Mathematics And TOK

MATHEMATICS & ToK

INTRODUCTION

Mathematics is a subject in school that everyone has to take.

• Why is that, and why is so much time in school allocated to the study of mathematics?

• In what ways is mathematics helpful to you? What have been the advantages, for you, of learning math?

• Do you like math? Is it easy for you? Why is it difficult for so many students?

• What would you consider to be the characteristics of those people you know that are good at math?

And when we compare it to other subjects we study in school, still more questions are raised...

• What does it mean to get the "wrong" answer in math? Can you get "wrong answers" in other subjects?

• Mathematics has been described as both the "queen" and the "servant" of Science. What could "queen" and "servant" mean in this context?

In general, Mathematics is an Area of Knowledge (AoK) that is often viewed as a very rigid one - i.e. there is little that's "uncertain" or "flexible". And when we look at it in light of our ToK issues, there are a number of questions that come to mind. Examples might be:

• Is mathematics a unique aesthetic experience? In other words, does it possess a sense of beauty, like art? Or is it more closely related to language than art?

OR

• What do we mean when we call mathematics a 'language'? Does mathematics function in the same way as our daily written and spoken language?

Many famous thinkers over the ages have contemplated these and many other questions relating to mathematics. A small sampling follows. Be aware that in this webpage, there is material for you to read as well as questions for you to ponder and answer. The text for all questions will be in pink so you won't be able to miss them!

You'll also see underlined hyperlinks. In some cases they'll be the source for the information you need. In other cases, they're there simply for you to consult if you'd like to "read up" on some of these important mathematicians.

WAS MATHETMATICS DISCOVERED OR CREATED?

Pythagoras (571-496 B.C.), a mathematician (perhaps best known for his "Pythagorean theorem") and philosopher born in Samos, Greece, believed that the answer to the big question "What is reality made of?" lay in mathematics. Pythagoras is famously quoted as having said "All is number." He even went as far as to declare that justice was the number 4 as it was a square number.

1. What do you think Pythagoras meant by "All is number."?

2. Do you agree with him? To what extent is it possible to explain the concept of reality through mathematics?

While most of us would find it difficult to entirely agree with Pythagoras, there is little doubt that mathematics has played a large role in our understanding of the world around us, and those we have yet to discover and explore.

When trying to define mathematics, several questions are raised. The big question concerning math, as far as ToK is concerned, is whether mathematics is a natural phenomenon or merely a game of logic invented by the human mind.

3. What is your "gut reaction" to this question? Was mathematics discovered or invented?

While Pythagoras believed that mathematics held all the answers,, PLATO,another Greek philosopher and mathematician whom we've discussed before (remember K=JTB?), held the belief that numbers held some kind of mystical existence, separate to the rest of the world.

Read the brief biography of Plato paying particular attention to the section on mathematics. In it you're reminded that Plato founded "The Academy" (an institution devoted to research and instruction in philosophy and the sciences) over which he presided from 387 BC until his death in 347 BC. Over the door of the Academy was written:

"Let no one unversed in geometry enter here."

4. What does this quotation reveal to us about Plato's view of mathematics? Summarize his belief about mathematics. 5. Why were so many of Plato's friends and students significant contributors to the field of mathematics despite the fact that Plato, himself, made no important mathematical discoveries?

In contrast to Plato, the German philosopher Immanuel Kant (1724-1804) viewed mathematics as an example of "synthetic a priori", in other words, mathematics will always be true for humans as that is the way our brains work.

6. Kant devoted serious time pondering the very ToK question, "What can we know?" What was his answer, and why did he believe this? (Click on the link above to find out).

However, the English Victorian philosopher, John Stuart Mill (1806-1873) claimed that mathematics was nothing more than a highly probably truth based on human experience.

Then along came the German philosopher, mathematician and logician, Gottlob Frege (1848-1925) who demystified mathematics by showing that numbers are not objects, that mathematical "facts", such as 2 + 2 = 4, have nothing to do with either the construction of our minds or our observations of the world, but are rather "logical truths".

