Let me pause for a minute before jumping into talking about Gunther Laird’s book “The Unnecessary Science” to talk about this post by Stephanie Zvan, since it actually relates to my post in this series defending conceptualism. Basically, there was a set of Twitter posts defending the idea that 2+2 = 5 and then another set that called that idea ridiculous, and Zvan decided to weigh in on the side saying that 2+2 can equal 5 or 2+2 can equal 3 and that those decrying that as a proof that postmodern ideas are ridiculous are completely wrong in the most obvious way. Since my post defending conceptualism against nominalism and realism about concepts insisted that 2+2 is always 4, I definitely wanted to see a really good argument before considering that this might be possible and that at a minimum people who think that 2+2 must equal 4 are being too rigid (because she parlays it into a discussion of Social Justice points, Zvan is actually quite a bit harsher on critics of her idea than that). And when she gets into talking about 2+2=4 being a Social Construct, I’m obviously going to want to pay a lot of attention because that seems only consistent with a nominalist position, and I think nominalism is completely misguided. So I’d be looking for a really good argument for why that’s right.

She starts, by her own admission, with committing a rounding error:

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.

So let’s start with significant figures and precision, because it will be far more important later. I *also* had to do physics for a significant amount of time — I even did a full year Astrophysics course — and the thing about precision is indeed as Zvan says: you can’t come up with a number that is true to more significant digits than your instruments can measure. In general, that isn’t much of an issue because all you do is take the numbers your instruments give you, and that’s that. Even if you add or subtract measurements you aren’t likely to end up with more significant digits than were in the original measurements (it is difficult to add something like 2.45 and 1.76 and get a number like 4.213, for example). But if you multiple or divide measurements, you can end up with that sort of situation. It’s very easy, for example, to divide 1 by 3 and get 0.333333…, and so you could get something like 4.213 from two numbers that were only measured to two significant digits. So that last 3 is reflecting a purported accuracy that your measurements could achieve. After all, there could have been numbers in that spot already that you couldn’t have measured, the existence of which would obviously have changed that number. So you can’t say that that last 3 is accurate because that digit wasn’t accounted for in your measurements, and so if you use it you would be claiming that your measurements were more precise than they actually were, which is an error.

The same thing, then, applies to rounding. If you round a number, then you are dropping the extra digits off the end to whatever you round it to. That means that your number, then, is at best only precise to that number of digits, so anything that would depend on or reference extra digits would run into the same problem as noted above. But even worse is the fact that you would explicitly be creating an approximation of the number, and in general deliberately doing it. It is possible in the above cases that what came after those digits was in fact a 0 and so that was a precise measurement to that number of digits, but if you round it is *clear* that that’s not the case or else you would have felt no need to round it at all. So when you round a number, you make an approximation, and a rounding error, contra to Zvan’s assertion, is a real problem where you treat the rounded numbers as if they actually *were* those numbers and so can be used as those numbers in all ways. Obviously, if you are willing to round it, you think that it’s going to be close enough for most of your purposes. The problem is when the rounding to an approximation is then used in a case where it *isn’t* close enough, and you get things wrong. So instead of calling it a rounding error, we can probably call it an approximation error instead, and note that what we always have to do is make sure that we understand how much error we’ve introduced with our approximations to make sure that when we try to use them we aren’t using them in a way that matters.

Let me give an example here. Imagine that someone is picking out things to put into their bathroom as they are redoing it, and they have to decide what things will fit through the door. They note that they are 5’6″ tall, and that the top of the doorway seems to be about 6″ taller than they are, so there’s a clearance of about 6′. Since this is an approximation, though, they have to be careful not to insist that the height really is 6′, because it might not be. If the tallest thing they are moving in is 5’9″, they know that it really should fit because they are unlikely to be off by that much. But if they instead say that they should be able to move in something that is 6′, they may well end up unpleasantly surprised when it doesn’t fit because the height was *actually *5’11” and their approximation was off by a bit. In science and anything that calculates error, we calculate error precisely so that we don’t say that one value is larger than another if the possible error we could have might mean that the other is actually larger than it. That’s what error bars are, in fact, designed to do: show us the potential overlap so that when we are drawing conclusions we don’t draw them based on assumptions that our known errors could overturn.

