“What is math?” for both Marxists who don’t know math and Marxists who do
When utilized correctly, mathematics, like Marxism, is a powerful tool for analyzing complex systems. Like dialectical and historical materialist analysis, it gives the tools for one to decompose any problem into more simple rules that allow one to reason in manageable steps. This series of articles is designed to guide a reader with any level of (or no) mathematical knowledge to understand why math is important to the study of Marxism. Additionally, it aims to give the reader a sense of what math is and, more importantly, how to utilize it effectively for analysis and prediction of material events. By the end of the series, we will discuss several mathematical toolkits and their usage in analyzing labor and other historical material developments, including some physics and higher level math, such as the free energy principle, synthetic differential geometry, and measure theory. Additionally, we intend to teach immediately useful skills, such as the fundamentals of computing and various bits of computer science, as well as how to use those skills to do data analysis or tool development.
The way mathematical concepts are usually taught in schools in America seem disconnected from material reality. Questions, very validly, are often asked by students along the lines of: “what is -1,” “why should I care about adding and multiplying ‘imaginary’ numbers,” or “what is sin(x), I’m a christian, there should be no sin here?” The reason for this is because the system of teaching in America is designed for getting one ready to function as a proletarian laborer, including traditionally labor-aristocratic fields such as STEAM. The method of communicating mathematical thought is divorced from its material and historical roots, and it’s treated like a system one just memorizes and works with like a machine. There is no human element or philosophical thought given to the concepts beyond the most basic necessary introductions needed to get one through the mechanistic memorization and application of formula. These articles intend to additionally fill these gaps for those who have always wondered why they were being forced to sit through fields like arithmetic or algebra, as that’s what was missing for many that never “clicked” with the subject.
Mathematics, like any system, has contradictions and flaws: it is not the universal truth as is often portrayed by liberal academics. Instead, its predictive power lies in its precision and the way it documents contradictions via formal definitions and the relationships between these definitions. This formalism makes contradictions harder to resolve, as one has to learn a way to communicate the syntheses of these contradictions formally as well (an activity called proving). The result of this labor, however, is that these syntheses, assuming the tools are used properly, become automatically and mechanistically applicable to many other problems1 . Additionally, the reason these contradictions and syntheses exist is fundamental to any system that’s powerful enough to use on any problem in our material universe2 , but these contradictions create philosophical debate within the community. As a result, the interpretation and use of these tools and systems is also often not fully understood by textbook authors and teachers.
These articles, therefore, will be written in a non-standard style from most academic Marxist or mathematical writing, and will contain a running narrative to motivate the discussion that follows after. The reader is encouraged to stop reading if they become confused or overwhelmed, and either ponder and play with the ideas in their head to get a sense of how to use them (a skill that aims to be hopefully taught as well), or to look up any terminology that seems incompletely explained. While we hope these moments of frustration are sparing and brief, we acknowledge both our limitations as communicators, as well as the fact that everybody learns differently and needs different ways of having things explained to make it make sense. We encourage the reader to treat these moments as temporary setbacks towards understanding additional powerful tools of reasoning and analysis. Furthermore, one result of material being dialectical in nature, part of the fundamental mechanical way our brains learn requires understanding the process of resolving contradiction, including contradictions in our own understandings. Overcoming these hurdles functions as practice for dialectical reasoning as a whole, and those reasoning skills will translate to Marxist thought, even if one never uses a number again.
Finally, and more personally, the primary author of this series of articles is passionate about mathematics, and they wish to convey a bit of their passion for the subject to others; especially fellow Marxists who may not have had an opportunity to see it as the creative, useful, and historically rich tradition that it is. Even though it was marred by the throes of unequal and classed development that it survived for millenia, mathematics contains incredible amounts of both use-value and deep inherent beauty as a field of study.
