The joy of abstract mathematical thinking - with Eugenia Cheng

The speaker begins by expressing gratitude to the audience and the Royal Institution for the invitation to speak. They reflect on their previous virtual appearance during lockdown and appreciate the in-person gathering. The speaker also acknowledges the online participants and thanks the technical crew. They mention their early connection with the Royal Institution through master classes and how it influenced their journey as a mathematician. The talk is set to focus on the joy of abstraction, as highlighted in the speaker's book, 'The Joy of Abstraction,' and the importance of abstract thinking in daily life.

The speaker discusses the common emphasis on the utility of mathematics and argues for a broader perspective that includes enjoyment and the development of abstract thinking skills. They critique the focus on boring tests in schools and the resulting negative impact on students' perception of math. The speaker uses a personal anecdote involving a jacket at the MARS Conference to illustrate the practical and fun aspects of abstract mathematical thinking. They emphasize that abstract thinking is widely applicable and often more relevant to everyday life than specific mathematical knowledge.

The speaker introduces category theory as a field of research that is highly abstract even by mathematical standards. They explain that category theory is about understanding connections between different ideas and putting things into context. The speaker argues that there is no absolute truth, only relative truths dependent on context. They discuss the importance of relationships in understanding the world and how category theory provides a framework for identifying and discussing nuances. The speaker also highlights the role of category theory in fostering more nuanced and less polarized discourse.

The speaker challenges the traditional view of mathematics learning as a linear progression of increasingly difficult hurdles. They argue that different areas of mathematics do not necessarily depend on one another and that category theory can be understood independently of other mathematical fields. The speaker advocates for a more inclusive approach to mathematics education that does not filter people out based on their ability to overcome each successive hurdle. They propose a web-like model where everything is connected and accessible via various paths.

The speaker shares their experience teaching category theory to art students, who typically had negative experiences with mathematics in the past. Despite lacking traditional mathematical background knowledge, these students were successful in learning category theory because they appreciated abstract thinking and enjoyed finding different perspectives on problems. The speaker emphasizes the importance of teaching abstraction and analogy in mathematics, arguing that these skills are inherent and can be developed in students of all backgrounds.

The speaker delves into the concept of levels of abstraction, using mathematical expressions as examples to illustrate how abstraction can include or exclude certain elements based on the level of generality. They highlight the importance of finding a good level of abstraction for understanding a particular concept and maintaining the ability to move between levels. The speaker also discusses the role of binary operations in abstraction and how these operations can be generalized to study a wide range of mathematical structures.

The speaker discusses how abstraction can be used to clarify disagreements and facilitate more productive discussions. They use the analogy of COVID-19 and seasonal flu to demonstrate how understanding the level of abstraction can lead to a more nuanced perspective. The speaker also touches on the concept of metrics, explaining how different metrics can be applied to measure various aspects of life, such as the obstacles overcome by individuals, rather than just the distance traveled.

The speaker explores the geometric representation of numbers and their factors, particularly focusing on the structure created by prime factors and their multiples. They use the example of the number 30 to illustrate how factors can be visualized as a cube, with each layer representing a different level of factorization. This geometric approach to numbers helps to reveal the underlying structure and relationships between different mathematical elements.

The speaker explains how visual abstraction through diagrams can make complex mathematical structures more understandable. They discuss how these visual representations can be applied to various situations beyond pure mathematics, such as analyzing social hierarchies or wealth distribution. The speaker also touches on the concept of enriched categories in category theory, which allows for a continuum of elements rather than a simple hierarchy.

The speaker uses the example of braiding hair to illustrate the concept of commutativity and the importance of direction in understanding mathematical operations. They explain how the ability to slide elements around each other and change perspectives is a higher-dimensional concept that can be taught even to young children. The speaker emphasizes the nuance that additional dimensions can provide in understanding complex mathematical and real-world scenarios.

In conclusion, the speaker reflects on the joy they find in category theory and the importance of abstraction in mathematics. They argue that while the utility of mathematics is important, the enjoyment and satisfaction derived from understanding abstract concepts should not be overlooked. The speaker shares their hope to spread the joy of abstraction to others and encourages a perspective that values relationships and context over intrinsic characteristics.