When Things Are and Aren't the Same
Nothing is exactly identical except itself; sameness and difference depend on the sense in which things are compared. Equations and analogies rely on a balance of sameness in one sense and difference in another, allowing us to pivot and gain insight.
THE IDEA
WHEN THINGS ARE AND AREN’T THE SAME
ONE LONG-STANDING MYTH about mathematics is that it is all about “getting the right answer”. That everything is simply right or wrong. Another pervasive myth is that it’s all about equations.
These myths both have some truth in them, but are very far from being completely true. Equations do arise a lot in school maths, but as the objects we study become more interesting than numbers, the issues become more interesting than equations too.
But there is an important idea at the heart of equations, which is that they are about finding things that are the same as each other. However, nothing is actually the same as anything except itself. As we saw in Chapter 8, equations are all lies apart from those of the form x = x, which is not at all illuminating. Other equations have some truth in them, so, there is a sense in which the two sides are the same, but, crucially there is a sense in which they are not. We saw that in this equation: 10 + 1 = 1 + 10 the two sides are the same in that they produce the same answer, but they are not the same in the sense that technically they describe a different process.
The point of equations in maths, where they do arise, is to find two things that aren’t the same in one sense and that are the same in another sense. We can then use the sense in which they are the same to pivot between the ways in which they aren’t the same, and thus gain some understanding, as we described in the previous chapter. This idea continues into research-level maths, where the senses of “sameness” become more and more subtle, and increasing amounts of technical effort have to be put into finding and describing appropriate notions of sameness.
We saw in the previous chapter that analogies involve finding situations that aren’t exactly the same, but that are the same in some sense – and necessarily different in another. We can then pivot using the sense in which they’re the same, and perhaps land in a place that is logically similar but more emotionally engaging, or more extreme and thus easier to judge morally.
But there are different levels of abstraction that produce different analogies with different quantities of sameness. Which one should we pick? This is another place where grey areas arise all the time. There are different ways in which things can be equivalent and not equivalent. A better question than “Are these the same or not?” is “In what sense are these the same and in what sense are they different?”
When to use it
Using the commutative property in arithmetic to simplify calculations while recognizing the different process orders.
Transferring a successful problem-solving strategy from one field (e.g., software engineering) to another (e.g., project management) by identifying the underlying principle (sameness) and adapting to the new context (difference).
In negotiations, finding common ground (sameness of interests) while acknowledging distinct positions (differences) to craft a solution.
In scientific modeling, using a simplified model that captures the essential dynamics (sameness) while ignoring irrelevant details (difference) to make predictions.
FROM THEORY TO EVERYDAY LIFE
In practice
Equation Example: 10 + 1 = 1 + 10
The two sides produce the same numerical result (11) but describe different processes (adding 1 to 10 vs. adding 10 to 1). This shows sameness in outcome but difference in operation.
Realistic Analogy: Business Strategy and Military Tactics
A business leader might draw an analogy between market competition and military warfare. Both involve resource allocation, strategy, and opponent reactions (sameness), but differ in stakes, rules, and ethical considerations (difference). The sameness allows pivoting to apply strategic principles, while the difference prevents over-identification.
A MOMENT TO REFLECT
Put it to work.
Think of a recent disagreement. In what sense were the two perspectives the same, and in what sense were they different?
Identify an analogy you use frequently. What are the specific similarities and differences between the two domains?
How can asking 'In what sense are these the same and different?' improve your problem-solving approach?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)