Note that, so far, I’ve said nothing about the universality of 2+2=4. Nothing in this in any way relates to whether or not 2+2=4. So we need to move on to her examples to try to get at this:

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.

So here’s what we’d do. We’d approximate the values as being 2, and note that they are indeed a bit larger than 2, as that’s why we had to round it. We just don’t know how *how much* bigger they are, and we really don’t care about that (or else we’d use a more accurate ruler). So then we add 2+2 and get 4, and so determine that the total length of the two objects is approximately 4. Then we put the two of them together and measure then and come up with an approximate answer of 5. In order for Zvan to make her point here, what she would have to argue is that by that second measurement we would be proving that, in that case in the real world, that 2+2=5. But, of course, we know that we didn’t actually *have* 2. We had an approximation of 2. We *knew* that the two numbers were both slightly larger than 2, and so knew good and well that the two of them combined *would* be larger than 4, and could very well be closer to 5 than to 4. But we considered the approximation error to be well within the bounds for our purposes, and so went with those approximations anyway. Otherwise, what we really should have done was indeed measure the two of them together in the first place to get the more accurate number, ignoring the approximations.

And, in fact, that’s what strikes against Zvan’s interpretation here, because in order to generate the puzzle that 2+2=5 what we had to assume in the first place was that 2+2=4 and do that calculation, and then measure the two objects together to say that, no, in this case the *real* outcome is that it equals 5. But what we are doing in the first case is appealing to a general, abstract rule that 2+2=4 and is always 4. So in this case instead of overturning the universal rule we’d be more likely to look for the error we made that led to the strange result, and fortunately that error is pretty obvious: the number was not actually 4 and was only *approximately* 4, and so adding approximate numbers can result in larger or smaller numbers when rounded than we expected. What makes this so obvious is that Zvan does *not* do what most people would do here, and say that we are *wrong* to say that 2+2=4. Instead, she merely notes that we got 5 and did everything correctly. But if this is true, then we could *never* say that 2+2=4, because it would depend entirely on what operation we are doing. And if it can change based on our approximations, then why couldn’t someone way that 2+2=1 based on an argument that that’s how they define the terms? If we allow for specific cases to actually change what it *means* to say 2+2, then why can’t we change what the terms themselves mean willy-nilly?

That was my argument for them being set concepts:

Mathematics also reflects this. One example used frequently by realists is that 2+2=4, clearly equals 4, and equals 4 even if there are no minds to understand that 2+2=4. Except that that isn’t quite true. 2+2=4 in the standard base 10 mathematical system, but I can indeed come up with a valid mathematical system where 2+2=5. All I need to do is invent one where the “+” operator means that you add the two numbers and then add one to them, and then in that system 2+2 clearly equals 5. So if someone says that 2+2=5, are they right, or are they wrong? Well, it depends what mathematical system you’re using. If someone came across two people doing work in that new system, and the one person notes that the other one got the wrong answer because they added 2+2 and got 4, if the observer then jumped in saying that 2+2

doesequal 4 and so that person is wrong to say otherwise, it would be theobserverwho is wrong. They would literally be not talking about the same things as the others, and so would be applying the wrong concepts to the discussion. This is, of course, a different sort of error than someone who thinks that 2+2=5 is true in the standard base 10 mathematical system, which only highlights the distinction: the former is using the wrong concept, the latter is understanding the concept incorrectly.

In mathematics, we can indeed change the definitions of the terms so that they come out differently. So, yes, I could build a mathematical system where 2+2=1. The cost of this, though, is that I wouldn’t be talking about base 10 standard anymore, but would be talking about a new mathematical system. As base 10 standard is defined, 2+2=4. If you challenge that, then you aren’t talking about base 10 standard anymore. And as mathematicians know, we don’t go out and measure things to try to prove that. It follows precisely from the axioms and definitions of the mathematical system. So you can’t overturn that by measuring things in reality and seeing that the measurements don’t come up as 4, but instead sometimes as 5 and sometimes as 3. At worst, all you’d end up doing is creating an argument that base 10 standard is not a very mathematical model for our reality. And that’s clearly not the case.

