FOM: What is mathematics? was intuition and rigor was arbitrary objects

[Ayan Mahalanobis]

In the current ongoing debate, which started from considering "arbitrary
objects" and now stands as a debate of intuition and rigor. I have a
more basic and elementary question; What is mathematics? or rather What
should be Mathematics?

As I understand this understanding is subjective as there were/are
mathematicians who were on opposite poles, and still there is no known
understanding as "What is intuition?" or intuitively clear. 

Is the existence of infinitesimal in real number system intuitively
clear? If so, should they be part of our mathematical rigor? 
I would like to ask this question to get some start on my earlier bigger
question.

Cheers
--Ayan