I think you’re right that what we disagree about is
the distinction here between epistemology and ontology.
The dichotomy you’ve provided seems to me to be an excellent definition of the difference between mathematical epistemological proof and empirical epistemological proof...it happens all the time that we may not be able to rigorously show N, but we nevertheless have extremely good reason to believe N with near-certainty, and even stronger reason to act as if we believed N.
If I hear you correctly, you think that we could plug in “the Universe is merely a mathematical object” for N.
I disagree. For me, the difference between epistemology and ontology is that there is a difference between what we can know and what exists. There might be things that exist about which we know nothing. There could even be things that exist about which we cannot know anything. One could reasonably call for scientists to ignore all such hypothetical objects, but, philosophically speaking, it doesn’t stop the objects from existing.
It boggles my mind to hear the claim that a mathematical object, as you have just defined it in your last comment, “exists in this second, ontological sense. The mandelbrot set expresses a relationship among points. If several small spheres exist and it turns out that the points approximate the relationship defined by the Mandlebrot set, then we might say that a Mandlebrot-ish shape of spheres exists. But the set itself doesn’t have any independent existence. This result doesn’t seem to me to depend on whether we use spheres or rays or standing waves—you still have to be vibrating something* if you want to talk about things that actually exist. I’m not the sort of nut that believes in good old-fashioned aether, but mathematical relationships alone won’t get you a flesh-and-blood universe where things actually exist...they’ll just get you a blueprint for one. Even if, epistemologically, we can know everything about the blueprint and model all of its parameters, it still won’t exist unless it’s made of something.
That, at any rate, is my modestly informed opinion. If you can see any flaws in my analysis, I would be grateful to you for pointing them out.
It continually amazes me that people think “physical existence” is somehow less mysterious and more fundamental than the existence of a mathematical object!
Er, no, it’s not less mysterious—we understand mathematical objects better than we understand physical existence; mathematical objects can be treated with, well, math, and physical existence gets dealt with by jokes like philosophy.
I’m not sure what you mean by more fundamental, but physical existence does seem to be roughly as important as mathematical objects...at any rate, it matters a lot to me whether things exist in fact or merely in theory.
We’ve been given special evidence in our own case, but if we step away from that for a moment, what I mean should be clear. Let’s take a hypothetical Universe X, which is very different from ours.
Saying “Universe X is a simple mathematical object” is pretty well comprehensible.
Saying “Universe X exists in some special way, distinct from just being a mathematical object, and in fact it might not be describable as a simple mathematical object” is just plain mysterious. It’s up for debate whether it’s even a meaningful statement.
But, but, you don’t understand. Math isn’t reeeeeeeaaaaaaaaaaaaaal!
I have an idea. Maybe it’d be more convincing if you said “Universe X is a simple computation.” People feel like computations are more real, and who knows, maybe they’re right. Maybe reality is computation, just a subset of mathematics. It seems a lot easier for people to envision that, at any rate. Or take Eliezer who (I think?) seems to think (or at least seemed to think) that reality juice is magically related to acyclic causal graphs.
You’ll still probably get the same objections, though: “Computations aren’t reeeeeeeaaaaaaaal, they have to be computed on something! Where’s the something coming from?” But that seems a little bit more silly, because the Something that is computing can be infinitely far back in the chain of computation. All of a sudden it feels more arbitrary to be postulating a Something that is Real. And real metaphysicists know that things shouldn’t feel arbitrary.
I am? I only read the decision theory chapters of Good and Real, the day before he showed up at SIAI house for the decision theory workshop. I’ll definitely read the last chapter when I get back to California.
I think your intuition is relying a little too much on the absurdity heuristic (e.g., “It boggles my mind...”) and flat out assertion (e.g., “But the set itself doesn’t have any independent existence.”). Metaphysical intuition is really misleading. I think most people underestimate that, especially because the absurdity heuristic is strong and therefore it’s easy to reach a reductio ad absurdum that is nonetheless true. I’ll give an example.
Once upon a time I didn’t think copies ‘counted’ in a multiverse, either morally or for purposes of anthropic reasoning. 200 Jacks had the same weight as 1 Mary. The opposite was absurd, you see: You’re claiming that 3 copies of the exact same computation are worth more than 2 computations of 2 different people, leading separate and diverse lives? Absurd! My moral and metaphysical intuition balks at such an idea! I came up with, like, 3 reductio ad absurdums to prove my point. Eliezer, Wei Dai, Steven Kaas, Nick Bostrom, what did they know? And there was some pride, too, because they way I was thinking about it meant I could easily deal with indexical uncertainty, and the others seemed clueless. … Well, turns out those reductios weren’t absurd: I just hadn’t learned to think like reality. I had to update, because that’s where the decision theory led, and it’s hard to argue with mathematics. And it came to my attention that thinking doubled computations had the same measure had a lot of problems as well. Since then, I’ve been a lot more careful about asserting my intuition when it disagrees with people who seem to have thought about it a lot more than I have.
In the case of the Mathematical Universe Hypothesis or permutations thereof (Eliezer seems to think the mysterious ‘reality fluid’ or ‘measure’ has a lot to do with directed acyclic graphs, for instance), there’s a lot of mental firepower aimed against you. Why do you believe what you believe? If it turns out the reason is metaphysical intuition, be on guard. Acknowledge your intuition, but don’t believe everything you think.
