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Carnegie Mellon University, Department of Philosophy,
Spring 2001
The Nature of Mathematical Reasoning
Dirk Schlimm
Overview
History and motivation

The History of Mathematics
Stone age, Babylonians, Egypts
Greeks: Thales, Pythagoras, Euclid, Aristoteles
What is mathematics about?

Some early proofs
Thales' theorem
Pythagoras' theorem
The square root of 2 is irrational
There are infinitely many prime numbers
Kinds of reasoning/arguments

inductive/deductive
direct/indirect
formal/contentful
fallacious
The structure of mathematical theories

Aspects of axiomatizations
completeness
independence

Axiomatic theories
Geometry: Euclid, Hilbert
Natural numbers: Peano
Case study I

Probability theory
Let's Make a Deal
HIV Testing
The nature of mathematical proof

Laws of reasoning: logic
Aristotle's syllogisms
Propositional logic
Predicate logic

Methods of proof
direct/indirect
mathematical induction
Case study II

Cantor's theory of the infinite
sets
sets of numbers: N, Z, Q, R
equinumerosity
diagonal argument
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© Dirk Schlimm, Last modified: Tue May 1 10:32:47 EDT 2001