Dear Irek,
Thanks for your comment.
You say that "The first problem is that theory requires real numbers in full, e.g. quantum mechanics fall apart without real numbers."
All the maths work out just fine with the framework of real numbers that we have. It is a good approximation. But I don't see why quantum mechanics would fall apart without it. I'm very familiar with how QM works and I am quite convinced that at the basic level it would be just fine regardless. Could you elaborate on that criticism?
My main point is that, essentially, 'nothing changes' at the calculation level by supposing finitism. And it's quite easy to show that they don't; if they do, we are not able to know it anyways. In terms of interpretation, many things change.
"But there is another level here if one asks why this value of charge and mass and how it interacts with surrounding space"
First, by definition, a free electron is not bound. Therefore, the surroundings don't matter. This is obviously just an example to make things more understandable, it is not relevant for the case I was making.
I also dedicated a paragraph of the essay to explain the second part of this comment. It relates to the ontology of space-time. If space-time isn't real, only relative position matters, my idea evidently holds. If it is real, the Universe cannot really keep track (in terms of storing information) of the 4-positions of every particle at every moment. Either way, within the finitist framework presented, things can be simplified to the degrees of freedom of particles and the way they interact (which is not different of how things are currently done, by the way).