How many here believe that our mathematical skills can be nurtured without learning it from the environment? I don’t mean learning it from academic exercises or text book questions, I mean learning it from observations about how the things in the world work in accordance with mathematical principles. For example, do we learn that 2 + 2 = 4 because we’ve consistently experienced that whenever we have 2 things and we added them to 2 other things, we always get 4 things, or do we learn this solely based on the inherent meaning of ‘2’ and ‘+’ (such that we could do it all in our heads)?
One thing I’ve considered, if we go with the learning-from-the-environment perspective, is a hypothetical scenario in which one learns a different set of mathematical rules if his/her environment worked radically different that the one we take for granted. For example, suppose one developped in an environment made of nothing but water (this doesn’t have to be realistic - just as long as the principle of the argument is revealed by the scenario). Every time the individual takes 2 water droplettes and adds them to two other water droplettes, he/she gets not 4 but 1! Water droplettes congeal together when they come in contact with each other, so you just get a bigger water droplette. If this is consistently the environmental experiences this individual has, and no other sorts of experiences are had, would he/she end up understanding the rules of mathematics in a radically different way?