Philosophy Phriday: If There’s a Number of Real Ants, Is There a Real Number of Ants?

The Daily Ant hosts a weekly series, Philosophy Phridays, in which real philosophers share their thoughts at the intersection of ants and philosophy. This is the forty-second contribution in the series, submitted by Dr. David Faraci. We apologize for yet another Monday posting.

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If There’s a Number of Real Ants, is There a Real Number of Ants?

Many people believe in ants. Most of those people believe, more specifically, that ants exist independently of what goes on in our heads. Philosophers who believe this more specific thing might say that ants are mind-independent, and they might call themselves anty realists.

Diacamma ants are well-known anty realists. Photo: Alex Wild

Of course, ants’ continued existence might depend on what goes on in our heads. If the right (or wrong) people think things like “wouldn’t it be fun to start a nuclear war?” then they might start that war and blow everything up, including all the ants. Or the radiation might turn all the ants into giant monsters like in Them!, and then are they really ants anymore? But that’s not what we mean when we say that ants are mind-independent. We mean that ants’ existence doesn’t depend on us in any deep metaphysical sense; their existence isn’t a function of what goes on in our heads.

What about things that are not ants? In many cases, the answer is the same. We should be realists about praying mantises, bees, and lots of other things that are not insects. Or, at least, we should be realists about those things insofar as we are realists about anything. Some pretty famous people have held the view that everything’s existence is a function of what goes on in our heads (or, sometimes, in God’s head). But no one endorses certain kinds of restricted anti-realism. No one is a bee realist and a praying mantis realist, but not an anty realist. What about numbers? That came out of nowhere. But that’s something people do here where they start talking about the thing they’re interested in that’s not ants without any sensible transition from talking about ants; so I’ll do it, too, but also mention that I’m doing it. This is a tried-and-true way of doing the same thing as everyone else but feeling cooler about it.

Should we be number realists, just like we should be anty realists? Actually, there are lots of interesting reasons to be number anti-realists that don’t apply to ants. I’m going to talk about one that has to do with knowledge.

How do we know about ants? For most of us, the answer is that we see them walking around, preferably not inside our houses. For some Daily Anters, there is probably a more complicated answer about myrmecology methodology, which is a phrase I find quite pleasing. Either way, there’s no deep philosophical worry. Ants, like other insects and quite a few non-insects, are physical things that we interact with using our senses. Mostly we see them. I don’t really know whether ants make sounds other than with their little feet but we can add hearing to the list. I’ve never tasted an ant but that’s a thing. And so on. The point is that there are nice plausible stories to tell about how we know about mind-independent ants.

An Argentine ant attempting to taste a tiny fire ant foot. Photo: Alex Wild

You might think the same about numbers. If there are three ants, I can see that by counting them. But I’m not really seeing the number three. I just see one ant and then another and then another, perhaps in one of those little marches they do that are sort of charming when they’re not across your kitchen counter. Also, I know ahead of time that if I see two ants and then I see one more I will have seen three ants. I don’t know this because it’s happened a bunch of times and I recognized a pattern. I know it because I know, totally independently of my experience of ants, that two plus another equals three. So I can only count the ants because I already know stuff about the number three.

What is knowledge, anyway? Well, for one thing, knowing something means more than just believing it and being right. Compare three characters who all believe that some ant species are red. Barbatus believes this because ants are his favorite insect and red is his favorite color. Weaver believes it because he just saw a bunch of red-colored ants walk by. But, actually, no red ant species live anywhere near Weaver; what he just saw were a group of brown ants that narrowly escaped drowning in a bowl of Cherry Kool-Aid. Azteca believes it because she just saw a bunch of red-colored ants walk by; they were in fact red harvester ants.

Barbatus clearly doesn’t know that there are red ant species. The obvious problem is that he formed his belief in a silly way. He had no reason to think that his favorite insect would sometimes be his favorite color. Barbatus doesn’t know because his belief isn’t justified.

Unlike Barbatus, both Weaver and Azteca arguably have justified beliefs (believing what you see, except in certain circumstances, is a pretty uncontroversially reasonable way of coming to believe things). But Weaver still doesn’t know that there are red ant species, while Azteca plausibly does. (Or, anyway, that’s the intuition most philosophers expect you to have.) What’s the difference between them? One common thing to note is that, unlike Weaver, Azteca believes that there are red ant species because there are red ant species. She just saw an actual group of ants from an actual red ant species, and that’s why she believes in them. But Weaver doesn’t believe the truth because it’s the truth; Weaver just happens to have seen something that coincides with the truth. If there were no red ant species, he would still believe in them, given of what he saw.

