I see this type of statement crop up a lot everywhere.
I used to agree wholeheartedly with the above sentiment; however, in recent times, I have come to disagree. There is a way to formalise as a number; many ways, actually.
You may have heard of the hyperreals . A particular subset of contains those numbers strictly greater than all numbers of the form . That is certainly one way to bring the concept of infinity into a formal context.
Another way is to define it as . Used purely in the context of limits and allowing for substitution (ie ), this is also a legitimate way to rigourise infinity. I even saw this in a respected book on analysis.
There are still many other ways to define infinity; some are compatible, others not so. We have cardinals, ordinals, surreal numbers: a plethora of axioms to choose from. I simply picked the above two examples for the sake of brevity and intuition.
The problem most people have comes in when you have indeterminate forms, like or . The two seem at first to be and respectively, from rules of algebra, and then you quickly jump into the sobriety of rigour and note that rules of algebra only apply to fields compatible with those rules. The monster group acts very much differently from the additive integers.
So you dismiss it as nothing more than a symbol, placeholder, idea.
Sure it is. But it's really so much more.
The reason we reckon them is simply because - I quote, if memory serves me right, Mr Brian Charlesworth - not all infinities created equal. In all formalisations of infinity, we don't treat every infinity the same. So, has no real answer because you have many different paths of and to take. (That's what I meant by different infinities.) Or rather, it is not defined since it can be as much equal to as as the cat you saw on the street this morning. (You didn't see a cat, you say? Exactly.)
Stop treating infinity like it's just a concept. It's not. As a number, it has real (pardon the pun) value and is very important in maths and science.