# toward a new Marxist mathematics
most of us understand mathematics as simple equations like "1 + 1 = 2" or "y = x + 1"
in these equations there is a small and predictable set of operations.
if you are doing arithmetic, you may see addition, multiplication, subtraction, and division.
if you are doing algebra you may see variables, trigonometry functions, powers and roots, and equations that result in groups of different solutions such as square root or absolute value equations.
if you are doing set theory, you may see groups of miscellaneous things called sets and various ways that sets are combined into new sets.
if you are doing matrix mathematics, you may see tables of numbers called matrices and operations specific to them such as transposition, dot product multiplication, and linear transformations across a coordinate system.
what if you could add together _any_ two objects you could think of
what if instead of adding integers or sets we were doing almost-absurd equations such as:
```
x = apple + knife
```
```
apple + y = apple pie
```
the moment we see these "equations", we know they have answers. it is not difficult to figure out the answer logically to what things need to be combined with "apple" to reach "apple pie", assuming somebody has ever seen a recipe book. there may be many possible logical answers to what other ingredients need to be added to make a pie, but that still means there is always at least one answer.
we can further say that, even in a world where every individual physical object is separate and unique from other physical objects, many ingredients are equivalent to each other. a red apple and a green apple are equivalent ingredients if you only want to make an apple pie in general without specifying further characteristics of the apple pie. a roll of cinnamon bark and a jar of ground cinnamon are equivalent. this makes it possible for any number of different households which each have slightly different forms of ingredients to make the same recipe and all end up with an apple pie as long as they possess suitable ingredients to begin with.
at the same time, the apple pie equation is not quite a proper math problem. given only the equation and not any ability to test the ingredients in some kind of real-world kitchen or oven, we cannot quite predict how to solve the equation using only the information in the "mathematical objects". with arithmetic, algebra, calculus, set theory, and even matrix mathematics, we can merely begin with the information contained in each of the objects in the equation; we do not need to take the objects to the kitchen just to calculate an answer.
perhaps we can take some guidance on defining operations from existing mathematics. whenever a new field of mathematics is created, it begins with first creating a new mathematical structure, and then defining how this structure is read for the purposes of doing operations.
in vector mathematics, vectors are introduced as collections of coordinates. once vectors exist, it is possible to define vector addition by reading the vectors as a sort of transformation against each other across a grid. assuming a coordinate system where the y axis points north and the x axis points east, a north vector plus an east vector is read as the end of the north vector proceeding east to a new point, and produces a northeast vector.
```
↑ + → = ↗
[0,1] + [1,0] = [√(2),√(2)]
```
in matrix mathematics, matrices are introduced as two-dimensional collections of coordinates. once there are matrices, matrix addition can be defined for matrices of the same size by reading the cells of each matrix as individual values to be summed separately.
```
⎡ 1 2 ⎤ + ⎡ 4 3 ⎤ = ⎡ 5 5 ⎤
⎣ 3 4 ⎦ ⎣ 2 1 ⎦ ⎣ 5 5 ⎦
```
as much as these two kinds of mathematical structures seem abstract, there are many real-world applications of both of them. a vector in a physics problem can represent a force in a particular direction which will be compared against forces in other directions to determine the resulting force that is able to accelerate an object from rest into motion.
applying matrices to physics is a lot more complicated, but for one example, matrices can be used to solve systems of equations in situations such as two objects of a given mass on oscillating springs to figure out the position equation of each object and derive how the waves may be interfering with each other. a matrix involving the values inside multiple equations can be turned into a new equation and the matrix can be collapsed into its determinant to solve for the relationship between the original equations.
vectors enable us to take a concrete real-world motion and turn it into a math problem where motions can be added or subtracted. matrices aid us in taking real-world periodic motions — waves — and computing how they affect each other to produce more complex waves.