7. Frege's work is intimately linked to one of our IB Ways of Knowing (WoKs). Which one, and how is Frege often viewed?

All was going well with the theories of Frege, until the Austro-Hungarian Kurt Goedel (1906-1876) came along and declared that "a consistency proof for [any] system...can be carried out only by means of modes of inference that are not formalized in the system...itself." In other words, in any system containing arithmetic, there are true statements which cannot be proved within the system; this is known as Godel's Incompleteness Theorem.

8. Most of Godel's mathematical study revolved around the concept of axiomatic systems and their consistency. What is an axiom? Provide a definition and an example.

Much of what you've read above suggests a problem with Mathematics. At first glance a very logical AoK, mathematics, at times, appears to escape from logic! Perhaps it isn't as CERTAIN or INFLEXIBLE as we thought...

DEFINITIONS

Above you were asked to define an axiom. There are several more "must know" terms in math - complete the table below with definitions of your own or, if required, appropriate dictionary definitions.

TERM	DEFINITION
An axiom
A conjecture
A theorem
A proof
A corollary to a theorem

AND NOW FOR SOME FUN...

Here are some mathematical puzzles for you to solve. Some of them you may have seen (and solved) before - if so, no worries - just move on to something that's looks new and interesting!

A) What's wrong with this proof?

Given:	A = B
Multiply both sides by A:	A2 = AB
Subtract B2 from both sides:	A2 - B2 = AB - B2
Factor both sides:	(A + B) (A - B) = B(A - B)
Divide both sides by (A - B):	A + B = B
Since A = B,	B + B = B
Add the B's:	2B = B
Divide by B"	2 = 1

B) Appearing Area
Consider the figures below. Both triangular figures have been built up from the same four parts. The parts with the same color have exactly the same shape and size! They are only moved around, which resulted in an appearing area in the lower figure, marked with a question mark ('?').

The Question: Where does the '?' hole come from?

The next puzzle is a good one to remind us that mathematics and logic have strong ties with one another. It also involves some lateral thinking - think back to our unit on logic!

C) Abracadabra with Apples
In Miss Miranda's class are eleven children. Miss Miranda has a bowl with eleven apples. Miss Miranda wants to divide the eleven apples among the children of her class, in such a way that each child in the end has an apple, but one apple still remains in the bowl.

The Question: Can you help Miss Miranda?

E) Seven Rows, Sixteen Numbers
In the figure below, you can fill in each of the sixteen numbers 1 up to 16, in such a way that the sum of the numbers in each of the seven rows is 29.

The Question: How should this be done?

F) Gas, Water & Electricity
There are three houses (A, B, and C) and three utilities (gas (G), water (W), and electricity (E)). Each house must get a direct, uninterrupted connection to each utility, but the various connections should not cross each other.

The Question: How must the connections be made?

G) Elegant Equation (this is the one of the tougher problems...)
There is a whole number n for which the following holds: if you put a 4 at the end of n, and multiply the number you get in that way by 4, the result is equal to the number you get if you put a 4 in front of n. In other words, we are looking for the number you can put on the dots in the following equation:

4... = 4 × ...4

The Question: Which number must be put on the dots to get a correct equation?

H) Unusual Paragraph
This is a most unusual paragraph. How quickly can you find out what is so unusual about it? It looks so ordinary that you would think that nothing is wrong with it at all, and, in fact, nothing is. But it is unusual. Why? If you study it and think about it, you may find out, but I am not going to assist you in any way. You must do it without any hints or coaching. No doubt, if you work at it for a bit, it will dawn on you. Who knows? Go to work and try your skill. Good luck!

The Question: What is unusual about the above paragraph? (OK, so this one really doesn't involve any math - give it a shot anyway...)

PROOFS in MATHEMATICS

To understand fully the nature of "rigorous proof" in mathematics, you will view the film "The Proof" in which Andrew Wiles' describes the seven-year-long process that led to the proof of Fermat's Last Theorem.

FINAL NOTES

1. If you need them to be provided, you can get the solutions to the puzzles next day in class.

2. Be sure that you have answered all questions and completed all required (pink) sections of this webpage. Follow-up discussion will take place next day in class!

