Philosophy of Numbers (part 1)
This post is the first in a series of things that I think would be fun to discuss on LW. Part two is here.
It seems like there are (at least) two kinds of things we make statements about: physical things, like apples or cities, and logical things, like numbers or logical relations. And it’s pretty interesting to question how accurate this seeming is. Are numbers really a “kind of thing,” and what do we mean by that anyways? Can we unify these multiple kinds of things, or kinds of statements, into one kind, or not?
For a light review of standard answers, see this nice video. For more depth, you might see the SEP on abstract objects or philosophy of mathematics.
Compare the statements “There exists a city larger than Paris” versus “There exists a number greater than 17.” It seems like we use much the same thought patterns to evaluate both these statements, and both seem to be true in the same ordinary sense. Yet the statement about cities seems true because of a correspondence to the external world, but there is no “17” object in a parsimonious predictive model of the world.
To this you might say, “What’s the big deal? Even if I don’t think numbers are physical objects, it’s perfectly reasonable to make this tight analogy between cities and numbers in our reasoning. How is making a big issue out of this going to help us do anything practical?”
Well, in logical decision theory, a recent formulation of some ideas from TDT/UDT, the agent wants to make a causal model of the world that includes (in the model) “causal” effects of a fixed mathematical statement (speficially, the output of the agent’s own algorithm). First of all, this is pretty novel and we don’t really know how to formalize learning such a model. Second, it’s pretty philosophically weird—how is a piece of math supposed to have something like a causal effect on trees and rocks? If we want to solve the practical problem, it might help to be less confused about numbers.
Plus, you know, it’s interesting! Why do we think there’s such a thing as “numbers,” how come the same reasoning works for both numbers and cities, and what are the limits to this analogy, if any?
When one wants to outdo an entire branch of philosophy, it’s nice to have some sort of advantage. And the sign of such an advantage is often a bunch of philosphers being loudly wrong about some related issue. But this case, I don’t see the signs of an easy advantage. Modern philosophy of numbers doesn’t seem to have a bunch of sharp divides or false confidence. Instead, most everyone seems pretty aware that they’re confused, despite some fairly interesting ideas being available.
But, okay, I do have some ideas.
See, if you ask philosophers about something that might exist, their first instinct is to try to find a necessary-and-sufficient definition of this thing, more or less on its own terms. Over here, we’re much more likely to think of how things are represented in people’s models of the world, and ask what chain of events led people to have that representation, which I think is some important philosophical technology.
This wouldn’t be a proper post without a pile of links. So here are some options we might want to keep in mind: Taboo your words. Focus on origin or function, like in the example of “truth.” Imagine what cognitive algorithm you’re using. Keep your eye on the reductionist ball.
Since this post is labeled “part 1,” you might expect that I’m going to end this without telling you exactly what I think about numbers. You’d be right!
But I do want to prompt you with some questions I think are more key than “what is math, really?”, and corresponding things I think might be hints.
Why do we say that numbers “exist?”
Why do we need a property called “existence” in the first place, even just for trees and rocks? Those Eliezer-posts about truth may hint at one point of view.
Why would we want to say that certain abstract sentences are “true?”
Do statements about math have the same properties Eliezer outlined as making “true” a useful word? Why would we want talk about labels on mathematical sentences if math is just a bunch of tautologies?
Does it make sense to evaluate “There exists a city larger than Paris” and “There exists a number greater than 17″ the same way?
What cognitive algorithms could we be using? What are their disadvantages?
Does this line of reasoning actually help us implement LDT?
I got nothing.
Just for fun, I’ll take a stab at your final questions (apart from the last one, where I too have nothing).
