It’s common knowledge on lesswrong that “0 and 1 are not probabilities”. This means that if you update using Bayes law, your prior is 0 or 1 iff your posterior is 0 or 1. In other words the only way to be absolutely certain, is if you already were certain before considering any evidence. You obviously shouldn’t be certain/unconvincable without evidence so you shouldn’t ever be.
But only this week, I thought of an important corollary: precise measurement is impossible. If X is a continuous random variable, then P(X=x)=0 for any x. Because 0 and 1 are not probabilities, this means you’ll never know the precise value of X. You can never measure something continuous precisely.
I think it’s suspicious to get this conclusion from an “armchair” argument. I’ve deduced something important about the world without taking into account reality, without evidence. Which I said you shouldn’t ever do! So where did I go wrong?
You’re not wrong but you may be overinterpreting the significance of what you report. It is true that you can never measure something truly continuous with infinite precision. But, this isn’t that interesting because there are physical limitations on measurement that are much more relevant in any given context. The probability theory issue here (and the relevant mathematical terms if you want to follow up on this are “almost surely” and “measure zero) doesn’t say anything about measuring as precisely as you’d like; it just rules out truly infinite precision.
I’m not a physicist, so this may not be quite right, but my basic understanding is that there are inherent physical limits on measurement precision and not merely apparatus-dependent ones. That is one can’t, in principle, measure anything shorter than a Planck length, for example.
It’s common knowledge on lesswrong that “0 and 1 are not probabilities”. This means that if you update using Bayes law, your prior is 0 or 1 iff your posterior is 0 or 1. In other words the only way to be absolutely certain, is if you already were certain before considering any evidence. You obviously shouldn’t be certain/unconvincable without evidence so you shouldn’t ever be.
But only this week, I thought of an important corollary: precise measurement is impossible.
If X is a continuous random variable, then P(X=x)=0 for any x. Because 0 and 1 are not probabilities, this means you’ll never know the precise value of X. You can never measure something continuous precisely.
I think it’s suspicious to get this conclusion from an “armchair” argument. I’ve deduced something important about the world without taking into account reality, without evidence. Which I said you shouldn’t ever do! So where did I go wrong?
You’re not wrong but you may be overinterpreting the significance of what you report. It is true that you can never measure something truly continuous with infinite precision. But, this isn’t that interesting because there are physical limitations on measurement that are much more relevant in any given context. The probability theory issue here (and the relevant mathematical terms if you want to follow up on this are “almost surely” and “measure zero) doesn’t say anything about measuring as precisely as you’d like; it just rules out truly infinite precision.
I’m not a physicist, so this may not be quite right, but my basic understanding is that there are inherent physical limits on measurement precision and not merely apparatus-dependent ones. That is one can’t, in principle, measure anything shorter than a Planck length, for example.