Mathematics has had a great impact on the way I think. Not just the way I think about mathematics, but the way I think about everything. The tools mathematicians use to think about mathematics transfer beyond mathematics, and even beyond academia. They are just good tools for thinking about anything, and they definitely have relevance to practical life. The trouble is, most people don’t know about them, because the mathematics that is taught in schools literally isn’t mathematics. This non-mathematics that most people are familiar with not only isn’t helpful for thinking, but it is actively unhelpful, because it is centered around following orders and saying the things that authorities want to hear. This is the very opposite of the free creative thinking that real mathematics brings. Most people don’t even know that real mathematics exists, since the non-mathematics in school calls itself “mathematics” and thus eclipses real mathematics.
So how can you learn real mathematics? Start with the books in the bibliography ;;;;;;;;;;insert link
Anyway, as I asserted at the beginning, real mathematics helps you think about everything, not just mathematics. So then, we would expect that people who go to college and graduate school for pure mathematics and get years and years of training would come out and help the rest of us think. But they don’t! Sadly, the only thing they apply all that training to is more mathematics. It is completely self-serving!
The isolation of mathematics from the rest of life is due to the dysfunctional way that our culture treats mathematics. As already mentioned, school gives most of the population a warped image of what mathematics actually is. In this warped image, math is not about creativity and understanding, but rather about memorization and following directions. People who get good at it are geniuses who just think better than ordinary people: that way they can follow more complicated directions and keep more things in their head at once. For non-mathematicians, math is inaccessible and irrelevant to the world.
Though mathematicians are saddened by the public’s misperception of mathematics, they are complicit in this misperception. This is because they write and speak in a highly cryptic style which only they understand. As long as they communicate this way, they have not the means to disabuse the public of their confusion. On the contrary, anything they say reinforces the image of mathematics as cryptic esoterica that only really smart people can understand. The dysfunctional communication style of mathematicians also alienates newcomers and prevents mathematical culture from ever making it into the general culture, which would be amazing if it happened.
As a consequence of all the obstacles to learning mathematics that our society sets up, the only people who ever learn it are those who do almost nothing else with their life. Thus they become a much happier version of MMORPG addicts, since they are living their whole life not for the real world but for the mathematical world. They are happier than MMORPG addicts because 1) mathematical addiction is much more highly valued in our society than MMORPG addiction and 2) the mathematical world is not designed by capitalists to maximize for addictiveness, but rather by the very same people (mathematicians) who live there. But it still does not rival the real world as a place to spend one’s life. It should be possible to be a mathematician, but live in the real world.
Now I said at the beginning that the tools of mathematics are useful for everything, not just mathematics. Despite this, mathematicians generally only apply these tools outside mathematics for fun, to make a joke, or to make an analogy for better understanding something mathematical. Professionally, they only apply the tools of mathematics to mathematics itself, or to very closely related fields such as physics and computer science. Modern mathematics is like avant garde art, which no longer attempts to mirror the world, but instead mirrors art itself. Just like art, mathematics used to be applied to the world. But now it is applied almost exclusively to itself, and furthermore it is applied to the application of itself to itself, ad infinitum. It is reflected into itself ad infinitum just like two mirrors facing each other. Mathematics sees infinitely far into itself, and this great self-knowledge is largely responsible for its great power. But its mistake is that it doesn’t see beyond itself. Mathematics does not exist as a celestial hall of mirrors surrounded by vacuum. It has a social existence. The mathematical world lives in the communities of mathematicians who think that they live in it. Mathematics sees infinitely far into itself, but it neglects the social sphere in which its actual existence resides. For mathematics to have true self-knowledge, it must be applied to society.
Social Math is a proposal to do this. In Social Math, mathematics will be applied to the world that humans live in. It will be presented accessibly, and its relevance to the real world and everyday life will be immediately apparent.
Mathematics has helped me think about a lot of things outside math. I’m sure other mathematicians will relate to this. The tools and ways of thinking that we use in mathematics are very helpful for making sense of life in general. But the mathematics we learn has nothing to do with life in general! It is all about manipulating shapes, numbers, and symbols abstractly. You have to learn a lot of mathematics, which is really difficult in our society given how it’s taught, before you see its relevance to life in general. Social Math, however, aims to talk about math in such a way that its relevance to life in general is immediately apparent. I also aim to make this math easily accessible to the people it can help, which is anyone, not just people who are already mathematicians.
This is very different than what mathematicians usually do. They usually go to school for years and years to learn the powerful system of conceptual techniques that is mathematics, and what do they apply these great tools to? Instead of applying them to something that can help people, most mathematicians apply them to the system of mathematics itself! It’s completely self-serving!
In the last paragraph I was primarily talking about pure mathematicians. There do exist applied mathematicians, and even mathematicians who apply mathematics to social science, but they are not doing what I want to do. There are a number of differences:
In the way math is usually used in social science, it is used as a black box. The concepts that social scientists come up with are put in one side of the box, and the mathematical results come out the other side. The social concepts (such as gender, crime rate, income) are accepted by the mathematician uncomprehendingly and uncritically: they simply become abstract variables for mathematical analysis, and they lose all social content. The results of mathematics are used by the social scientist, but the concepts of mathematics, the ways of thinking which are so helpful for me, are never applied to society. Sure these ways of thinking are used to produce the results which the social scientists then use, but these results are taken as pure results, stripped of the conceptual process that created them. In this approach, mathematics never actually makes contact with social science: they talk past each other.
