MAT336: History of Mathematics

Study Guide: Quiz 6

Hellenic Mathematics before Euclid

Note: This is a study guide. The quiz will consist of three or four questions covering the material below. If you understand the ideas in the conceptual questions and can work through the reasoning in the others, you will be well prepared.

Part 1: Quiz-Style Questions

Greek Mathematics: Context and Origins

  1. What features became especially prominent in Greek mathematics compared with the Egyptian and Mesopotamian mathematics we studied?
  2. Greek intellectual life often involved public debate and persuasion. How might this environment have encouraged mathematicians to value rigorous proof?
  3. Why do historians say that Greek mathematics did not develop in isolation? Give one example of influence from earlier cultures.

Plato and Aristotle

  1. According to Aristotle, what is an axiom, and what role do axioms play in mathematical reasoning?
  2. Aristotle distinguishes between two types of quantity: magnitude and number. Define each one and give an example.
  3. Why did Plato view mathematics as a model for knowledge? Your answer should address the relationship between mathematical truth and sense experience.

The Pythagoreans

  1. According to Aristotle's report, what was the central Pythagorean claim about numbers and the universe? What does this claim mean?
  2. Give one example from Pythagorean work that supports the idea that numbers reveal patterns in nature.
  3. Why did the Pythagoreans consider the number 10 "perfect"? What does this example illustrate about how they interpreted numbers?
  4. Why does the discovery of the incommensurability of √2 challenge the Pythagorean belief that "all things are number"?
  5. Why can't the Pythagorean theorem be reliably attributed to Pythagoras? What later evidence helped associate the theorem with Pythagoras, and how certain is that attribution? Where is the first extant proof within an axiomatic system?
  6. Recall that among the figurate numbers conceived by the Pythagoreans, a triangular number can be represented as dots arranged in a triangle. Explain how this representation helps show that the nth triangular number equals n(n+1)/2.
    Hint: Arrange dots in a rectangle.

The Three Classical Problems

  1. Most Greek geometric constructions, including the three classical problems, were posed with only a straightedge and a compass. Why might these two tools have been considered fundamental to geometry?
  2. What does it mean for a number to be constructible? Explain it geometrically and algebraically.
  3. Explain what doubling the cube means and why it is impossible using straightedge and compass. Use constructible numbers in your explanation. You may use the fact that a certain number is not constructible, but you must say which number it is and why it is the relevant one.
  4. Explain what squaring the circle means and why it is impossible using straightedge and compass. Use constructible numbers in your explanation. You may use the fact that a certain number is not constructible, but you must say which number it is and why it is the relevant one.
  5. Explain what trisecting an angle means. A 90° angle can be trisected with straightedge and compass. Why doesn't this contradict the impossibility of the trisection problem?

Part 2: Reflection Questions

  1. The Pythagoreans mixed mathematics, philosophy, and mysticism. Do you think this mixture helped or hindered mathematical development?
  2. The discovery of incommensurable magnitudes forced mathematicians to rethink their assumptions. What does this episode suggest about how mathematics progresses?
  3. The three classical construction problems turned out to be impossible. How can studying impossible problems still lead to important mathematical discoveries?
  4. Plato did not prove new theorems, yet historians say he influenced mathematics. In what ways can someone influence mathematics without doing mathematics themselves?
  5. Does it matter who first discovered a mathematical idea? Consider questions of credit, collaboration, and how mathematical knowledge spreads.
  6. The Greeks asked "why" alongside "how." Does this change what mathematics is, or only how it is presented?
  7. In what sense did the Greek obsession with straightedge-and-compass constructions generate new mathematics rather than limit it?
  8. Greek mathematics is often treated as the origin of rigorous proof. What does the evidence of Mesopotamian influence suggest about this narrative?

Quiz Problem Rubric

Points Criteria
3 Correct answer with reasoning/work shown
2 Partially correct with some reasoning shown
1 Correct answer without reasoning/work OR significant attempt with some understanding
0 Incorrect or blank

Notes