As a quick heuristic, Edward Nelson has done some very impressive stuff. But a) his investigation into this question is by his own account motivated by pre-existing beliefs and b) he was 84 at the time he wrote this essay. The two of those don’t give me a great amount of confidence in the likelyhood that this is correct.
As a more direct issue, saying that induction doesn’t hold in general is akin to saying that I can have an infinite chain of dominoes such that each one will cause the next to fall if it falls over, and that the first has fallen, but all the dominoes won’t eventually fall. That’s an essentially physical representation of induction. If that’s not valid then things are very weird.
The mathematics he is summarizing is utterly correct. Actually the small things I singled out I think are due to Godel and not to Nelson. But my summary is much more likely to have errors.
The dominoes analogy is revealing. If you lay out one million actual dominoes as usual and push over the first one I think it’s very unlikely that the millionth one will fall.
Yes, the basic math is correct. That’s not the point. Nelson’s objection is to say that the axiom of induction is not supported in the same way as the others. That’s not a math issue, that’s an issue of what intuition and evidence tell us. In that sense, the evidence is overwhelming that induction is fine. Almost anyone who thinks for even a short period of time will be ok with it, unless, like Nelson, they have an external motive to be uncomfortable with it.
Incidentally, if one started with the assumption that PA has a contradiction, and asked me given that assumption who would I think is most likely to find the contradiction, Nelson would be very high up on the list simply given his work in foundations. The fact that he has a special motive to find such and still hasn’t found it is additional evidence that the system is really consistent.
The only real upshot of this essay is that a contradiction in PA might not result in a contradiction in a sufficiently weakened form of PA that we can still do most forms of useful arithmetic.
The only real upshot of this essay is that a contradiction in PA might not result in a contradiction in a sufficiently weakened form of PA that we can still do most forms of useful arithmetic.
I hope you are referring to my essay and not Nelson’s, which is a gem.
No, I was talking about Nelson’s essay. There’s nothing in that essay that wouldn’t be covered in a basic course in foundations, aside from his ramblings on the nature of monotheistic faith and the meaning of “I AM WHO I AM” (capitalization in the original). The point that there’s a distinction between how addition and multiplication behave and how exponentiation and higher analogs behave is not new either, although some sections of the essay might be worth explaining some concepts if one removes the theology.
I’m open to the possibility that PA might be inconsistent, although I assign this claim a very low probability. If one asked the question about some broader useful foundational system, such as ZF, I’d assign a much higher but still low probability.
If you think that either of these claims is wrong, I’d be happy to discuss making some form of wager over the likelyhood of an inconsistency being found within some fixed timespan (say 5 years or a decade?).
I’ve heard that Nelson has a standing bet with a colleague: he pays one dollar each year until an inconsistency is found in ZF, is paid one hundred dollars each year after an inconsistency is found.
Do you mean one that focuses on PA? Do you care to suggest a specific bet? (For what matters, I’d estimate around a 10^-6 chance that PA is inconsistent.)
I am fairly certain that no inconsistency will be found in the next 10 years.
Incidentally, I’m a little put off by your bringing up betting so early in the conversation. Isn’t it clear that I’m interested in talking about this stuff? That should be enough, especially before you’ve even located a place where you and I disagree.
I’ll mention that if PA is inconsistent, then a consistent prior probability distribution must have P(PA is inconsistent) = 1. (This might not be true if PA is consistent.) Developing a formalism for handling uncertainty about mathematical truths is a line of research in the same ballpark as developing mathematics without induction.
Incidentally, I’m a little put off by your bringing up betting so early in the conversation. Isn’t it clear that I’m interested in talking about this stuff? That should be enough, especially before you’ve even located a place where you and I disagree.
Hmm, I’m not sure why that would be off-putting. There must be some theory of mind issue I’m missing (possibly some sort of implication by suggesting bets that has some sort of negative connotation that I’m not picking up).