I would be remiss if I didn’t note that Zvan did try to address the issue of there being extra digits:

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.

But this is just pushing the issue back one step further in terms of digits. We still end up approximating, and still end up treating the approximation as the real number, and then hit the approximation errors. And she’s right that keeping track of the error is important, and yet those who work in those system usually don’t insist that 2+2=4, but instead insist that we need to carefully note our error bars and note when our approximation errors might impact our actions and decisions. So the people who are being incredulous on Twitter are, in fact, complaining about something that the people who work in fields that depend on math wouldn’t usually argue with them over: the idea that the odd results that follow from approximations would actually overturn the mathematical definition of the base 10 standard mathematical system that mean that 2+2=4.

Zvan continues:

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.

No, actually, that statement is more for people who are *advanced* in mathematics and know how it works and what it all means. They then understand that it clearly means that 2+2=4 in base 10 standard and that that is indeed absolutely and completely true and undeniable. If you want to insist otherwise, you are either wrong or talking about a completely different mathematical system.

But this also reveals the flaw in her argument, which is that she focuses on *measurement* but ignores the main application of addition to our lives, which is for *counting*. While measuring things can introduce approximation errors based on how accurately our instruments can measure things, that’s not true for counting. If Edmund Blackadder tells Baldrick that he has two beans, and then two more beans are added to it, there is no possibility of there being an approximation error. There are two beans, and then two more beans. There is no way of getting 2.3 beans and adding them to 2.3 beans to get 4.6 beans which we can then round up to 5 beans. There are 2 beans, and 2 more beans, and so a total of 4 beans. Full stop. This is not some sort of juvenile or inferior example to the *real* mathematics of measurement. *This is the paradigm of how addition is used to model reality.* To dismiss it as she does here is ridiculous, and to do so in favour of the examples that she needs to build her own case is suspicious to say the least.

Let me address the social convention idea here, because there *is* a social convention involved here, just not the one that Zvan thinks. We could decide to teach other mathematical systems by default instead of base 10 standard, and because our default is base 10 standard people will, rightly, insist that if someone simply says “2+2=5” that they are wrong, because if they mean that that is true in base 10 standard then they are, indeed, wrong. We pick that one, however, because it seems to work the best for us in the reality that we are in, so it’s a perfectly reasonable default for us to choose.

Also, this *does* allow us to outline an error that people could make that might even relate to Zvan’s Social Justice issues at the end of the post. If we measured those two objects and noted that they were approximately 2 and so said that their total length must be 4, and then measured the two of them and noted that in this case the actual value was 5, if someone insisted that they must be 4 because 2+2=4 instead of what we measured then *they’d* be the ones making the error and that would be wrong. It wouldn’t be the error of insisting that 2+2=4, but instead the error of assuming that the approximations of the numbers must behave exactly the same as the numbers themselves, and of ignoring the error bars.

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.)

I also assume that those better communication skills would involve not insisting that your opponents are childish if they don’t agree with you, or by assuming that a rather simplistic analysis of mathematical systems somehow counts as “more complex”. But this idea of complexity will carry on to her last points:

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.

No, that’s not a heuristic developed for children. It’s a *folk* heuristic, one developed for people who don’t have the need or interest to delve deeply into evolution but that works well-enough for pretty much all practical purposes. The creationists she refers to here are like those who want to insist that the length must still be 4 even after we measure both objects together, as they are trying to apply a limited — by design and purpose — concept in a case where we need to look at the more complete concept to make that all work. And it is interesting to note that that is *precisely* the mistake Zvan makes when it comes to 2+2=4: using the limited concept of the approximations to attempt to disprove the idea that 2+2=4 universally, ignoring that that is a universal truth because of the deeper meaning that it is part of the base 10 standard mathematical system.

