Framework for Analogies

Framework for Analogies

Whenever mathematicians feel like they're doing the same thing over and over again, they look for an abstract version that represents the situation. I find myself making analogies repeatedly, so what about the abstract version of an analogy? The general situation is that we are making an analogy between concepts A and B, via an abstract principle X that is often implicit rather than explicit.

As with monkeys swinging from branches, we should think about what level of abstraction we choose to pivot from. The more details we ignore about a situation, the more things become the same. Maths tends to get more and more abstract as it goes on, gradually losing people at each stage as they become uncomfortable with the level of abstraction.

The source’s diagrams of these abstraction levels are not included here. At the level of general addition, a and b can represent numbers such as 1, 2, 3, or something else. But there's a level even more abstract than that, where we draw an analogy between addition and multiplication themselves, and think about "binary operations", which include addition and multiplication but also many other things.

At the first level, 1 + 2 is analogous to 2 + 3 but not to 1 × 2. At the second level addition itself is analogous to multiplication, so sums and products can be treated analogously as applications of binary operations. For some purposes the top level is a good level, but for other purposes we should only go as far as the middle level.

One of the important lessons my PhD supervisor Martin Hyland taught me was the importance of finding the right level of abstraction for the situation. In a way, for abstraction, this consists of forgetting as many details as possible while still retaining the truth of what you are trying to study. If we forget details that are relevant, we might forget something that is critical to the situation.

However, if we don't go abstract enough, we might miss the chance to draw some links between more things. If we only go up to the level of "2 fruits" we will get an analogy between 2 apples and 2 bananas, but will omit 2 chairs. In order to include "2 chairs" we need to go up further, to the level of "2 things". This is what happened when we found the cube of privilege. We started with the cube for factors of 30, and the one for factors of 42, and we found that they had the same shape because both numbers are a product a × b × c of three different prime numbers.

But then we realised that if we go one level more abstract and think of it as a cube of subsets of {a, b, c} then the analogy applies to many more things, including any three types of privilege.

Thinking more abstractly can take us further from the details of a particular case while letting us connect it to a wider range of concrete cases. Much of my argument about maths comes from my view that maths is a little removed from normal life, so if we pick low pivots we will not get very far out of maths, perhaps only as far as physics. But if we get more abstract we can pivot far from maths and apply our analogies to very real situations in life.

The excerpt’s final comparison refers to people in a diagram that is not included. The general question remains: at what level of abstraction are their experiences analogous? Unfortunately, divisive arguments often arise because everyone picks the level of abstraction that best suits their argument and refuses to consider the possibility that other levels could in any way be valid.