I think you’re right that what we disagree about is
The dichotomy you’ve provided seems to me to be an excellent definition of the difference between mathematical epistemological proof and empirical epistemological proof...it happens all the time that we may not be able to rigorously show N, but we nevertheless have extremely good reason to believe N with near-certainty, and even stronger reason to act as if we believed N.
If I hear you correctly, you think that we could plug in “the Universe is merely a mathematical object” for N.
I disagree. For me, the difference between epistemology and ontology is that there is a difference between what we can know and what exists. There might be things that exist about which we know nothing. There could even be things that exist about which we cannot know anything. One could reasonably call for scientists to ignore all such hypothetical objects, but, philosophically speaking, it doesn’t stop the objects from existing.
It boggles my mind to hear the claim that a mathematical object, as you have just defined it in your last comment, “exists in this second, ontological sense. The mandelbrot set expresses a relationship among points. If several small spheres exist and it turns out that the points approximate the relationship defined by the Mandlebrot set, then we might say that a Mandlebrot-ish shape of spheres exists. But the set itself doesn’t have any independent existence. This result doesn’t seem to me to depend on whether we use spheres or rays or standing waves—you still have to be vibrating something* if you want to talk about things that actually exist. I’m not the sort of nut that believes in good old-fashioned aether, but mathematical relationships alone won’t get you a flesh-and-blood universe where things actually exist...they’ll just get you a blueprint for one. Even if, epistemologically, we can know everything about the blueprint and model all of its parameters, it still won’t exist unless it’s made of something.
That, at any rate, is my modestly informed opinion. If you can see any flaws in my analysis, I would be grateful to you for pointing them out.
It continually amazes me that people think “physical existence” is somehow less mysterious and more fundamental than the existence of a mathematical object!
Er, no, it’s not less mysterious—we understand mathematical objects better than we understand physical existence; mathematical objects can be treated with, well, math, and physical existence gets dealt with by jokes like philosophy.
I’m not sure what you mean by more fundamental, but physical existence does seem to be roughly as important as mathematical objects...at any rate, it matters a lot to me whether things exist in fact or merely in theory.
We’ve been given special evidence in our own case, but if we step away from that for a moment, what I mean should be clear. Let’s take a hypothetical Universe X, which is very different from ours.
Saying “Universe X is a simple mathematical object” is pretty well comprehensible.
Saying “Universe X exists in some special way, distinct from just being a mathematical object, and in fact it might not be describable as a simple mathematical object” is just plain mysterious. It’s up for debate whether it’s even a meaningful statement.
Apply that to discussion of our own universe.
But, but, you don’t understand. Math isn’t reeeeeeeaaaaaaaaaaaaaal!
I have an idea. Maybe it’d be more convincing if you said “Universe X is a simple computation.” People feel like computations are more real, and who knows, maybe they’re right. Maybe reality is computation, just a subset of mathematics. It seems a lot easier for people to envision that, at any rate. Or take Eliezer who (I think?) seems to think (or at least seemed to think) that reality juice is magically related to acyclic causal graphs.
You’ll still probably get the same objections, though: “Computations aren’t reeeeeeeaaaaaaaal, they have to be computed on something! Where’s the something coming from?” But that seems a little bit more silly, because the Something that is computing can be infinitely far back in the chain of computation. All of a sudden it feels more arbitrary to be postulating a Something that is Real. And real metaphysicists know that things shouldn’t feel arbitrary.
Now you’re just ripping off the last chapter of Drescher’s Good and Real. You know he comments here sometimes—he’d be so hurt at such plagiarism. :)
I am? I only read the decision theory chapters of Good and Real, the day before he showed up at SIAI house for the decision theory workshop. I’ll definitely read the last chapter when I get back to California.
I think your intuition is relying a little too much on the absurdity heuristic (e.g., “It boggles my mind...”) and flat out assertion (e.g., “But the set itself doesn’t have any independent existence.”). Metaphysical intuition is really misleading. I think most people underestimate that, especially because the absurdity heuristic is strong and therefore it’s easy to reach a reductio ad absurdum that is nonetheless true. I’ll give an example.
Once upon a time I didn’t think copies ‘counted’ in a multiverse, either morally or for purposes of anthropic reasoning. 200 Jacks had the same weight as 1 Mary. The opposite was absurd, you see: You’re claiming that 3 copies of the exact same computation are worth more than 2 computations of 2 different people, leading separate and diverse lives? Absurd! My moral and metaphysical intuition balks at such an idea! I came up with, like, 3 reductio ad absurdums to prove my point. Eliezer, Wei Dai, Steven Kaas, Nick Bostrom, what did they know? And there was some pride, too, because they way I was thinking about it meant I could easily deal with indexical uncertainty, and the others seemed clueless. … Well, turns out those reductios weren’t absurd: I just hadn’t learned to think like reality. I had to update, because that’s where the decision theory led, and it’s hard to argue with mathematics. And it came to my attention that thinking doubled computations had the same measure had a lot of problems as well. Since then, I’ve been a lot more careful about asserting my intuition when it disagrees with people who seem to have thought about it a lot more than I have.
In the case of the Mathematical Universe Hypothesis or permutations thereof (Eliezer seems to think the mysterious ‘reality fluid’ or ‘measure’ has a lot to do with directed acyclic graphs, for instance), there’s a lot of mental firepower aimed against you. Why do you believe what you believe? If it turns out the reason is metaphysical intuition, be on guard. Acknowledge your intuition, but don’t believe everything you think.