Core Concepts
Mathematicians have transitioned from fixed axiomatic systems to exploring various coherent starting points for proofs, emphasizing the importance of simplicity and consistency in mathematical reasoning.
Abstract
The evolution of mathematical proof systems has shifted from rigid axioms to flexible starting points. Aristotle's deductive system laid the foundation, but issues with Euclidean axioms led to new approaches. David Hilbert proposed exploring any coherent starting point, leading to a plurality of axiomatic systems like ZFC. Gödel's incompleteness theorem highlighted the limitations in proving all true statements within a system, emphasizing the need for practicality in mathematical work.
Stats
"proving 2 + 2 = 4 took a vast amount of space"
"most of us work with the same 10 axioms, a system called ZFC"
"There must be an 11th axiom"
Quotes
"We’re working people." - Peter Sarnak