Is $\approx$ an equivalence relation? If $\approx$ is transitive, then does the error inherent in the approximation accumulate?

Question

I was doing some physics calculations that involved approximations such as the small-angle approximation. I then started to wonder about how the relation $\approx$ can be used in comparison to the relation $=$: firstly, whether treating $\approx$ in the same way as $=$ is mathematically valid, and, secondly, in treating $\approx$ in the same way as $=$, whether continuously setting various expressions $\approx$ to each other increases the error inherent in the approximation.

If my understanding is correct, then I think what I'm trying to ask is

1. Is $\approx$ is an equivalence relation?

2. If $\approx$ is transitive, then does the error inherent in the approximation accumulate?

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I want to expand upon question 2., because, after thinking about this, I cannot see how $\approx$ could reasonably be transitive.

To illustrate what I mean here, let $A \approx B$, $B \approx C$, $D \approx A$, and $E \approx C$. We have that $A \approx B$ and $B \approx C$ so, assuming transitivity, we have that $A \approx C$. We also have that $D \approx A$, so now we can set that $D \approx C$. And, finally, we have that $D \approx A$, so we can say that $D \approx E$. When we had that $A \approx B$, $B \approx C$, $D \approx A$, and $E \approx C$, all of these approximations had, by definition, some error inherent in them. But, since they are approximations rather than equivalences (which are, in the logical sense, true by definition and therefore, in some (crude) sense "100% accurate with no error"), it seems reasonable to me that, when we start mixing-and-matching approximations as if they are transitive, then, since these approximations all have different amounts of error inherent to them, and these errors are in relation to certain values and not necessarily others used in the transitive calculation, then treating $\approx$ as if it were transitive is not sensible. For instance, the error inherent in $A \approx B$ is in relation to $A$ and $B$ specifically, and not necessarily in relation to $C$, even if it is true that $B \approx C$. So what happens when if we use transitivity between $A \approx B$ and $B \approx C$ to get $A \approx C$? If we are able to do this, then it seems that "approximately" loses meaning.

As I understand error accumulation due to approximations in mathematics, these things have the potential to quickly balloon from being relatively minor errors (and, therefore, reasonably accurate approximations) to tremendously large errors that make any "approximation" useless and meaningless. So if the errors in this case do accumulate, then I cannot see how it is reasonable to treat the relation $\approx$ as transitive.

Answer 1

It depends on how you define $\approx$. You are right that the accumulation of “negligible” errors can lead to a non-negligible error, which is why the definition of $\approx$ is very important.

If you define the statement $A\approx B$ to mean that $|A-B|\le\epsilon$ for $A,B\in\mathbb R$ and $\epsilon$ some “negligibly small” but fixed positive constant, then the relation is clearly not transitive. To see why, notice that $A\approx A+\epsilon$ and $A+\epsilon\approx A+2\epsilon$, but it is not the case that $A\approx A+2\epsilon$.

You might extend this definition and capture the “loss of significance” that you mention in the question by defining a measure of approximateness relative to the error, defining a relation $\approx_\epsilon$ that takes an argument $\epsilon$, saying, for instance, that $A\approx_\epsilon B$ if $|A-B|\le \epsilon$ for any $\epsilon\in\mathbb R^+$. Then it still wouldn’t be transitive, but you would have the nice transitive-like relationship $$A\approx_{\epsilon_1} B\space\space\text{and}\space\space B\approx_{\epsilon_2} C\space\implies\space A\approx_{\epsilon_1+\epsilon_2} C$$ which is a weaker version of transitivity, but it captures the subtlety that you were talking about.

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As a side note, this actually came up in a philosophy discussion that I recently had. The topic of the discussion was identity, and we were discussing the following fallacious argument:

Surely it is true that you are not a fundamentally different person than you were $1$ second ago. By the same reasoning, your self from $1$ second ago is not fundamentally different from your self from $2$ seconds ago, so transitively, you are not fundamentally different from yourself from $2$ seconds ago. By continuing this argument, we can show that you are not fundamentally different from any version of yourself arbitrarily far in the past, including when you were an infant or even an embryo.

This argument makes the same mistake of treating the relationship of “not fundamentally different” as transitive... but as you noted, the negligible differences can add up.