80-110 The Nature of Mathematical Reasoning
Spring 2002
Overview
History and motivation
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The History of Mathematics
Stone age, Babylonians, Egypts
Greeks: Thales, Pythagoras, Euclid, Aristoteles
What is mathematics about?
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Some early proofs
Thales' theorem
The square root of 2 is irrational
There are infinitely many prime numbers
Kinds of reasoning/arguments
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valid/invalid
inductive/deductive
direct/indirect
formal/contentful
fallacious
The structure of mathematical theories
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Aspects of axiomatizations
completeness
independence
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Axiomatic theories
Geometry: Euclid, Hilbert
Natural numbers: Peano
Case study I
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Probability theory
Let's Make a Deal
HIV Testing
The nature of mathematical proof
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Laws of reasoning: logic
Aristotle's syllogisms
Propositional logic
Predicate logic
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Methods of proof
direct/indirect
mathematical induction
Case study II
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Cantor's theory of the infinite
sets
sets of numbers: N, Z, Q, R
equinumerosity
diagonal argument
© Dirk Schlimm, Last
modified: 1/28/02