I’ll mention that if PA is inconsistent, then a consistent prior probability distribution must have P(PA is inconsistent) = 1. (This might not be true if PA is consistent.) Developing a formalism for handling uncertainty about mathematical truths is a line of research in the same ballpark as developing mathematics without induction.
Yes, hence making time-based estimates makes more sense.
I am fairly certain that no inconsistency will be found in the next 10 years.
Ok. So what is your reasoning behind this conclusion?
some sort of implication by suggesting bets that has some sort of negative connotation that I’m not picking up
That you don’t think the other party is capable of marshaling arguments to make you update, or of updating on your arguments.
What is your reasoning behind this conclusion?
An estimate of how much progress is made on this sort of problem per man-hour, and of how many man-hours will be devoted to the problem in the next ten years. But I am simply agnostic about whether or not a contradiction can be found “in principle.”
some sort of implication by suggesting bets that has some sort of negative connotation that I’m not picking up
That you don’t think the other party is capable of marshaling arguments to make you update, or of updating on your arguments.
The LW community probably considers betting on a disputed proposition to be much more normal, natural, and non-confrontational than most people do. This is likely because of our Overcoming Bias heritage and Robin Hanson’s work on prediction markets. Betting seems like a good quick way to get people to publicly quantify the probabilities that they assign to propositions. And this, ideally, could help the disputants approach Aumann agreement more quickly.
That you don’t think the other party is capable of marshaling arguments to make you update, or of updating on your arguments.
Ah, I see. That’s not an intended implication. I prefer constructing bets because it forces one (myself) to think carefully about how confident I actually am for a claim. But I see how one might think that.
An estimate of how much progress is made on this sort of problem per man-hour, and of how many man-hours will be devoted to the problem in the next ten years.
Ah, that makes a lot of sense.
But I am simply agnostic about whether or not a contradiction can be found “in principle.”
That make a lot of sense. Presumably this issue is connected to the problem that no one seems to have any idea how one would go about finding such a contradiction.
Incidentally, seriously thinking about these sorts of issues brings up strange issues of equiconsistency. I’m particularly now wondering about the equiconsistency statuses of Robinson arithmetic, the arithmetic hierarchy and, and PA. I don’t know of any result that says something morally like contradictions in PA can be imported into contradictions in Robinson arithmetic (or some extension via the arithmetic hierarchy), but this is pushing the bounds of my knowledge on these issues. Does anyone know if there are any results of that flavor or results in the other direction?
As a quick heuristic, Edward Nelson has done some very impressive stuff. But a) his investigation into this question is by his own account motivated by pre-existing beliefs and b) he was 84 at the time he wrote this essay. The two of those don’t give me a great amount of confidence in the likelyhood that this is correct.
As a more direct issue, saying that induction doesn’t hold in general is akin to saying that I can have an infinite chain of dominoes such that each one will cause the next to fall if it falls over, and that the first has fallen, but all the dominoes won’t eventually fall. That’s an essentially physical representation of induction. If that’s not valid then things are very weird.
The mathematics he is summarizing is utterly correct. Actually the small things I singled out I think are due to Godel and not to Nelson. But my summary is much more likely to have errors.
The dominoes analogy is revealing. If you lay out one million actual dominoes as usual and push over the first one I think it’s very unlikely that the millionth one will fall.
Yes, the basic math is correct. That’s not the point. Nelson’s objection is to say that the axiom of induction is not supported in the same way as the others. That’s not a math issue, that’s an issue of what intuition and evidence tell us. In that sense, the evidence is overwhelming that induction is fine. Almost anyone who thinks for even a short period of time will be ok with it, unless, like Nelson, they have an external motive to be uncomfortable with it.
Incidentally, if one started with the assumption that PA has a contradiction, and asked me given that assumption who would I think is most likely to find the contradiction, Nelson would be very high up on the list simply given his work in foundations. The fact that he has a special motive to find such and still hasn’t found it is additional evidence that the system is really consistent.
