I voted you up, but I’m genuinely confused here—does the concept of probability/possibility apply to a strict, axiomatic, isolated (yet human created and thus fallible) system like mathematics?
Not all logical contradictions have to do with mathematics. For instance, it’s impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered. Pancakes tend to have a very poor grasp of math.
Your confusion may have to do with epistemic v. metaphysical possibility. I can imagine that I am so deeply, profoundly confused about the universe that I could be mistaken about arithmetic; therefore, it’s sort of epistemically possible that two and two be five. However, as it happens, it’s not actually possible that two and two be five. Because I am part of a philosophy department in which David Lewis is practically worshiped, I’ll put it this way: I can think about two and two making five, but there’s no possible world in which that thought is reality.
Not all logical contradictions have to do with mathematics. For instance, it’s impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered.
It does have to do with mathematics, though. A buttered pancake is a pattern, an unbuttered pancake is a different pattern. Each pattern can be expressed as a series of bits, and the first series will not equal the second.
but there’s no possible world in which that thought is reality.
I find your distinction between “epistemic” and “metaphysical” possibility to be fairly useless. Are you using a different brain when you consider “epistemic” possibility than when you consider “metaphysical” possibility? Does your concept of “metaphysical” possibility somehow not reside within your own mind, but rather “out there” somewhere in the realm of Platonic logic? That seems suspicious. And even if it were somehow “out there”, how do you know you’re not mistaken about what it is?
More to the point: can you name a single “metaphysical” possibility that does not reduce to an “epistemic” possibility when considered from the context of your own brain?
It is a metaphysical possibility that the universe is actually entirely random and only seems to us as if it were lawful. But it is not epistemically possible for me to ever be 100% certain that the universe is random, or that it is lawful.
So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map.
The fact that I represent both concepts within my own brain doesn’t seem to change this.
So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map. The fact that I represent both concepts within my own brain doesn’t seem to change this.
This would seem to make it useless to talk of metaphysical possibilities, seeing as there is no way to directly access the territory.
I don’t think so. I cannot directly access reality, but it still seems very useful to me to speak of it. Even if only to have something for my beliefs to try to correspond to.
In that case, you can refer directly to your map. Instead of saying, “The sky is blue,” you can say, “My map of the territory contains a blue sky.” (Naturally, this is only necessary when context requires; if you’re in an ordinary conversation, there’s no need to go that far.) To me, it seems that the only time you need to really refer to the territory is when you’re talking directly about the map-territory relationship, e.g. “As research continues, our understanding of quantum physics will hopefully increase.” But to speak descriptively of the territory is to commit the Mind Projection Fallacy. After all, there’s no difference between saying, “I believe X,” versus just “X”; the two statements convey exactly the same information, and this information pertains only to the speaker’s map, not the territory. In my view, then, all possibilities are of the “epistemic” sort. To add a second type, “metaphysical”, seems wholly unnecessary.
I am in complete agreement with what you said that I need only talk about reality to have something to check my map against.
But I believe we may be using the term “epistemic possibility” differently.
It appears to me that when you say that X is epistemically possible, that it is possible that your map can contain X (and your map is correct in this aspect). I.e. that one can have a true belief that X.
What I mean when I say that X is epistemically possible is that I have a justified true belief that X. In that sense epistemic possibility is stronger than metaphysical possibility for me in that I require the justification of the belief as well as its truth.
Corresponding to this notion, there are stochastic models of (generally-agreed) necessary truths. The best known is probably the “probability of n being prime”:
or the question of who wins (first or second player) an integer-parametrized family of combinatorial games.
More exotically, Neal Stephenson’s “Anathem” and Greg Egan’s short stories “Luminous” and “Dark Integers” explore the possibility that what we think of as “necessary truths” are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence. (Note: I think this might sound ridiculous to a logician, but moderately reasonable to a cosmologist.) It is quite difficult to tell the difference between a necessary truth and a contingent truth which has always been true.
More prosaically, we do make errors and (given things like cosmic rays and other low-level stochastic processes) it seems unlikely that any physical process could be absolutely free of errors. We might believe something to be impossible, but erroneously. Your answer to the question “Are there any necessary truths?” probably depends on your degree of Platonism.
Greg Egan’s short stories “Luminous” and “Dark Integers” explore the possibility that what we think of as “necessary truths” are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence.
Nice stories, but the author didn’t find the optimal solution at the end. The red arithmetic should have kept a second small island with smooth borders, inside which a small blue patch with rough borders were maintained. This would have allowed communications without any risk of war.
I voted you up, but I’m genuinely confused here—does the concept of probability/possibility apply to a strict, axiomatic, isolated (yet human created and thus fallible) system like mathematics?
