History

1
Introduction

• History
• What is a Proof?
• The roles of Proof?

2
Where did the idea of proof come from?

• The early Egyptians, Babylonians and Chinese used
  observational evidence to justify mathematical
  statements.
• The classical Greek mathematicians were the first
  to try and make absolute statements.

3
The roles of Proof

• Verification- proving absolute statements

4
Axiomatic Systems

• Axioms- fundamental statements
• Rules of inference- laws of logical reasoning
• Theorems- all other propositions

5
Euclidean Geometry

• 23 Definitions
• A point is that which has no part.
• A line is a length without a breadth.
• The extremities of a line are points.
• A circle is a plane figure contained by one line
  such that all the straight lines falling upon it
  from one point among those lying within the
  figure are equal to one another.

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Euclidean Geometry

• Five Common Notions
• Things that are equal to the same thing are also
  equal to one another.
• If equals are added to equals, the wholes are
  equal.
• If equals are subtracted from equals, the
  remainders are equal.
• Things that coincide with one another are equal
  to one another.
• The whole is greater than the part.

• Five postulates
• A straight line segment can be drawn joining any
  two points.
• Any straight line segment can be extended
  indefinitely in a straight line.
• Given any straight line segment a circle can be
  drawn having the segment as its radius and one
  endpoint as its centre.
• All right angles are congruent.
• Through a point not on a given line there passes
  exactly one line parallel to the given line.

7
Rules of inference

• Modus Ponens
• P implies Q
• 2. P is true
• 3. Therefore Q is true
• Modus Tollens
• 1. P implies Q
• 2. Q not true
• Therefore P not true

• Elimination of disjunction
• T is true or S is true
• P implies R
• S implies R
• Therefore R is true

8
What is a Proof?

• Ways of thinking about a proof
• Formalist view A finite number of logical steps
  from what is known to a conclusion, using
  accepted rules of inference.
• Platonist view An abstract object which is
  discovered by mathematicians.
• Intuitionist A mental construction- a sequence
  of acts that is or could be carried out by a
  mathematician.
• Wittgenstein A type of linguistic practice or
  usage.
• Cognitive Provides evidence or justification for
  a mathematical proposition. Where to have
  evidence we have to have some kind of
  understanding.

9
Proof vs. Intuition

• For 2000 years Euclids Elements was seen as a
  model for finding truth and structuring results.
• However, a lot of work was done without using
  this kind of argument.
• Patterns and analogy were widely used to validate
  results.
• As mathematics progressed, mathematicians found
  weaknesses in more intuitive arguments

10
The roles of Proof

• Verification- proving absolute statements.
• Explanation- Answering why something is true.
• Conviction- Convincing someone something is true.
• Systematization- Fitting results in to the
  mathematical body of knowledge.
• Discovery- Finding new results.

11
Revisiting Euclids Elements

• The fifth postulate was thought to not be
  independent of the other postulates.
• The fifth postulate states Through a point not
  on a given line there passes exactly one line
  parallel to the given line.
• In fact it is independent.

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Newer Insights into the Axiomatic System

• Mathematician tried using other postulates.
• Led to greater understanding of axiomatic
  systems. Dont necessarily need axioms to be
  intuitive.
• Euclids Elements used unstated assumptions.

13
Newer Insights into the Axiomatic System

• Hilberts Foundations of Geometry used six
  primitive terms and 21 axioms.
• Started a trend towards axiomatization.
• People tried to axiomatize the number system.

14
Gödels incompleteness theorem

• Definition- A set of axioms is said to be
  complete if every properly constructed statement
  can be determined to be true or false on the
  basis of the axioms and the rules of inference.
• But Gödel proved that any set of axioms strong
  enough to describe the number system would have
  some undecidable statements.
• So truth is not the same as proof.

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