The only real upshot of this essay is that a contradiction in PA might not result in a contradiction in a sufficiently weakened form of PA that we can still do most forms of useful arithmetic.
Thinking about motives and contradictions, you might find this interesting:
http://video.ias.edu/voevodsky-80th
I hope you are referring to my essay and not Nelson’s, which is a gem.
No, I was talking about Nelson’s essay. There’s nothing in that essay that wouldn’t be covered in a basic course in foundations, aside from his ramblings on the nature of monotheistic faith and the meaning of “I AM WHO I AM” (capitalization in the original). The point that there’s a distinction between how addition and multiplication behave and how exponentiation and higher analogs behave is not new either, although some sections of the essay might be worth explaining some concepts if one removes the theology.
I’m open to the possibility that PA might be inconsistent, although I assign this claim a very low probability. If one asked the question about some broader useful foundational system, such as ZF, I’d assign a much higher but still low probability.
If you think that either of these claims is wrong, I’d be happy to discuss making some form of wager over the likelyhood of an inconsistency being found within some fixed timespan (say 5 years or a decade?).
I’ve heard that Nelson has a standing bet with a colleague: he pays one dollar each year until an inconsistency is found in ZF, is paid one hundred dollars each year after an inconsistency is found.
I might take a more domino-oriented bet.
Do you mean one that focuses on PA? Do you care to suggest a specific bet? (For what matters, I’d estimate around a 10^-6 chance that PA is inconsistent.)
I’m teasing. I don’t think your domino argument can be used to support induction.
Ok. So joking aside, do you want to make a bet on an inconsistency being found in PA in the next five years? 10 years?
I am fairly certain that no inconsistency will be found in the next 10 years.
Incidentally, I’m a little put off by your bringing up betting so early in the conversation. Isn’t it clear that I’m interested in talking about this stuff? That should be enough, especially before you’ve even located a place where you and I disagree.
I’ll mention that if PA is inconsistent, then a consistent prior probability distribution must have P(PA is inconsistent) = 1. (This might not be true if PA is consistent.) Developing a formalism for handling uncertainty about mathematical truths is a line of research in the same ballpark as developing mathematics without induction.
Hmm, I’m not sure why that would be off-putting. There must be some theory of mind issue I’m missing (possibly some sort of implication by suggesting bets that has some sort of negative connotation that I’m not picking up).
Yes, hence making time-based estimates makes more sense.
Ok. So what is your reasoning behind this conclusion?
That you don’t think the other party is capable of marshaling arguments to make you update, or of updating on your arguments.
An estimate of how much progress is made on this sort of problem per man-hour, and of how many man-hours will be devoted to the problem in the next ten years. But I am simply agnostic about whether or not a contradiction can be found “in principle.”
The LW community probably considers betting on a disputed proposition to be much more normal, natural, and non-confrontational than most people do. This is likely because of our Overcoming Bias heritage and Robin Hanson’s work on prediction markets. Betting seems like a good quick way to get people to publicly quantify the probabilities that they assign to propositions. And this, ideally, could help the disputants approach Aumann agreement more quickly.
Ah, I see. That’s not an intended implication. I prefer constructing bets because it forces one (myself) to think carefully about how confident I actually am for a claim. But I see how one might think that.
Ah, that makes a lot of sense.
That make a lot of sense. Presumably this issue is connected to the problem that no one seems to have any idea how one would go about finding such a contradiction.
Incidentally, seriously thinking about these sorts of issues brings up strange issues of equiconsistency. I’m particularly now wondering about the equiconsistency statuses of Robinson arithmetic, the arithmetic hierarchy and, and PA. I don’t know of any result that says something morally like contradictions in PA can be imported into contradictions in Robinson arithmetic (or some extension via the arithmetic hierarchy), but this is pushing the bounds of my knowledge on these issues. Does anyone know if there are any results of that flavor or results in the other direction?