A common concept in mathematics is generalizing definitions, theorems, and pretty much everything. There are several examples of this: for example, the Chinese Remainder Theorem is a statement about when you can simultaneously solve a system of congruences, but it has many generalizations to algebra. Another example, is the Riemann Hypothesis which has many generalizations to not only other areas of mathematics but also physics and statistics.
Generalizing arguments is not limited to mathematics; It is usually referred to as making an analogy. The real question that arise when making analogies (in mathematics and elsewhere) is “what are the underlying reasons which make the argument valid?”
There a couple of reasons why this has been on my mind lately: (a) I’m currently writing a thesis in mathematics where a generalize some results; (b) Over at Atheist Revolution, vjack has a valid argument that shows why I hate analogies in the real world:
We often talk as if another person has the power to offend us.
That [thing that you did or said] is offensive!
Your use of profanity offends me. Please stop it.
But the reality is that the other person’s speech or behavior is only half the story, maybe even less. The person taking offense is at least an equally important part of this tale. The lesson our easily offended people have for us is that we all differ in our thresholds for taking offense. The person taking offense is at least as important as the one accused of doing the offending, perhaps more so.
If this argument is taken to the extreme, it starts to become blame the victim. Just replace “being offended” with “being raped” or “being assaulted” or “being swindled out of your money.”
So now the task becomes determining what the major differences between the above examples and being offended are. This will help us illuminate how far the analogy can go. The major difference is that being offended is completely subjective while being raped or assaulted is objective. Taking offense is entirely based on opinion. Let’s look at swastikas as an example. At a children’s museum in California, they are causing a controversy mainly because people are unaware of its historical use throughout the world. In fact, there’s even a town in Ontario called Swastika. (Aside: How awesome are the people of Swastika?:
During World War II the provincial government sought to change the town’s name to Winston in honour of Winston Churchill, but the town refused, insisting that the town had held the name long before the Nazis co-opted the swastika symbol (卐). Residents of Swastika used to tell the story of how the Ontario Department of Highways would erect new signs on the roads at the edge of the town. At night the residents would tear these signs down and put up their own signs proclaiming the town to be “Swastika”.
Christopher Macaulay, a direct descendant of Thomas Babington Macaulay, was instrumental in fighting to keep the name of the town unchanged despite the association with National Socialism.
Swastika has periodically been subject to derision for retaining the name. Local residents, however, have continued to resist a change.)
Teaching children about a symbol that has been used throughout most of the world and still in use by three major religions will not make them hate Jews or become Nazis. And if you don’t teach them about it, they will likely learn it online, and we know how objective and factual the internet is.
So vjack’s argument works so long as we stay in the subjective realm. There may be a way to make this argument more general but I will leave that for another post.