P. barbatus holds unjustified beliefs and thus knows so little. Photo: Alex Wild

These lines of thought hint at the two most popular philosophical answers to the question of why Azteca knows there are red ant species, but Weaver doesn’t. The first is that the truth makes a difference to what Azteca believes, but not to what Weaver believes. The second is that Azteca would still believe the truth if things were different in important ways, but Weaver wouldn’t.

Now suppose we try to tell another story, except this time our characters all believe that 89 is a prime number (which it is). Silly old Barbatus believes this because 89 is his favorite number and prime numbers are his favorite kind of number. Once again, he doesn’t know because he’s not justified. To echo the original cases, we next need to describe things so that both Weaver and Azteca are justified in believing that 89 is a prime number, but only Azteca knows it.

First, suppose you favor the idea that Weaver doesn’t know there are red ant species because that fact made no difference to his belief. It’s easy to echo that case. Weaver sees 89 on a list labeled “prime numbers” in a well-regarded textbook. But there was a printing error; he’s actually looking at a mislabeled list of Fibonacci numbers (which 89 is one of). Weaver doesn’t believe that 89 is prime because 89 is prime, so he doesn’t know that it is.

The hard part here is describing Azteca’s case, where the fact does make a difference. Of course, we can come up with one where it seems she knows that 89 is prime. Perhaps she divided 89 by each of the integers between 1 and 89, and only got an integer when she divided it by 1 and 89. But as suggested earlier, there’s no easy story to tell about how a mind-independent fact that 89 is prime really made a difference to what she believed. Unlike with the ants, we didn’t need to mention that 89 is prime to explain her belief at all! If the truth has to make a difference to what we believe for us to know about it, but number facts can’t make any such difference, then we can’t know any number facts. That’s a bad result.

What about the appeal to how things might have been? If 89 weren’t prime, would Azteca still believe it is? Would Weaver? It’s tempting think that these answers get us the results we want. Given how Azteca got her result, it might seem that if 89 weren’t prime—if it had some factor other than 1 and 89—then she would have ended up with a different belief. On the other side, if 89 weren’t prime, Weaver would still have seen it on the list on Fibonacci numbers. So she knows; he doesn’t.

Intuitive as this might seem, it is notoriously difficult to offer a theory that captures it. Why? Because it’s impossible for 89 not to be prime, and that means we need an account of what would happen if things were different in ways they couldn’t have been different. The problem may be easiest to see in Weaver’s case. I just suggested that if 89 weren’t prime, Weaver would still have seen it on the list of Fibonacci numbers. But that seems to assume that if 89 weren’t prime, it would still be a Fibonacci number. Is that true? Maybe not; maybe there are important relations between 89’s being a Fibonacci number and its being prime. Or, perhaps more radically, maybe 89’s being prime is required for the existence of life in the universe and so if 89 weren’t prime, neither Azteca nor Weaver would believe anything at all!

Finally, then, we come to the appeal of number anti-realism, and mathematical anti-realism more broadly. Suppose that 89’s primeness is just a function of how we think about the world. There isn’t any real sense in which 89 is prime, independently of us. That makes it a whole heck of a lot less surprising that we can get at the truth, since the truth is up to us! That doesn’t mean we get to just make things up willy-nilly. We can’t wake up tomorrow and decide that 89 isn’t prime anymore. But it’s still not a fact out there that we need to have some access to, in anything like the way we see those ants.

At the same time, mathematical anti-realism faces its own challenges. For one, mathematics isn’t an isolated area of study. We use mathematical facts to do all sorts of amazing things in the world. How can mathematics be so useful in helping us interact with the real world if numbers aren’t real, too? It’s a puzzle! Nevertheless, mathematical anti-realism continues to be a prominent contender in philosophy. Anty anti-realism, not so much.

Dr. David Faraci teaches at Georgetown University. He works in metaethics, epistemology, applied ethics, and the experimental philosophy of moral responsibility. What do you call a soft-spoken ant zoologist? A murmurcologist.