I would say 0 can be used to describe abstract concepts, but I do not think it can be observed in the real world. All measurements have a finite sensitivity, so measuring zero only means the variable of interest is smaller than the sensitivity of the measurement. For example, if a termometer of sensitivity 0.5 K, and range from 0 K to 300 K indicates 0 K, we can only say the temperature is lower than 0.5 K (we cannot say it is 0).
I agree 0 should not be used for real probabilities. Abstractly, we can use 0 to describe something impossible. For example, if X is a uniform distribution ranging from 0 to 1, the probability of X being between −2 and −1 is 0.
If I say I have 0 apples in my hands, I just mean 0 is the integer number which most accurately describes the vague concept of the number of apples in my hands. It is not indended to be exactly 0. For example, I may have forgotten to account for my 2 bites, which imply I only have 0.9 apples in my hands. Or I may only consider I have 0.5 apples in my hands because I am only holding the apple with one hand (i.e. 50 % of my 2 hands). Or maybe having refers to who bought the apples, and I only contributed to 50 % of the cost of the apple. In general, it looks like human language does not translate perfectly to exact numbers.
I would say 0 can be used to describe abstract concepts, but I do not think it can be observed in the real world. All measurements have a finite sensitivity, so measuring zero only means the variable of interest is smaller than the sensitivity of the measurement. For example, if a termometer of sensitivity 0.5 K, and range from 0 K to 300 K indicates 0 K, we can only say the temperature is lower than 0.5 K (we cannot say it is 0).
I agree 0 should not be used for real probabilities. Abstractly, we can use 0 to describe something impossible. For example, if X is a uniform distribution ranging from 0 to 1, the probability of X being between −2 and −1 is 0.
If I say I have 0 apples in my hands, I just mean 0 is the integer number which most accurately describes the vague concept of the number of apples in my hands. It is not indended to be exactly 0. For example, I may have forgotten to account for my 2 bites, which imply I only have 0.9 apples in my hands. Or I may only consider I have 0.5 apples in my hands because I am only holding the apple with one hand (i.e. 50 % of my 2 hands). Or maybe having refers to who bought the apples, and I only contributed to 50 % of the cost of the apple. In general, it looks like human language does not translate perfectly to exact numbers.