Fundamentals of Mathematics I -- Paper

Due 4:00 p.m., Thursday, November 17, 2011

Write a 5 to 6 page essay on some topic related to the mathematical subjects studied in MA160 or to the general character of mathematics. Possibilities include the relevance of beauty to mathematics, the epistemology of mathematics or a relevant topic in the history of mathematics. This list of topics is not meant to be exhaustive. However, if you have another idea for a paper, come discuss it with me before writing.

Alternately, you may write a pair of 3 to 4 page responses to each of two chapters from Mathematics Through the Eyes of Faith, Bradley and Howell. You should write in response to chapters that are relevant to the material in MA 160 (chapters 2, 3, 6, 7, 9, and 10).

Prior to giving your paper to me, submit it to Turnitin . You can submit your paper multiple times to clean up plagiarism problems. (Class ID: 4277017; Enrollment Password: rosentrater; Instructions)

Read Westmont's Plagiarism Policy. Then print out a copy of Westmont's brief plagiarism statement, sign it, and turn it in with your paper.

Technical requirements
General suggestions for a good paper
Possible initial sources

You can view a copy of the sheet that I will use to evaluate your paper here.

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Technical Requirements:

Pages should be numbered and set up with 1 inch margins top and bottom and 1.25 inch side margins.
Use a standard, readable, 12-point font such as Times New Roman.
Your paper should be double spaced.
If you want to insert equations, use italics for variables and subscript and superscript fonts (under the Format menu) to create the subscripts and powers. Alternately you can insert an object of type equation. You may need to install Equation 3.0 from your MS Word or MS Office CD. Your formulas should look like

x2 + y2 = 1 or x1 + x2 =12 not
x^2 + y^2 =1 or x1 +x2 =12.

All paragraphs should be well organized and have a clear relationship to the main ideas of the paper.
Use complete sentences with proper grammar and punctuation.
Proof read your paper and run it through a spell checker before you submit it.
Your paper should use reviewed print sources (though you might find and access them via the web) and not just internet sources.
Carefully evaluate your sources -- particularly those from the internet.

Plagiarism and Crediting Sources:

To plagiarize is to present someone else's work-his or her words, line of thought, or organizational structure-as your own. This occurs when sources are not cited properly, or when permission is not obtained from the original author to use his or her work. Another person's "work" can take many forms: printed or electronic copies of computer programs, musical compositions, drawings, paintings, oral presentations, papers, essays, articles or chapters, statistical data, tables or figures, etc. In short, if any information that can be considered the intellectual property of another is used without acknowledging the original source properly, this is plagiarism.

Please familiarize yourself with the entire Westmont College Plagiarism Policy. This document defines different levels of plagiarism and the penalties for each. It also contains very helpful information on strategies for avoiding plagiarism. It cannot be overemphasized that plagiarism is an insidious and disruptive form of academic dishonesty. It violates relationships with known classmates and professors, and it violates the legal rights of people you may never meet.

Westmont College Plagiarism Statement

Plagiarism has severe consequences both inside and outside of academia. See for example the following stories (story 1, story 2). At its heart, plagiarism is a dishonest act that has no place in the life of a Christian or at an institution of higher learning.

Be sure to provide references for all ideas, information, graphics and diagrams you have gained from other sources. It is not sufficient to simply provide references for quotes. Failure to properly credit your sources is plagiarism and will result in severe penalties. Depending on the severity of the omissions, you may receive an F on the assignment or for the course or be expelled from the Westmont. You should read the Westmont College plagiarism policy before you begin writing your paper and again before you turn it in.
For questions about plagiarism and referencing consult the two sites below. Consider carefully the examples of plagiarism and non-plagiarism on these sites. www.ccc.comment.edu/mla/plagiarism.shtml and www.indiana.edu/~wts/pamphlets/plagiarism.shtml.
Rutgers University has produced an entertaining set of educational videos on plagiarism.
Include the full citation of all works referenced in a bibliography at the end of your paper.
When you cite your sources, you should either use one of the two following forms:
1. [Jones, 50], where “Jones” is replaced by the author you are citing and “50” is replaced by the number of the page you are referencing. When needed to distinguish between multiple references, include the year of publication in the reference. [Jones 1998, 50]
2. Use a standard footnote with a superscript number and the reference at the bottom of the page. If you use footnotes, be sure to also use the shorthand notations such as “Ibid” to reduce the size of the footnotes.

