Welcome to Numberholic! I’m your host, Sejari, and I’m excited to introduce a brand-new series on this channel: “Proven Yet Hard to Believe.” In this series, we’ll explore some of the most important proofs in the history of mathematics—proofs that not only shaped the field but also challenged our ability to believe them. Some truths are so surprising that even after proving them, they still feel unbelievable.
Today marks the very first episode, and we’ll begin our journey by uncovering the origins of mathematical proof itself. Let’s dive in and discover the story behind one of humanity’s most groundbreaking ideas!
What Was the First Mathematical Proof?
The first mathematical proof in recorded history emerged from ancient Greece, the cradle of Western civilization. The philosopher and mathematician credited with this milestone is Thales of Miletus, one of the Seven Sages of Greece. Thales lived around 600 BCE in Miletus, a city in Ionia—what is now modern-day Turkey.
For context, Thales’ life overlapped with that of Confucius in the East, who was active around 500 BCE. While Confucius shaped philosophy, Thales was laying the groundwork for mathematical reasoning.
The Legacy of Thales’ Proofs
Thales is attributed with several early proofs, but details about how he formulated them remain scarce. His proofs were likely documented in a now-lost text, History of Geometry by Eudemus, written in the 4th century BCE. Later accounts by Proclus, a Neoplatonist philosopher writing 1,000 years after Thales, referenced these lost works.
The proof we’ll explore today is one Thales is believed to have discovered during his travels in Egypt: the theorem of vertical angles.
The First Recorded Proof: Vertical Angles Are Equal
Thales’ theorem states: Vertical angles are equal.
Here’s how this historic proof might have unfolded:
- Observation: All straight angles (angles formed by a straight line) are equal.
- Logical deduction: Subtracting or adding equal angles leads to equal results.
- Conclusion: The opposite angles at an intersection of two lines are always equal.
This seemingly simple result represents a monumental leap—it was the first time logical reasoning and general principles were used to derive a mathematical truth.
In contrast, we see a different approach in the works of the Indian mathematician Bhāskara, who was active around 1150 CE. Bhāskara is well-known for his contributions to the Pythagorean theorem, describing the relationship between the three sides of a right triangle. He reportedly explained this concept to his students by simply saying, “Behold!”
While his explanation may have been understandable, Bhāskara did not provide a formal proof. This highlights the fundamental difference between his approach and Thales’ groundbreaking use of logical reasoning and structured proofs.
Mathematics Before Thales
Mathematics as a tool for calculation predates Thales by millennia. Civilizations like Mesopotamia and Egypt, flourishing as early as 4000 BCE, developed mathematical methods for practical purposes. The Babylonians, for instance, solved quadratic equations and devised rules for right triangles.
However, their mathematics was based on examples and empirical results. There were no generalizations or proofs—concepts we now associate with rigorous mathematics.
Thales’ Revolutionary Contribution
Thales didn’t invent geometry, but he invented the concept of proof. His approach had two key aspects:
- Precise Language: Thales expressed mathematical properties with clarity, emphasizing accuracy even for seemingly obvious truths.
- Logical Structure: He connected these properties through a logical framework, creating the foundation for axioms, theorems, and proofs.
This marked the birth of formal mathematics.
Thales’ contributions remind us that mathematics isn’t just about numbers—it’s about reasoning, structure, and uncovering truths that can feel almost impossible to believe. If you enjoyed this journey, don’t forget to like this video and subscribe to Numberholic! I’m Sejari, and I’ll see you next time on “Proven Yet Hard to Believe”, where we continue to explore the wonders of mathematics.