It is generally only quantitative math, such as statistics, that is applied to social science. I plan to apply qualitative math. This is more apt, since the things humans care about in society are qualitative. What math is qualitative, you ask? Isn’t math all numbers and quantities? No. As I see it, math has four big things that it is about: number, space, structure, and argument. Geometry and topology study space, not number. Algebra studies structure. Mathematical logic studies argument. Math is so much more than just number. Society is also so much more than just number. Why not apply the mathematical analysis of structure to social structures, or the mathematical analysis of space to social spaces?
When mathematics is applied to the analysis of society, society is generally simplifed and reduced in order for it to be described with mathematics. Most mathematicians, if you asked them, would say it is necessary to make such a simplified model if you hope to analyze anything as complicated as society. Social Math doesn’t simplify society. Rather it complicates math. (If you think mathematics is already complicated, let me remind you of a quote by von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”) This is what has to be done in order for mathematics to be correctly applied to society. Mathematics is just a language and a conceptual scheme, and it must be subservient to the reality that it describes, not vice versa. Math must be bent to society, not society to math.
Applied mathematicians often want their models to have predictive power. Basically, they want to make effective calculators for predicting the phenomena they are studying. Social Math strives to have descriptive power rather than predictive power. Society is much too complicated to put into a calculator, unless you are analyzing a very restricted range of phenomena. For some purposes, like insurance and marketing, this is adequate, because you are analyzing a very restricted range of phenomena. But my purpose is to cultivate healthy relationships and change society, and in this it is wholly inadequate. Moreover, I think that for these purposes human beings should always be the last agents in predicting outcomes of actions in order to decide on actions. In order to do such predictions, humans need to understand the situation. Thus, it is much more important for Social Math to create human understanding through its descriptive power than for it to supplant the role of humans by predicting outcomes all on its own. Even if it did have predictive power, conscientious humans would not trust its predictions without first understanding the reasoning behind them, so predictive power only really matters insofar as there is descriptive power to support it.
The bending of math to society is not unprecedented. In fact, it has already happened in the application of mathematics to chemistry. This is my model for how I want to apply mathematics to society.
Hence I will talk about the application of mathematics to chemistry.
If I was told about how chemical theory worked before I had ever done it, I would have thought it was ridiculous and I would not have believed that anyone could be confident in its results. A chemical law is only true in a special case. It has tons of exceptions, and it directly contradicts other chemical laws. It requires an experienced practioner of chemistry to sort through all these laws, figure out which ones apply, account for the exceptions, etc. When two laws predict contradictory results, they must be compared. Yet other laws answer the question “Which law has the bigger effect?” To come to a conclusion in chemical theory often requires discussion and debate among multiple chemists.
I learned how to think in this messy, complicated system, and I can do it and even come to conclusions that convince me, but I am baffled that it works at all, and that the conclusions it produces are as convincing as they are. This system seemed highly exotic to me when I learned it, since I was used to mathematics, but really this kind of messiness and complexity is the rule rather than the exception, and the polished logic of mathematics is the exception. Social things especially are complex like this. Social things are more complex and messy than chemical things. So all the more, if we apply math to society, we should expect a mess of this nature. We need to stop thinking of the application of math to society as something that will reduce the complexity of society to something more manageable. If we attempt to reduce the complexity of society, we will no longer be talking about society, but rather a simplified version. We will have bent society to math instead of math to society. Math will help us understand society, but not through reducing society’s complexity. Rather than supplanting our existing tools for understanding society, mathematical tools will enhance them.
Why does chemical reasoning work when it is so messy? It is because the mess is grounded in reality. Without the grounding of actual experiments and other chemical things that happen in the real world, chemical theory could not sustain itself. Really, chemical theory is a language for interpreting experimental results after they already happened, not a language for predicting new ones. It is a descriptive system, not a predictive one. Chemists can use the system to make predictions, but the system itself does not predict. This is just how Social Math will have to be if it wants to describe the complexity of social life without simplification.
So how exactly must math complicate itself in order to adhere to society? Let’s use chemistry as a model. In chemistry, all concepts are open for revision. For example, the concept of acids and bases is revised to give the concept of an acidity scale. The acidity scale, first a qualitative thing, is given a precise definition in terms of the concentration of solvated protons (pH), but this is later further precisified by replacing “concentration” with “activity”. This goes on and on. Later refinements of a concept are not given different names and considered different concepts, but are rather considered different moments, or aspects of the same concept. No concept is fixed. All concepts adapt and bend themselves to fit what they try to describe. Similarly, in Social Math, what is now something that is just true or false may later turn out to have degrees of truth, and what is now a function may turn out later to be just a partial function, etc. Social Math will never become attached to a model. Any model will give way to another model without hesitation if by doing so it can more accurately describe reality.
The key driving forces behind mathematics are the conceptual techniques it comprises. These conceptual techniques are what go into the creative aspect of mathematics: creating systems and creating arguments through which we know the systems. However, these conceptual techniques are not disclosed in mathematical writing. Mathematical writing customarily states only the results of such creativity. This constitutes a lack of self-awareness in mathematics. The conceptual techniques of mathematics are generally only applied to the results of mathematics, not enough to these conceptual techniques themselves. In order for mathematics to bend itself to society or whatever else it is correctly applied to, it really needs to become more self-aware. In general, self-awareness is the key prerequisite for self-change. Thus writing in Social Math will address all aspects of mathematics as well as of society. It will not only discuss the results of mathematics, but the methods used to attain those results as well. This self-consciousness is different from what is done ordinarily in mathematics, but it is essential if we want to bend it to society.
Social Math is a confluence of the following:
