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.

I loved this video ▶ this video "An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math."

3blue1brown has an earlier, kind of meta, video about “What does it feel like to invent math?” where Grant tries to show how one might invent, and so by define, p-adics. Complete with the customarily excellent visual aides. ▶ this video

It's also not a full introduction, but for those who want to have a taste of p-adic numbers, see 3b1b's video ▶ this video which motivates 2-adic numbers. There's also a neat connection between twos complement and 2-adic numbers.

3lue1brown has a great visualization of constructing p-adic numbers and calculating their metric through the lens of 2-adic numbers: ▶ this video This article fills in some bits omitted from the video: link

You add infinite number of positives numbers to each other and you get -1 as a result. p-adic numbers are adding some sense to this seemingly contradictory result. Great intro: ▶ this video