The title of this question is the exact title of one of the sections of a book written by Alexandre Borovik: Mathematics under the Microscope. Under the title, we read:
How should we approach the position of those mathematicians who refused to accept actual infinity? Henri Poincare is a prominent example. Weller et al. quote him on a par with modern day undergraduate students:
"There is no actual infinity; and when we speak of an infinite collection, we understand a collection to which we can add new elements unceasingly."
If I had in my class a student who made a similar statement and if the student's name happened to be Henri Poincare, I would not be in a hurry to correct him but would try to check whether Henri had a consistent vision of mathematics which was compatible with his thesis...Also, I have a gut feeling that Henri would still be able to solve every problem in the advanced calculus/analysis class. Indeed, within undergraduate calculus, every problem is solvable within the framework of the potential infinity of processes and sequences-you need only to be sufficiently attentive to detail.
Have you got the same gut feeling? Indeed, it is for years that a fundamental part of my course on the foundation of mathematics is to help my students to realize the actuality of actual infinity. That aside, next term I'll work with a group of mathematicians teaching Calculus, and today I have realized that the book they have chosen for the course is a kind of "soft analysis" that starts with real numbers and include different versions of the completeness of real numbers including the one that is based on decimal representation of numbers. That means, from very early on students are faced with the distinction between potential and actual infinity, and such a "famous" thing as $0.9999...$. For many, that thing just represents an infinite process. Shall we let them live with that conception providing that they have a consistent vision of what they do? (of course, this is a big assumption unless one of them be Henri!). Is there any problem in calculus that the distinction between potential and actual infinity indeed matters? That is to say, if one hasn't got the latter conception, the problem would become hard (or perhaps, impossible) to solve. If you think so, please give concrete examples.