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July 25, 2008


Hmm, Eliezer likes Magic the Gathering (all five basic terrains?)...

Math is just a language. I say "just" not to discount its power, but because it really doesn't exist outside of our conception of it, just as English doesn't exist outside of our conception of it. It's a convention.

The key difference between math and spoken language is that it's unambiguous enough to extrapolate on fairly consistently. If English were that precise we might be able to find truth in the far reaches of the language, just like greek philosophers tried to do. With math, such a thing is actually possible.

So, 2+3=5 corresponds to your dots or sheep, and that's the whole fact of the matter. Cats are called cats because that's what we feel like calling them and calling them dogs won't change their cat-ness.

It FEELS like there should be more because of the way we are accustomed to extrapolating math. There is no additional fact to account for, though.

The only time this isn't really the case is with exotic math which corresponds to a basically "counterfactual" world like "What if the world were made of city blocks?" (Taxi Cab Geometry). It's true that we can imagine false worlds and invent precise language to describe those worlds, but such a description does not make them less false, just more vivid fiction.

Why do you have to say the math is "outside" the brain? I do understand that the model of the natural numbers is particularly useful in making elegant predictions about our physical universe, but why does that say something about the numbers or the math? The integers are an example of a formal system, but we can construct other formal systems where the formula 2+3=6 holds (I don't know of any *interesting* such formal systems, though). I can easily see that we have these formal systems, and we also have inductive arguments that they describe the world well. I get the sense Eliezer that you posit a third thing "exists". But, wouldn't this be a case of the "mind-projection fallacy"? Why do we need a third thing exist when the formal system and the inductive argument account for everything (or, perhaps they don't, and I'm missing the point...).

This might be stupid, but it's probably more intelligent than the 'subjunctive mood' grammar-joke I was going to tell.

Suppose I say, "Even if my mother were kidnapped by terrorists, I would still consider all terrorists freedom-fighters."

Suppose I believe that with such conviction that I'm unable to imagine a reality in which, regardless of whether the physical state of my brain changes, it would not still be true that terrorists+mom=freedom fighters. (The "terms" of this "equation" don't necessarily correspond with anything in the OP. The analogy is still functional).

In other words, I can dream up a scenario where terrorists are just terrorists, but I cannot fathom such a state of affairs actually coming to be.

So would this be subjunctively objective like your numerical epistemology or would it simply mean that my imagination is defective?

BTW, I don't truly believe anything I just wrote.

Math isn't a language, mathematical notation is a language. Math is a subject matter that you can talk about in mathematical notation, or in English, etc.

2+3=5 is an outcome of a set of artificial laws we can imagine. In that sense, it does exist "purely in your imagination", just as any number of hypothetical systems could exist. "2+3=5" doesn't stand alone without defining what it means - ie. the concept of a number, addition etc. It corresponds to the statement that IF addition is defined like so, numbers like this, and such-and-such rules of inference, then 2+2=5 is a true property of the system.

In a counterfactual world where people believe 2+3=6, in asking about addition you're still talking about the same system with the same rules, not the rules that describe whatever goes on in the minds of the people. (Otherwise you would be making a different claim about a different system.)

So yes, 2+3=5 is clearly true and has always been true even before humans because its a statement about a system defined in terms of its own rules. Any claims about it already include the system's presumptions because those are part of the question, and part of what it means to be "true".

2 rocks + 3 rocks is a different matter - you're talking about the observable world rather than a system where you get to define all the rules in advance. To apply mathematical reasoning to the real world, you have to make the additional claim "combining physical items is isomorphic to the rules of addition", and you're now in the realm of justifying this with empirical evidence. (Of which there is plenty)

I think a better phrasing of your final question then is to ask why do physical systems seem to correspond to the rules of *this* particular system, but there is a degree of circularity there - obviously we haven't just made up the rules of mathematics arbitrarily - we've based the lowest levels on recognised concepts, and then found that the same laws seem to apply at very deep levels with very high degrees of congruence with the world. If the universe were somehow different and nothing ever acted in any way corresponding to our model of "addition" or "numbers" though, then we'd not attach any special significance to it. Our "mathematics" would be quite different, and we'd be asking the same question about *that* system.

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"

A mathematical truth can be formalized as output of a proof checking algorithm, and output of an algorithm can be verified to an arbitrary level of certainty (by running it again and again, on redundant substrate). When you say that something is mathematically true, it can be considered an estimation of counterfactual that includes building of such a machine.

