I remember telling my tutor in the first year of a (pure) mathematics degree that I didn’t really enjoy analysis because I “couldn’t see why anyone should believe in the real numbers”. He looked at me as if I was slightly mad. At the time I’m not sure I realised I wasn’t totally alone in having such worries.
It seems that most professional mathematicians have never really given it much thought (or at least no more than other ontological, ‘non-mathematical’ questions). Of course, when it comes to fundamental physics it all starts to become more relevant.
edit: My comment shouldn’t be taken as a recommendation. I’m not suggesting mathematicians should worry about such things (it may well be a waste of their time!). I’m simply surprised that more don’t find it as unnerving as I did at the time.
It seems that most professional mathematicians have never really given it much thought (or at least no more than other ontological, ‘non-mathematical’ questions). Of course, when it comes to fundamental physics it all starts to become more relevant.
edit: My comment shouldn’t be taken as a recommendation. I’m not suggesting mathematicians should worry about such things (it may well be a waste of their time!). I’m simply surprised that more don’t find it as unnerving as I did at the time.