Criticized for using formal mathematical manipulations, without understanding how they worked: Should I refuse a good dinner simply because I do not understand the process of digestion?

...there is no absolute scale of size in nature, and the small may be as important, or more so than the great.

Why should I refuse a good dinner simply because I don't understand the digestive processes involved?

Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?

It is shocking that young people should be addling their brains over mere logical subtleties in Euclid's Elements, trying to understand the proof of one obvious fact in terms of something equally .. obvious.

Waves from moving sources: Adagio. Andante. Allegro moderato.

Logic can be patient, for it is eternal.