from the example of vectors and matrices, it would seem like in order to do operations on some arbitrary set of real objects we would need to already have prior knowledge of what ways objects interact and apply mathematical equations to objects rather than going in the other direction and transforming objects into mathematics. when objects are so complicated, how could we predict what kind of mathematics will apply to them and know we have chosen the correct one?
the answer is that we must start from interactions rather than individual objects. if we observe the interaction partially, we can likely already predict what kind of interaction it will be by the time it completes.
let us solve some more logic problems:
```
distance + distance = f
force + force = g
object + force = h
```
a distance plus a distance will generally be another distance, because two spatial areas by themselves have no real other way to interact than to combine into a larger spatial area.
a force plus a force will generally be another force, because two accelerations-per-mass have nothing else to combine into than another acceleration-per-mass in some changed direction or simply at a nonexistent force of zero.
an object plus a force can have many possible results. one of the simplest results is acceleration. another potential result could be the destruction of the object, if the surrounding forces pointing into the object overcome the natural resistance of the object. if we were solving the "apple + knife" problem, the normal force from the apple against the knife is only so strong before the knife overcomes it and slices apart the apple. yet, if the side of a knife hits an apple with no obstructions the apple will simply go rolling. we can still predict which of these outcomes is more likely to happen as soon as we see the apple and the knife approach each other. if the knife and cutting board are both pointing into the apple, it will be a problem of structural integrity; if the knife is moving toward the apple but nothing else is, it will be a problem of acceleration.
without an impending interaction between the apple and the knife we would not be sure what kind of mathematics applied to them, but as soon as that interaction begins we come to a very important realization: every single mathematical equation is an interaction. just as there is no "2" without an addition interaction of "1 + 1", there is no concrete physics equation without an initial interaction of "apple + knife".
the interaction of separate objects across material reality's gaps is the key to extracting physics out of objects.
arguably, this is the reason chemistry was able to transform what may first seem like opaque material objects into nearly-mathematical equations. [*atp]
```
water + air + light = plant sugar + air
```
```
chemical + chemical = chemical + possible waste (or no change)
```
```
6H₂O + 6CO₂ + ~600 kJ energy ⟶ C₆H₁₂O₆ + 6O₂
```
whenever we do a chemistry problem, we always begin with the interaction of two chemicals. because a chemistry problem always involves a limited range of objects such as atoms, molecules, and ions, and we have already specified which of these are about to interact in what general conditions, we can usually predict that the result will be another chemical — unless this particular combination of substances simply does not react and produces no change. because we know that a chemical equation is an interaction of two chemicals in what is likely to be a chemical reaction, we can then begin to work with the known rules of electrons and orbitals to determine what resulting substance might be produced.
after we have been through this overview of mathematics, physics, and chemistry, in what way can these concepts immediately apply to Marxism?
at the end of the day, the task of Marxism is nothing more than another "apple pie" equation.
```
apple + n = apple pie
```
```
x(union) + z(theorist) = party-nation
```
```
x(union) + z(theorist) + y = workers' state
```
the "apple + knife" and "spring oscillator + spring oscillator" exercises show us that to a certain extent, any interaction between two things has similar characteristics by virtue of being an interaction. any interaction between material things with the potential to create a larger or smaller material thing can be considered as if it were a kind of physics, and in some cases even as if it were a kind of mathematics.
this is to say, the behavior of assembling together many separate carbon molecules into a glucose molecule or large organic polymer is loosely comparable to the behavior of assembling together people into a workers' state.
specifically, a molecule must always have local structures in order to materially exist, such as monomers that assemble into a polymer. the same is true of people assembling into political or economic structures which compose an entire country and thus constitute a workers' state.
every single named Marxism contains these internal structures. if any particular named Marxism wishes to realize itself into a material workers' state which is to remain standing, it will inherently realize itself through creating particular arrangements of people which are then linked together in larger arrangements. the small-scale arrangements may include such structures as co-ops, town soviets, or local party branches. the large-scale arrangements may include regional soviets, ministries, union republics, or a party congress.
