If we miss one dimension (in my opinion, along hyperdimensions created or causing tetration of (e^pi/2) = -I,) we have no way to calculate properly h(3), because You will never get to infinity by direct calculation. It is just an axiom, that divergent series reach REAL infinity-another symbol. Axiom needed and well grounded, but still possible to improve or change.
Euler. Borel etc . summations, just a simple Symbolic or formal geometric series application to infinite series gives results which seems not to be sums, but are true since they lead to correct results obtained via usage of them. And there has to be logic, not magic, why it is so, but initial definitions have to made less definite.
I think You know that in non-standard analysis things are treated differently, the results are the same. But non-standard does not go far enough. So it does not harm anyone to have new axioms. What harms is that only 1000 people in the world may be understand string theory, tetration etc. -theories become more complex, which is because we are chasing reality not trying to create it. But that is not what knowledge is about- to have more complex theories- or it should not be about that. Knowledge has to be simple, inputs may wary.
I can not promise stopping differentiating "constants" like i, pi, because they are just symbols and any symbols can be combined with another symbol ( e.g. differentiation) and attributed a different symbol as an outcome. The less definitions You have in the beginning, the more chance there to reach something new, and not necesserily changing true results, but just changing dogmas. If you start with axioms and definitions which are never to be challenged, You will always end within the logical limits they impose. Which is all what dogma is about.
The true constants, like 1/2 , 1/3 etc are not differentiable.
Euler. Borel etc . summations, just a simple Symbolic or formal geometric series application to infinite series gives results which seems not to be sums, but are true since they lead to correct results obtained via usage of them. And there has to be logic, not magic, why it is so, but initial definitions have to made less definite.
I think You know that in non-standard analysis things are treated differently, the results are the same. But non-standard does not go far enough. So it does not harm anyone to have new axioms. What harms is that only 1000 people in the world may be understand string theory, tetration etc. -theories become more complex, which is because we are chasing reality not trying to create it. But that is not what knowledge is about- to have more complex theories- or it should not be about that. Knowledge has to be simple, inputs may wary.
I can not promise stopping differentiating "constants" like i, pi, because they are just symbols and any symbols can be combined with another symbol ( e.g. differentiation) and attributed a different symbol as an outcome. The less definitions You have in the beginning, the more chance there to reach something new, and not necesserily changing true results, but just changing dogmas. If you start with axioms and definitions which are never to be challenged, You will always end within the logical limits they impose. Which is all what dogma is about.
The true constants, like 1/2 , 1/3 etc are not differentiable.