Notes on Proofs and Mathematical Reasoning

Introduction to Proofs

• Definition of Proof:
  • A proof is a method of establishing truth.
  • In mathematics, it is a verification of a proposition through logical deductions from axioms.
  • Proofs exist beyond mathematics; they can be seen as methods for ascertaining the truth in various fields.

Ascertaining Truth

• Methods for Establishing Truth:
  • Observation and Experimentation:
    • Example: Observing a piece of chalk fall to the ground (gravity).
  • Establishing Falsehood:
    • Finding counterexamples helps narrow down what is true.
  • Judicial Systems:
    • Truth can be established by judges/juries (e.g., legal cases).
  • Authority Figures:
    • Trusting the word of a boss, professor, or religious leader as a means of establishing truth.

Mathematical Proofs

• Components of a Mathematical Proof:
  • Propositions: Statements that can be true or false (e.g., "2 + 3 = 5").
    • Example of an interesting proposition:
      • "For all n in the natural numbers, n^2 + n + 41 is a prime number."
  • Logical Deductions: Steps that lead from axioms to conclusions.
  • Axioms: Propositions assumed to be true without proof (e.g., a = b and b = c implies a = c).

Important Concepts

• Predicate: A proposition whose truth depends on a variable's value (e.g., in the example, it depends on n).
• Quantifiers: Symbols indicating the scope of a variable (e.g., "for all" and "there exists").
  • Universal Quantifier: "For all" (symbol: upside down A).
  • Existential Quantifier: "There exists" (symbol: backward E).

Example of a Proof and Counterexamples

• The proposition about n^2 + n + 41 being prime is disproved by finding a counterexample:
  • Counterexample: For n = 40, it results in 1681 (not prime).
• Many famous mathematical propositions have been conjectured but not proven (e.g., Goldbach's Conjecture).

Axioms in Mathematics

• Definition: Axioms are propositions assumed to be true.
• Examples:
  • Axioms can differ in various fields of mathematics (e.g., Euclidean vs. spherical vs. hyperbolic geometry).
• Consistency and Completeness:
  • A set of axioms should be consistent (no contradictions) and complete (capable of proving every proposition as true or false).
• Gödel's Incompleteness Theorem:
  • Proved that no consistent set of axioms can be both complete and consistent.

Conclusion

• Understanding proofs, propositions, and axioms is essential in mathematics.
• A critical approach is needed to identify and scrutinize assumptions and truths in math and beyond.