(Epistemic status: sufficiently abstract that I can’t be very confident without more familiarity with the topic)
(1) the objective of science is, or should be, to increase our ‘credence’ for true theories
I would suggest that it should also decrease our credence in false theories, and allow us to correctly estimate the likelyhood of conjectures not yet proven or disproved.
However, if T is an explanatory theory (e.g. ‘the sun is powered by nuclear fusion’), then its negation ~T (‘the sun is not powered by nuclear fusion’) is not an explanation at all.
Well, no—it’s a set of explanations. A very large set, consisting of every explanation other than ‘the sun is powered by nuclear fusion’, but smaller than T | ~T, and therefore somewhat useful, however slightly.
Therefore, suppose (implausibly, for the sake of argument) that one could quantify ‘the property that science strives to maximise’.
Per the first line, we are supposing this property to be ‘our credence in true theories’
If T had an amount q of that, then ~T would have none at all, not 1-q as the probability calculus would require if q were a probability.
All else being equal, if we come to believe T, our credence in true theories will be higher by 1 - p, where p is our previous credence in T. If we come to believe ~T, our credence in true theories will be lower than if we were uncertain by p.
I’m not sure that it makes sense in this context to assign a value of ‘the property that science strives to maximise’ to a statement. It’s not a property of statements alone but of our belief in them.
If you want to assign a value of q to near-absolute confidence in T, I would say that it’s 1 - ϵ. Thus, the ~t has near-zero value as far as the objective of science is concerned, and also has 1-q + ϵ = 0 + ϵ as the laws of probability demand.
Also, the conjunction (T₁ & T₂) of two mutually inconsistent explanatory theories T₁ and T₂ (such as quantum theory and relativity) is provably false, and therefore has zero probability. Yet it embodies some understanding of the world and is definitely better than nothing.
(Assuming for the sake of example that quantum theory and relativity mutually inconsistent, but both likely,) T1 & T2 is provably false, and indeed idea that quantum theory and relativity are both true is nonsense. T1 | T2, on the other hand, embodies some understanding of the world and is definitely better than nothing.
Well, no—it’s a set of explanations. A very large set, consisting of every explanation other than ‘the sun is powered by nuclear fusion’, but smaller than T | ~T, and therefore somewhat useful, however slightly.
Infinity minus one isn’t smaller than infinity. That’s not useful in that way.
It may be useful in some way. But just ruling a single thing out, when dealing with infinity, isn’t a road to progress.
indeed idea that quantum theory and relativity are both true is nonsense
He’s saying we use them both, and that has value, even though we know there must be some mistake somewhere. Saying “or” misrepresents the current situation. Both of them seem to be partly right. The situation (our current understanding which has value) looks nothing like we’ll end up keeping one and rejecting the other.
Infinity minus one isn’t smaller than infinity. That’s not useful in that way.
The thing being added or subtracted is not the mere number of hypotheses, but a measure of the likelihood of those hypotheses. We might suppose an infinitude of mutually exclusive theories of the world, but most of them are extremely unlikely—for any degree of unlikeliness, there are an infinity of theories less likely than that! A randomly-chosen theory is so unlikely to be true, that if you add up the likelihoods of every single theory, they add up to a number less than infinity.
It is for this reason that it is important when we divide our hypotheses between something likely, and everything else. “Everything else” contains infinite possibilities, but only finite likelihood.
Well, no—it’s a set of explanations. A very large set, consisting of every explanation other than ‘the sun is powered by nuclear fusion’, but smaller than T | ~T, and therefore somewhat useful, however slightly.
This was talking about set sizes, which is what I replied about.
You can’t quantify your fallibility in the sense of knowing how likely you are to be mistaken in an unexpected way. That’s not possible.
He’s saying we use them both, and that has value, even though we know there must be some mistake somewhere. Saying “or” misrepresents the current situation. Both of them seem to be partly right. The situation (our current understanding which has value) looks nothing like we’ll end up keeping one and rejecting the other.
