Why are there irresolvable arguments?


One of the nice things about the modern world is that arguing over matters of fact has become less of a thing. With the internet at your fingertips, it is possible to look up a lot of things in a flash. On the other hand, it is still possible to argue motive and interpretation and we are still having a lot of arguments. In this post Occupy Math looks into a type of stalemated argument that cannot be influenced by logic or fact. This type of argument is closely connected to a primary foundation of mathematics. One nice thing — once you notice the wrinkle that shows why some arguments cannot be resolved, you get a tool for understanding that the other person may not be an idiot. The problem is, rather, that their assumptions are different from yours.

Quite some time ago, Occupy Math tuned in to a Usenet discussion group called sci.math. Someone was demanding in an aggravated tone to know how mathematicians knew there was any such thing as infinity. Occupy Math gave the correct and standard answer: we do not know there is such a thing as infinity. The seventh axiom of set theory, the axiom of infinity assumes there is an infinity. The correspondent on sci.math did not like that answer at all.

Given that Occupy Math has a whole blog about how there are different sizes of infinity this may seem a bit odd. “There is an infinity” is a standard assumption and Occupy Math just went with it; once there is one infinity it turns out that there must be many. Here is the scoop: mathematics starts with a list of assumptions that cannot be proved. These are called axioms. You can start with different axioms and get a different type of math.

A wonderful example of this appears in Euclid’s Elements. Euclid states postulates that form the foundation of Euclidean or plane geometry. These postulates are the axioms of Euclidean geometry. It turns out that by denying Euclid’s postulates, you don’t get a contradiction, you get non-Euclidean geometry.

Wait just a pea-picking minute! There are multiple kinds of math?


Those among you that have struggled with math may be appalled to learn that we over in the ivory tower cook up new types of math. Here are words of comfort: we found almost all the math you will ever encounter before we even noticed the dragon of axioms hiding deep in the ocean of logic. This will not cause you, personally, any problems. Occupy Math raises this whole topic not only because it’s an interesting part of the culture of mathematics, but because our culture seems to be descending into a state of polarized and fairly mindless opposition to one or more other sides. Occupy Math has already tried to make a small dent in this very unfortunate situation; this post continues that effort.

The big reveal

Here is the key point. Math has no content or existence until the assumptions (axioms, postulates) on which it is based are clearly and unambiguously stated. The most intractable arguments arise when the sides have very different sets of assumptions. In social and political life, these assumptions are often unstated, which means that both sides can quite honestly see the other as immoral, moronic, clue-less, and opposed to what is obviously right. In Shaw’s play Cesar and Cleopatra the following appears:

CAESAR (recovering his self-possession).


Pardon him. Theodotus: he is a barbarian, and thinks that the customs of his tribe and island are the laws of nature.

The character Britannus finds the Egyptian custom of brother and sister marrying to be horrifying and wrong. In ancient Egypt it was a long-standing custom to preserve the royal line. Arguments about genetics (unknown at the time) aside, the disagreement is at the level of assumptions.

The pro-choice, pro-life debate, the creation-evolution debate, the various debates on the distribution of wealth, all of these angry, violent arguments arise from strong differences in the assumptions that the two sides make. Once the argument has started, of course, discourse is often replaced by name-calling and vilification. Can you think of an argument you’ve been sucked into that made no sense? Was the problem a difference in fundamental assumptions?

What can you do?

If you think someone, particularly someone taking a controversial position, is obviously correct or obviously idiotic, take a breath. Do your best to figure out what the assumptions underlying the position are. Remember that a person holding a position — your esteemed self included — may not understand or have examined their assumptions. Examining your own assumptions can be a painful and sobering experience and it may strengthen your resolve or cause self-doubt that leads to change.

The huge danger in all this is that some people desire to win at all costs, to gain power for its own sake. Another big problem in the modern world is that some people lie to support a position they have already decided is correct. These are people who are not aware of their assumptions and uninterested in examining them. Both types are both potentially dangerous and philosophically crippled.

Occupy Math urges you to be aware of your own assumptions and, beyond that, to be aware that most positions on most issues rely not only on facts and logic (or lies and illogic) but on an often unconscious choice of assumptions. Ask yourself: are these assumptions I hold ones that I am comfortable with? Remember that civility is a virtue (one of Occupy Math’s own assumptions).

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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