AREAS OF KNOWLEDGE:
MATHEMATICS
Collatz Conjecture* visualization using Houdini
*Start with any positive integer n. Obtain the next number by first checking if the number is even or odd. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. And so on…
ABSTRACT BEAUTY — COLD AND AUSTERE
Mathematics is our language of quantity and space—the science of pattern. Mathematics is a symphonic edifice of rigorous deduction, built on self-evident axioms that are both abstracted from nature, and imagined. Mathematics can be a creative, inchoate, human enterprise in its inception, but when formally set down—manifest as incontrovertible proof—it is austere, ideal, cold and pure.
The unforgiving, analytic nature of Mathematics—at crude first glance—seems to situate it as disjoint from the other Areas of Knowledge. The Natural Sciences, Human Sciences, History, and the Arts are less exact, more interpretive, and more obviously rooted in the messiness of the real world. Math (and logic) certainly merit special status, but in truth they are inextricable, often in very nuanced ways, from the rest of Knowledge.
In the class activities that follow, students will have the opportunity to transcend the familiar routine of “doing classroom math,” and will glean some meta-perspective on the nature of math itself. A glance at the menu below will evoke the wealth of Knowledge Questions that will be explored. As so often in TOK—delight and a sense of wonder have been sitting there all along—albeit hidden in plain sight.
Indian postage stamp depicting Indian mathematician Srinivasa Ramanujan (1887 - 1920)
MENU OF CLASS ACTIVITIES
Proof
Solve a quadratic
Sum of the angles in a triangle
The Monty Hall problem
Thinking about proof and intuition
Imagining geometry—a thought experiment!
Ideal gas law compared to Euler’s relation
Pure and applied mathematics
The path from metaphor to algorithm
Mathematical induction
Revisit Pascal's triangle
Build a house of cards
The special case of proof by mathematical induction
House of cards resolved
This Statement is False
The liar's paradox
The barber's paradox
Non-Euclidean geometry
Infinities
Beguiling with statistics (in progress)
Correlation vs. causation
Disingenuous distortions
Data dredging
Platonists and Formalists
Written assignment
Why is ethics like math and not like math?
A preposterous question
Prove it!
Inventing math thought experiments
Attempting a calculus of felicity
The golden mean is not the arithmetic mean
A self-evident, axiomatic foundation for all ethics?
“Look… three deer!”
”No, you’re wrong… it’s four deer!”
We have been abstracting from nature for a long time. Here is a 20,000 year old cave painting from Lascaux IV, Dordogne, France
KNOWLEDGE QUESTIONS
The new Theory of Knowledge Guide (2020) provides no less than 385 Knowledge Questions for student exploration. Here are my personal favorites from the Mathematics section.
SCOPE
How have technological innovations, such as developments in computing, affected the scope and nature of mathematics as an area of knowledge?
Is absolute certainty attainable in mathematics?
Should mathematics be defined as a language?
Does mathematics only yield knowledge about the real world when it is combined with other areas of knowledge?
PERSPECTIVE
What is the role of the mathematical community in determining the validity of a mathematical proof?
Is mathematical knowledge embedded in particular cultures or traditions?
METHODS AND TOOLS
What is meant by the term “proof” in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?
What does it mean to say that mathematics is an axiomatic system?
Do mathematical symbols have meaning in the same way that words have meaning?
ETHICS
If mathematical knowledge is highly valued, does this place special ethical responsibilities on mathematicians when they are making claims?
How are unethical practices, such as “data dredging”, used by statisticians to deliberately manipulate and mislead people?
CONNECTING TO THE CORE THEME
What steps can we take to help ourselves avoid being misled by statistics used in unclear or disingenuous ways in the media?
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.”
Image created by Dominic Walliman