When 2 + 2 = 5 and Other Ways to Be Wrong with Heuristics

I recently finished my first set of classes toward a BS in data analytics. It’s not very useful advice now, I hope, but I wouldn’t necessarily recommend attempting this in the academic quarter containing the most critical election of your lifetime. That notwithstanding, this means I’ve spent the past several weeks immersed in discussions of deriving meaning from numbers.

As I was gearing up to start my term, a “debate” broke out on Twitter. I put the term in scare quotes, because what actually happened is that one group of offered reasoning and explanations and another group pointed to the first and had vapors about the end of Western civilization. The question at hand? “Does 2 + 2 = 4?” The answer some people found objectionable? “Sometimes. Not always.”

Surrounded as I’ve been this term with issues of data quality, making assumptions explicit, the limits of the most common statistical tests, and error terms, this “debate” has never been far from my mind. I saw an echo of it again today, and lo and behold, I finally have some time to write about it.

The boys who cry “Postmodernism!” without much understanding of the history of philosophy are all but background noise these days, so I mostly noted their existence once again and moved on. Funnily enough, though, this actually is a postmodernism question. This is all about deconstructing the meaning of the equation. Are we talking about some ideal of “2” and “4”, or are we communicating about something else, where “2” and “4” are abstractions of reality that may be more or less reflective of that reality?

Also, does anyone else laugh when someone claims that questioning the perfect, inherent two-ness of “2” will be the end of civilization as we know it?

No, what struck me about this particular debate was how wrong it is to claim that philosophy adds anything new to the practical math on this one. Engineering got there first.

Once upon a long time ago, when I was in high school, I took a bunch of physics. I enjoyed physics so much I majored in it when I went to college, and only when I was faced with the prospect of choosing a career in physics did I realize I didn’t want to work in physics. I just like doing math and logic puzzles. Mmm, word problems.

Anyway, one of the earliest things I remember from physics was learning about significant figures. Essentially, we were told that our calculators could produce a lot of digits after the decimal point, and most of them were garbage because our initial measurements weren’t that precise. This is, to put it lightly, not an unusual thing to be taught in the physical sciences.

Nor was it difficult to grasp. We measured things. We experienced doing estimation that was reasonable to our tools. We experimented and re-measured, and we saw how impossible it was to verify results more precise than our tools could measure directly.

If someone complaining about the postmodernism of 2 + 2 = 5 or 3 has taken a lab class that involved basic physical science, they have also done this. They know. Beyond that, they’ve almost certainly also experienced measuring 2 + 2 = 5 or 3 directly. We call those “rounding errors”, even though the problem isn’t in the rounding, which was executed correctly, but in the assumption that 2 + 2 = 4 is a universal truth.

If you’re having trouble visualizing this, picture a ruler where the smallest increment marked is centimeters. You measure one block and see the length falls closer to 2 centimeters than 3. If we had millimeters marked off, we would put the length at 2.3, but without that, we’d just be guessing. We note a length of 2. We do the same with another block of the same length.

Then we put them together and measure again. We note, entirely accurately, that the combined length is closer to 5 than 4. We note a length of 5. We’ve done everything correctly.

If you find yourself saying, “Oh, but the real length is 4.6!”, is it? If we were measuring in millimeters, we might be reading 2.33 as 2.3. Then the “real” length is 4.7 cm. Or 4.6667 cm, depending on where we hit the limit of our instruments.

This isn’t just some quibble, either. Losing track of the error in mathematics has real consequences. People who work in fields that depend on math know this, even if the perpetually or professionally incredulous on Twitter don’t.

Photo by Ben White on Unsplash

The statement “2 + 2 = 4” is for people new to math. It’s a heuristic for teaching people who still need to see their examples laid out in blocks or pieces of fruit. There’s nothing wrong with that. Learning math isn’t automatic, and heuristics like these are the social conventions we’ve developed to teach it.

That, however, is precisely why it’s absurd to fall back on “2 + 2 = 4” as some kind of deep truth. That’s why it’s ridiculous to hyperventilate if someone points out that the equation is a social construct. And that’s why it’s childish to throw a tantrum if someone exposes you to a more complex idea. (Worse than childish, really. Part of growing up is supposed to be developing better communication skills than stamping your feet.)

Mathematics isn’t the only discipline that has to contend with this, of course. It’s just one of the simpler ones to debunk with basic examples. There are plenty of regressive groups that use this tactic.

There are creationists who ask, “If we’re descended from monkeys, why are there still monkeys?” We’re not. We’re descended from primates, and we have no living primate relatives who haven’t also evolved. “We’re descended from apes” is a heuristic we developed for children who aren’t ready to think about evolution more abstractly and completely.

There are transphobes who say, “Large gametes = female; small gametes = male. End of story.” It isn’t, of course. Even in nonhuman organisms, it’s the beginning of a story that also includes hormones, environmental and social influences, additional sexes, chimerism—and I’m leaving things out because I’m not a biologist. Then humans come along and add several societies worth of gender roles. “Large gametes = female; small gametes = male” is a heuristic developed from the practice of teaching people a subject by discussing the history of study on that subject.

There are racists who say, “We’re just recognizing human subspecies.” They aren’t. The question of how to define the boundaries of even a species is still being debated among taxonomists in biology. Subspecies as a concept is far from universally considered to be useful. And humans don’t genetically sort into subgroups that look anything like our conception of races. “These are the subspecies of humans” is a heuristic developed and maintained to justify colonization and enslavement.

The complaints about the complexity of science are a tactic. They are an industry. That doesn’t mean that science has everything right at any particular point in time, but people falling on their fainting couches exclaiming, “Can you believe someone said that?!” aren’t engaging in debate themselves, much less science.