I haven’t much knowledge of physics, and though that he was discussing the idea of two mutually exclusive theories which we use both of. From what you’re saying, it sounds more like the crucial point is that they are presumably false, but still useful. Is that a good description of the situation?
As far as partly right theories that have value: if we know quantum theory is not completely right, then we’ve ruled out the hypothesis ‘quantum mechanics’ and are now dealing with the hypothesis space of theories relevantly similar to quantum theory. So I agree that I theory known to be inaccurate in some cases can be useful, but by treating it as a piece of evidence towards the truth, which is rather different than how we treated it when we thought it could be true in its own right.
Relativity and QM contradict but we don’t know which is mistaken or why. Either one, individually, could be true in its own right.
The situation (our current understanding which has value) looks nothing like we’ll end up keeping one and rejecting the other.
I don’t see how these two statements can be consistent. If either one, individually, could be true in its own right, then why wouldn’t we won’t end up keeping one? If they contradict, then why wouldn’t we reject the other?
As far as partly right theories that have value: if we know quantum theory is not completely right, then we’ve ruled out the hypothesis ‘quantum theory’ and are now dealing with the hypothesis space of theories that share some parts with quantum theory.
T in this case is not atomic; it is itself a conjunction of a lot of statements. So I agree that I theory known to be inaccurate in some cases can be useful, in that it may contain some true components as well as some untrue ones. But this is rather different than how we treated it when we thought it could be true in its own right.
In general, I agree that there are certain ideas in science that aren’t propositions in Bayesian sense, and that treating them as if they were is a serious mistake. I don’t think that this means that there’s something wrong with the probability probability calculus, however.
But just ruling a single thing out, when dealing with infinity, isn’t a road to progress.
We don’t deal with infinities. When asked “what sun is powered by?”, humans formulate a finite, typically small, set of hypotheses, e.g.
By nuclear fusion
By a burning woodpile
By elven magic
By something else
Ruling even a single thing out from this small set is quite useful.
If you manage to rule out everything but something else, that’s the most exciting time in science because you’re now in uncharted territory (where every true scientist wants to be) and might be on a verge of a major breakthrough.
However, if T is an explanatory theory (e.g. ‘the sun is powered by nuclear fusion’), then its negation ~T (‘the sun is not powered by nuclear fusion’) is not an explanation at all.
Ideas don’t negate to all the alternatives humans are currently interested in. That isn’t how logic works.
It is not an explanation, but it is a (potentially) useful statement which leads you closer to an explanation. And I don’t see any logical problems here (notice the something else alternative).
In any case, the underlying issue is hypothesis generation and any purely Bayesian view of science is necessarily incomplete because St.Bayes says absolutely nothing about how to generate hypotheses.
I agree that ruling statements like you talk about out is useful – I just don’t think it’s useful in the Bayesian model. The use is due to the Critical Rationalist approach.
(Epistemic status: sufficiently abstract that I can’t be very confident without more familiarity with the topic)
I would suggest that it should also decrease our credence in false theories, and allow us to correctly estimate the likelyhood of conjectures not yet proven or disproved.
Well, no—it’s a set of explanations. A very large set, consisting of every explanation other than ‘the sun is powered by nuclear fusion’, but smaller than T | ~T, and therefore somewhat useful, however slightly.
Per the first line, we are supposing this property to be ‘our credence in true theories’
All else being equal, if we come to believe T, our credence in true theories will be higher by 1 - p, where p is our previous credence in T. If we come to believe ~T, our credence in true theories will be lower than if we were uncertain by p.
I’m not sure that it makes sense in this context to assign a value of ‘the property that science strives to maximise’ to a statement. It’s not a property of statements alone but of our belief in them.
If you want to assign a value of q to near-absolute confidence in T, I would say that it’s 1 - ϵ. Thus, the ~t has near-zero value as far as the objective of science is concerned, and also has 1-q + ϵ = 0 + ϵ as the laws of probability demand.
