Foundations of mathematics under neuroscience conditions of lateral inhibition and lateral activation

In neuroscience we know that there exist the following two basic effects in perceiving information: (i) the lateral inhibition, responsible for a cognitive blindness in seeing the whole picture; (ii) the lateral activation, responsible for a cognitive blindness in seeing the details. In this paper, we show that the same effects can be considered in the proof cognitions performed by mathematicians in proving sophisticated theorems, such as Fermat’s Last Theorem. Hence, we insist that there can be two different foundations of mathematics: (i) the discrete foundations, dealing with a logical way of automatic proving from some axioms (the lateral inhibition in math); (ii) the analogue foundations, combining proof trees on tree forests by using the analogies as inference metarules (the lateral activation in math). We propose a kind of analogue logic for analogue reasoning in mathematics.