Short Answer:


  1. What is the Golden Section? Why was it important to the Pythagoreans?


  1. What does it mean to say that the diagonal of a unit square is incommensurable?


  1. What is Zeno's Achilles paradox? What is its significance?


  1. What is the difference between w (the first infinite ordinal) and 1 .


  1. Explain the difference between Plato's ways of Reason and Necessity from the  Timaeus.


  1. What is the Principle of  Relativity?


  1. Why is squaring a circle a problem?


8.     What were the points of each of Galileo’s three famous experiments?


You must be prepared to explain any given observable phenomena of the stars, sun, or planets using either the Ptolemaic system or the Copernican system.


Short Essay:


1. Explain the difference between the methodologies of Aristotle and of Galileo and the scientific revolution.


2. Explain the connection between the Principle of Relativity, the Copernican Revolution, and Galileo's distinction between Primary and Secondary properties?


3. Why are the concepts of absolute space and time important for Newton? What is his argument for them?


4. Describe Plato's account of the elements and explain its importance.


5. What did the Pythagoreans mean by the claim that all is number?


6. Explain the implications of the fact that a number of different geometrical systems seemed to be able to save the astronomical appearances.


7. What was the significance of Cantor’s theory of infinite and trans-infinite numbers?


8. How did Ancient scientists attempt to handle the problem of  continuity? What problems did this encounter?


9. Why was the discovery of  incommensurable quantities important?


10. What was the point of Zeno’s paradoxes? What is the scientific response to them?


11. What is the basic model  or schema which science follows in the application of form to phenomenal appearance? Give an example from Ancient science.


12. How was ancient astronomy an application of the Pythagorean method of constructing the world from number?


13. How do the phrases “purifying the appearances” and “separating the relative from the absolute” summarize the scientific method. Give examples form Galileo.