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That's a Fact

Entry 2301, on 2023-10-31 at 12:54:56 (Rating 2, Philosophy)

I recently read an article about the nature of maths, and by extension reality itself, and it got surprisingly fractious given the actual apparently mundane nature of the subject.

It was about whether 2 + 2 equals 5 or not. Of course, our immediate reaction would be that simple addition problems like this only have one solution, and that the solution is obvious given that most people learned basic maths quite early in their education, but there is some unexpected subtlety here, so let's have a look at it.

One reason for this particular statment being controversial is that it has been used for years as an example of misinformation and propaganda. It was notably used by both Dostoevsky and Orwell, who said that if the leaders of Nazi Germany had said 2 + 2 = 5, the people would have had to believe it, implying it is obviously false.

Additionally, it is used as a way to criticise the modern relativist and postmodernist views that maths is a matter of opinion or oppression, rather than anything with intrinsic truth, and that insisting that only one answer is possible subjugates alternative views.

So almost everyone would say that the equation in question is untrue, but let's look at a few examples of where it (and similar equations) might not be.

First, we need to define the symbols. In normal maths the symbol "+" means addition, and "=" means evaluate the preceding symbols to give a final answer. And "2" and "5" are symbols mapped to specific numbers (whatever numbers might be, because no one seems to know).

But in some representations, such as certain programming languages, the symbol "+" could mean something else, like concatenation, so "2 + 2" in this case would be equal to "22". Plus, some languages use "=" to mean test for equality rather than evaluate a statement, so "2 + 2" might evaluate to 4 but "4 = 5" would equate to false because 4 doesn't equal 5 (or does it?)

In fact, in certain programming languages, which don't have strong typing (meaning that when you store a value, which might be a number or a string of characters, it isn't enforced to be a certain, predictable type) it is easy to make an error that when you expect the answer 4 you might unexpectedly get "22" instead (note my use of quotes there: one way languages handle this problem). I don't know of any natural languages with this characteristic, but I'm not a linguist and maybe they exist.

So I might say that "2 + 2" evaluates to "22" or that "2 + 2 = 5" evaluates to false are correct answers under some circumstances.

But let's assume the symbols have all the usual meanings and see where else things might go wrong. In the real world many measurements have arbitrary precision. For example, if I measure the amount of coffee in my cup I might get 100 ml. But when I use a more accurate measure it might be closer to 99.1, or more accurately still 99.095781. In fact, there is no correct answer, unless I count the molecules (a bit impractical, and with many ancillary issues), so many answers are possible.

When measuring a continuous quantity, like volume, mass, length, etc we are always estimating, so multiple answers are all correct, or all incorrect, depending on your perspective.

And there's a corollary to this too. If we measure the volume of one cup of coffee and get 2.3 we might round that to the nearest whole number and say 2. Then we might ask, how much is in two cups that size? The answer is 5, because 2.3 x 2 is 4.6, which rounds to 5, not 4. So in that case 2 + 2 does equal 5.

Here'a another place where we might get unexpected results: modular or "clock" arithmetic. Imagine it is 22:00 (or 10 PM at night; I will use a 24 hour clock here for simplicity) and we ware going to take 3 hours to wrote a blog post. When will I finish? According to simple math the answer is 22 + 3 which is 25, but 25:00 isn't a time, because at 24 we revert back to 0. The real answer is 1:00, so in this case 22 + 3 = 1.

Here's another example. I'm flying my plane at a heading of 270 degrees (west) and I change my heading by 135 degrees. What angle am I flying at now? Well 270 + 135 is 405, but angles are usually only measured up to, but not including, 360. At 360 we go back to 0, so the real answer should be 45 degrees, or north-east.

Again, we can get anomalies like this in programming too. In many languages whole numbers (integers) are stored in a certain space and when the number is too big for this space and overflows it can give unexpected results, including adding 2 positive numbers giving a negative number. This is more of a bug than a feature though, so doesn't really fit in with my argument here of there being several *correct* answers.

In the original article there were several comments, some of which got quite heated and even threatened to stop reading the publication because it so obviously dealt in propaganda. But, of course, my comment was more reasonable and nuanced: aren't they always?

Here it is: "I think it is safe to say 2 + 2 doesn't equal 5 given the usual assumptions about how the calculation is done. In general we would assume we are talking about normal integers without rounding, non-modular arithmetic, etc. If the usual assumptions aren't being made they should be stated."

In other words, we all know that 2 + 2 doesn't equal 5 given the usual way we treat those symbols and operations. If the person making the assertion that it is true is using some other, less standard, way to do the maths, then they should make that clear in the original statement of the problem.

For example, I could say that 22 + 3 equals 25, but 22:00 + 3 hours equals 1:00; or 270 + 135 = 405, but 270 degrees + 135 degrees equals 45 degrees.

In fact, when you look at almost any statement about anything, there are always assumptions inherent in any statment, no matter how truthful it might seem. If even maths can be subject to so many potential sources of confusion, then we would expect other, less rigorous, forms of expression might be subject to many more.

Always check the assumptions, definitions, and methodologies for any subject under discussion. Actually, that isn't really practical, or we would never be able to discuss anything in a reasonable time, but at least be aware of these issues.

Does 2 + 2 = 5? Well, it depends on a number of factors, but in general, using the assumptions most people agree on, and the usual interpretation of the question, the answer is no.

By the way, I wrote this post partly as the result of a criticism someone once made about me. A person told me that "you would argue about anything, even that 1 + 1 = 3". Well yes, I would. I chose a slightly different question here because of its historical significance, but I'll argue with anyone about anything, because certainty never really exists. On the other hand, to engage in the world in a practical way, we do have to make some assumptions about how we communicate.

We can't prove that facts exist, but we should act like they do, and that's a fact.

Comment 1 (7518) by Anonymous on 2023-10-31 at 16:34:03:

Pah - sounds like DoubleThink to me.

Comment 2 (7519) by OJB on 2023-10-31 at 20:47:58:

Doublethink is an Orwellian concept too, of course. Seeing this that way is what got a lot of people upset. When used maliciously the denial of facts can be harmful. I just wanted to show it’s not that simple.

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