Toward a Mathematical Theory of Anarchism, Part 2: Goals and Limitations

 

The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find  our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.

 

(Gottfried Wilhelm Leibniz in “The Art of Discovery”)

 

In the first post of this series, I discussed the motivation for a mathematical theory of anarchism. I will say immediately that I do not, at that time, know even the beginnings of such a theory. I believe I have some idea of what it should look like, and what its goals should be. This is what I will discuss in this post.

I start from the following two premises:

1. (Most) People share a common set of core values or beliefs, e.g. (almost) everybody believes that they should not kill or harm their neighbor without reason.
2. Mathematical reasoning is an effective way to derive “non-obvious” conclusions from “obvious” axioms.

I am aware that these premises are not beyond discussion, especially the first one. They are stated in the most general way and could be weakened in some cases. However, at least in this post, I will take them for granted and discuss their consequence.

The idea would be to formalize some of these core values inside some mathematical system or logic. This begs two questions: first, what should be the system or model used, and second what should be these axioms. Without an answer to the first question, it is hard to answer the second, since what axioms should be chosen depend in part on the deductive strength of the system (i.e. on what consequence one is able to derive from the axioms).

Here is an example: suppose one wants to study happiness of people in a society. Model a society as a graph, where the vertices are the people, and an edge between two people mean that they are friend. The happiness of a person could e.g. be a function taking as argument the person and returning a real number. An axiom could be that the more friend a person has, the happier it is. Of course, this is very simplistic, and not a good model, but I hope it illustrates the kind of thing I have in mind.

Going more into the mathematical details, one would need to choose a logic in which to express the various axioms, and an underlying language specifying the various objects one is talking about. In the example above, the language contained a unary “happiness” function, and a binary “friendship” relation. The logic was not really specified, but first order might be enough in this case (one would need infinitely many axioms; one for each possible vertex degree).

Of course, what model to choose should depend on what kind of questions one wants to answer. One does not want the model to be too simple as in the example above, and on the other hand making it too complicated would not necessarily make it more accurate, but would definitely make it harder to study and reason about.

One in my opinion worthy goal would be to find such a system capable of modeling various types of societies (capitalistic,, marxist, anarchist, etc.) and show that in some sense the “best” society is an anarchist one (e.g. one could have some partial orderings between societies, and an axiom stating when a society is “better” than another one).

A word of caution is in order: the goal here is not to build a machine capable of answering any political (or even non-political) question in our place from a very stupid model. Rather, the objective is to be able to study more closely, and in a non-ambiguous setting, various arguments for or against an anarchist society (and more generally, many political argument). Non-ambiguous is key here: a machine should be able to check the argument. That would only imply correctness of the argument within the particular model, not in the “real” world. Also, even if the model is perfect, you still have to believe the underlying axioms in order to agree their consequence are true.

An interesting example I have in mind is Gödel proof of the existence of God . It is unclear whether Gödel himself believed in God, but this proof shows it is possible to bring some of the discussion down to a formal level. The proof can be machine-checked, so is definitely correct in that respect. Of course, one might still disagree with the axioms, or on whether the statement proven really “means” that there is a God.

At least such proofs have the merit of bringing the discussion back to the fundamental assumptions. If two people agree on those, and believe the proven statement really corresponds to the fact that there is a God, then they should be forced to agree on the conclusion.

In the next part, I will discuss the premises above and the extend to which they are valid.

Toward a Mathematical Theory of Anarchism, Part 1: Language and Truth

Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.

(Winston Smith in “Nineteen Eighty Four”)

I adopt the platonic view of truth: to me, every clear statement must be either true or false, in some absolute sense.

By a “clear statement”, I mean a declaration in which all the concepts
involved are well-defined. A well-defined concept is one that can
explicitly be defined using simpler well-defined concepts.

I am well-aware that the above definitions are unclear. I would go as far as to say that the words “well-defined”, “true”, “false”, “absolute sense”, are not well-defined… Here are some examples to show you what I mean:

• The statements: “two plus two make four”, “two plus two make five”, “Near-Earth unsupported objects fall toward the Earth’s [1] are what I would consider clear: there is no ambiguity, and one could write simple definitions for all the concepts involved. The first and third statement would be true, the second one false.