(Assuming for the sake of example that quantum theory and relativity mutually inconsistent, but both likely,) T1 & T2 is provably false, and indeed idea that quantum theory and relativity are both true is nonsense. T1 | T2, on the other hand, embodies some understanding of the world and is definitely better than nothing.
Infinity minus one isn’t smaller than infinity. That’s not useful in that way.
It may be useful in some way. But just ruling a single thing out, when dealing with infinity, isn’t a road to progress.
He’s saying we use them both, and that has value, even though we know there must be some mistake somewhere. Saying “or” misrepresents the current situation. Both of them seem to be partly right. The situation (our current understanding which has value) looks nothing like we’ll end up keeping one and rejecting the other.
The thing being added or subtracted is not the mere number of hypotheses, but a measure of the likelihood of those hypotheses. We might suppose an infinitude of mutually exclusive theories of the world, but most of them are extremely unlikely—for any degree of unlikeliness, there are an infinity of theories less likely than that! A randomly-chosen theory is so unlikely to be true, that if you add up the likelihoods of every single theory, they add up to a number less than infinity.
It is for this reason that it is important when we divide our hypotheses between something likely, and everything else. “Everything else” contains infinite possibilities, but only finite likelihood.
This was talking about set sizes, which is what I replied about.
You can’t quantify your fallibility in the sense of knowing how likely you are to be mistaken in an unexpected way. That’s not possible.
I haven’t much knowledge of physics, and though that he was discussing the idea of two mutually exclusive theories which we use both of. From what you’re saying, it sounds more like the crucial point is that they are presumably false, but still useful. Is that a good description of the situation?
As far as partly right theories that have value: if we know quantum theory is not completely right, then we’ve ruled out the hypothesis ‘quantum mechanics’ and are now dealing with the hypothesis space of theories relevantly similar to quantum theory. So I agree that I theory known to be inaccurate in some cases can be useful, but by treating it as a piece of evidence towards the truth, which is rather different than how we treated it when we thought it could be true in its own right.
Relativity and QM contradict but we don’t know which is mistaken or why. Either one, individually, could be true in its own right.
I don’t see how these two statements can be consistent. If either one, individually, could be true in its own right, then why wouldn’t we won’t end up keeping one? If they contradict, then why wouldn’t we reject the other?
i expect we’ll keep parts of both.
As far as partly right theories that have value: if we know quantum theory is not completely right, then we’ve ruled out the hypothesis ‘quantum theory’ and are now dealing with the hypothesis space of theories that share some parts with quantum theory.
T in this case is not atomic; it is itself a conjunction of a lot of statements. So I agree that I theory known to be inaccurate in some cases can be useful, in that it may contain some true components as well as some untrue ones. But this is rather different than how we treated it when we thought it could be true in its own right.
In general, I agree that there are certain ideas in science that aren’t propositions in Bayesian sense, and that treating them as if they were is a serious mistake. I don’t think that this means that there’s something wrong with the probability probability calculus, however.
But, again, we don’t know that. QM could be right.
I don’t see how these statements can be consistent.
...if relativity and QM contradict, and QM turns out to be right, I’d expect us to reject relativity. Do you agree?
We don’t deal with infinities. When asked “what sun is powered by?”, humans formulate a finite, typically small, set of hypotheses, e.g.
By nuclear fusion
By a burning woodpile
By elven magic
By something else
Ruling even a single thing out from this small set is quite useful.
If you manage to rule out everything but something else, that’s the most exciting time in science because you’re now in uncharted territory (where every true scientist wants to be) and might be on a verge of a major breakthrough.
DD:
Ideas don’t negate to all the alternatives humans are currently interested in. That isn’t how logic works.
It is not an explanation, but it is a (potentially) useful statement which leads you closer to an explanation. And I don’t see any logical problems here (notice the something else alternative).
In any case, the underlying issue is hypothesis generation and any purely Bayesian view of science is necessarily incomplete because St.Bayes says absolutely nothing about how to generate hypotheses.
I agree that ruling statements like you talk about out is useful – I just don’t think it’s useful in the Bayesian model. The use is due to the Critical Rationalist approach.