My response would be that this unfairly (and even absurdly) maligns “theory”! The “theory” here seems like a blatant straw-man, since it pretends probabilistic reasoning and cost-benefit tradeoffs are impossible.
This is because mathematics plays with evidence very differently than most other fields. If the probability is not equivalent to 1 (I.E it must happen), it doesn’t matter. A good example of this is the Collatz conjecture. It has a stupendous amount of evidence in the finite data points regime, but no mathematician worth their salt would declare Collatz solved, since it needs to have an actual proof. Similarly, P=NP is another problem with vast evidence, but no proof.
And a proof, if correct, is essentially equivalent to an infinite amount of observations, since you usually need to show that it holds up to infinite amounts of examples, so one Adversarial datapoint ruins the effort.
Mathematics is what happens when 0 or 1 are the only sets of probabilities allowed, that is a much stricter standard exists in math.
To nitpick, a mathematician won’t even accept that probability 1 is proof. Lots of probability 1 statements aren’t provable and could even be false from a logical perspective.
For example, a random walk in one dimension crosses any point infinitely many times, with probability one. But it could go in one direction forever.
So, proof could be said to be nontrivially stronger than even an infinite number of observations.
I feel like this is a difference between “almost surely” and “surely”, both of which are typically expressed as “probability 1”, but which are qualitatively different. I’m wondering whether infinitesimals would actually work to represent “almost surely” as 1−ϵ (as suggested in this post).
Also a nitpick and a bit of a tangent, but in some cases, a mathematician will accept any probability > 1 as proof; probabilistic proof is a common tool for non-constructive proof of existence, especially in combinatorics (although the way I’ve seen it, it’s usually more of a counting argument than something that relies essentially on probability).
A good example of this is the Collatz conjecture. It has a stupendous amount of evidence in the finite data points regime, but no mathematician worth their salt would declare Collatz solved, since it needs to have an actual proof.
It’s important to distinguish the probability you’d get from a naive induction argument and a more credible one that takes into account the reference class of similar mathematical statements that hold until large but finite limits but may or may not hold for all naturals.
Similarly, P=NP is another problem with vast evidence, but no proof.
Arguably even more than the Collatz conjecture, if you believe Scott Aaronson (link).
Mathematics is what happens when 0 or 1 are the only sets of probabilities allowed, that is a much stricter standard exists in math.
As I mentioned in my other reply, there are domains of math that do accept probabilities < 1 as proof (but it’s in a very special way). Also, proof usually comes with insight (this is one of the reasons that the proof of the 4 color theorem was controversial), which is almost more valuable than the certainty it brings (and of course as skeptics and bayesians, we must refrain from treating it as completely certain anyway; there could have been a logical error that everyone missed).
I think talking about proofs in terms of probabilities is a category error; logical proof is analogous to computable functions (equivalent if you are a constructivist), while probabilities are about bets.
I think talking about proofs in terms of probabilities is a category error; logical proof is analogous to computable functions (equivalent if you are a constructivist), while probabilities are about bets.
I agree that logical/mathematical proofs are more analogous to functions than probabilities, but they don’t have to be only computable functions, even if it’s all we will ever access, and that actually matters, especially in non-constructive math and proofs.
I also realized why probability 0 and 1 aren’t enough for proof, or equivalently why Abram Demski’s observation that a proof is stronger than an infinite number of observations is correct.
And it essentially boils down to the fact that in infinite sets, there are certain scenarios where the probability of an outcome is 0, but it’s still possible to get that outcome, or equivalently the probability is 1, but that doesn’t mean that it doesn’t have counterexamples. The best example is throwing a dart at a diagonal corner has probability 0, yet it still can happen. This doesn’t happen in finite sets, because a probability of 0 or 1 implicitly means that you have all your sample points, and for probability 0 means that it’s impossible to do, because you have no counterexamples, and for probability 1 you have a certainty proof, because you have no counterexamples. Mathematical conjectures and proofs usually demand something stronger than that of probability in infinite sets: A property of a set can’t hold at all, and there are no members of a set where that property holds, or a property of a set always holds, and the set of counterexamples is empty.
