[DRMacIver]

I think mathematicians have mostly moved past the idea of fighting over whether something is or isn't a number. There are lots of interesting number systems that never made it into widespread use in the same way that the "core" number systems did, and there are lots that see niche usage, but people are pretty relaxed about whether they are "actually" numbers - they might or might not typicaly be referred to as numbers, but the strongest negative answer to "Is this a number?" you'd typically see would probably be to shrug and say "sure I guess, if you like?"

One example of previous attempts which we'd now consider to be invalid is a lot of operations on infinite series. In early days of analysis you'd get people concluding things like "1 - 1 + 1 - 1 + ... = 1/2", which gets you into more hot water the more you look at it. The problem here isn't that they are philosophically unsound per se - you can define all sorts of notions of "infinite sum" that make this work, like cesaro summation - but they don't behave as nicely as people intuitively expected them to and the naive versions of them don't really work.