MathematicsMaths is the subject where we never know what we are talking about,

nor whether what we are saying is true.--Bertrand Russel--

Scope & Applications, Language, & Methods

• Identify & Describe nature & methodology of mathematics; included in this process will be key language terms that are important to mathematics

• Discussion

• Socratic Dialogue Activity

Using the Knowledge Framework—by the end of these first 2-3 lessons, you

should be able to answer the questions below• Scope & Applications: what is math about and

what practical problems can be solved through applying this knowledge?

• Language & Concepts: what role does language play in the accumulation of knowledge in math? Key terms? Key concepts?

• Methodology: what methods or procedures are used in math and how do they generate knowledge? What assumptions underlie these methods?

Objectives Today • Identify & Describe the Characteristics of

Math

• Describe what mathematicians study & what problems they are trying to solve

• Understand the methodology of mathematics

• Define key terms & concepts related to math

Solve This1. Start with the four-letter English Word “SHIP”

& write down a list of four letter words given the following rules:

A. Each word in the list is a genuine English word as can be found in standard dictionaries

B. Each word is identical to the word above it in the list except in one letter

2. The aim of the game is to write the word “DOCK”

As You Work:• Pay attention to your thought processes• Consider the rules you used: what is the logic

employed? How does this relate to math?• Consider your emotions: How did you feel if you

solved it? If you didn’t?• Are there “better” solutions?• Why does every solution to the problem contain

at least one word that has 2 vowels? (can you prove this?)

Tennis Tournament

• Solve this Problem: There are 1,024 people in a knock-out tennis tournament. What is total number of games that must be played before a champion can de declared?

Prove that a=b

a° b°

• What are the rules that make the conclusion so convincing?

• What are the right starting points (axioms)?

Truth is Beauty: Beauty is Truth

1. Axioms: starting points, basic assumptions

• Elegant, independent, begin with the fewest number of axioms

• Euclid started with 5 simple, clear axioms, e.g. all right angles are equal to one another)

• Simple, Fruitful, Useful

Truth is Beauty: Beauty is Truth

2.Deductive Reasoning3.Theorems are derived from

deductive reasoning using the axioms

• Proofs are used to show that a theorem follows logically from the relevant axiom

• Conjectures are hypotheses that seem to work but have not been shown (proven) to be true

• Goldbach’s Conjecture: every even number is the sum of two primes

Axiomatic System• We have to start somewhere

• So we choose the “safest” axiom but have to accept consequences

EXAMPLE: THE CONTROVERSIAL “AXIOM OF CHOICE”

• Ernst Zemelo’s “axiom of choice” states that if you have a collection of non-empty sets, you can make a new set by choosing elements of the original sets or as wikipedia says it: “given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin:”

• Seems innocuous? Right? However one of the implications of this is the following:

• Banach-Tarski Paradox: you can take a mathematical sphere the size of a tennis ball, cut it up into pieces, and simply by re-arranging these pieces, without changing their size, make a sphere the size of the earth

Socratic Dialogue• Read your portion of the

dialogue • Highlight important concepts,

ideas• Discuss the questions /

concepts• Using the Google Slide, type

up your key points from the dialogue on the slide

• Present your findings to class & explain what you learned to the class from the dialogue

Characteristics of Mathematical Knowledge that students should identify from Socratic Dialogues

• Math exists in the mind of the mathematician

• Study of numbers & shapes

• Math provides certainty b/c the mathematician’s mind sets all parameters & definitions, logic rules

• Concepts can be applied to the real world

• Math as a map, guide to reality (a language?)

• Tools & concepts are invented to discover math reality

• Math applies rigorous logic; avoids self-contradiction

• Starts from “first principles” (axioms)

Characteristics of Math Knowledge & Methods• Maths is certain but not real (where do numbers exists?) (creation of the

mathematician)

• Maths is the study of study of numbers and forms and patterns

• Math is certain b/c is exists exclusively in the mind (imagination) of the mathematician (real-world observations are always limited)

• Mathematicians study the properties of numbers themselves (not quantities) (numbers do not equate with their symbols)

• Independent math truths exists (external to mind of mathematician)

• Tools and concepts are invented by mathematicians to reach the destination of truth

• Math relies on high standards of logical thinking: math cannot be self-contradicting

• Clearly and precisely defined terms and concepts

• Math is a map which allows us to navigate the real world (a guide? a language?)