Why do we say that numbers “exist”? I think there are two separate questions here. (1) Why do we treat numbers as real? (2) Why do we bother with terminology like “exist”, (2a) at all and (2b) for numbers in particular? I think at least part of our answer to (1) is that our best explanations for our observations have numbers in them, and this seems to be robust in the sense that it applies across a wide range of explanations of a wide range of things at a wide range of sophistication levels. Dunno whether it’s the whole reason. For (2a) … well, actually mostly we don’t talk about things “existing” except in marginal cases where there’s disagreement or doubt (only when doing philosophy or drugs does anyone ask whether tables exist). Our language has nouns, and our thinking has things, because the most convenient approximations to how the world works (at most levels—this one does break down a bit in some of our very best models) treat it as made out of things of various kinds; we not infrequently make mistakes about what there is, and to talk about such questions—which we need to do given our fallibility—we need a term like “exists”. We need something similar to state generalizations: “for all x, …” is the same as “there doesn’t exist an x such that not-...”. For (2b), our best models of the world involve mathematical structures in which propositions like “there exist infinitely many prime numbers” make sense. It’s not clear (to me, at least) how much of all this is dependent on details of human brains and minds—to what extent we should expect intelligent aliens, AIs, archangels, etc., to have notions of “things” and to use them in their mathematics. My guess is that we should expect both.
Why would we want to say that certain abstract sentences are “true”? Of course “true” is like “exists”. Most of the time we don’t say that X is true, we just say X; just as most of the time we don’t say Y exists, we just talk about Y. We need the term “true” (and its opposite “false” and other terms like “meaningless” and “unclear”) mostly because sometimes we err, and we need ways to think and talk about whether we’re erring. (E.g., we have some logical argument; it starts with innocuous premises and ends somewhere surprising; it’s useful to be able to go through step by step and ask at each step “well, is this true?”.) We can err just as easily when doing mathematics as when doing anything else.
Does it make sense to evaluate “There exists a city larger than Paris” and “There exists a number greater than 17″ the same way? I’m not sure what “the same way” really means here. The steps I would go through to try to convince a thoroughgoing skeptic of those two propositions would diverge quite quickly. It’s clear that there’s something in common between my mental representations of those propositions, but I don’t think I’d want to describe it by saying that I evaluate them the same way. Does it make sense for them to have as much in common as they do? Yup. Numbers have this comparability-structure that resembles that of city-sizes; why? partly because one thing we use numbers for—one thing that picks the number systems we use out of the vast universe of possible abstract structures—is comparing things. And then we use parallel language to describe these things with parallel logical structure. (I don’t want to overstate the extent to which we get to choose our abstract structures. E.g., if we want a minimal structure in which we can count indefinitely then we automatically get an ordering. This isn’t a coincidence because you can think of the ordering as arising from the counting process. In some cases it may be better to say “we can compare city-sizes because we can compare numbers” than “we use numbers rather than something else because we can compare them as we can city-sizes”. But whichever comes first, there are correspondences between things in the world and the abstractions we use, and this suffices to make it appropriate to use similar language and similar thoughts to handle both.)
Note 1: In the above, I’ve repeatedly claimed things whose rough form is “we do X because doing X is useful”. That shouldn’t be taken as meaning that someone sat down, figured out that doing X would be useful, and convinced everyone else to do it. The processes by which we ended up doing X are mysterious and complicated, and probably include biological evolution (perhaps our brains have structure in them that predisposes us to use nouns and numbers because our candidate-ancestors who didn’t were thereby disadvantaged in figuring out the world) and memetic kinda-sorta-evolution (if there were ever languages without anything nounlike, I guess they weren’t helpful to their users) and who knows what else.
Note 2: Nouns and things are so deeply built into our language and cognition that there’s no getting away from them. So there’s some circularity here.
Thanks! I think this is pretty darn good.
You say
without any explicit justification: perhaps it’s meant to be obvious? It’s not at all obvious to me. The simplest model of the world might very well have numbers (and sets and functions and manifolds and who knows what else) in it. Our current best models of the world certainly have those things in them.
I wrote a response to this here. Basically, I believe that physical existence and mathematical existence are different things, but we can also define a common interface that both kinds of objects fulfil, which is why we are tempted to use a common word. Unfortunately, this definition of existence becomes reified and given ontological existence, when there are at least two distinct types that need to be considered separately.
In the numberphile video you linked to, I was disappointed that when the speaker was asked about how a nominalist can think about kinds of numbers that are difficult to describe in physical terms, he went to the example of pi rather than i. It seems to me that how to think about complex numbers is a much trickier problem for mathematical nominalists than how to think about pi is, and I’m curious if anyone here has any thoughts on that matter.