as well, a workers' movement leading up to the day of a workers' state will have its own large- and small-scale structures, which may include such things as unions, party branches, and propaganda groups. collectively, the combined set of proposed structures in each of these stages of development forms the named Marxism's _particle theory_.
it is entirely possible, much as with the "apple pie" equation, that there are many viable Marxist particle theories and many viable named Marxisms. however, it is not critical to get deep into meta-Marxism and comparing different named Marxisms simply for the purpose of constructing a basic mathematics of workers' movements.
here most Marxisms in existence should be able to agree around the same fundamentals: any workers' state or proletarian civilization must be composed of workers, and they must be arranged into larger structures to produce a proletarian civilization. if the structures do not successfully hold together, the movement or workers' state will in some way shatter into a disorganized array of smaller and lesser things. this is as true for generic Marxism-Leninism as it is for Trotskyism as it is for Juche-socialism, Maoism, Gramscianism, and all would-be Marxism replacements such as the Structuralist tradition. [*t]
the final question is how to transform such seemingly-complex processes as society and democracy into mathematics.
can it even be done?
of course it can be done. even some of the densest Liberals already believe that society itself can be represented by mathematics. in _The Excessive Subject_, Lacanians attempted to represent many of Hegel's theses in the _Logic_ in the form of set theory, and ultimately to use this in a confused argument for a model of society based entirely on the stochastic movements of individuals while discarding all notions of greater structure. it is not difficult to see how the culmination of Hegel really is in realizing that the distinction of things into separate compartments of "Being", the exclusion of one substance in the presence of another, and the interactions of things and emergence of their inner potential, as well as everything else, expresses itself in reality in the shape of analogies and overlaps between different physical or mathematical behaviors. the statements made by Hegel are statements of physical macrocosm and microcosm, of repeated physical behaviors which can be discovered and used not only for interpreting the stochastic motions of individuals in various ways, but for the benefit of creating a better society.
with that said, what are the most basic structures of a society?
in various parts of this book, we have covered that one of the most basic characteristics of a group of people is its internal connections.
culture and beliefs (the _philosophical system_ or _symptoms of the system_) are almost always affected by the basic social graph of people they exist within (the _social system_). this remains true all the way to politics and political theories. if there is a chunk of people containing 500 people and 50 of them associate together but 10 of these are Trotskyists, the entire local chunk of 50 people is more likely to believe in Trotskyism. if 150 other people associate together but 30 of these are mainstream Marxist-Leninists, it becomes more likely there will be a local chunk of Marxism-Leninism and a local chunk of Trotskyism. if 5 of the staunch Trotskyists turn against the other 5, the actions of the surrounding people determine the final result; if all 40 stay loyal to the 5 with the best theories, the group may not go through much material change, while if 20 of them become loyal to the 5 with the worst theories, the group will likely divide into two Trotskyist groups of different Trotskyist ideologies.
in the end, particularly when we are speaking of political groupings aimed at realizing some set of material objectives onto society, any group of people or chunk of society is easily transformed into an undirected graph.
for a real-world situation there may be complexities beyond this, such as solving whether a particular chunk of people truly counts as one graph, two graphs competing with each other, or numerous graphs, but this does not change the overall pattern that graphs are the basic mathematical unit of society.
knowing that society is made of graphs, how do we begin using these graphs in a logic problem? we know that graphs are already mathematical structures, but in the abstract it can be difficult to visualize exactly what people are in which graphs or what these graphs are advised to do.
this is the point at which geographical space and maps come into the picture. throughout this book we have discussed how nearly all political concepts are spatial — conflict, borders, war, class territories, imperialism, structural racism, marginalization in general, democracy, ideologies and ideological factions, sectarian conspiracies, "terrors", the internal structure of workers' states, the phenomenon of diverging named Marxisms, international conference structures, real and effective internationalism versus conflict between regions, and so on. to understand any social or political problem is to learn to visualize it spatially as the flow of coherent graphs across space and the collisions of different coherent graphs or separations that form new graphs.