3. Don't miss the class in which "The Proof" is shown - an assignment sheet will be distributed at that time.

MATHEMATICS & ToK

INTRODUCTION

Mathematics is a subject in school that everyone has to take.

• Why is that, and why is so much time in school allocated to the study of mathematics?

• In what ways is mathematics helpful to you? What have been the advantages, for you, of learning math?

• Do you like math? Is it easy for you? Why is it difficult for so many students?

• What would you consider to be the characteristics of those people you know that are good at math?

And when we compare it to other subjects we study in school, still more questions are raised...

• What does it mean to get the "wrong" answer in math? Can you get "wrong answers" in other subjects?

• Mathematics has been described as both the "queen" and the "servant" of Science. What could "queen" and "servant" mean in this context?

In general, Mathematics is an Area of Knowledge (AoK) that is often viewed as a very rigid one - i.e. there is little that's "uncertain" or "flexible". And when we look at it in light of our ToK issues, there are a number of questions that come to mind. Examples might be:

• Is mathematics a unique aesthetic experience? In other words, does it possess a sense of beauty, like art? Or is it more closely related to language than art?

OR

• What do we mean when we call mathematics a 'language'? Does mathematics function in the same way as our daily written and spoken language?

Many famous thinkers over the ages have contemplated these and many other questions relating to mathematics. A small sampling follows. Be aware that in this webpage, there is material for you to read as well as questions for you to ponder and answer. The text for all questions will be in pink so you won't be able to miss them!

You'll also see underlined hyperlinks. In some cases they'll be the source for the information you need. In other cases, they're there simply for you to consult if you'd like to "read up" on some of these important mathematicians.

WAS MATHETMATICS DISCOVERED OR CREATED?

Pythagoras (571-496 B.C.), a mathematician (perhaps best known for his "Pythagorean theorem") and philosopher born in Samos, Greece, believed that the answer to the big question "What is reality made of?" lay in mathematics. Pythagoras is famously quoted as having said "All is number." He even went as far as to declare that justice was the number 4 as it was a square number.

1. What do you think Pythagoras meant by "All is number."?

2. Do you agree with him? To what extent is it possible to explain the concept of reality through mathematics?

While most of us would find it difficult to entirely agree with Pythagoras, there is little doubt that mathematics has played a large role in our understanding of the world around us, and those we have yet to discover and explore.

When trying to define mathematics, several questions are raised. The big question concerning math, as far as ToK is concerned, is whether mathematics is a natural phenomenon or merely a game of logic invented by the human mind.

3. What is your "gut reaction" to this question? Was mathematics discovered or invented?

While Pythagoras believed that mathematics held all the answers,, PLATO,another Greek philosopher and mathematician whom we've discussed before (remember K=JTB?), held the belief that numbers held some kind of mystical existence, separate to the rest of the world.

Read the brief biography of Plato paying particular attention to the section on mathematics. In it you're reminded that Plato founded "The Academy" (an institution devoted to research and instruction in philosophy and the sciences) over which he presided from 387 BC until his death in 347 BC. Over the door of the Academy was written:

"Let no one unversed in geometry enter here."

4. What does this quotation reveal to us about Plato's view of mathematics? Summarize his belief about mathematics. 5. Why were so many of Plato's friends and students significant contributors to the field of mathematics despite the fact that Plato, himself, made no important mathematical discoveries?

In contrast to Plato, the German philosopher Immanuel Kant (1724-1804) viewed mathematics as an example of "synthetic a priori", in other words, mathematics will always be true for humans as that is the way our brains work.

6. Kant devoted serious time pondering the very ToK question, "What can we know?" What was his answer, and why did he believe this? (Click on the link above to find out).

However, the English Victorian philosopher, John Stuart Mill (1806-1873) claimed that mathematics was nothing more than a highly probably truth based on human experience.

Then along came the German philosopher, mathematician and logician, Gottlob Frege (1848-1925) who demystified mathematics by showing that numbers are not objects, that mathematical "facts", such as 2 + 2 = 4, have nothing to do with either the construction of our minds or our observations of the world, but are rather "logical truths".