This disingenuous froth is the work of people who understand the cultural cachet of science but aren’t fond of the results. They’re not standing up for mathematics or science. If they were, they’d be championing the methods that do a better job of describing reality as it really is. What they’re championing is their “right” to substitute rules—their rules—for the complexity that is reality. They’re championing their ability to be wrong.

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When 2 + 2 = 5 and Other Ways to Be Wrong with Heuristics
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12 thoughts on “When 2 + 2 = 5 and Other Ways to Be Wrong with Heuristics

  1. 1

    Great to see you back here with another well-reasoned argument. And to think I see 2 + 2 = 4 as a truism mathematically – and now also don’t. Interesting….

  2. 4

    “We experimented and re-measured, and we saw how impossible it was to verify results more precise than our tools could measure directly.” This is something computer programmers have a hard time learning. I see complaints that a result printed to 12 significant digits varies in the last digit when a compiler is changed, or similar, and they ignore that their inputs had at most three digits of precision.

    I really liked that you presented this in a larger context – great writing!

  3. 6

    You make an excellent point, which should really be driven home, namely that syntax and semantics are distinct, and one should always clarify what the precise meaning of the terms and operations used is.
    That said, the rest of the post appears to me to be rather confused. Breaking it down:
    – > This is all about deconstructing the meaning of the equation.
    DEconstructing implies a top-down approach, whereas the correct approach would’ve been bottom-up: we assign meanings to terms and operations and construct the meaning of a sentence from there, not the other way around.
    – > “Are we talking about some ideal of “2” and “4”, or are we communicating about something else, where “2” and “4” are abstractions of reality that may be more or less reflective of that reality?
    This is crucial; mathematics has nothing to do (per se) with reality. It’s only the bijections (mappings) we derive experientially that map reality to mathematics. Hence, in math, we’re _always_ talking about some abstraction. How you map it to/fro reality is up to you, and that’s an entirely separate question from the math at hand.
    – > “Then we put them together and measure again. We note, entirely accurately, that the combined length is closer to 5 than 4. We note a length of 5. We’ve done everything correctly.”
    In the context of (potentially) inaccurate measurements, you’re talking about an algebra of numbers that have a certain number of significant digits. This is absolutely legitimate. Then you speak of repeating some measurement that (in some sense) is the sum of the previous measurements. Could you legitimately claim that the sum of some independent measurements must equal some other measurement? It certainly can’t be part of your underlying algebra, accounting for error or not. You could ostensibly _define_ equality this way, but it wouldn’t be consistent.
    In other words, in stating “2 + 2 = 5”, you need to clarify what you mean by “2”, “5”, “+”, and “=”, in a manner that forms a consistent system.
    – > “The statement “2 + 2 = 4” is for people new to math. It’s a heuristic for teaching people who still need to see their examples laid out in blocks or pieces of fruit.”
    No, it is a syntactic sentence that has varied meanings depending on the formal language in which you choose to evaluate it. It is a valid expression in, say, arithmetic over natural numbers, in which is almost definitially true.
    It may also be a statement in some other language, which syntax and semantics we have either already defined, or have yet to define. But that needs to be done before this sequence of symbols gains any meaning at all.
    It is emphatically NOT a heuristic.
    – > “That’s why it’s ridiculous to hyperventilate if someone points out that the equation is a social construct.”
    It’s only a “social construct” in the sence that all mathematics is a social construct — an abstract yet consistent system that is only applicable to reality in cases when one could provide a valid bijection between the mathematics at hand and the real context one wishes to model. Agreed, though, that there is absolutely no need to hyperventilate.

  4. 7

    I disagree with your post and I hope not to be stomping my feet too hard.

    2+2=4 stays true in mathmatics and your example does not show anything socially constructed in math. Ironically, you are making the a similar mistake to that the creationists are making.
    In short, you are using science to build a simplified model of reality and find that it does not have perfect predictive power. But instead of going back to improve your model (by including the measurement error or the uncertainty of the measurement), you conclude that the science had to be the issue, that the symbols of “2” and “4” are not exact as mathmatics claims.

    The longer version: you build a model for predicting the combined length of blocks. Since you are restricted in your measurement accuracy, your choice of model (adding the lengths) is actually good, because it will give you the right answer in many cases and minimizes prediction error (under some weak assumptions). If you trained some machine learning predictor, it would probably end up being this exact model of adding the length. When you evaluate your model, 2+2=4 still holds true. And that doesn´t change because the model doesn´t have perfect predictive power or because your unwilling to describe your model correctly, which then isn´t 2+2=4 anymore.
    The basic mathmatics are still true, simply because they are not concerned with real world examples. The simple equation 2+2=4 is taught because it is true and without it, it is impossible to understand advanced topics like modelling, which deal with the complexity that you describe.

    Your creationist example follows the same structure. The creationists use a simplified model of evolution by throwing the tree of life concept over board and assuming that each species evolves to exactly one other species and then goes extinct (so, more a chain of life). Of course, this simplistic model fails to explain most of the evidence. Now the creationists claim science must be wrong, because their simplistic model failed.

  5. 8

    I’m so sorry I missed these comments when they came in. This is hilarious. 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.

    There really are people who are upset when others note that 2 + 2 does not always = 4 in the real world because 2 is an approximation with an unspecified error. I think they’re the people you want to talk to.

  6. 9

    > 2 + 2 does not always = 4 in the real world because 2 is an approximation with an unspecified error

    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)!

  7. 10

    You have misunderstood my post: I am not criticizing that you do not account for the uncertainty. The problem is your conclusion from your observation:

    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.

  8. 11

    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.

  9. 12

    I may be living in a bubble, but no, that’s now how we language. Not if we want to make ourselves understood and not just be provocative.
    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.

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