Honors Interdisciplinary Seminar (175)

 Proofs and Uncertainty

Assignment following Week #10 (Due 11/23 in class) 
(Section 004 - Due 10/12 in class)

1. Finish the final draft of Response Paper #1.  Email it to Steve (szydliks@uwosh.edu) by 11:00 am on Wednesday, November 18.
2. Report your Final Project group project topic preference list to me by 11:00 am on Thursday, November 19.  Remember that the earlier you submit your list, the more likely you are to get a top preference, and the more preferences you list, the less likely you are to get a topic you do not want. Preferences are *not* guaranteed.
3. Read Melvyn B. Nathanson's editorial essay "Desperately Seeking Mathematical Truth" from the Notices of the American Mathematical Society, August 2008. (online)
4.  Read the excerpt from Lakatos’s text Proofs and Refutations. (ereserve).
   (Remember that if you print any of these excerpts, the reproduction is not to be used for any purpose other than private study, scholarship, or research. If you use a photocopy or reproduction for purposes in excess of fair use, then you may be liable for copyright infringement.)
5. Complete the Study Questions below.
6. Read any emails I send you, and respond if requested.
7. There may be a brief quiz at the start of the next class.

Study Questions 

Please carefully write out your answers to these questions.  Make a copy of your answers and be prepared to hand that copy in at the start of class.  Look at the Study Questions Main Page for general guidelines on study questions in Steve's section.

1. Carefully look at the "Teacher's Proof" of Euler's formula on p. 7-8.  Study the proof until you think you understand it.  Notice the pictures that the teacher draws to illustrate the proof in the case of a cube.  I want you to illustrate the proof in the case of a triangular prism.  Draw the corresponding pictures for how the proof would go in in the case of the prism.
   
   
   

2. Pupil Sigma says on p. 7, "...As however the truth of it has been established in so many cases, there can be no doubt that it holds good for any solid."  How would a mathematician react to this statement?

3. On p. 10, the teacher says, "Conjectures ignore dislike and suspicion, but they cannot ignore counterexamples?"  What does the teacher mean?

4. Carefully explain the difference between "local counterexample" and "global counterexample" as Lakatos sees it.

5. Carefully explain why V-E+F=4 for a "hollow cube."

6. The teacher comments on p. 16:  "I didn't define 'polyhedron.'  I assumed familiarity with the concept, i.e. the ability to distinguish a thing which is a polyhedron from a thing which is not a polyhedron."  Now reread Euclid's definition of "point," "line," and "plane" as they are given in the Elements.  Are they real definitions, or do they fall into the teacher's category?  Explain why or why not.

7. Alpha leaves the classroom on p. 21.  Why does he do so?  Do you have any sympathy for his position?  Why or why not?

8. At the bottom of p. 21, Delta says, "...monstrostities never foster growth, either in the world of nature or in the world of thought.  Evolution always follows an harmonious and orderly pattern."  What does Delta mean by this?  Do you agree with this position?  Why or why not?

9. Throughout the excerpt, Lakatos speaks of the "monster-barring" method, or the "exception-barring" method.  What does he mean by this?  What does Lakatos think of this as a means of developing mathematics?

10. What is the connection between Lakatos and Nathanson's article?  Explain carefully.

11. Write a paragraph identifying what you see as the main point of the excerpt from Lakatos.  Be insightful!
    

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