To me there is very big difference between 0 probability and an exact infinidesimal probability and I disagree that it is obvious they suffer from the same problems.
For example if I have a unit line and choose some particular point the probability of picking some exact point is epsilon. If I where to pick a point from a unit square the probability would be yet epsilon times smaller, for a total of epsilon*epsilon. If I where to pick a point from a line of lenght 2 the probability would only be half for a total of epsilon/2.
When usage of infinidesimal probabilites often fail is not spesifying which one and treating them all as the same one. It is not so that If you can’t multiply an amount finite times and end up with a finite amount then all such amounts must be equal. If I multiply epsilon by the first order infinite I get a finite 1. If I multiply epsilon*epsilon by the first order infinite I get a non-finite positive amount (exactly epsilon).
What impact infinite or infinidesimal probabilities make can largely be adopted by using rules to the same effect. An Example could be distinguishing between “pure” 0 and “almost never” and “pure” 1 and “almost always”. For what practical effect they might make consider darts. There are various probabilities conserning in which sector the dart lands and for example whether it it lands on a line dividing areas. However the numbers being passed around conserning lines will live a life largely separated by the math done for the areas. Now I can either take the separatedness as known fact outside the analysis or have the analysis show the separtedness of them. And there will be multiple types of zero probabilities. For example given that the board was hit, the probability for the dart not hitting any spesific area, line separating areas or a connection between lines is zero. However if I throw a dart I know I should not expect to hit that exact spot during the evening, the probability of it’s recurrence is an “impure” zero. The dart can still land there and it won’t magically avoid that spot. And no matter how many darts I throw the probability of hitting an old spot increases but I am not expecting to actually hit one. However if I notice that my probability of hitting an area divider or a line intersection is vanishing, in practise I know to focus on the area ratios, but I won’t accuse of someone of lying if they report a single such occurrence during the time I know such a man. However if they report 2 such occurrences I have reason to be suspicious.
I am aware of how infinitesimals work. However, consider Bayes’ theorem: If you have an infinitesimal prior, you have to find evidence weighted ω:1 in order to end up with a real posterior probability.
To me there is very big difference between 0 probability and an exact infinidesimal probability and I disagree that it is obvious they suffer from the same problems.
For example if I have a unit line and choose some particular point the probability of picking some exact point is epsilon. If I where to pick a point from a unit square the probability would be yet epsilon times smaller, for a total of epsilon*epsilon. If I where to pick a point from a line of lenght 2 the probability would only be half for a total of epsilon/2.
When usage of infinidesimal probabilites often fail is not spesifying which one and treating them all as the same one. It is not so that If you can’t multiply an amount finite times and end up with a finite amount then all such amounts must be equal. If I multiply epsilon by the first order infinite I get a finite 1. If I multiply epsilon*epsilon by the first order infinite I get a non-finite positive amount (exactly epsilon).
What impact infinite or infinidesimal probabilities make can largely be adopted by using rules to the same effect. An Example could be distinguishing between “pure” 0 and “almost never” and “pure” 1 and “almost always”. For what practical effect they might make consider darts. There are various probabilities conserning in which sector the dart lands and for example whether it it lands on a line dividing areas. However the numbers being passed around conserning lines will live a life largely separated by the math done for the areas. Now I can either take the separatedness as known fact outside the analysis or have the analysis show the separtedness of them. And there will be multiple types of zero probabilities. For example given that the board was hit, the probability for the dart not hitting any spesific area, line separating areas or a connection between lines is zero. However if I throw a dart I know I should not expect to hit that exact spot during the evening, the probability of it’s recurrence is an “impure” zero. The dart can still land there and it won’t magically avoid that spot. And no matter how many darts I throw the probability of hitting an old spot increases but I am not expecting to actually hit one. However if I notice that my probability of hitting an area divider or a line intersection is vanishing, in practise I know to focus on the area ratios, but I won’t accuse of someone of lying if they report a single such occurrence during the time I know such a man. However if they report 2 such occurrences I have reason to be suspicious.
I am aware of how infinitesimals work. However, consider Bayes’ theorem: If you have an infinitesimal prior, you have to find evidence weighted ω:1 in order to end up with a real posterior probability.