Mathematics and TOKExploring the Areas of Knowlege

Keith J. Devlin

“Mathematics is the abstract key which turns the lock of the physical universe.”

Reuben Hersch

“What kind of thing is a number?”

“What is mathematics? It’s neither physical nor mental, it’s social. It’s part of history. It’s like law, like religion, like money, like all those other things which are very real, but only as part of collective human consciousness. That’s what math is.”

Bertrand Russell

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

“The mark of a civilized man is the ability to look at a column of numbers and weep.”

Descartes

“To speak freely, I am convinced that (mathematics) is a more powerful instrument of knowledge than any other.”

“Math is an island of certainty in an ocean of doubt.”

Albert Einstein

• “Mathematics is not waiting to be discovered but instead exists as a ‘product of human thought, independent of experience.’”

Video Clip

• Taken from “The Ascent of Man” • A BBC Documentary from 1973• J. Bronowski (Mathematician and

Narrator) looks at the interlocking of numbers and nature in descriptions of musical harmony and Pythagoras’ Theorem

TOK and Mathematics

• What is Math?- The science of rigorous proof by1. Deductive Reasoning2. Exhaustive Proof3. Proof by Contradiction4. Proof by Mathematical Induction

TOK and Mathematics

What are the tools for proofs?

• Axioms – like premises (can be algebraic such as 5 + 0 = 5 or geometric as in all right angles are equal to one another). Assumed to be true.

• Theorems– like conclusions. Can be used to further other proofs.

TOK and Math: Proof

• What does it mean to prove something mathematically?

• Mathematical Proof– A collections of logically valid steps or

demonstrations that form an argument which serves as a justification of a mathematical claim. Steps within the argument normally make the use of definitions, axioms, properties and previous claims that are consistent (theorems).

TOK and Math

• How do proofs come about? – We identify problems (like experimentation,

develop a step by step procedure, and produce a conjecture)

• Conjecture– A conclusion made from a reasonable number of

individual cases which are nonetheless insufficient to form substantial proof

– Leads to proofs which involves making sure every possibility will work (unlike science)

Is Mathematics Invented or Discovered?

Two types of opposing views:

1. Formalist: Mathematics is an abstract activity governed by rules (like a logical game such as chess)

2. Realist: Mathematics is fundamentally describing the way the universe actually works

Activity

1. Work in groups at your table. 2. Select either the formalist (invented) or the

realist (discovered) point of view.3. Come up with 2-3 arguments why you think

mathematics is supported by this point of view

4. Rely on your observations in science/nature, games, math symbols and language, math concepts (imaginary numbers, quadratics, equations, geometry,etc.).

TOK Math Questions to Consider

1.

Is infinity a number? Is it ever correct to write x = infinity? Do some infinite sets have more elements than others (i.e. If there are an infinite amount of distances between 1cm and 2cm on a ruler, how many are there between 1 cm and 10 cm?)

More TOK Questions to Consider

2.

How important is it to be exact? Discuss this with regard to different scenarios in mathematics, science, medicine, architecture, etc.

More TOK Questions to Consider

3.

What is the difference between the empty set, the number zero, and nothing at all? Can there be such a thing as a complete vacuum? Do you think we can consider the universe to be the ultimate universal set U containing everything?

More TOK Questions to Consider

4.

We must be very careful with brackets in Mathematics. Are brackets in Mathematics more important than brackets used in other areas of knowledge? (i.e. History, Literature, Science, etc.)

More TOK Questions to Consider

5.

It is very important for the timing to be as accurate as possible in the Olympic Games. However, the same degree of accuracy is not necessary when timing the cooking of a pot of rice. So, how important is it to have “exact” values? What do you understand by an “exact value”? Is it more important to be exact in Math & Science than in other Areas of Knowledge?

More TOK Questions to Consider

6.

When conducting a survey how can you ever be sure that the replies given are true? In a survey the people questioned could lie about the information they give. How do both the type of questionnaire and to whom it is distributed affect the results of your survey?

More TOK Questions to Consider

7.

How do we know when two things are related? Can two sets of data have a very strong correlation and yet not be related? Try to think of your own examples. Can you find any examples in advertising?

More TOK Questions to Consider

8.

If mathematics is about following rules and we know that 10/10 = 1 and 4/4 = 1, then why does 0/0 not = 1 ?