When asked to give an example of an analytic truth, something that simply must be true no matter what, a clear go-to example is 2+2=4. No matter what happens, no matter what experience may tell us, we can rest assured on the unassailable truth that putting two together with two will always yield four. The only problem is, it isn’t always the case that 2+2=4. The world in which we live does not always yield itself to such simplistic precision.

For example, you can add 2 cups of liquid with another 2 cups of liquid and get a grand total of 3.8 cups of liquid. (You can see a simple illustration of this here.) How is this possible? It can happen when the first two cups of liquid are water and the second two cups are 99% rubbing alcohol. The alcohol molecules are smaller than the water molecules, and hence fit in the 'negative space' between the water molecules (think of sand filling in the negative space in a mason jar full of ping-pong balls. 2 cups, plus 2 cups equals 3.8 cups.

Now I know what you're thinking: 'of course 2 and 2 of DIFFERENT THINGS won't necessarily equal 4 of the same thing. 2 + 2 is only guaranteed to equal 4 when you are adding the SAME THING together.' Okay, fair enough, but that just gives rise to a key question: what, exactly, does it mean for two things to be 'the same thing'? Clearly two cups of water and two cups of alcohol are not 'the same thing' for our current purposes, but when can we say that two things are, indeed, the same thing?

Consider just water for a moment. Any two cups of water will be different in myriad ways from any other two cups of water; they will have slightly different levels of trace minerals and other atoms besides hydrogen and oxygen. No matter what your filter company might tell you, in the real world, there is no such thing as 'pure water', and as such, no two cups is ever 'the same' as another two cups.

I suspect that some of you still will be thinking that I've missed the point. When we say '2+2 = 4', we're not talking about two OF anything; we just mean TWO, the abstract, the number, not a measurement of things in the world, but simply the concept of 'one and one.'

Again, this is a fair retort, but also again this idea needs to be unpacked. What exactly IS 'the number two' when it is abstracted away from things in the physical world? There is a huge and historic literature on the ontology of numbers, which I cannot begin to summarize here. Suffice to say, the view of numbers expressed in the above objection is likely some kind of 'mathematical realism', the most popular variety of which is 'mathematical Platonism'. This view, roughly stated, claims that numbers are real, just as real (if not more so) than material objects, and they endure in their own 'plane' of existence separate from the physical.

According to it's champions, mathematical Platonism is the only way we can truly make sense of the idea that '2+2=4 is true', precisely because the non-Platonist alternatives are susceptible to the kind of counterexample that I opened up with. Non-Platonist views are limited to saying '2+2=4' is only 'mostly true', 'approximately true', or 'true in some contexts'.

Personally, I don't find this a terribly hard bullet to bite. For me, it's a lot easier to bite than the idea that there really is an eternal, immaterial realm of numbers (and possibly other categories of non-physical things) that transcends, yet at least loosely applies to the physical world. This is, of course merely an appeal to my personal intuition, an argumentative strategy that I don't place too much stock in, but if nothing else it should suffice to give the Platonist-sympathizer pause. Platonists have more than just their appeal to incredulity in their quiver, of course, but I suspect that said incredulity drives more people towards Platonism than are probably comfortable admitting.

I think this temptation towards mathematical Platonism should be resisted. One way to do so is to reflect on how science education often sweeps situations where 'the math doesn't add up' (like my opening example) under the rug. Anyone who has taken a high school science class is familiar with the concept of 'significant digits', which allow scientists to ignore potential differences in measurement that go beyond either their instruments' capacities, or the particular needs of the given experiment.

There's nothing wrong with appealing to significant digits, of course; infinite precision of measurement isn't possible, so for practical purposes it makes sense to just, at a certain point, round things off. But when we do that we need to recognize that our inability (or our not caring) to measure infinitesimally small differences doesn't mean they're not there. Richard Feynman famously compared the accuracy of the predictions of quantum physics to specifying the width of North America to within the length of a single human hair. That is an utterly astounding finding, of course, but it only underscores the point I'm trying to make: if you add 2 hairs and 2 hairs to another 2 and another 2 and another 2... pretty soon your margin of error will add up. In quantum mechanics 2+2 might equal 4, but 2 trillion + 2 trillion might only equal 3,999,999,999,999.

Perhaps things are different in the world of Plato's forms, but the world in which we live is not mathematically precise. Most of the time this impercision doesn't matter, and we can safely ignore it. But just because we can ignore it doesn't mean it's not there. In this world, 2+2 only equals 4 when we decide that we don't really care about the microscopic difference between the substances in question.

