While this new article is written in a slightly more clear and precise way, Dr Anderson still tries to play some games on semantics by saying things like: “NaN is, as it says, not a number. Nullity is a number - that makes a difference.” I still, however, don't see any difference here. Both things are just symbols which can be used in arithmetical expressions. The actual difference lies within the axiomatic systems that define how these symbols are supposed to be interpreted and evaluated within those arithmetical expressions.
There are also some of his claims that worry me a bit, he says for example that: “the work has been proved consistent twice, by hand, by [him], and has been checked at another university by computer.” I don't know exactly what he means by proved consistent but, as far as I know, it has not been possible to prove the consistency of Peano Arithmetic (which only deals with natural numbers). How did he managed then to prove the consistency of his system that is supposed to work for reals (plus a bit more)?
Moreover, Dr Anderson himself has accepted that his theory actually needs to go under some revision: “I have examined all of the comments and over a hundred counter-proofs to my work. Each was incorrect except one.” Well, in mathematics one counter-example is enough to show that your proof is wrong.
It might be the case that actually Dr Anderson's idea is quite brilliant and revolutionary. However I had been put off to spend some more time actually reading his work and checking whether his claims are right or wrong, mostly because of his arrogant attitude. If he really wants his work to be taken seriously, he should be worrying about having it peer-reviewed by the mathematical community and then, if it turns out that it actually contains something revolutionary, publicize it and get all the press that he might want.
I think that one comment from a reader of the new article better explains my view about this whole situation
To be honest, this is a lot of fuss and bother about nothing. So, Dr Anderson has defined a new system comprising the real numbers plus three extra points [...], and he's shown that his system satisfies various basic algebraic constraints. Well excuse me for not being blown away - mathematicians play these sort of games the whole time. [...] Assuming that it's right, this research should not be controversial per se. The reason it's attracting so much negative criticism is the extraordinary gulf between the overinflated and arrogant claims being made (the suggestion that he's solved an ancient problem, or that this research constitutes a “paradigm shift”), and the reality, which any decent mathematician will recognise as being of marginal interest at best, or banal and trivial at worst.