Some of the following passages are literally quoted or translated, others are paraphrased (condensed, modernised, …). [Square brackets are used to indicate inserted words in the quotations and comments following the quotations.] I tried not to distort too much the meaning intended by the authors. At any rate, as I read them, these quotations greatly helped me to clarify my opinions and they may find responsive readers.

1 The ideas should be born in the student's mind and the teacher should act only as midwife.

Socrates

2 … we should give no small share of the credit to Democritus who was the first to state the result though he did not prove it [just guessed it] … The method I used did not furnish an actual demonstration [just a suggestion, a guess … Yet] I foresee that this method, once understood, will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.

Archimedes

[First guess, then prove – that's the way to do it.]

3 Intuition is the conception of an attentive mind, so clear, so distinct, and so effortless that we cannot doubt what we have so conceived.