I would guess that most nominalists would stand their ground with only a little pragmatic redefinition. There are things in the world (like waves) we can model using complex numbers. Therefore, they might say, complex numbers are objects that we abstract out of waves in merely a slightly more complicated way than we abstract numbers out of sets of objects.
I do not evaluate “there exists a city larger than Paris” and “there exists a number greater than 17″ in the same way.
The first statement is a proposition about the world.
The second statement is a proposition about my map of the world.
Given a set of axioms, we say a mathematical statement is true if it’s true in some model of the axioms.
If the statement is derivable from the axioms, it’s true in all models of the axioms.
However, given a set of axioms A, we can add another set of axioms B (such that B is consistent with A) to produce A’.
A model is an implementation of the axioms.
Let M be a model of A.
Let M’ be a model of A’. M’ is also a model of A. It is possible for a given statement X to be true in M’, but not in M. This means that X is not derivable from A.
When we say X is true in M, we can reason about “truth” in a model the same way we reason about truth in reality.
Presumably you answered the question about Paris without needing to go look at Paris, or even its wikipedia page. If this is the case, then I would argue that the question about Paris and the question about 17 were both resolved, in your head, as propositions about your map of the world.
I agree that this is more or less common usage of the word “true,” when applied to mathematical statements and given some axioms. But we could just as well call this “the fleem property”—a mathematical statement has the fleem property, given some axioms, if it appears in a model of the axioms. After all, the word “true” is already used to talk about correspondences between our map of the world and the world—why would humans mix up the fleem property with truth?
I’ll make a series of stubs that present my answer soon.
The second stub is this.
On of the big difference between the case of cities that are bigger than Paris and numbers that are greater than 17 is that there’s an infintive number greater than 17.
On Wikidata you can easily ask for both the case of cities bigger than Paris and about numbers bigger than 17. The problem comes when you focus on bigger numbers. Wikidata doesn’t have items for all numbers and you are going to get wrong answers when you get past a certain point. Wikidata also doesn’t contain all cities but it does contain all the cities larger than Paris.
I can recommend going through practical discussions inside Wikidata if you care about the subject of how things differ.
One reason why it makes sense to say certain mathematical statements are “true” is because if you do, then you can build airplanes. Mathematical structures seem to be really intertwined with reality, and math seems to be built into how we think about things (to some degree). If I tried to claim that there are only 17 cities in the world, you’d call bullshit becuase… well duh, there’s clearly more than 17 cities. But if I was being obstinate and kept arguing from some wierd relatevisitc perspective, the counter arguments you’d come up with would probably start to look like formal mathematics after a point. (dislaimer: Have not actually tried this)
This is well written, but I honestly got the feeling that there is nothing worth talking about here. What is the number 4? Easy, 4 is {{}{{}}{{}{{}}}{{}{{}}{{}{{}}}}}. The only thing left to decide is “what is the empty set,” to which the answer is “the unique a such that ∀b: ¬(b ϵ a)”. And I understand how the system which defines what those things mean works… which is just based on defintions and axioms. Maybe this is stupid, but I don’t feel any need to go deeper, and I don’t feel confused about numbers. Set theory has already provided me with all the answers I want.
I think there is something left to be said about how “there exists a city larger than paris” could also be modeled in set theory and ultimately corresponds to a logical formula ranging over quanutm fields – or rather, a set of formulas which may have different truth values because we don’t have a perfect mapping from natural language to formal language. But that’s more of a different topic.
The formal set definition is only one of many ways of defining the number 4.
Rather than trying to immediately jump to what things are, I strongly recommend trying to understand why people believe what they do about it. I can claim that 2 is {{}{{}}} or {{}{}} or SS0, but this doesn’t help much more than just calling it 2 - it’s just substituting a symbol for another symbol. What’s needed is to understand the function of 2 in peoples’ thoughts.
Yeah, the statement about cities could be formulated much more mathematically. But ultimately, it’s going to have to refer to empirical facts about the universe in order to be evaluated.