Sheep, Rocks, Counting, and Representation
Prelude
What follows is the first of our narratives. An important point to note is that we are assuming our protagonist has never even encountered the concept of a formal number. This means that the first few narratives might seem a bit obtuse to those who already have an understanding of even basic arithmetic from school. However, the point of these first narratives is not necessarily to go via the easiest route, but, rather, to transfer over how the mathematical reasoning process looks as a whole. Additionally these narratives aim to give one a personal sense of both some of the history behind the meaning of the tools and how they are based in a material system. For the sake of completeness, we will not necessarily be using the exact historical development of mathematical reasoning. However, the order in which things will be presented will give enough substituted philosophical foundations to serve the same purpose, and will therefore be much more streamlined. This will, hopefully, also make connecting with much more advanced concepts in the future easier as well for those with shakier backgrounds and those with stronger backgrounds alike.
Narrative
One day, in an ancient town in a time before numbers or counting were discovered, a shepherd is sitting on a hill. Her normal work partner is out sick, and she’s deep in thought, contemplating how she can deal with a complicated lost sheep problem that she’s dealing with: after she went out to find a lost sheep in the pasture, she got back and was pretty sure the flock seemed slightly smaller than when she left it. A pair of sheep that constantly get in trouble together haven’t been seen since she went out, either, but it’s hard to say for sure. “Too many sheep to keep track of,” she mumbles, and, fearing even more sheep getting lost, starts to get frustrated. She kicks a small rock a few feet; hearing it strike another rock, she suddenly has an idea.
Excitedly, she starts picking up small rocks as she quickly rounds up the sheep in a nearby corner of the pasture. She starts letting them pass one at a time, placing a rock in one of her empty pouches on her waist every time one goes by her. After they all pass and she has a rock in her bag for every sheep, she runs off to find the oft-missing duo, and places a pair of additional rocks in her pouch as she brings them back to the herd. Finally, she rounds them all back into the corner, and, this time, takes a rock out of the bag for every sheep she lets pass by her, one at a time. When the last sheep passes and she takes the rock out of her bag, she lets out a sigh of relief, as it is the final rock, and there was a rock for every sheep. “Well,” she thinks, “assuming I didn’t drop any rocks while searching for the other two,” making her suddenly a little worried again, but less so than last time: the flock seems about the right size and she doesn’t notice any missing faces. “A problem for later, I guess,” she muses.
She heads home, proud of her achievement of handling the sheep alone for the day without any lasting issues, and her discovery of this new rock trick. Later that night, while sitting and thinking a bit, she starts to consider what else that rock trick might be useful for. Knowing that they often lose sheep going from the barn to the pasture, she realizes that she can start bringing a bag with her every morning, and load up rocks into it for every sheep that leaves the barn. “What if I lose sheep, though, won’t I have extra rocks,” she thinks, and, with another bit of insight, realizes she can just bring a pair of bags for rocks: if she ever has to leave the herd to find sheep, she can just do the rock trick again, moving the necessary rocks from one bag to another and keeping the spares safe for later.
The next day, her work partner is still sick, so she again tends the fields alone. Suddenly, multiple sheep bolt in different directions. Thinking quickly, she’s able to round up most of the lost ones, but is unsure if she got them all. She quickly applies the rock trick, and finds that she has a triplet of rocks left. Cursing, she starts scanning the field, and, luckily, spots the small group of sheep together in the distance. Quickly rounding them up and moving the remaining trio of rocks into the mostly full bag, she brings them back to the main herd. She quickly does the trick again for all the sheep, and confirms that there was exactly one rock for every sheep. Relieved, she sits and takes a break from all the running and searching. While sitting and contemplating, she thinks over what just happened: “that worked better than I thought, and there were just enough rocks left in the other bag to match to the missing sheep when I found them, as well!” She thinks for a bit and realizes that that would obviously always have to be the case, as there is no way there could be more rocks than total sheep or more sheep than total rocks, and, despite moving some to another bag, the rocks that remain still had to be paired to another sheep somewhere. She is happy with that realization, and wonders if she can make use of this some day…
Interlude: Marxism and Mathematics
We see in the above narrative someone discovering the basics of both dialectical materialism and mathematical reasoning: she finds use-value in the rocks, and it’s immediately tied to her ability to do her labor effectively. Additionally, she realized a contradiction: she could only manage her flock by keeping track of the count of the sheep, but her memory wasn’t enough on its own. This contradiction drove her to invent the ‘rock trick,’ resolving the contradiction between her natural limits and the demands of her labor. It was a small synthesis of thought and action, and a dialectical step forward.