Come on, everyone knows 2 + 3 = 11!

I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.

There are two complementary answers to this question that seem right to me: Quine's Two Dogmas of Empiricism and Lakoff and Núñez's Where Mathematics Comes From. As Quine says, first you have to get rid of the false distinction between analytic and synthetic truth. What you have instead is a web or network of mutually reinforcing beliefs. Parts of this web touch the world relatively closely (beliefs about counting sheep) and parts touch the world less closely (beliefs about Peano's axioms for arithmetic). But the degree of confidence we have in a belief does not necessarily correspond to how closely it is connected to the world; it depends more on how the belief is embedded in our web of beliefs and how much support the belief gets from surrounding beliefs. Thus "2 + 3 = 5" can be strongly supported in our web of beliefs, more so than some beliefs that are more directly connected to the world, yet ultimately "2 + 3 = 5" is anchored in our daily experience of the world. Lakoff and Núñez go into more detail about the nature of this web and its anchoring, but what they say is largely consistent with Quine's general view.

Math isn't a language, mathematical notation is a language. Math is a subject matter that you can talk about in mathematical notation, or in English, etc.

What is the useful distinction here? Are you claiming that Math has a reality outside the notation? If Math isn't defined by the notation we use, then what is it?

I think it doesn't make sense to suggest that 2 + 3 = 5 is a belief. It is the result of a set of definitions. As long as we agree on what 2, +, 3, =, and 5 mean, we have to agree on what 2 + 3 = 5 means. I think that if your brain were subject to a neutrino storm and you somehow felt that 2 + 3 = 6, you would still be able to verify that 2 + 3 = 6 by other means, such as counting on your fingers.

I think once you start asking why these things are the way they are, don't you have to start asking why anything exists at all, and what it means for anything to exist? And I'm pretty sure at that point, we are firmly in the province of philosophy and there are no equations to be written, because the existence of the equations themselves is part of the question we're asking.

But I mean, this question has been in my mind since the beginning of the quantum series. I've written a lot of useful software since then, though, without entertaining it much. Do you think maybe it's just better to get on with our lives? It's not a rhetorical question, I really don't know.

It seems to me that when I say "every Hilbert space is convex", I'm not saying something in math; I'm saying something about math, in English. Yes, I might talk about the world by saying "the world has the structure of a Hilbert space". But then I might talk about blog commenters (not the ones here at OB) by saying they are like a horde of poo-throwing chimpanzees, and yet that doesn't make primatology a language.

I would encourage Peter's route related to Quine. A formalist in Phil of Math would say that a mathematical statement is true if it can be derived from axiomatic set theory. That is, the truth of the statement is then grounded in formal logic.
This does, of course, beg the question of what grounds our formal logic, but at least it puts basic arithmetic on more firm footing ... in Peter's words, even more deeply imbedded in our belief system.

What Would Plato Do?

Thomas: which set theory? There are lots of them.

Math isn't supposed to be some sort of universal truth, but I also don't think it's quite accurate so say it's just a language. It just happens to be a useful abstraction. Granted, an apparently universally useful abstraction, but it's still an invention of humans, the same as boolean logic or physical models.

I'm not convinced that it makes sense to talk about visualizing two dots and three dots that are six dots. I would say that the physical event of visualizing two dots and of visualizing three more dots IS the event "visualizing five dots". There is then a separate event, lets call it "describing what you have visualized", that can be mistaken. You can visualize five dots and as a result of interference in the information flow to your mouth end up saying "I see six dots". For that matter, you can visualize five dots, and as a result of either noise or the fact that it is hot out and other parts of your brain are competing to control your vocal apparatus, end up saying "it sure is hot out" instead of "I see five dots". In either of these cases you would say that the vocalization is not about the visualization, but rather, about the other mental and physical processes that caused it. In that case, why not say that "I see six dots" is about your visualization AND the neutrinos, not about the visualization.

It seems to me that math is a set of symbolic tools for clarifying the tautological nature of non-transparently tautological assertions.

"...then where is it?"

Same place all the other true counterfactuals are.

That was me at July 25, 2008 at 02:15 PM.

Can we taboo the words "math", "maths", and "mathematics"? I think there are mathematical facts and then there is the study of mathematical facts, and these two things are as different in the same sense that the universe isn't cosmology, crops aren't agronomy, minds aren't psychology, and so on.

