AREAS OF KNOWLEDGE: MATHEMATICS
CLASS ACTIVITY I: THE LIAR'S PARADOX
Certain logical conundrums, like this version of the liar's paradox, sabotage conventional ideas of absolute certainty, and perfect consistency, in the edifice of pure mathematics.
After the initial discomfort generated by this mind-bending paradox fades, and students see for themselves that this is not a puzzle of logic easily resolved, some lively interchange will ensue. The following guiding questions may or may not add value.
- Well, is the statement true or false?
- This is a self-referential statement? What is self-referentiality and why is it problematic here?
- We commonly associate the analytical realms of pure mathematics and logic with certainty. To what extent does the existence of occasional paradoxical results undermine the assumption of certainty?
THE BARBER'S PARADOX
Continue the class conversation with: If Barbers are those who shave men if and only if they do not shave themselves, do barbers shave themselves? Ask students what they think? Of course, a contradiction is soon reached; but just saying the barber in question must be a woman who does not shave, or the that the sentence just doesn't make sense, are shrewd responses, but they just jump out of dealing with the paradox.
Inform students that at the beginning of the 20th Century, famous mathematicians (including Russell, Whitehead, Frege and Hilbert) were aiming for a rock solid foundation for the whole of mathematics based on logical certainty. Paradoxes undermined their towering ambition, and in 1930 such noble but, ultimately, hubristic attempts were permanently laid to rest by Gödel's famous Incompleteness Theorem.
The following quotes are very revealing and students should be told to keep them in mind when tackling the Platonists vs. Formalists written assignment which follows this unit.
CLASS ACTIVITY II: NON-EUCLIDEAN GEOMETRY
Remind students that all of Euclidean geometry is built on five self-evident axioms and postulates. Over the centuries it became apparent that proving Euclid’s fifth postulate (the postulate of parallels) was problematic. Recognizing that the postulate of parallels was not as self-evident as the other four postulates, led to the discovery of non-Euclidean geometry, in which the parallel postulate is assumed to be false.
HANDS ON TASK WITH A REAL GLOBE
For this "hands-on" activity students should work in trios. Well in advance of the class arrange to have on hand sets of school globes, rulers, string, tape and protractors.
1. Lead students through the following thought experiment:
First ensure that you know the difference between lines of latitude and lines of longitude. The most famous line of latitude is the equator. All the other lines of latitude are parallel to the equator and get smaller as they approach the poles. The lines of longitude are all "great circles" like the equator and pass through the North and South poles. The lines of longitude are conventionally 15 degrees apart. Why?
Imagine a line of longitude intersecting with the equator. What is the size of the angle formed? Now mentally shift around the equator to where the next line of longitude intersects with the equator. Again what is the size of the angle formed between them? Next, picture in your mind an isosceles triangle with a base that is the portion of the equator between two lines of longitude fifteen degrees apart. The sides of the triangle are precisely the two lines of longitude which converge the further away they get from the base, eventually intersecting at the North pole forming an angle of 15 degrees. What is the sum of the angles of the triangle?
2. On a globe mark 3 random crosses on three different continents, (to allow significant distance between them). Use a stretched string and tape to define the triangle made by joining up the points. Use a protractor to measure carefully sum of the angles of the triangle.
3. First choose your favorite European city. Next identify two different flight paths between San Francisco and your chosen destination based on the Google map above
A. Going directly West to East.
B. Going East to West but veering North along the way, so you go over arctic Canada Greenland and Iceland.
Use stretch string and tape to mark the two journeys. Finally using a ruler measure the two lengths of string that you used.
CLASS ACTIVITY III: INFINITIES
This activity consists of nothing more than observing a humorous video about the Hilbert Hotel which seems to have infinite possibilities. Invite students to respond to the Toy Story graphic. Like logical paradoxes and non-Euclidean geometries, the counterintuitive consequences of infinities will inform the Platonists vs. Formalists written assignment which will be the culmination of our brief meta-view of mathematics as a Way of Knowing.