The philosophy of Marxism is grounded in dialectical and historical materialism. This is a powerful reasoning tool which allowed Marx, our shepherd, and many after still, to have insight into systems that seem, on the surface, unable to be understood in isolation. It allows one, from basic principles of labor and value, to describe (and affect) the broad unfolding of social and economic development. Additionally, it naturally allows one to gradually refine their understandings over time: one just needs to look for contradictions between their own expectations and material reality, and from there one knows that they need only come up with an explanation that encompasses both. At that point, they know they are honing in on a more complete understanding of material reality itself.
With the ‘rock trick,’ the shepherd discovered several basic mathematical principles, the first and foremost of which was representation: the most powerful tool used in mathematical reasoning. Additionally, she discovered counting. Given the insights of thousands of years of philosophical development, many elementary schoolers in the modern age could feasibly come up with the idea of using numbers and addition to count instead of rocks and storage. However, the important lesson was not the numbers or the adding, it was the representation of sheep into another, easier to manage, medium. Representation, therefore, is the ability to modify and act upon a placeholder for something else in a way that somehow stands for “the essence” of the thing one is reasoning about. In her case, she cared about (and represented) the count of sheep with the count of rocks, and used the act of moving a rock into or out of a bag to represent the action of a sheep being counted. In doing so, she represented the entire part of the problem she cared about in the movement and storage of rocks.
Additionally, one sees another physical power of material representation: she didn’t have to remember how many sheep there were, even if she could count with numbers; the rocks are material items that stay with her in material reality, freeing up her mind to worry about the immediate problem at hand. In a real way, it compressed the idea of the count of the sheep into the physical count of the rocks. Just as dialectical reasoning uses contradictions to refine our understanding of material systems, mathematical representation allows us to abstract (or compress some quality of something into something else in a way that is both structured and repeatable) and manipulate material relationships: both are tools for navigating complexity, and uncovering hidden structures: grasping complex systems, as Marx said, by the root.
The shepherd’s discovery of the ‘rock trick’ reminds us that the simplest tools can open the door to profound insights. At its heart, mathematics is a means of representing, describing, and resolving contradictions in the material world, just as dialectical materialism helps us grasp and transform the social systems that shape, and are shaped by, labor. These tools are not separate: they are deeply intertwined, born from the labor of navigating complex systems and resolving contradictions. To put it another way, there’s a reason Marx occasionally borrows mathematical notation in Kapital, such as when he describes commodity value as \(C=c+v+s\) (The value of a commodity is the sum of the constant capital, the variable capital, and the surplus value). We will discuss the reasoning behind the power of expressing ideas like those with that notation in later articles.
Furthermore, as the shepherd’s journey continues, we’ll follow her through the evolving challenges of her material world: from managing flocks and trading wool to navigating the development of social systems and labor itself. Along the way, we’ll uncover the mathematical and dialectical tools she uses to understand and reshape her reality. For example, we’ll explore how numbers and value function, and how to develop tools that allow us to structure and automate the analysis of complex data; especially as her society rapidly advances through computing technology. Ultimately, the goal is to equip ourselves to analyze and resolve the contradictions we face in our own time.
Mathematics and Marxism are both tools of liberation when wielded with purpose. Together, they help us see the hidden structures of the world and give us the means to transform it. With each step forward, let’s hold on to the spirit of the shepherd: curious, resourceful, and always ready to find the tools needed to grasp complexity and move beyond it.