I would suggest to read the George Berkeley's critics on the Infinitesimal calculus
Berkeley was one of the most important phylosopher and scientist of the 17 th century

He claimed that no right minded person might accept such an abstruse notion, because

a) An infinitesimal quantity should be something smaller than even the smallest finite quantity but not null
b) The calculus disregards some infinitesimal quantities nevertheless it should produce , according to its supporters, exact results

If you read the Newton's reply , you feel his embarassement
Newton himself was not able to provide a " logic " explanation to the above mentioned critics
Even nowadays it would not be that easy

Newton simply said that his method works, despite the Berkeley's sharp remark, so he will keep using it

The idea that science , including math , is based on the common sense is rather naive