In short, Eliezer isn’t describing how he could come to belief 2 + 2 = 3, but how new evidence might show 2 + 2 would truly equaled 3.
From the beginning of his post:
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations”, and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
So on the point of interpretation, I’m pretty sure you are wrong.
On the substantive point, I think reliance on traditional philosophical distinctions (a priori/a posteriori, analytic/synthetic) is a recipe for confusion. In my opinion (and I am far from the first to point this out) these distinctions are poorly articulated, if not downright incoherent. If you are going to employ these concepts, however, an important thing to keep in mind is the hard-won philosophical realization, stemming from a tradition stretching from Kant to Kripke, that the a priori/a posteriori distinction is orthogonal to the necessary/contingent distinction. The former is an epistemological distinction (propositions are justifiable a priori or a posteriori), and the latter is a metaphysical distinction (propositions are true/false necessarily or contingently).
My position (and, I believe, Eliezer’s) is that mathematical truths are necessarily true. A world in which 2 + 2 = 3 is impossible. This does not, however, entail that it is impossible to convince me that 2 + 2 = 3. Nor does it entail that empirical considerations are irrelevant to the justification of my belief that 2 + 2 = 4.
I am sure there is some proposition (perhaps some complicated mathematical truth) that you believe is necessarily true, but you are not certain that it is true. Maybe you are fairly confident but not entirely sure that you got the proof right. So even though you believe this proposition cannot possibly be false, you admit the possibility of evidence that would convince you it is false.
First, I do reject the analytic/synthetic distinction. It always seemed like Kant was trying to make something out of nothing there. But I do think that math lacks empirical content, which is why I label it a priori.
I am sure there is some proposition (perhaps some complicated mathematical truth) that you believe is necessarily true, but you are not certain that it is true. Maybe you are fairly confident but not entirely sure that you got the proof right. So even though you believe this proposition cannot possibly be false, you admit the possibility of evidence that would convince you it is false.
But if math is not empirical, then this way of talking about math makes it seem less certain than it really is. I may be fallible, and thus not know every mathematically or logically provable statement, but that doesn’t show anything about the nature of provable statements. A proof of the Pythagorean Theorem is not (empirical) evidence that the theorem is true. The proof (metaphysically) is the truth of the theorem.
That said, I would certainly appreciate suggestions on a deeper overview of the necessary/contingent distinction.
A proof of the Pythagorean Theorem is not (empirical) evidence that the theorem is true. The proof (metaphysically) is the truth of the theorem.
Consider the four color theorem. We have a proof by computer of this theorem, but it is far too complex for any human to verify. Would you agree that the fact that a computer built and programmed in a certain way claims to have proven the theorem is empirical evidence for the truth of the theorem? If yes, then why treat a proof computed by a human brain differently?
Consider the four color theorem. We have a proof by computer of this theorem, but it is far too complex for any human to verify. Would you agree that the fact that a computer built and programmed in a certain way claims to have proven the theorem is empirical evidence for the truth of the theorem?
No. The computer output is a strong justification for behaving as if all maps are four-colorable.
But if the “proof” cannot be understood, then the truth of the theorem is simply beyond human comprehension. We could petition the evolution fairy for a better brain. Then again, dogs don’t seem to mind that they can’t comprehend that the derivative of e^x is e^x.
No. The computer output is a strong justification for behaving as if all maps are four-colorable.
Would you feel differently if the proof were verified by a general AI? If not, how is this not just carbon chauvinism?
Also, if you want another example, consider the classification of finite simple groups. Here the combined proofs run into the 1000s of pages, and it is likely that no single human being has checked the entire thing. Is your analysis for that case different from that of the four color theorem?
Can the proof be understand by a motivated, human-intelligence Cartesian skeptic who is protected from errors of carelessness? Because a Cartesian skeptic will never derive true physics statements, no matter how much effort is applied, since the skeptic is cut off from empirical data by definition.
And I think that is an interesting distinction between math and physics.
