You can’t get continuity without real numbers. Technically the required property is completeness, but the intuition is the same. For example, suppose you have some idea that some random number is “equally likely” in the range from 0 to 1 in some sense. You want to get a number that expresses the plausibility that its square is less than 1⁄2. You’ll find that every rational number is either too large, or too small.
One of the intuitions of continuity is that if you have a continuous curve that goes from too small to too large, there must be a point somewhere in the middle where it is just right. If you apply this requirement to rational numbers (via the Dedekind Cut construction), you get the real numbers. Real numbers are the smallest extension of the rational numbers in which you have this sort of continuity, and so if you want a mathematical theory of plausibilities that allows rational values and continuity then you need real numbers.
I am a little confused. I was working with a definition of continuity mentioned here https://mathworld.wolfram.com/RationalNumber.html : “It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.”
I understand that Rationals aren’t complete, and my question is why this is important for scientific inference. In other words, are we using the reals only because it makes the math easier, or is there a concrete example of inference which completeness helps with?
Specifically, since the context Jaynes is interested in is designing a (hypothetical) robot’s brain, and in order to achieve that we need to associate degrees of plausibility with a physical state, I don’t see why that entails the property of completeness which you mentioned? In fact, we mostly use digital and not analog computers, which use rational approximations for the reals. What does this system of reasoning lack?
The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.
You can’t get continuity without real numbers. Technically the required property is completeness, but the intuition is the same. For example, suppose you have some idea that some random number is “equally likely” in the range from 0 to 1 in some sense. You want to get a number that expresses the plausibility that its square is less than 1⁄2. You’ll find that every rational number is either too large, or too small.
One of the intuitions of continuity is that if you have a continuous curve that goes from too small to too large, there must be a point somewhere in the middle where it is just right. If you apply this requirement to rational numbers (via the Dedekind Cut construction), you get the real numbers. Real numbers are the smallest extension of the rational numbers in which you have this sort of continuity, and so if you want a mathematical theory of plausibilities that allows rational values and continuity then you need real numbers.
Thank you.
I am a little confused. I was working with a definition of continuity mentioned here https://mathworld.wolfram.com/RationalNumber.html : “It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.”
I understand that Rationals aren’t complete, and my question is why this is important for scientific inference. In other words, are we using the reals only because it makes the math easier, or is there a concrete example of inference which completeness helps with?
Specifically, since the context Jaynes is interested in is designing a (hypothetical) robot’s brain, and in order to achieve that we need to associate degrees of plausibility with a physical state, I don’t see why that entails the property of completeness which you mentioned? In fact, we mostly use digital and not analog computers, which use rational approximations for the reals. What does this system of reasoning lack?
The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.