Before before you turn in your paper, run it through Turnitin to check for plagiarism/citation problems. You can submit your paper multiple times to clean up plagiarism problems. (Class ID: 4277017; Enrollment Password: rosentrater) Instructions for Turnitin

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General Suggestions for all papers:

Select a single approach/theme/thesis and stick to it. Moving between ideas without tying them to a central point makes for a poor paper.
Make sure that you support your claims with clear arguments and examples. Never simply make a claim and then move on to the next point. Make sure that you develop and support the arguments rather than skipping from one point to another.
Answer or address any obvious questions that your paper raises. In particular, always address the "why" question.
Avoid clichés and contractions.
Be specific; give examples.
Think about what you are going to write and make an outline before you begin putting words on paper. It will keep you from wandering.
Give yourself plenty of time to think about what you write. Don't try to throw something together at the last minute. This will also reduce the likelihood of plagiarism.
Have someone proof read your paper who will give you honest feedback. Use a person who does not already know what you are trying to say, but who is a good writer. Make use of Writer's Corner.
Make sure that your reader will understand how each point fits into the overall plan of your paper.
If you include algebraic derivations, assume that your reader is algebraically competent. Instead of including all the steps, include only major steps while explaining what you are doing and why.
Before using internet sources you should spend some time at Internet Detective and The Virtual Training Suite learning how to evaluate sites. Make sure that you use sources other than internet sources.
The use of first person is usually inappropriate for this type of paper.
Diagrams can be very effective in communicating an idea.

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How do beauty, creativity, and intuition relate to the mathematical enterprise?

Beauty, creativity, and intuition/inspiration are sometimes viewed as falling exclusively within the realm of the arts --- music, theater, visual arts, literature. Consider the degree to which mathematics can be viewed as an artistic endeavor.

Possible approaches: (It is not wise to mix approaches.)

1. First consider the essential characteristics. Then discuss how aspects of beauty relate to the mathematical questions, results, and arguments. Can portions of mathematics ever be called beautiful or ugly? Or are such terms really irrelevant to mathematics? Support your thesis with examples and arguments. (Please! No bromides about "beauty is in the eye of the beholder.")

2. What role does creativity play in solving mathematical problems or in answering mathematical questions? Is this the same as what is intended when speaking of creativity in the humanities? This could be approached from a personal point of view or from a historical perspective. Support your claims with carefully explained examples.

3. Is there such a thing as mathematical intuition? Is it a good or bad thing; or can it be either? If it exists, how can it be developed or avoided?

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The philosophy of mathematics:

How or when does one know a fact in Mathematics? What is it that we know about?

Possible approaches: (Resist the temptation to mix approaches.)

1. In mathematics we constantly behave as if the numbers 1, 2, ... are objects with a concrete existence. We have symbols and sets to represent them, yet one cannot touch or move them. In what sense can we say that numbers (and other mathematical objects) exist? Or are they only useful fictions?

2. Discuss Gödel's Incompleteness Theorem and its implications for the effectiveness of mathematics for settling all mathematical questions.

3. What makes a question a mathematical question as opposed to a question in philosophy, history, literature or another academic area?

4. In what types of facts or data are mathematicians interested? How is this different from (or the same as) the facts and data in chemistry, political science, history, etc.?

5. What methods do mathematicians use that distinguish them from scholars in other disciplines?

6. What philosophies of mathematics undergird current mathematical teaching strategies? How do the two interact?

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History of Mathematics:

Research a mathematical idea and write a paper tracing its development over time or discuss the contributions made by a significant mathematician. You should investigate a mathematician or mathematical idea that is relevant to the topics we are studying this term: logic, sets, number systems, number theory, algebra. If you investigate a mathematician, you should spend most of your energy discussing the mathematical contributions of the mathematician and the influence the mathematician had on the direction of development of mathematics rather than discussing biographical information. Since the level of creative and analytical thinking exhibited in these papers is typically significantly lower than that required for the other types of papers, it is more difficult to obtain an A writing an historical paper.

Possible mathematicians and ideas:
1. Infinity
2. George Cantor.
3. David Hilbert.
4. Kurt Gödel.
5. Diophantus.
6. The beginnings of Algebra.
7. Fermat's "Last Theorem"
8. Theorems about primes.
9. The development, use, and acceptance of negative numbers in various cultures.
10. Fractions in the Egyptian numeration system.
11. Numeration systems in cultures other than the ones we have studied.

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Possible Initial Sources:

A History of Mathematics, Boyer and Merzbach.
The Mathematical Experience, Davis and Hersch.
Fleeting Footsteps, Yong and Se.
The World of Mathematics, Newman.
Metamagical Themas, Hofstadter.
From Five Fingers to Infinity, Swetz.
The Number Sense, Debaene.
Women in Mathematics, Henron
Women in Mathematics, Osen.
Why Beauty is Truth, Stewart.
The MacTutor History of Mathematics archive
The ACMS On-Line Journal