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.

So, here’s the thing: that “heuristic” is an approximation, like the ones Zvan uses to argue against the universality of 2+2=4. Since those exceptions are, I am given to understand, less that 1% of all cases, for practical purposes we can treat them as simple exceptions and indeed rely on that heuristic. What that would mean is that for the specific argument listed here, Zvan would need to show that they are making approximation mistakes of the sort listed above: treating the gametes as being determinate in those cases where those exceptions might actually matter. But this analysis actually kills most trans-positive arguments, because most transgender people, at least, are *not* in these exception cases. So we can see here that this argument actually makes the *opposite* error to those who would insist that the length of the objects together must be 5 because 2+2=4, as they would insist that since some measurements will come out as 5 we could *never* really say that the result 2+2 is actually 4. And note that even *Zvan *shies away from that argument above, only going so far as to deny that it always comes up that way. But since, again, most transgender people *aren’t* exceptions we cannot simply say that they might be exceptions and so we cannot assume that they fit into the standard categories. In terms of biology, almost all of them *do* fit into those categories. So, then, it’s obvious that the purpose of the argument is to say that because there are *some* cases that aren’t clear, we cannot do that sort of classification at all, *even if the people that they don’t want classified that way actually do fit neatly into the existing categories.* And just as obviously that’s a bad argument.

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.

For the most part, races were developed on the basis of what seemed like obvious physical differences. Advancing science has shown that what it means to be a subspecies or species is not that simple, so that’s correct. But that has nothing to do with the reason it was developed in the first place, and if there *were* going to be “subspecies” of humans the original categories would actually be pretty good for most practical purposes. The error here is, again, an approximation error, as while there are obvious physical differences as far as we can tell those differences didn’t matter in the cases where people engaged in egregious racism. So their error was assuming that the physical differences that they could see and readily identify matter for those cases as well. And they didn’t.

So, finally, what about postmodernism? To return to earlier in the post:

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?

Well, we don’t *need* to deconstruct them. The ideals themselves *are* the abstractions, as per mathematics. But as an analytic philosopher who did deal with postmodernists in his philosophical career, I don’t dismiss postmodernism (or at least existentialism) out of hand. I just think that any errors that the analytic approach would make that the postmodern approach would catch are ones that the analytic approach can catch as well, as would be demonstrated by my entirely analytic analysis of the issues over 2+2=4. I didn’t need a postmodernist analysis to determine what we were doing wrong and what we were doing right, and neither did Zvan, as her argument is *not* a postmodernist one. So, no, the question is not postmodernist, and Zvan herself doesn’t actually deconstruct the terms and so doesn’t make a postmodernist argument.

So properly understood, 2+2=4 and anyone who says that 2+2=5 is wrong. However, anyone who insists that if we take two measurements and approximate them as 2 that therefore the two objects measured when placed together *must* be 4 are wrong as well. While the debate of 2+2=4 didn’t result in people taking the latter tack, it is possible that in the cases that Zvan cares about some people are making that sort of argument. It would be much better for her to argue that that is what’s happening rather than making the argument she does here.

January 8, 2021 at 4:56 am |

Mathematics also reflects this. One example used frequently by realists is that 2+2=4, clearly equals 4, and equals 4 even if there are no minds to understand that 2+2=4. Except that that isn’t quite true. 2+2=4 in the standard base 10 mathematical system, but I can indeed come up with a valid mathematical system where 2+2=5. All I need to do is invent one where the “+” operator means that you add the two numbers and then add one to them, and then in that system 2+2 clearly equals 5.ISTM that this example of Zvan’s just misses the point. When realists say “2 + 2 always = 4”, they aren’t talking about the symbols, but about the operation which those symbols represent. Saying, essentially, “Yes, but we can redefine the symbols so that they represent a different operation” is true, but also irrelevant to the actual point.

January 8, 2021 at 9:36 am |

Yeah, while I’m not a realist even my view, as outlined in that post, is that while you can say 2+2=5 at a minimum you aren’t talking about the same thing as they are anymore, whatever that is.