Almost everyone agrees that our mathematical talk is practically helpful. Unlike astrology, doing math helps us build bridges. But how is math practically helpful? And does the way in which talking about numbers is practically helpful give us any reason to think numbers actually exist?
In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.
1. How talking about abstracta like numbers is helpful
Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.
For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "
Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:
"if Ex7 cupcakes today Ex14 cupcakes tomorrow."
"if Ex8 cupcakes today Ex16 cupcakes tomorrow."
"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."
2. What role do abstract objects play in explaining why talk of abstract objects is helpful?
Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...
- lets us track and predict what cupcakes there are and will be
- lets us track *the pattern in* what cupcakes there are and will be
- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.
It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.
3. Are there numbers? A good and bad reason for believing in numbers.
If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?
I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.
Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.
The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.