The discussion on this site seems to have many strands, but concerning Calculus - I was trying to figure it out for years and then came across smooth infinitesimal analysis. This is a version of analysis based on microstraightness and nilsquare infinitesimals. It reduces differential calculus to simple algebra, it does not lead to contradictions (such as the Banach-Tarski paradox), it facilitates physical derivations based on microadditivity, and it does not employ the dubious 'taking the standard part' trick of non-standard analysis (which is just Limit theory in disguise). The best book on the subject is A Primer of Infinitesimal Analysis by J L Bell.
The discussion on this site seems to have many strands, but concerning Calculus - I was trying to figure it out for years and then came across smooth infinitesimal analysis. This is a version of analysis based on microstraightness and nilsquare infinitesimals. It reduces differential calculus to simple algebra, it does not lead to contradictions (such as the Banach-Tarski paradox), it facilitates physical derivations based on microadditivity, and it does not employ the dubious 'taking the standard part' trick of non-standard analysis (which is just Limit theory in disguise). The best book on the subject is A Primer of Infinitesimal Analysis by J L Bell.