7. Frege's work is intimately linked to one of our IB Ways of Knowing (WoKs). Which one, and how is Frege often viewed?

All was going well with the theories of Frege, until the Austro-Hungarian Kurt Goedel (1906-1876) came along and declared that "a consistency proof for [any] system...can be carried out only by means of modes of inference that are not formalized in the system...itself." In other words, in any system containing arithmetic, there are true statements which cannot be proved within the system; this is known as Godel's Incompleteness Theorem.

8. Most of Godel's mathematical study revolved around the concept of axiomatic systems and their consistency. What is an axiom? Provide a definition and an example.

Much of what you've read above suggests a problem with Mathematics. At first glance a very logical AoK, mathematics, at times, appears to escape from logic! Perhaps it isn't as CERTAIN or INFLEXIBLE as we thought...

DEFINITIONS

Above you were asked to define an axiom. There are several more "must know" terms in math - complete the table below with definitions of your own or, if required, appropriate dictionary definitions.

TERM	DEFINITION
An axiom
A conjecture
A theorem
A proof
A corollary to a theorem

AND NOW FOR SOME FUN...

Here are some mathematical puzzles for you to solve. Some of them you may have seen (and solved) before - if so, no worries - just move on to something that's looks new and interesting!

A) What's wrong with this proof?

Given:	A = B
Multiply both sides by A:	A2 = AB
Subtract B2 from both sides:	A2 - B2 = AB - B2
Factor both sides:	(A + B) (A - B) = B(A - B)
Divide both sides by (A - B):	A + B = B
Since A = B,	B + B = B
Add the B's:	2B = B
Divide by B"	2 = 1

B) Appearing Area
Consider the figures below. Both triangular figures have been built up from the same four parts. The parts with the same color have exactly the same shape and size! They are only moved around, which resulted in an appearing area in the lower figure, marked with a question mark ('?').

The Question: Where does the '?' hole come from?

The next puzzle is a good one to remind us that mathematics and logic have strong ties with one another. It also involves some lateral thinking - think back to our unit on logic!

C) Abracadabra with Apples
In Miss Miranda's class are eleven children. Miss Miranda has a bowl with eleven apples. Miss Miranda wants to divide the eleven apples among the children of her class, in such a way that each child in the end has an apple, but one apple still remains in the bowl.

The Question: Can you help Miss Miranda?

E) Seven Rows, Sixteen Numbers
In the figure below, you can fill in each of the sixteen numbers 1 up to 16, in such a way that the sum of the numbers in each of the seven rows is 29.

The Question: How should this be done?

F) Gas, Water & Electricity
There are three houses (A, B, and C) and three utilities (gas (G), water (W), and electricity (E)). Each house must get a direct, uninterrupted connection to each utility, but the various connections should not cross each other.

The Question: How must the connections be made?

G) Elegant Equation (this is the one of the tougher problems...)
There is a whole number n for which the following holds: if you put a 4 at the end of n, and multiply the number you get in that way by 4, the result is equal to the number you get if you put a 4 in front of n. In other words, we are looking for the number you can put on the dots in the following equation:

4... = 4 × ...4

The Question: Which number must be put on the dots to get a correct equation?

H) Unusual Paragraph
This is a most unusual paragraph. How quickly can you find out what is so unusual about it? It looks so ordinary that you would think that nothing is wrong with it at all, and, in fact, nothing is. But it is unusual. Why? If you study it and think about it, you may find out, but I am not going to assist you in any way. You must do it without any hints or coaching. No doubt, if you work at it for a bit, it will dawn on you. Who knows? Go to work and try your skill. Good luck!

The Question: What is unusual about the above paragraph? (OK, so this one really doesn't involve any math - give it a shot anyway...)

PROOFS in MATHEMATICS

To understand fully the nature of "rigorous proof" in mathematics, you will view the film "The Proof" in which Andrew Wiles' describes the seven-year-long process that led to the proof of Fermat's Last Theorem.

FINAL NOTES

1. If you need them to be provided, you can get the solutions to the puzzles next day in class.

2. Be sure that you have answered all questions and completed all required (pink) sections of this webpage. Follow-up discussion will take place next day in class!

3. Don't miss the class in which "The Proof" is shown - an assignment sheet will be distributed at that time.