Usual mathematicians and phyisicists do not really care seriously about the validity of logic itself, the true entity of concepts such as sets, functions, particles, or effectiveness of our cognitive ability. However these kinds of things always make me feel uncomfortable; not settling them perfectly and cleanly seem to ignore the importance of the basis on which a building stands....
So I have two questions.
What exactly is the usefulness of delving into those fundamental issues regarding our cognition and thinking?
Are there introductory books on these issues to read? Could anyone recommend me some?