3 + 2 = 6 for me if I choose to define 6 to signify five. 3 + 2 = 5 only for common mathematical definitions of 2, 3, 5, + and =. Otherwise everything is fine, your opponent agreed somewhere at the beginning, that a group of three objects (such as sheep) and two objects will make five objects for our definitions of two, three and five weather we exist or not.

Is it useful to say that "2+3=5" is our shorthand for referring to the infinite number of statements of this form:

2 sheep and 3 sheep make 5 sheep
2 rocks and 3 rocks make 5 rocks
2 dinis and 3 dinis make 5 dinis

and so forth? And that the external truth of the statement depends in principle on all these various testable sub-statements?

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"

The truth-condition for "There are five sheep in the meadow" concerns the state of the meadow.

My guess is that the truth condition for "2 + 3 = 5" concerns the (more complex, but unproblematically material) set of facts you present: the facts that e.g.:
*It's easy to find sheep for which two sheep and three sheep make five sheep
*It's fairly easy to build calculators that model what happens with the sheep
*It's fairly easy to evolve brains that model what happens with the calculators and the sheep.
*It's fairly easy to find "formal mathematical models" that can run on these evolved brains, that model what's going on with the sheep *and* the rocks *and* various other systems, with axioms and rules of inference that can be briefly described in English.

We have good reason to claim that "2+3 = 5" has an existence outside your mind. We have such reason because, as you point out, we see many different material systems that "correlate with each other and successfully predict one another".

But... do we have any reason to claim that "2+3=5" has an existence outside of these correlations between material systems? My guess is "no". My guess is that we should say that "2+3=5" is about these correlations. Once we say this, we can go ahead and investigate these correlations the way we'd investigate other aspects of material systems: we can try to spell out just what systems do "correlate with each other and successfully predict one another" in the ways we summarize with addition, and then what systems correlate with one another in the ways we summarize with Euclidean geometry, and then look for the meta-level pattern that unites the two sets of correlations.

These questions about the correlations are interesting and partially unsolved. But my guess is that they aren't gaps in our understanding of what math is about, just gaps in our understanding of the correlated material processes that math is about. The lines of questioning needed to explain these correlations are different from the lines of questioning that tend to be invoked when someone asks "where" math is.

I've been wondering. The conventional wisdom says that it's a problem for mathematical realism to explain how we can come to understand mathematical facts without causally interacting with them. But surely you could build causal diagrams with logical uncertainty in them and they would show that mathematical facts do indeed causally influence your brain?

Also, I would say the problem (if any) is the location of 2, 3, and 5, not the location of 2+3=5, unless the location of "Napoleon is dead" is also a problem.

Isn't this George Berkeley's issue? Isn't math just the structural part of another sort of language? Isn't 2 + 3 = 5 the same as red and blue make purple in the sense that each observer has a sense of red, blue, purple, 2, 3 and 5 all his/her/its own?

If space aliens find Voyager and read 1 *, 2 **, 3 ***, 4 ****, 5 *****, etc do they see those ***s in any context other than the three tentacles on their second heads?

How then is "2" in any sense different than "red"? How then is "2" any more independently real than "red"?

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"

For some reason most mathematicians don't seem to feel this sort of ontological angst about what math really means or what it means for a mathematical statement to be true. I can't seem to articulate a single reason why this is, but let me say a few things that tend to wash away the angst.

* it doesn't matter "where it is", it is a proven consequence of our axioms.

* it is in every structure in the universe capable of representing integers and performing arithmetic on them.

* there are many ways you can define the real numbers, but they're all isomorphic. When making statements like "2 + 3 = 5" we don't need to worry about which version of the reals we're talking about; it's true for all of them.

* there's a hierarchy of types of mathematical questions. At the bottom are recursive ones: questions we could answer with a big enough computer and enough time. Then there are R.E. questions: questions that if-the-answer-is-yes, we can confirm with a big enough computer and enough time (also, co-R.E., for if-the-answer-is-no). R.E. + co-R.E. is exactly the questions you can write in first-order logic (with the variables taking on integer values) with symbols for all recursive functions and only one quantifier. More quantifiers move you further up the hierarchy. Past that there are questions like the continuum hypothesis that aren't even *about* numbers, and don't seem to be constrained by anything physical. So even if you feel quite uneasy about what some mathematics means, remember that the stuff low on the hierarchy can be on solid ground even if the higher stuff isn't.

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