Characteristics of Mathematics

1. Mathematics
Maths is the subject where we never know what we are talking about, nor whether
what we are saying is true.
--Bertrand Russel--

2. 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

3. 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?

4. 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

5. Solve This
1. 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”

6. 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?)

7. 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?

8. Prove that a=b
a° b°

9. • What are the rules that make the conclusion so
convincing?
• What are the right starting points (axioms)?

10. 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

11. Truth is Beauty: Beauty is
Truth
2. Deductive Reasoning
3. 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

12. 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

13. 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

14. 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)

15. 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?)