Exploring the Areas of Knowledge Math and TOK Exploring the Areas of Knowledge

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

Reuben Hersh “What Kind of a Thing is a Number?” What is mathematics? It's neither physical nor mental, it's social. It's part of culture, 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 “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 “Contrary philosophical positions view this differently; mathematics is not waiting to be discovered by instead exists as a ‘product of human thought, independent of experience.’”

Math and TOK What do we want to accomplish by looking at Math in a TOK way?

TOK and Math What is Math? -The science of rigorous proof -Axioms: Postulates, Self evident truths (??)

Math and TOK Does the fact that Math is built upon self evident truths make this subject more or less certain? Deductive reasoning Process using rules to arrive at a specific conclusion drawn from general statements. Theorems statement proved on the basis of previously accepted or established statements The Pythagorean Theorem

Math and TOK 1. All human beings are normal. 2. Socrates is a human being. 3. Therefore, Socrates is mortal. 1 & 2 are premises. 3 is the conclusion. In mathematics axioms are like premises and theorems are like conclusions.

Practical Math and TOK In groups of two create a math problem, identify the axiom(s), be able to present the deductive reasoning on the board, list out the theorem(s), and identify a knowledge claim imbedded within the problem

Hint: Euclid’s 5 axioms It shall be possible to draw a straight line joining any two points. A finite straight line may be extended without limit in either direction. It shall be possible to draw a circle with a given center and through a given point. All right angles are equal to one another. There is just one straight line through a given point which is parallel to a given line.

TOK and Math: Proof What does it mean to prove something mathematically? Mathematical Proof a collection 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 use of definitions, axioms, properties and previous claims that are consistent.

Buchberger’s Model: Conjecture Knowledge Spiral Identify Problem (phase of experimentation) Develop algorithm or step-by-step procedure (phase of exactification) Conjecture (phase of application) Conjecture = a conclusion made from a reasonable number of individual cases which are nonetheless insufficient to form substantial proof Quasi-empirical = having a likeness to a scientific method which requires careful observation and a gathering of relevant factual support

Practical Exercise Proof vs. Conjecture *In groups of 2 create two problems one that shows proof is possible and another that gives an example of conjecture.

Practical Questions Do axioms, theorems, deductive and inductive reasoning, proofs, conjectures apply in real everyday life? Is it actually real or do we make it real only in our minds?

Math and TOK: Fibonacci Sequence Do we discover math or is it already there? 1,1,2,3,5,8,13,21,34,55 Examples in nature Male bee Chambered nautilus Number of Flower petals Spiral pattern in broccoli Parthenon in Athens

TOK and Math How should we approach understanding math? -empirically -analytically: true by definition -synthetically through reason alone

Klein’s Quartic Curve

Escher’s Tessalations

The Droste Effect a picture with a smaller version of itself within the picture. Mathematicians call this self-similar recursion.

TOK and Math How certain is math? How does it compare to other disciplines when it comes to providing knowledge as Plato defined (k=jtb)?

Math and TOK Is Math consistent? -Riemannian Geometry (playing with axioms) -Problem of consistency (you can’t make up the rules as you go) -Godel’s incompleteness theorem (it is impossible to prove that a formal mathematical system is free from contradictions)

Real World Math Project: Successfully approach some of the more abstract questions that naturally arise through looking at this subject in a TOK fashion. Answer the question(s) assigned to your group. Decide on mathematical concept(s) to teach to a child that will prove your position concerning your questions. (The concept has to be above the children’s natural abilities, and it MUST attempt to prove your position concerning your questions.) Organize gathered information into an essay that you will present to the class. TOK Rubric Evidence from experiment Summarize how your group responded to the knowledge claim(s) imbedded in the original question(s). Any Questions????

TOK and Math: Questions for project What does it mean to say that mathematics can be regarded as a formal game devoid of intrinsic meaning? If this is the case, how can there be such a wealth of applications in the real world? We can use mathematics successfully to model the real world processes. Is this b/c we create mathematics to mirror the world or b/c the world is intrinsically mathematical? What do mathematicians mean by mathematical proof, and how does it differ from good reasons in other areas of knowledge? Can math be characterized as a universal language? To what extent then might math be different in different cultures? How is math the product of human social interaction?

TOK and Math: Questions Continued Are all mathematical statements either true or false? Can a mathematical statement be true before it has been proven? Is it correct to say that math makes truth claims about non-existent objects? Explain It has been argued that we come to know the number 3 through examples. Does this support the existence of the number 3 and, by extension, numbers in general? If so, what of number such a 0,-1, i, and a trillion? If not, in what sense do numbers really exist? What counts as understanding in math? Is it sufficient to get the right answer to a problem and then say that one understands the concept and it relevance to the world. To what extent can math be beautiful? Can it be understood in a similar fashion like the arts? If math has an element of elegance does this jeopardize it level of certainty? Explain.

TOK and Math: Questions for panelists Is math built upon self evident truths? Is math the only discipline that has self evident truth? How do we choose the axioms underlying math? Is it an act of faith? If math is discovered not invented do we ever really have new mathematical knowledge? How does mathematical proof differ from proof in other areas of knowledge? Can mathematics be characterized as a universal language? Since languages are determined in community what role does community play in determining mathematical certainty? Might this mean that math is different from culture to culture? Why do you think different cultures value math in different ways? In what ways does our American culture hinder our understanding of math? What counts as understanding in math? Is it sufficient to get the right answer to a problem and then say that one understands the concept and it relevance to the world. What does it mean for math do be beautiful? Are numbers real tangible things that exist outside our minds? If not them how can math be applicable to the real word? Are there aspects of math that one can choose whether or not to believe? Does intuition play a role in Math? Create some of your own questions!!!