Harristotelian Logic

The problem I outlined in a previous post with the way mathematicians talk about the prime probability is rather huge, though it may be hard to see that at first.

To recap, you can figure out things in an exact way about the prime distribution itself because the naturals are well-ordered, so you can do things like use floor() to get exact counts, like the exact count of naturals that have 3 as a factor up to and including a natural number x is given by floor(x/3).

That is an absolute that works for any prime p, so floor(x/p) is the exact count of naturals up to and including a natural number x that have p as a factor.

So you can get an EXACT prime probability by using simple equations for each prime up to and including the square root of x.

In contast, with the question of values of p_1 mod p_2 where p_1 and p_2 are primes, it is easily shown that the residue of p_1 modulo p_2 CANNOT show any particular preference for a non-zero residue of p_2, so it is random.

And notice there is NO COMPARABLE WAY to get an exact count, to what you can do with counting primes out of the naturals.

So you have TWO SYSTEMS: one where you have exact methods and probability does not apply, and the other where you can't do things exactly and can prove random behavior so only probabilistic methods will work.

But then, there needs to be a rigid separation of approaches to the two different types of problems, but the current math field does not show that separation, and does not admit that probabilistic methods are the only ones that will work in certain areas, like questions on twin primes.

Do some digging if you don't believe me. The prime probability can be related to continuous functions that are well-studied like x/ln x, while in contrast look for similar equations with the twin primes probability, or even more dramatically, for the probability of an arbitrary prime gap g.

The reason this is a huge problem is that there are a lot of people doing research in areas like twin primes with techniques that cannot work, because it is a probabilistic area, and mathematicians have it now as an open question as to whether or not probability applies with primes.

Primes are not THAT mysterious as it's easy to explain behavior that appears to show that probability doesn't work well is just misapplication of probability to an ordered system, and also to prove random behavior like with p_1 mod p_2 in the other areas.

Simple it is mathematically, but politically the impact is immense, as if it is acknowledged it would shift any number of people in various fields out of certain lines of research as they're using methods that cannot succeed.