the basic form of a major coherent graph, such as "all people of the United States", "all Democratic party voters", or "all members of Communist Party USA", is the same thing we often colloquially refer to as a "mass".
in the time of Liberal capitalism, most large-scale coherent graphs are especially devoid of meaningful structure. they typically do not contain any of the finer-scale formations characteristic of a Marxist particle theory which enable a smooth transition from "masses" to "workers' movement". this is to say, the act of assembling a workers' movement out of graphs requires the creation of new graphs within the old graphs.
the new graphs must not shatter the old graphs — which would lead to sideways Hegelian conflict and problems — yet at the same time, they must become locally coherent on their own level. the process of forming a new graph at a small scale and forming a new graph at a large scale are broadly similar, to the point they can more or less be considered microcosm and macrocosm to each other.
to add graphs, the first rule is to add graphs across a map.
on the large scale, small local graphs may be considered similar to points on a Cartesian grid. this enables us to add local graphs together by drawing a line across space from one point to the other.
```
point A + point B = line AB
```
```
west graph + east graph = west-to-east graph
```
starting from this basis, all local graphs must begin drawing paths to each other.
they will often find themselves obstructed by their surroundings and having to make decisions between several different spatial directions in order to find the one that works. nonetheless, they must do whatever they can to cut across wider areas and expand the overall graph while maintaining the minimum acceptable standards which began the creation of new graphs in the first place. if people understand that maintaining the new graph as uncompromisingly distinct from the old yet not in full opposition to it brings the new graph power and agency the greater it expands, this can entice people to continue the new graph in spite of everything, and allow the new graph to become the basis for permanent revolution in one country — or perhaps even open a small possibility for a cascade of movements across two or more countries, assuming they all have at least the faint beginnings of a Lattice.
the mathematics of graph addition across a wide space is really just the mathematics of a social system, and does not fully touch on the mathematics of a philosophical or material system. to add the realization of philosophies to the model will bring more detail and constraints to the basic process, allowing for a spatial analysis of whatever small- and medium-scale structures are to be realized and how these might conflict or fit together.
understanding that people are grouped into graphs and it is the action of graphs that creates organizations
we have a firm mathematical basis for the Lattice model
let's imagine we have a country the size of Germany.
as of 2023, Germany contained about 84 million people (84,000,000). the 84 million people of Germany are geographically divided into sixteen local states with a median population of around 3 million and a minimum of around 0.6 million.
in order to build a workers' state in Germany, there must be at least one Lattice center somewhere within each chunk of 0.5~3 million. if there are only Lattice centers in about six of sixteen major chunks and not in the other chunks, we know historically that this caused Germany to become divided into two countries without successfully uniting into a single workers' state.
to ask what results in a workers' state, we have to ask what combination of local graphs in what regions and at what scales connects into a country-wide graph extending spatially across all 84 million people in Germany.
in one sense we need to find the minimum number of individual people that covers the greatest area, because we can never rely on any single spatial division we pick out of Germany to contain only Marxists or usable theorists. every arbitrary region of Germany will contain unusable chunks of people and usable chunks that must route around them or somehow absorb them.
---
[*atp] ATP holds 30.5 kJ energy in its terminal bond as measured under laboratory conditions. this means that a bit over 549 kJ energy is required to assemble each glucose molecule through photosythesis, as this requires at least 18 ATP molecules.
[*t] the fact that some versions of Trotskyism claim to have particle theories working all the way up to a world government in short periods of time does in fact put a significant wrinkle in this statement because of the inherent differences between the processes of creating a regional government and a gigantic global-scale government. I have attempted to ignore this in this particular chapter in order to focus more on the notion of Trotskyism in one country or the small local or regional Trotskyist movements which currently exist in reality. for a more proper discussion of world-government Trotskyism, see "4.3/4441 All Trotskyisms". [*t2]
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