Homework 1

Notes

• In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
• Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
• It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
• These questions are designed to exercise your amazing human mind. Unless specifically allowed, please do not use AI (ChatGPT or similar) to answer them.
• Recall that the slides of the lectures can be found here.
• Follow the AI policy.

Problems

1. Find a quotation about mathematics that includes or implies a definition (of mathematics) and explain what you think it means. Discuss whether you agree or disagree and explain your reasons.
2. Let your imagination fly and then reflect on your findings.
   1. Create a hypothesis as to the meaning of the scratches on the Ishango bone or why they might have been made. Be creative. The only requirement is that your hypothesis matches the pattern exhibited in the bone.
   2. Discuss whether your hypothesis fits in with what we know of math history at that point.
3. Pythagoras: Myth vs. Modern Scholarship: Most people today have strong beliefs about who Pythagoras was and what he discovered (your instructor held these beliefs before starting teaching this class). Using the Stanford Encyclopedia of Philosophy entry on Pythagoras (https://plato.stanford.edu/entries/pythagoras/), discuss:
   1. One common belief about Pythagoras that you held before reading this source
   2. What current scholars actually think about this same topic, based on the evidence
   3. Why this difference exists - what accounts for the gap between popular belief and scholarly understanding?
   Your response should be 100-200 words and include at least two specific references to the Stanford Encyclopedia article with proper citation format: (Author, "Title," date accessed).
4. Units of time, such as a day, a month, and a year, have ratios. In fact, you probably know that a year is about 365 1/4 days long. Imagine that you were never taught this fact.
   1. How would you -how did people originally- determine how many days are are in a year?
   2. What do you think was the purpose of a calendar for very ancient societies? Write down two possible uses of calendars for them.
5. In what sense is it possible to know the exact value of a number such as √2. Obviously, if a number is known only by its whole infinite decimal expansion, nobody does know and nobody will ever know the exact value of this number. What immediate practical consequences, if any, does this fact have? Is there any other sense in which one could be said to know this value exactly? If there is no direct consequences of being ignorant of its exact value, is there any practical value in having the concept of an exact square root of 2? Why not simply replace it by a suitable approximation such as 1.41421? Consider also other "irrational" numbers, such as π and e. What is the value of having the concept of such numbers as opposed to approximate rational replacements for them.
6. Find one peer-reviewed secondary source related to your presentation and paper topic. It should be different from the one given on the course website. Then complete the following:
   1. Topic
      Write the title or focus of your presentation.
   2. Complete bibliographical information
      Include author, article title, journal name, volume/issue (if applicable), year, and page numbers.
   3. Link to the article (if available)
      Paste a direct link or a stable JSTOR, Project MUSE, or DOI link. It must be freely accessible or accessible through your Stony Brook account.
   4. Answer in 3–4 sentences (maximum 100 words):
      • What is this source about?
      • Why might it be useful for understanding your topic?
      • How do you know it is peer-reviewed?

   Example Problem 6

   Topic: The beginnings of counting

   Bibliographical information:
   Barras, Colin. "How did Neanderthals and other ancient humans learn to count?" Nature 594, no. 7861 (2021): 22–25.

   Link: https://www.nature.com/articles/d41586-021-01429-6

   Answer:
   This paper discusses different hypotheses about the invention of numbers. It is useful because it tackles exactly what my topic is about. I know that it is peer-reviewed because Nature only publishes peer-reviewed articles, and the Stony Brook Library website also marks it as peer-reviewed.

Note: Problems 4 and 4 are adapted from the Roger Cook's book History of Mathematics: A Brief Course, by Roger Cooke, 2005, Wiley-Interscience.

Grading Rubric

Problem 1: Mathematics Definition Quotation (10 points)

Component	Points	Criteria
Quotation provided	3	Clear quotation with source identified
Interpretation	4	Explains what the definition means
Critical analysis	3	Takes a clear position with reasoning
Total	10

Problem 2: Ishango Bone Hypothesis (10 points)

Component	Points	Criteria
Creative hypothesis (Part A)	4	Hypothesis matches bone patterns
Historical context (Part B)	3	Connects to known math history of the period
Reasoning	3	Logical connection between evidence and hypothesis
Total	10

Problem 3: Pythagoras - Myth vs. Modern Scholarship (10 points)

Component	Points	Criteria
Personal belief identified (Part A)	2	Clearly states a specific belief they held about Pythagoras
Scholarly perspective (Part B)	3	Accurately explains what current scholars think, with evidence from source
Analysis of difference (Part C)	3	Thoughtful explanation of why gap exists between popular belief and scholarship
Citations	1	At least two specific references in correct format
Length and focus	1	100-200 words, stays on topic
Total	10

Problem 4: Calendar Development (10 points)

Component	Points	Criteria
Method explanation (Part A)	4	Clear method for determining days in a year
Calendar uses (Part B)	4	Two distinct uses for ancient societies
Historical reasoning	2	Shows awareness of ancient societies' needs
Total	10

Problem 5: Exact Values & Irrational Numbers (10 points)

Component	Points	Criteria
Conceptual understanding	4	Understands difference between exact and approximate
Practical consequences	3	Discusses real-world implications
Extension to π and e	3	Meaningful discussion of other irrational numbers
Total	10

Problem 6: Peer-Reviewed Source (10 points)

Component	Points	Criteria
Topic provided	1	Topic clearly stated
Complete citation	2	All required elements present (author, title, journal, volume, year, pages)
Working link	1	Link works and leads to the source
Source appropriateness	2	Source is peer-reviewed and relevant to topic
Content explanation	2	Clearly explains what source is about
Relevance justification	1	Explains why source is useful for their topic
Peer review evidence	1	Shows understanding of how they identified peer review
Total	10