What does it feel like to invent math?

So many things I could say about this, but I"ll let Alan Kay speak for me - [link] Alan talks about how ideas in Calculus could be manifested from an early age and a richness in understanding built up as they aged up: - at very young ages, kids really respond well through "doing" / the enactive channel. When asked to draw a circle, kids in Papert's group would first emulate what a LOGO turtle would do by rotating their body in a circle (making tiny increments in x and y). - as they got a bit older, the visual / iconic channel was more developed and they could understand the abstraction of a circle on pencil/paper and how the concepts carried over there - closer to early teens, symbols were much easier to grasp and relate to, etc. With this context in mind, there have been some cool efforts to mix the second and third channels I just mentioned to communicate advanced math concepts. Vi Hart and Grant Sanderson's youtube channels come to mind. Here are my favorite videos by Grant: - "What does it feel like to invent math": [this video] - On the visual intuition behind a hard problem on the Putnam exam - "The hardest problem on the hardest test": [link] - From vectors to matrices to vector spaces to higher-level ideas like the link between linear algebra and calculus: "Essence of linear algebra" series: [link] What I especially like about Grant's videos, is that he often walks through what a mathematician would do, the questions she would ask, etc.