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.

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

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.

If you had a Turing machine that perfectly simulated the physical laws of our universe, could an external person use that machine's source code to derive the laws of arithmetic as they are within our universe, even if the laws of arithmetic for the external person's universe were different?

Suppose we think about it the opposite way: what if we built a machine that simulated the physical laws of a universe where 2+3 = 6, where if you stick 2 whatsits by 3 whatsits you get 6 whatsits total. What would that universe be like? Could it even be built?

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...).

I'm quite unconfident about this whole line of argument, and concerned that we're heading for some moral conclusions based on appeals to this argument. If you have to get into odd discussions about the truth and meaning of mathematical entities to make a metaethical l argument, I doubt you have a good metaethical argument.

The funny thing is I consider morality subjective objective, just like yummyness. What is subjectively yummy to you is an objective fact about you, just as what is moral to you is an objective fact about you. If we run the You algorithm to evaluate yummyness of a root beer float, we'll get an answer, depending on a lot of state variables about you. Similarly, we can run your moral algorithm to evaluate a person's actions, and depending on a lot of state variables, we'll get an answer there too. Both answers are objective facts about the subjective you.

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.

"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.

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.

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.

"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'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.

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.

In college, I made the observation that math majors tended to think that math itself was something real, while physcis majors, studying the exact same math in the same classes at the same time, tended to think that math was just a conceptual tool that was sometimes useful when trying to discover things about reality, but that math wasn't itself real. I'm not sure which view is more valid then the other, or how you even distinguish the two views.

Why not call the set of all sets of actual objects with cardinality 3, "three", the set of all sets of physical objects with cardinality 2, "two", and the set of all sets of physical objects with cardinality 5, "five"? Then when I said that 2+3=5, all I would mean is that for any x in two and any y in three, the union of x and y is in five. If you allow sets of physical objects, and sets of sets of physical objects, into your ontology, then you got this; 2+3=5 no matter what anyone thinks, and two and three are real objects existing out there.