(Sometimes mathematics conjectures that something must exist, but this doesn’t change the answer to why probability!=proof in math, or why probability!=possibility in infinite sets.)
Unfortunately, infinite sets are the rule, not the exception in mathematics, and this is still true of even uncomputably large sets, like the arithmetical hierarchy of halting oracles. Finite sets are rare in mathematics, especially for proof and conjecture purposes.
This is because mathematics plays with evidence very differently than most other fields. If the probability is not equivalent to 1 (I.E it must happen), it doesn’t matter. A good example of this is the Collatz conjecture. It has a stupendous amount of evidence in the finite data points regime, but no mathematician worth their salt would declare Collatz solved, since it needs to have an actual proof. Similarly, P=NP is another problem with vast evidence, but no proof.
And a proof, if correct, is essentially equivalent to an infinite amount of observations, since you usually need to show that it holds up to infinite amounts of examples, so one Adversarial datapoint ruins the effort.
Mathematics is what happens when 0 or 1 are the only sets of probabilities allowed, that is a much stricter standard exists in math.
To nitpick, a mathematician won’t even accept that probability 1 is proof. Lots of probability 1 statements aren’t provable and could even be false from a logical perspective.
For example, a random walk in one dimension crosses any point infinitely many times, with probability one. But it could go in one direction forever.
So, proof could be said to be nontrivially stronger than even an infinite number of observations.
I feel like this is a difference between “almost surely” and “surely”, both of which are typically expressed as “probability 1”, but which are qualitatively different. I’m wondering whether infinitesimals would actually work to represent “almost surely” as 1−ϵ (as suggested in this post).
Also a nitpick and a bit of a tangent, but in some cases, a mathematician will accept any probability > 1 as proof; probabilistic proof is a common tool for non-constructive proof of existence, especially in combinatorics (although the way I’ve seen it, it’s usually more of a counting argument than something that relies essentially on probability).
It’s important to distinguish the probability you’d get from a naive induction argument and a more credible one that takes into account the reference class of similar mathematical statements that hold until large but finite limits but may or may not hold for all naturals.
Arguably even more than the Collatz conjecture, if you believe Scott Aaronson (link).
As I mentioned in my other reply, there are domains of math that do accept probabilities < 1 as proof (but it’s in a very special way). Also, proof usually comes with insight (this is one of the reasons that the proof of the 4 color theorem was controversial), which is almost more valuable than the certainty it brings (and of course as skeptics and bayesians, we must refrain from treating it as completely certain anyway; there could have been a logical error that everyone missed).
I think talking about proofs in terms of probabilities is a category error; logical proof is analogous to computable functions (equivalent if you are a constructivist), while probabilities are about bets.
I agree that logical/mathematical proofs are more analogous to functions than probabilities, but they don’t have to be only computable functions, even if it’s all we will ever access, and that actually matters, especially in non-constructive math and proofs.
I also realized why probability 0 and 1 aren’t enough for proof, or equivalently why Abram Demski’s observation that a proof is stronger than an infinite number of observations is correct.
And it essentially boils down to the fact that in infinite sets, there are certain scenarios where the probability of an outcome is 0, but it’s still possible to get that outcome, or equivalently the probability is 1, but that doesn’t mean that it doesn’t have counterexamples. The best example is throwing a dart at a diagonal corner has probability 0, yet it still can happen. This doesn’t happen in finite sets, because a probability of 0 or 1 implicitly means that you have all your sample points, and for probability 0 means that it’s impossible to do, because you have no counterexamples, and for probability 1 you have a certainty proof, because you have no counterexamples. Mathematical conjectures and proofs usually demand something stronger than that of probability in infinite sets: A property of a set can’t hold at all, and there are no members of a set where that property holds, or a property of a set always holds, and the set of counterexamples is empty.
(Sometimes mathematics conjectures that something must exist, but this doesn’t change the answer to why probability!=proof in math, or why probability!=possibility in infinite sets.)
Unfortunately, infinite sets are the rule, not the exception in mathematics, and this is still true of even uncomputably large sets, like the arithmetical hierarchy of halting oracles. Finite sets are rare in mathematics, especially for proof and conjecture purposes.
Here’s a link on where I got the insight from:
https://en.wikipedia.org/wiki/Almost_surely