From: lexfridman
The age-old debate of whether mathematics is discovered or invented persists as one of the more intriguing philosophical discussions in the field of mathematical inquiry. This debate essentially questions the nature and origins of mathematical concepts: are they inherent truths found within the fabric of the universe, or are they merely constructs devised by humans to understand and communicate complex phenomena?
The Dual Nature of Mathematics
Grant Sanderson
“I think there’s a cycle at play where you discover things about the universe that tell you what math will be useful, and that math itself is invented in a sense. But of all the possible maths that you could have invented, it’s discoveries about the world that tell you which ones are.” [11:00]
Grant Sanderson, the mind behind the YouTube channel ThreeBlueOneBrown, articulates a perspective that harmonizes the notions of discovery and invention. He suggests that mathematics is born from a cyclical process where discoveries in the natural world inform the invention of mathematical concepts. In other words, while the mathematical forms we use might be invented, their usefulness and applicability are discovered through interaction with the universe.
Mathematics: A Blend of Discovery and Invention
Discovery
Throughout history, many have viewed core mathematical truths as discoveries. One classic example is the Pythagorean theorem. Initially, ancient civilizations may have observed geometric truths in the physical world that led to the formulation of the theorem. This observation might suggest that mathematics exists independently of humanity, awaiting discovery by curious minds.
Invention
On the other hand, the methodologies, notations, and systems we use to express mathematical truths can be viewed as inventions. Grant explains that the distance metric in 2D space, often discussed as ( \mathbb{R}^2 ), was “invented” to make the Pythagorean theorem naturally hold [13:55].
Mathematical Constructs: A Human Touch
Sanderson acknowledges that there are countless mathematical constructs one could theoretically invent, but it’s our understanding of the universe that directs which of these constructs are useful [11:12]. For example, certain mathematical systems are designed to aid in “discovering the same principles of mathematics,” regardless of who or what species might develop them [13:35].
The Role of Notation and Abstraction
Mathematical notation plays a significant role in this interwoven narrative. Sanderson argues that notation heavily influences our thought processes and consequently the journey of mathematical discovery [02:31]. Notation is, undoubtedly, an invented tool that supports the broader narrative of mathematical exploration.
Furthermore, abstraction in mathematics allows us to organize and understand complex ideas, creating a bridge between intuitive understanding and rigorous formalism [28:20]. Whether math is discovered or invented, abstraction is intrinsically tied to its utility.
Concluding Thoughts
The debate of whether mathematics is discovered or invented is unlikely to reach a definitive conclusion. Instead, it prompts a deeper appreciation of how we engage with the world through mathematical ideas. As Sanderson puts it, the cycle between discovery and invention is constantly feeding itself, evolving with each new insight into our universe. Whether it’s discovered truths leading to invented systems, or invented methods unveiling new discoveries, mathematics remains a testament to the beauty of human curiosity and imagination.