" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.
I admit, some of his ideas are a bit.. well, I don't like when people talk about God seriously, and he sometimes mentions it, very rarely. But aside that, everything I can understand from what he says is true. It's a philosophical debate and if you are on the "real numbers" bandwagon (where most people are), you would lose integrity and your reputation might suffer even if you would speak to Wildberger about real numbers. It's a shame really how people don't see why it's bad to use abstractions which are so general that they can be fit for any kind circumstances. Even if you real this article about the real numbers, there is wishful thinking (where he says that the real numbers would look like a line even at infinity but the rationals wouldn't. well, I don't see why the rationals would stop especially given what he says later...), cherry picking / the whole axiom selection stuff for proving it... and yeah, the axiom idea is generally bad anyway. etc.
So I have a phd in math and I do tend to think less of other mathematicians who argue against infinite sets or uncountable sets and such, the argument's been over for a hundred years, you lost, deal with it. It's mathematical geocentrism.