I'll try to summarize my finitist position, which I seem stuck with despite years of trying to accept the mainstream / Cantorian view. One criticism is that the mainstream treatment of infinity is more invented, and less grounded in nature, relative to other areas of math. Another is that it is rife with equivocation, especially between the notion of infinity, and the notions of number and quantity.

It's commonly debated whether math is discovered, or invented. I think it depends on which part of math you're talking about. I think 1 + 1 = 2 is way over on the "discovered" end of the spectrum. It's very strongly grounded in nature, reality, everyday experience, etc. The mainstream treatment of infinity, including infinite sets, "different sized" infinities, bijections, etc., rely more on definitions and consensus about what passes as an acceptable "proof". For example, the tangent function is cited as a mapping between [-pi/2, pi/2] and [-inf, +inf], which is supposed to show that a subset of the reals is the same "size" as the whole set. But this requires defining division by zero as infinity in this context, whereas that's commonly considered undefined. I also have a big argument with the use of bijections (mappings) to compare the supposed "sizes" of infinities, but I can't fit it in a comment. The summary is that I think the ideas of cardinality and infinity are inherently contradictory, and putting them together creates nonsense. The idea of different sized infinities is actually created by (rather than proved by) the conventional restrictions on the style of bijections / mappings that are proposed and considered. But that's just another way of saying I think this area of math is a lot more on the "invented by definitions" end of the spectrum, and that whole philosophical question is another area on which people are going to differ in their attitudes.

On equivocation: Infinity is neither a number nor a quantity. It's not a number, because you can't get there by counting. It's not a quantity, because it can't be measured. But in the mainstream treatment of infinity, all the common intuitions about number and quantity get mixed in, for example the idea of different sized infinities. Infinity means "in this place where a number belongs, the value is unlimited". It's not itself an unlimited number, because as soon as it becomes unlimited, it no longer refers to any number. Similarly, I believe a more reasonable treatment of sets would say that "infinite" and "set" are incompatible, and attempting to force the concept of infiniteness onto a set makes it no longer a set, but rather something else like an abstract category. I think it was a mistake to generalize sets to include infinite sets.

As to how I treat infinity, it simply means something is boundless, inexhaustible, unlimited. I have no problem saying there are infinite reals, while minding that "infinity" is not a "count" of the reals. There are also infinite natural numbers. It doesn't make any sense to say there are more reals than naturals, in spite of "bijection theory". Reals, integers, natural numbers, etc. can all be considered different ways of naming items plucked from an infinite bag. When laid on a number line, reals and integers acquire one difference - integers can be adjacent, and reals can't.

One last note: I'm fully aware that my whole argument can be easily refuted by saying that math is made of definitions. That's fine. My position is simply that when people say things like "Hey, did you know there are actually different sized infinities? Isn't that cool?" they should be mindful that they're talking about a convention within a theory based on conventions, and not a natural or logical fact.