Undecidability Tangent (History of Undecidability Part 1) - Computerphile

TL;DR

A historical journey into the concept of undecidability, its origins in Euclid's geometry, and its implications in mathematics and physics.

Transcript

in my younger days i found undecidability hard to get to grips with so what i thought i would do was to try and go back a little historically and do so much as i can about undecidability to lead back up to touring and hopefully make it easier for those of you who are struggling with the concept to find some way to internalize it and get to understa... Read More

Key Insights

• 🫥 Euclid's proposition on parallel lines sparked a long-standing undecidable problem in mathematics.
• 👾 Gauss and Lobachevsky's exploration of non-Euclidean geometries challenged the notion of Euclidean space.
• ❓ Undecidability has practical implications in physics, specifically in Einstein's theory of general relativity.
• 🥺 The concept of undecidability has sparked debates and led mathematicians to strive for rigorous foundations in mathematics.

Questions & Answers

Q: What was the earliest undecidable problem and how did it emerge?

The earliest undecidable problem can be traced back to Euclid's proposition on parallel lines. Mathematicians struggled for centuries to prove it, leading to the exploration of non-Euclidean geometries.

Q: How did Gauss and Lobachevsky contribute to the understanding of undecidability?

Gauss and Lobachevsky independently explored non-Euclidean geometries, showcasing that parallel lines can behave differently in curved spaces. This challenged the idea that Euclid's geometry was the only valid one.

Q: What role does undecidability play in physics?

Undecidability became relevant in physics with Einstein's theory of general relativity. The theory suggests that the presence of mass distorts the curvature of space-time, leading to non-Euclidean geometries.

Q: How did mathematicians react to the concept of undecidability?

Some mathematicians believed that undecidability was an isolated issue, while others, like Hilbert, wanted a rigorous mathematical foundation where all mathematical problems can be proven true or false.

Summary & Key Takeaways

• Undecidability is a concept that has puzzled mathematicians for centuries, starting from Euclid's proposition on parallel lines.

• Gauss and Lobachevsky explored non-Euclidean geometries, challenging the idea of Euclidean space.

• The concept of undecidability became relevant in physics with Einstein's theory of general relativity and its implications on the curvature of space-time.