I certainly admit that there are physical processes that could cause me to believe a false mathematical statement was true. But that is properly understood as a fact about me, and does not mean that math has any empirical content.
Can the proof be understand by a motivated, human-intelligence Cartesian skeptic who is protected from errors of carelessness?
To the same extent the proof of the four color theorem can be. It will just take orders of magnitude more time than any human has. So do you consider it to be proven in the same sense? Do you need to wait until such a person exists and does it? If so, why is that different?
From the beginning of his post:
So on the point of interpretation, I’m pretty sure you are wrong.
On the substantive point, I think reliance on traditional philosophical distinctions (a priori/a posteriori, analytic/synthetic) is a recipe for confusion. In my opinion (and I am far from the first to point this out) these distinctions are poorly articulated, if not downright incoherent. If you are going to employ these concepts, however, an important thing to keep in mind is the hard-won philosophical realization, stemming from a tradition stretching from Kant to Kripke, that the a priori/a posteriori distinction is orthogonal to the necessary/contingent distinction. The former is an epistemological distinction (propositions are justifiable a priori or a posteriori), and the latter is a metaphysical distinction (propositions are true/false necessarily or contingently).
My position (and, I believe, Eliezer’s) is that mathematical truths are necessarily true. A world in which 2 + 2 = 3 is impossible. This does not, however, entail that it is impossible to convince me that 2 + 2 = 3. Nor does it entail that empirical considerations are irrelevant to the justification of my belief that 2 + 2 = 4.
I am sure there is some proposition (perhaps some complicated mathematical truth) that you believe is necessarily true, but you are not certain that it is true. Maybe you are fairly confident but not entirely sure that you got the proof right. So even though you believe this proposition cannot possibly be false, you admit the possibility of evidence that would convince you it is false.
Thanks for a really interesting reply.
First, I do reject the analytic/synthetic distinction. It always seemed like Kant was trying to make something out of nothing there. But I do think that math lacks empirical content, which is why I label it a priori.
But if math is not empirical, then this way of talking about math makes it seem less certain than it really is. I may be fallible, and thus not know every mathematically or logically provable statement, but that doesn’t show anything about the nature of provable statements. A proof of the Pythagorean Theorem is not (empirical) evidence that the theorem is true. The proof (metaphysically) is the truth of the theorem.
That said, I would certainly appreciate suggestions on a deeper overview of the necessary/contingent distinction.
Consider the four color theorem. We have a proof by computer of this theorem, but it is far too complex for any human to verify. Would you agree that the fact that a computer built and programmed in a certain way claims to have proven the theorem is empirical evidence for the truth of the theorem? If yes, then why treat a proof computed by a human brain differently?
No. The computer output is a strong justification for behaving as if all maps are four-colorable.
But if the “proof” cannot be understood, then the truth of the theorem is simply beyond human comprehension. We could petition the evolution fairy for a better brain. Then again, dogs don’t seem to mind that they can’t comprehend that the derivative of e^x is e^x.
Would you feel differently if the proof were verified by a general AI? If not, how is this not just carbon chauvinism?
Also, if you want another example, consider the classification of finite simple groups. Here the combined proofs run into the 1000s of pages, and it is likely that no single human being has checked the entire thing. Is your analysis for that case different from that of the four color theorem?
Can the proof be understand by a motivated, human-intelligence Cartesian skeptic who is protected from errors of carelessness? Because a Cartesian skeptic will never derive true physics statements, no matter how much effort is applied, since the skeptic is cut off from empirical data by definition.
And I think that is an interesting distinction between math and physics.
I certainly admit that there are physical processes that could cause me to believe a false mathematical statement was true. But that is properly understood as a fact about me, and does not mean that math has any empirical content.
To the same extent the proof of the four color theorem can be. It will just take orders of magnitude more time than any human has. So do you consider it to be proven in the same sense? Do you need to wait until such a person exists and does it? If so, why is that different?
Here’s a quick overview of the necessity/contingency distinction in philosophy. For a deeper overview, try Kripke’s Naming and Necessity.