"Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing"? You appear to think irrationals are numbers, so what physical thing can you have an irrational number of?
This is a dangerous path to walk down, but that doesn't make it an unimportant one. And that's why this article is ultimately flawed. The idea is important though. We need to understand how to operate formally on mathematical objects. If there's one thing I learned in my Philosophy of Mathematics class as a math undergrad, it is that we shouldn't struggle over defining what it means to be a number. We can just use them. I'll leave the philosophy to the philosophers.
Well, if your definition doesn't work for irrational numbers, then your argument isn't good enough: you exclude infinity on criteria that also exclude "good" numbers. In the context of the article, π is a number. And you cannot count things with an irrational number, by definition.
No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of. You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.
A lot of the comments here are getting hung up on trying to pin down the mathematics of what you can and can't do with infinity. That's fine, it's been keeping mathematicians busy for centuries, but this article is for the layman who doesn't know about rings, groups, algebras or any other mathematical structure which you might call 'numbers'. It's for someone who thinks the obvious when someone says 'number'.