Not all logical contradictions have to do with mathematics. For instance, it’s impossible (barring childish equivocation etc.) that a given pancake be both buttered and not buttered. Pancakes tend to have a very poor grasp of math.
Your confusion may have to do with epistemic v. metaphysical possibility. I can imagine that I am so deeply, profoundly confused about the universe that I could be mistaken about arithmetic; therefore, it’s sort of epistemically possible that two and two be five. However, as it happens, it’s not actually possible that two and two be five. Because I am part of a philosophy department in which David Lewis is practically worshiped, I’ll put it this way: I can think about two and two making five, but there’s no possible world in which that thought is reality.
It does have to do with mathematics, though. A buttered pancake is a pattern, an unbuttered pancake is a different pattern. Each pattern can be expressed as a series of bits, and the first series will not equal the second.
Are you 100% certain about that?
No. Do I need to explain epistemic possibility versus metaphysical possibility again?
I find your distinction between “epistemic” and “metaphysical” possibility to be fairly useless. Are you using a different brain when you consider “epistemic” possibility than when you consider “metaphysical” possibility? Does your concept of “metaphysical” possibility somehow not reside within your own mind, but rather “out there” somewhere in the realm of Platonic logic? That seems suspicious. And even if it were somehow “out there”, how do you know you’re not mistaken about what it is?
More to the point: can you name a single “metaphysical” possibility that does not reduce to an “epistemic” possibility when considered from the context of your own brain?
It is a metaphysical possibility that the universe is actually entirely random and only seems to us as if it were lawful. But it is not epistemically possible for me to ever be 100% certain that the universe is random, or that it is lawful.
So it appears to me that an epistemic possibility is something which exists only in my map, whereas a metaphysical possibility is something which exists in the territory and may be represented in my map. The fact that I represent both concepts within my own brain doesn’t seem to change this.
This would seem to make it useless to talk of metaphysical possibilities, seeing as there is no way to directly access the territory.
I don’t think so. I cannot directly access reality, but it still seems very useful to me to speak of it. Even if only to have something for my beliefs to try to correspond to.
In that case, you can refer directly to your map. Instead of saying, “The sky is blue,” you can say, “My map of the territory contains a blue sky.” (Naturally, this is only necessary when context requires; if you’re in an ordinary conversation, there’s no need to go that far.) To me, it seems that the only time you need to really refer to the territory is when you’re talking directly about the map-territory relationship, e.g. “As research continues, our understanding of quantum physics will hopefully increase.” But to speak descriptively of the territory is to commit the Mind Projection Fallacy. After all, there’s no difference between saying, “I believe X,” versus just “X”; the two statements convey exactly the same information, and this information pertains only to the speaker’s map, not the territory. In my view, then, all possibilities are of the “epistemic” sort. To add a second type, “metaphysical”, seems wholly unnecessary.
I am in complete agreement with what you said that I need only talk about reality to have something to check my map against.
But I believe we may be using the term “epistemic possibility” differently.
It appears to me that when you say that X is epistemically possible, that it is possible that your map can contain X (and your map is correct in this aspect). I.e. that one can have a true belief that X.
What I mean when I say that X is epistemically possible is that I have a justified true belief that X. In that sense epistemic possibility is stronger than metaphysical possibility for me in that I require the justification of the belief as well as its truth.
That’s a very interesting question. Philosophers have been arguing about the concept of possibility (and its dual, necessity) for some time.
There’s a sense of “necessary randomness”, that Chaitin has written very extensively about.
http://www.umcs.maine.edu/~chaitin/
Corresponding to this notion, there are stochastic models of (generally-agreed) necessary truths. The best known is probably the “probability of n being prime”:
http://primes.utm.edu/glossary/xpage/PrimeNumberThm.html
But there are plenty others—e.g. the 3n+1 problem
http://en.wikipedia.org/wiki/Collatz_conjecture
or the question of who wins (first or second player) an integer-parametrized family of combinatorial games.
More exotically, Neal Stephenson’s “Anathem” and Greg Egan’s short stories “Luminous” and “Dark Integers” explore the possibility that what we think of as “necessary truths” are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence. (Note: I think this might sound ridiculous to a logician, but moderately reasonable to a cosmologist.) It is quite difficult to tell the difference between a necessary truth and a contingent truth which has always been true.
More prosaically, we do make errors and (given things like cosmic rays and other low-level stochastic processes) it seems unlikely that any physical process could be absolutely free of errors. We might believe something to be impossible, but erroneously. Your answer to the question “Are there any necessary truths?” probably depends on your degree of Platonism.
Nice stories, but the author didn’t find the optimal solution at the end. The red arithmetic should have kept a second small island with smooth borders, inside which a small blue patch with rough borders were maintained. This would have allowed communications without any risk of war.