Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64

Discovery

Lex Fridman and Grant Sanderson explore whether mathematics is discovered or invented. Grant suggests a cyclical process where discoveries about the universe inform the invention of useful mathematical concepts. Lex adds that historical discoveries often stemmed from physical intuition, but modern mathematics has become more abstract.

It's not an either or. It's not that math is one of these or it's one of the others. At different times, it's playing a different role.

--- Grant Sanderson

They discuss how physical observations have historically influenced mathematical definitions, such as the Pythagorean theorem 1 2.

   

Abstraction

Grant describes mathematics as the study of abstraction over patterns, while physics aims to understand the physical world. He notes that different mathematicians have varying motivations, from pure puzzles to physical applications. Lex questions why the universe's fundamental laws are so elegantly compressible into simple equations.

We have to have some connection of reality to be able to take our potentially oversimplified models of the world but then actually twist the world to our will based on it.

--- Grant Sanderson

They also touch on the simulation hypothesis, pondering whether our universe could be a computation 3 4.

   

Simplicity

The discussion shifts to the complexity or simplicity of physical laws. Grant and Lex consider whether the simplicity of these laws is a result of human bias or an inherent feature of the universe. They ponder the possibility of a world with incompressible laws and how it would impact our understanding.

Would such a world with uncompressible laws allow for the kind of beings that can think about the kind of questions that you're asking?

--- Grant Sanderson

They also discuss the effectiveness of mathematical notation, particularly the exponential function 5 6.

   

Visualization

Grant emphasizes the importance of visualizations in understanding abstract mathematical concepts. He explains that visualizing concepts helps make them more concrete and accessible. Lex agrees, noting that visualizations often start with low-level examples that build up to more abstract ideas.

With every image, you're making a choice. With each choice, you're showing a concrete example. With each concrete example, you're aiding someone's path to understanding.

--- Grant Sanderson

They discuss the process of creating visualizations and the challenges involved in making abstract ideas clear 7 8.