Jordan Ellenberg: Mathematics of HighDimensional Shapes and Geometries  Lex Fridman Podcast #190  Summary and Q&A
TL;DR
Mathematician Jordan Ellenberg discusses the interconnectedness of mathematics and language, the role of geometry in understanding the world, and the concept of symmetry in mathematics.
Questions & Answers
Q: Is mathematical thinking as fundamental as language, or is it an abstract framework that is connected to language?
Mathematical thinking shares similarities with language, but it also has its own distinct features. While language is necessary for expressing mathematical thought, mathematics itself involves reasoning and problemsolving beyond mere linguistic expression.
Q: How does visualization of shapes and concepts play a role in mathematical thinking?
Visualizing shapes and concepts, especially in geometry, is an important aspect of mathematical thinking. It allows mathematicians to manipulate visual elements and discover new properties and relationships between objects. Visual proofs, such as dissection proofs, can provide intuitive understanding of mathematical concepts.
Q: What is symmetry, and how does it contribute to our understanding of the world?
Symmetry is a fundamental concept in mathematics that goes beyond the traditional notion of balance or reflection. It encompasses various transformations and patterns, both in physical objects and abstract spaces. Understanding symmetry helps uncover relationships and invariants, leading to deeper insights into the world around us.
Q: Can we conceptualize and understand dimensions beyond our threedimensional world?
While it may be challenging to visualize higher dimensions, mathematics provides tools to reason and explore concepts beyond our perceptual limitations. The study of higherdimensional spaces, such as topology, allows us to comprehend and reason about objects and relationships that exist in dimensions beyond our own.
Summary
In this conversation, Jordan Ellenberg, a mathematician at the University of Wisconsin and author, discusses various topics related to mathematics, including its relationship with language, visualizing shapes and concepts, and the importance of geometry. They also touch upon the work of mathematician Henri Poincare and his contributions to geometry and topology.
Questions & Answers
Q: Is mathematical thinking as fundamental as language?
Mathematical thinking is closely interconnected with language, although it may not be as fundamental as language itself. It is difficult to imagine doing mathematics without talking about it or thinking in propositions. However, it is possible that this is just the way we do mathematics, and there could be other ways to approach it.
Q: How does visualizing shapes and concepts relate to language?
Visualizing shapes and concepts, especially in the context of geometry, can sometimes go beyond language. There are geometric proofs, such as dissection proofs, that are purely visual and rely on observing shapes and their transformations. While it is hard to determine if visualization is considered language, the process of manipulating visual elements and the process of manipulating linguistic elements are similar in nature.
Q: What is the significance of geometry in mathematics?
Geometry holds a special place in mathematics as it allows for a visual representation of mathematical ideas. Geometry was what made many fall in love with mathematics because it demonstrated that something definitive could be stated about the world through intuitive visual proofs. Geometry helps us understand and prove relationships between shapes and structures, and it can provide a sense of order and understanding in a messy and complex world.
Q: How did Jordan Ellenberg fall in love with geometry?
Jordan Ellenberg did not initially fall in love with geometry, unlike many others. However, he recalls a specific moment as a child when he was looking at a rectangular array of holes in a wooden box and realized that the number of holes could be counted using different perspectives. This experience made him realize that mathematics could provide direct access to information and led him to appreciate the intertwining of algebra and geometry.
Q: What is symmetry and its use in mathematics and understanding our world?
Symmetry is an essential concept in mathematics that goes beyond the classical notion of symmetry, which involves axis or mirror symmetry. Symmetry refers to any transformation of an image, space, or object that has an input and output. It is a fundamental question in mathematics to determine what remains the same under different transformations or symmetries. The study of symmetry helps us understand the relationships and properties of objects and systems, and it plays a significant role in many fields, including art, science, and design.
Q: Why is the Poincaré conjecture important, and what does it entail?
The Poincaré conjecture is a question about curved threedimensional spaces and is a central problem in the field of topology. It deals with the notion of whether a space is "simply connected," meaning any loop drawn on that surface can be continuously shrunk to a point without leaving the surface. The conjecture states that any closed threedimensional space that is simply connected is homeomorphic to a sphere. Its solution has implications for understanding the shape and structure of our universe, but it remains an open question in mathematics.
Takeaways
Mathematics is deeply intertwined with language and can be considered a process similar to language. Visualizing shapes and concepts, particularly in geometry, plays a crucial role in understanding mathematical ideas and relationships. Geometry, with its ability to provide visual proofs, holds a special place in mathematics and can reveal the order and beauty in our complex world. Symmetry is a fundamental concept that helps us understand and analyze objects and systems across various fields. The Poincaré conjecture, a significant problem in topology, deals with the structure of threedimensional spaces and remains unsolved, offering insight into the shape of our universe.
Summary & Key Takeaways

Jordan Ellenberg reflects on the relationship between mathematical thinking and language, and how they are tightly interconnected in the search for meaning.

He discusses the role of geometry in his own love for mathematics, emphasizing how geometric proofs can reveal something definitive about the world.

Ellenberg dives into the concept of symmetry in mathematics, exploring its various forms and its use in understanding the world around us.