Not fully grokking how “Critical-Issue Fail” works as a label for that category, but the category makes sense to me.

]]>I was so looking forward to joining the Orbit server. I hope you found a good one to call home.

]]>Is it wrong to say 2 tiny apples are 1 normal apple? No. Does that make 2+2=4 wrong? No. It makes it unclear what 2+2=4 means or is used for in this case.

2 big apples + 2 tiny apples = 4 apples of different sizes.

Measuring your blocks is the same:

2 cm that are a bit bigger than 2 cm + more than 2 cm but less than 3 cm = 5 cm that are smaller than 5 cm (or almost 5 cm, or something between 4 and 5 cm, etc.)

Besides, numbers are just that. It is what you want that number to represent that is important. If you were to use rulers that measure in meters, you would end up (in your example) with: 0 + 0 = 0 and presto: you just proved that 2 blocks of approx. 2 cm do not exist.

Perhaps I am missing the point you try to make, if so I apologise. But what I get from your article is that being clear and precise is important and never generalise, use shortcuts or use vague, blanket statements.

]]>My whole point was that using mathematical notation for something that isn’t mathematical is _confusing_. That’s not prescriptivist, per se; use language as you will, there’s no semantic police that will come for you (nor should there be). But it hinders understanding, rather than enabling it.

Convention exists for a reason; it eases the cognitive burden on the recipient. Should you break with convention, you ought to have a good reason for doing so. Or at least provide a very clear glossary to clarify where your use of certain terms differs from conventional usage. Else it’s just muddying the waters.

Physicists can refer to flavor or charm because they’ve clearly defined them and made it clear in the discourse that this is what the meaning of these otherwise overloaded terms is. When we do modulo arithmetic, we don’t say “2 + 2 = 1”, we say “2 mod 3 + 2 mod 3 = 1 mod 3”; when we’re using a binary counting system, we don’t say “1 + 1 = 10”, we say “0b1 + 0b1 = 0b10”. Clarity matters. ]]>

Why would I use mathematical notation? Because this is how we function in this real world. Have you not noticed? This is how we language. And while it is entirely possible to get weirdly prescriptivist about the process, it does nothing to help people understand each other. We can see this in the fact that you dudes keep explaining to me the same thing I have explained to you. Or maybe that’s just your kink. Whatever.

]]>The uncertainty that you observe is not inherent to the numbers or the arithmetic that you are using, but it is a part of your model. The confusion happens because you do not explicitly describe this model, which I tried to explain in my first post.

Concerning talking to a different kind of person: discussing this topic with people who hold the same opinion is not interesting. I have read your post with interest and I think that you are making a mistake. I have tried to explain this mistake calmly and without stomping my feet. Now, if I failed at that, that is alright. If you still want to discuss your ideas, I am happy to do so, because, again, I find this topic interesting.

]]>I don’t know what this means. There is no “2”, “4”, “+”, or “=” in the real world. These are mathematical concepts. We model the real world with math, because we get neat properties, and can derive consequences that we can then map back to the real world. That is, so long as we’re doing math and not just abusing mathematical notation for informal reasoning.

Which brings me to the abuse of notation 🙂

> Maybe you missed the part where I explicitly accounted for the uncertainty, using an example of where it can come from? Wait, no. You cited it in the process of telling me I didn’t do it.

Maybe I misunderstood, but no one’s criticism was that you didn’t account for uncertainty. The point of contention is the bait-and-switch with definition of equality.

So, in your significant figures example, you take 2 approximate measurements (call them `a` and `b`). You perform some arithmetic on them (`a` + `b`) to obtain some other approximate quantity. So far, so good, we’re squarely in the realm of arithmetic with significant figures. Then you take some other approximate measurement `c`. And finally — here’s where the equivocation takes place — you declare that `a` + `b` = `c`. And you fail to explain what you mean by “=” in this context. In standard arithmetic with significant digits, 2 + 2 = 4. So clearly you mean something else by “=” here. You fail to specify precisely what. Ostensibly, what you mean is that `a` + `b` and `c` are measurements of the same _thing_. If so, that’s an idiosyncratic definition of equality that places you firmly outside the realm of math and into informal reasoning about your measurements. Which is all fine and dandy, but in that case, why use mathematical notation for it? It’s confusing and seems to serve no recognizable purpose. One could always redefine words and symbols in a way to make any statement tautologically true, but that gives no new insights or information. It certainly doesn’t reveal anything new about the real world or about mathematics; it’s just word games.

If you really wanted to account for uncertainty, this is very easily accomplished with middle school math. We measured, at 1 significant digit, `a` -> 2, `b` -> 2, and `c` -> 5. Hence:

1.5 ≤ a < 2.5

1.5 ≤ b < 2.5

=> 3 ≤ a + b < 5

and

4.5 ≤ c < 5.5

It’s not difficult to see that (a + b) ∈ [3, 5) and c ∈ [4.5, 5.5). And that’s all we can say about these quantities. Nothing in the inequalities above implies they’re equal. Nor are they even contained within the same interval (sure, there’s overlap, but a non-empty intersection of 2 sets doesn’t imply equality of these sets). What we _could_ do, using these two independent measurements, is refine our approximation, by taking the intersection. This is actually useful, because we’ve improved our guess to [4.5, 5)!

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