• The statements: “All men are born equal”, “every clear statement must be either true or false”, are not clear: in the first one, one would need to define men, born and equal. In the second one, one would need to define e.g. true and false, and “clear statement” is not well-defined either.

This is not to say that unclear statements are useless (this post is
full of unclear statements). Most of our thinking and conversations
are done in “unclear” language. In fact, one could argue that Science
is the only domain of human thought where clear language even
appears. Mathematics makes it a requirement, and it is I believe
really the right (and only known) tool to formulate and study clear statements.

However, even in Mathematics, it is very useful to think about “non-clear” ideas: concepts like numbers, shapes, geometry, are somewhat ill-defined, but indispensable words that convey very useful pictures.

Sometimes being absolutely clear would mean wasting too much time defining words, at the expense of being concise (and paradoxically easier to understand). Sometimes, one just does not know how to exactly define some word, and hope whoever we are communicating with share our internal ideas on what the word “should” mean. This is the case in political conversations, and of course it sometimes does not exactly work out.

For example, when I talk about Anarchy, most people just assume at first that I mean chaos and destruction. In general, I find that confusion is rampant whenever people talk about politics (and not only about Anarchy). It seems to me most people share the same core values, but not the same definitions [2] for various words. Couple this with the usual logical fallacies that everyday language makes easy to fall into, and it follows that political communication is almost impossible. So we obtain the interestingly common situation that two persons sharing the same core values completely disagree.

Yet in Mathematical logic, whenever two systems share the same axioms and rules of inference, exactly the same statements are true (provable) in both systems. To me, this suggests a radical approach to understanding Anarchism better, and make communicating about it very precisely possible: create a mathematical theory of Anarchism.

In the next parts, I will explain what I mean, and the goals and non-goals such a theory should have.

UPDATE: Part 2

[1] This is also taken from Nineteen Eighty Four, same paragraph as the above quote. It is interesting that some of Winston’s other truisms, e.g. “The solid world exists” are not even what I would call clear statements (but are still true !).

[2] Or, if a clear definition is missing, associated pictures and ideas.

On original ideas and citing sources

I am well aware (almost) none of my ideas are original. Look at this
from a theoretical point of view: If I have been able to come to a
conclusion, then another person could have been able to do it too.

Nevertheless, I often manage to come to a conclusion independently,
without reading it in a book, or hearing about it beforehand. Only
(much) later will I discover that somebody else wrote it down in a
book before me. Often, it is not only one book… It is a very special
feeling when you discover that somebody (possibly from a very
different background) reached the same conclusion as you did. You gain
confidence in your reasoning abilities, and feel less alone.

For example, when I first began thinking about Anarchy, and reached
the conclusion that it might be a viable way to go, I didn’t know
thousands of books had been written on the subject, and a lot of other
people had reached those same conclusions before me. When I talked
about it with my friends, I got the usual “Who will build the roads
?”, “Who will take out the garbage ?” kind of questions, and answered
the best I could, but I had never read any other answer in print. Only
recently did I discover the book Anarchy Works, and saw that
all those questions and more had been answered, with much more
research than I cared to make.

This is not to say that books do not shape my ideas. Clearly, my ideas
are shaped by what I read in books or on the web, but it is often very
hard to cite the single perfect reference. On “sensitive”,
non-mainstream subjects like Anarchy, I find it is often difficult to
find good sources without help. This is where the internet comes in,
but more on this in another post.

All this to say, if I know of a great source for a topic I am
writing about, I will try to mention it. However, I may not be aware
of its existence, partly because I am too lazy to search for it,
partly because it is hard to find in the first place.

There is much more to say about doing your own thinking versus
“Standing on the shoulders of Giants”. In this post, I simply wanted
to apologize in advance for not mentioning enough references. Feel
free to jump into the conversation and fix this particular shortcoming !

Under Construction

Friends once jokingly suggested I write a book about some of my weird ideas. Unlike a book, a blog is constantly changing: new posts, comments, edits… This has advantages and disadvantages.

The advantages are that I can write down and share my ideas quickly, without thinking too much about how to structure them, or worrying about global consistency.

The disadvantages are that I can write down and share my ideas quickly, without thinking too much about how to structure them, or worrying about global consistency.

Keep this in mind.