In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here.
"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.
I pretty much agree with you. I'm just worried that with all the comments about 'what is a number?', and 'sure, we can make infinity a well defined number', it will dilute the message of the article. Which is that infinity is not a 'number' (in the sense of real numbers). It's important to understand why it's not a number and what it means when we write it on one side of an equality.
"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.