of being symbols in our combinatorics and thus form an algebra. And as you have seen combinatorics might be the algebra of mathematics or of social sciences or of biology or chemistry.
Thus as Justus Grassmann pointed out, we must found all subjects in a more fundamental combinatorics, and then define an Algebra from these combinatorics rules and structures.
If we do so we remove Mathematics to its proper place in philosophy and computational sciences.
You may weep now understand why methods and systems are prominent in the sciences and mathematics and why Pythagorean scholars say first the Arithnoi and then the Geometree or Gematria or Numerology or Quabballah. It is general combinatorics of sequences and patterns of ideas, objects, processes and elements tha give rise to the algebras found scattered among all subjects.