Geometric algebra and the scientific method

I interpreted your question to be an ironic gesture, so I didn’t felt equipped to answer, but I feel, that the answer is already implied in the question? Depending on whether you want a philosophical answer or a mathematical answer one could say:
Quantities, as you say, only relates to what is identical, you cannot have a quantity of dissimilar objects, so then, if I interpret your question correctly we can reformulate the question like this: does there exist identical objects in the manifested/physical reality? And if so what predicate must be satisfied for two things to be identical with each other? Should they be in a identical state aswell as having an identical kind or class?

Grassmann thinks is that discrete mathematics (dealing with different quantities (the integers) and their combinatorial relations) are ‘thought forms’. He calls the first part in the book ‘a theory of forms’

The name “theory of magnitude” is inappropriate for all of mathematics, since one finds no use for magnitude in a substantial branch of it, namely combination theory, and even in arithmetic only in an incidental sense.** On the other hand the expression “form” might seem rather too broad, and the name “thought form” more appropriate;

My interpretation here is that the word Magnitude stand in relation to what is manifested (comparable effect), a magnitude of weight for example is only intelligible by means of comparison to another weight, therefore it is not pure mathematics but something else.

Another concept to keep in mind is to understand what the virtual/possibility/contingency is, compared to what is manifest. I think that we can say that mathematics is more than simple contingencies, yet mathematics still shares the same nature of indeterminacy or self-sufficiency/unrelated-ness depending on how you approach the question. Lets us first understand the meaning of what a simple possibility / contingency is. In modal logic, a contingency / possibility refers to nothing more than the name and is determined only by that agent who gave ‘it’ a name, but cannot in an objective sense be associated with anything definite at all, not even a name. Guenon states it well:

The contingent being may be defined as one not having in itself its own sufficient reason;consequently, such a being is nothing* *in itself, and nothing of what it is belongs properly to it.

We can also say that this lack of sufficient information / its indeterminacy / its unrelatedness is what makes it in-actual. What I’m trying to say is that Mathematics shares this nature but is yet something more than just simple contingency. (depending on your outlook you could say it in a better way by focusing on the quality of self-sufficiency mathematics has in contrast to the inter-relatedness/fused togetherness that something manifest has depending on how you approach the question.) Mathematics can be seen to be working according to internally consistent rules on how to construct and operate with identity. Mathematics therefore can only have relation to actuality by what extent we give it a determining or attributing power to control what is actual. Like the math in a program that controls a robot for example. the math in itself has no information about the indefinitude of complex relationships that makes up the circuit board and the motors moving the robot, and even less so, everything that made it even actually possible to construct the robot in the first place. something possible/virtual, is so precisely because of its lack of sufficient information or interface for it to ‘act’ or have act-tual presence in our sensible reality, where everything is interrelated

But also, I think perhaps there is a deeper dimension to what you’re asking in relation to what grassmann means by saying that something is retained.
I interpret it to mean that what exists has always been available to reorder and re-conjoin from eternity. Something new cannot be constructed from parts that hasn’t always been, and I think he means that this is also true in the virtual realm.
Also I think you are on the right track by questioning what we mean with the + and – operation in simple arithmetic. It should be regarded more in relation to what grassmann calls conjunction: 1+1 is not to add, it is to conjoin two equals which makes a difference, and makes subtraction possible, the ancient greeks did not have the concept of zero. And to make 1 from zero is questionable.

The opposition between the discrete and the continuous is (as with all true oppositions) fluid, since the discrete can also be regarded as continuous, and the continuous as discrete. The discrete may be regarded as continuous if that conjoined is itself again regarded as given, and the act of conjunction as a moment of becoming. And the continuous can be regarded as discrete if every moment of becoming is regarded as a mere conjunctive act, and that so conjoined as a given for the conjunction.

I interpret this to mean, that every mathematical operation, if it is to be related properly to the actual/manifested must also be linked to its moment or its instant, I.e that there is a state before and a state after.

This is interesting, I’m thinking about similar topics
but I’m not taking the empirical route, but rather the mythological in relation to the diapason and Platos nuptial number and the great year, and the idea of eternal recurrence.
Doesn’t the precession of mercury take a very long time?
much much more than a life-time?

I agree: “It should be regarded more in relation to what Grassmann calls conjunction: 1+1 is not to add, it is to conjoin two equals into a new identity.”
This is very true. Consider 2a and 2b. 2a + 2b = 2(a + b) a and b need not be the same but they can be conjoined into a new identity. This is the axiom n(a + b) =na + nb. However, a + b ≠ c and n is always a part to whole relationship where the part in this case is a + b. But is not one’s that are added but quantities. Note the 2’s were added but we Did Not get 4.

The axiom that I mention above is Euclid’s first axiom, by the way. Suddenly all of his axioms make perfect sense.
WhiteEyes what is your background? Mine is physics, math and cognitive science.

Have no academic background, but I want to study liberal arts program,I want to learn ancient Greek.

You have the perfect background! Would you like to learn some advanced math and physics? It is really quite simple, much simpler than any university course. And that is because the math will be a language that is descriptive of what is taking place. i.e. no imaginary numbers, no mysticism or magic. We might even convince chakravala to create some software to do many of the calculations. What do you think? Shall we embark on a modified version of geometric algebra? Would anyone else be interested? Please post!