Garret Merriam

Philosophy Department

Sacramento State

Garret, thanks for this. It's an important, fundamental issue for students to be able to think through. It also connects really well with my previous post on spoilers and keepers.

ReplyDeleteI'm pretty much in your camp with respect to your naturalistic considerations, as you know. I also pretty much accept Quine's characterization of analyticity, not as something that holds true no matter what, but as something that, like probability, admits of degrees. Basically, the less susceptible a proposition is to empirical disconfirmation, the more analytic it is. So, in this sense, 2 + 2 = 4 can remain an excellent example of an analytic truth. (Of course, the sense of 'plus' you use when when talking about concatenating or collecting physical objects is not, and never has been, regarded as analytic, which I think you mean to acknowledge.) When you go at analyticity in this way, then you get the same practical result as you get with probability. i.e., just as .999999999999 probable is certain for all practical purposes, .999999999999 analytic is true no matter what for all practical purposes.

This connects to my previous posts in the following way: When we do philosophy, we are not simply in the business of determining whether there is any such thing as X, in this case analytic truth. We are really in the business of determining whether a particular definition of X is useful, and whether a different one might be better. If we suspect the latter, we propose an explication of X. Quine did not argue that analyticity does not exist. He explicated the concept in a way that made it more epistemically defensible and useful.

Another small point: I'm sure you would want your readers to understand that Platonism isn't the only option for someone who wants to preserve the analyticity of 2 + 2 = 4. Mathematical equations are, in the end, just shorthand for counting. On Peano's axioms, their analyticity reduces to the claim that 0 is a number and that every number has a successor. So if you want to attack the analyticity of 2 + 2 = 4, you really need to be arguing that 0 might not be a number and the successor of 0 might not be 1, both of which are axioms in Peano’s system and generally held to be true by definition.

Hi Randy, thanks for the feedback. Yes, I think we're generally in agreement here. I was tempted to speak more directly to my own Quineian sensibilities, but I ran out of space. I'm thinking about appealing to him more directly in another post on the recent redefinition of the kilogram. But that's for next semester.

DeleteI probably could have been more clear about other, non-Platonistic forms of mathematical realism, but again I felt pressed on the word count. But I hadn't thought about the issue from Peano's perspective, that's an interesting take. I can't say I have any clear intuitions on that approach, but I'd love to hear more about it.

I'm also a naturalist on numbers. However, here are some thoughts: 1) What if someone says 1+1=2 when we're adding categories? Instead of adding particular things in the real world like cups of water or hair, one can jump up to a higher level of abstraction to superordinate categories. For example, if I add the category of hammers with the category of pencils, there will be 2 categories.

ReplyDelete2) Regarding whether we can say that 2 cups of water are the same thing despite some differences in trace minerals and atoms: What if we take something like Richard Boyd's homeostatic cluster theory of natural kinds? Water has properties like being a clear, odorless, tasteless, liquid that fills rivers and streams. These are not necessary and sufficient conditions. These properties are likely, however, to cluster together given an underlying chemical causal mechanism. There can be some disparate atoms in the different cups but as long as the core chemical mechanism is present, then that is fine. We use the cluster of properties and the underlying causal mechanism to determine that both cups of water are cups of water. Since the cluster of properties are not nec/suff conditions for classification, this also allows for some differences in the properties between both cups. In this fashion, we can conclude that both cups are the same in the sense that they both can be classified as cups of water.

Hi John, thanks for the feedback.

DeleteIn re (1): I think there are some difficult questions about categories, what they are, how to think about them, etc. Obviously Plato and Aristotle are going to be big dogs in that fight, but I don't have much sympathy for either of their approaches. Russell might take a more set-theoretic approach, but I have issues with that, too.

In short, I can't claim to have a general theory of categories that squares with my naturalistic thinking. But as far as the math goes, categories might be one of those instances (of which there are many) when 2+2 really does equal 4. It's only in very rare circumstances where that doesn't hold true.

(2) I think Boyd's approach is an interesting one, but I'm not sure it will resolve the problem. To use another example: if you add two cups of liquid water to two cups of frozen water you'll get four cups... until either the frozen water melts (in which case you'll have less than four cups) or the liquid water freezes (in which case you'll have more than four cups).

Now, does Boyd's theory treat liquid water and frozen water as the same 'natural kind' or different ones? If they're the same, his account will have trouble with the end-state (we added 2+2 and got more/less than four). If they're different then he has problems with the beginning-state (we added two of one kind to two of another kind, but nonetheless got four of the same kind.)

Of course, these 'problems' are only problems if Boyd wants to resist my conclusion, that 2+2 doesn't always equal 4. If he's willing to accept that, then these aren't problems, that's just the way the world works sometimes.