Gödel's Lasting Legacy  Summary and Q&A
TL;DR
Mathematics is a mysterious yet certain field, relying on proofs and formal systems to establish truth.
Questions & Answers
Q: What makes mathematics a mysterious field?
Mathematics is mysterious because it is not based on empirical evidence and relies on proofs and logical reasoning rather than observations or experiments.
Q: What is a formal system in mathematics?
A formal system is a purely mechanical and computational structure that follows stipulated rules and allows for mathematical reasoning without concern for the meaning or interpretation of the symbols or operations.
Q: What is Godel's first incompleteness theorem?
Godel's first incompleteness theorem states that in any formal system rich enough to express arithmetic, there will always be undecidable propositions, meaning they cannot be proven true or false within the system itself.
Q: What is Godel's second incompleteness theorem?
Godel's second incompleteness theorem states that the consistency of a formal system cannot be proven within the system itself. Inconsistent systems, which can prove contradictory statements, are considered useless.
Summary & Key Takeaways

Mathematics is not empirical and is based on a priori proofs from first principles.

Formal systems are mechanical and purely computational, allowing for mathematical reasoning.

Godel's incompleteness theorems state that there are undecidable propositions and the consistency of a formal system cannot be proven within the system.