7. mathematics as an area of knowledge

Mathematics Method

AIO: Climate ChangePP: Albert Einstein or Kurt Godel

Socratic Seminar: History (Thursday)

• Readings:1. P. 133-1372. P. 138-1433. P. 144-1474. P. 148-1505. P. 151-1526. A brief history of

Mathematics Podcast

http://www.bbc.co.uk/podcasts/series/maths

Activity: Mind Reading?• I will attempt to read

5 students’ minds with only the most basic of information.

• PoK question: How do we use math to anticipate and predict future “knowing”? Is math superior in this regard to the other areas of knowledge?

1089

Where we are Going?• Definitions and

Philosophy• Axioms and

Theorem• Truth vs.

Certainty• Math and

Culture• Art Connection

Let me see: four times five is twelve, and four times six is thirteen, and four times seven is–oh dear! I shall never get to twenty at that rate!

Mathematics and Prediction• Nate Silver used

computer simulations to accurately predict every State’s outcome during the recent presidential election.

• Was ridiculed by pundits, who believed it was “impossible” to know who was going to win such a close election.

What is Mathematics?1. Mathematics is founded on a

set of more or less universally accepted definitions and basic assumptions.

2. It proceeds from a system of axioms using deductive reasoning to prove theorems or mathematical truths.

3. These theorems have a degree of “certainty” that is unmatched by other areas of knowing

4. Question: Should Mathematics be an area, or a way of knowing? How is Math different from “Reason”?

Math as “Ding an Sich”• Kant posited that Math could

provide knowledge that is more than purely logical (a priori synthetic knowledge).

• How can math give us knowledge of the “thing in itself” beyond empirical knowing?

• Presupposes that Space and Time our “forms for our intuition” not a “Ding an Sich”

If two sides of a triangle are congruent, the angles opposite them are congruent.

How does this theorem give us “synthetic knowledge” of reality?How is it different than formal logic?

Pure Mathematics Crea

tive A

pplic

ation

Reality

a priori synthetic knowledge

Math and Academic Rigor1. A proof of a formula A is a finite

list of formula's in which A is the last formula, and all formula's are either axioms or follow from an formula or formula's earlier mentioned in the list by an rule of inference.

2. Direct proof, induction, transposition (logic), contradiction, construction, exhaustion, probablistic, etc.

3. Consider the visual proof of the Pythagorean Theorem, how is this a proof? What does this tell us about reality?

Truth or Certainty?1. List 5 “true”

mathematical statements/formuli

2. Are they “certain”? How can we know?

3. In what sense could we challenge the certainty of each formula?

1=0 “Proof”?x = y. Then x2 = xy. Subtract the same thing from both sides: x2 - y2 = xy - y2. Dividing by (x-y), obtain x + y = y. Since x = y, we see that 2 y = y. Thus 2 = 1, since we started with y nonzero. Subtracting 1 from both sides, 1 = 0.

Activity: Monty Hall Problem• Would you like to play a

game?– In one of the three cups,

there is 5 dollar bill. – In two of the three cups,

there is a “goat”1. Select a cup2. After revealing an

empty cup, you may either change your original choice, or keep it.

3. Did you win?

How might we “prove” this?• Let’s run a quick

experiment of “exhaustion” of this probability theorem.

• With a partner, complete two sets of 10 guesses.– One partner ALWAYS

changes cups– One partner NEVER

changes cups.• Who wins more?

http://blogs.discovermagazine.com/notrocketscience/2010/04/02/pigeons-outperform-humans-at-the-monty-hall-dilemma/#.UWCwXpPCZ8E

Activity: Color Theorem• Draw 5 shapes,

overlapping so as to create various fields of space.

• Begin numbering so as to do two things:– Have the least amount of

numbers– Never have one number

share a border with another.

• What is the least amount of numbers needed?

Math across Cultures• Language

– Integers and numerals based in India and Islamic civilization

– Consider “quipu” which records numbers on knotted strings.

– Consider imaginary numbers, e, or pi as a symbol corresponding to reality or logic.

• What is the purpose of our understanding of numbers?

Math across Cultures• Verification

– How may the verification, or thought process, change between cultures?

– What may the result of this diversity of thinking result in regarding theorems?

Math across Cultures• Education

– Skills-Based or Inquiry based?

– Differing methods of teaching skills vs. Inquiry

– Political dimension, competition and economy

– Value of the area of knowledge in relation to others.

Math and the Arts• Math interacts with the

philosophy of beauty• Some would argue that

mathematical principles are derived from a pursuit of music, art, etc.

• Consider the Fibonacci sequence, how does it operate in the world? Why is it “harmonious”?

Intuition, Emotion, and Faith• Many formulas, theorems

and theories concerning the scope and nature of Math seem partially based on intuited “knowing” (impartial)

• How does the intuition of order and consistency of axioms relate to presuppositions about God, faith, Platonic “realms of ideas” etc.?

Reading Discussion• Informal Socratic

Seminar (20 minutes)– Summarize your notes into

three main points (don’t just repeat a fact, make a general observation).

– Present an answer to the following question:

• In what ways is mathematics important to society?

– Write one question to present to the class

3 Ideas for the Week• Statistical data is the

interaction between – Science (what we look at)– Math (how we look)– Epistemology (why we

look)• Math is a counterweight

to expectation and intuition

• Our response to mathematical choices reveal underlying risk behavior.

Intro to Statistics: Climate Change

Figure 1: Berkeley Earth Surface Temperature (BEST) land-only surface temperature data (green) with linear trends applied to the timeframes 1973 to 1980, 1980 to 1988, 1988 to 1995, 1995 to 2001, 1998 to 2005, 2002 to 2010 (blue)

Figure 2: Berkeley Earth Surface Temperature (BEST) land-only surface temperature data (green) with linear treneds applied from 1973 to 2010 (red).

Figure 3: Berkeley Earth Surface Temperature (BEST) land-only surface temperature data (green) with linear trends applied to the timeframes 1973 to 1980, 1980 to 1988, 1988 to 1995, 1995 to 2001, 1998 to 2005, 2002 to 2010 (blue), and 1973 to 2010 (red).

What makes a “good” statistical reading?

• How does one collect data so as to:– Maintain ethical integrity?– Reflect the “real world”?– Answer your goals regarding science?– Avoid mistakes of history?

• How does one interpret data such that:

– Bias is reduced?– Understanding is progressing?– Dialogue is maintained?

• How does this use of data relate to the following:

– Ethics– History– Emotion– Reason– Language

Activity: Monty Hall Problem• Would you like to play a

game?– In one of the three cups,

there is 5 dollar bill. – In two of the three cups,

there is a “goat”1. Select a cup2. After revealing an

empty cup, you may either change your original choice, or keep it.

3. Did you win?

How might we “prove” this?• Let’s run a quick

experiment of “exhaustion” of this probability theorem.

• With a partner, complete two sets of 10 guesses.– One partner ALWAYS

changes cups– One partner NEVER

changes cups.• Who wins more?

http://blogs.discovermagazine.com/notrocketscience/2010/04/02/pigeons-outperform-humans-at-the-monty-hall-dilemma/#.UWCwXpPCZ8E

What is the Math?

On Wednesday• Discussion question:

– Come to class with a statistics or probability problem related to your Extended Essay topic

– Discuss how the competing views of the data affect your position or understanding.

– Write 3 questions about your topic as it might relate to math.