Some notes on format:

Your papers must be typed. You may write in mathematical symbols (neatly) if it is too hard to typeset them nicely. If you do that, of course, you will have to give me a hard copy of your paper rather than blitzing it to me.

Please use only one side of the page, and please double space. Approximately 1 inch margins with 12 point font is good. (I'm not going to measure, but I will have trouble reading a smaller font.) Make sure your name appears on each page. (I sometimes need to make multiple copies, and pages can become separated.)

Your first drafts as well as final versions must include correct acknowledgments and citations. I prefer a "note" (footnote or endnote) citation style, but any style is acceptable as long as it is consistent throughout your paper and you give all the necessary information. The MIT article on writing mathematics explains the citation format usually used in mathematics papers.

Paper One

Your first term paper is to be an essay about a philosphical topic connected with the concept of infinity. You may pick a topic from the list below or choose one of your own. If you choose your own topic be sure to consult with me to make sure it is appropriate. Recommended length, 5-8 pages.

Topics

1. State the positions of Achilles and the Tortoise in Carroll's article. Choose one of the two responses in our reader (or another one of the responses from the October 1995 issue of Mind) and reconstruct the argument. Some questions you might address: How does the article respond; whose side (if any) is it taking; what does it point out as important to understanding what is going on in the Carroll article? Do you agree? What do you think we learned about arguments from Carroll? (Do not choose this topic unless you feel you have something to say beyond simply repeating the arguments of the articles you discuss.)

2. Given that no one really agrees with Zeno's conclusion, how would you argue with him? What kind of evidence would you use --- mathematical, physical, intuitive, logical? Sketch out an approach and say why you think your evidence is legitimate or convincing. (I am thinking of Achilles and the tortoise, but you can respond to any one of Zeno's paradoxes from the Salmon article.)

3. What exactly is fishy (if anything) about proof by mathematical induction? Would it work if you were trying to show, e.g., that all swans are white? (The first swan I see is white; whenever I see a white swan, the one sitting next to it is also white...) Why or why not? If not, why doesn't this affect (mathematical) proofs by mathematical induction?

4. Discuss my "proof" by mathematical induction that no amount of coffee is enough to wake me up in the morning. Is this a paradox? If so, how would you classify it (using Quine's categories), and why? Can you explain the paradox? Can you come to any conclusion about when proof by mathematical induction is appropriate and when it is not?

The next topics require reading ahead:

5. Pick a big problem (global warming? well, maybe that's too big) and use Descartes's method on it. See how far you get, and say at what point you get stuck. In what ways is the method limited?

6. The title of Descartes's paper refers to "seeking truth in the sciences." How would a modern scientist describe his or her method of seeking truth? Is it different from Descartes's method? If so, what are their relative strengths and weaknesses in light of the assurances we want from a method of seeking scientific truth? Is there another area (human nature, religion, morality) in which you think we should use a different method for seeking truth? Why, and what properties should it have?

Grading

The grade for your first paper will be based on how well it meets these criteria:

(1.) Paper has something to say.

You don't have to have an ambitious goal like solving Zeno's paradox once and for all, but you should have something in particular to communicate. A good test is to see whether you can write a sentence or two summarizing what your paper has to say. It should look something like, "My paper argues that Zeno's paradox still has the status of an antinomy," or "My paper explains how Achilles can answer the tortoise using modern physics," rather than, "`My paper is about mathematical induction."

(2.) Examples and arguments support the points being made.

A particular note here: Claims about the nature of reality or the validity of empiricism or ... imported from your outside knowledge of philosphy need to be argued and supported to convince the reader. "Obviously, we don't need to discuss the existence of things we cannot perceive with our physical senses," doesn't do the job. Another note: Be careful, especially in your introduction and conclusion, to avoid the temptation to make sweeping claims (especially if you're not sure you can support them) in order to make your paper seem significant.

(3.) Language is used precisely and correctly.

This is extremely important when discussing philosophy, as shown by the care with which Moore explains Aristotle's use of the word "untraversable." Be especially careful with words that have special importance for our discussions. As an example, you can't "prove infinity" any more than you can "prove New Hampshire;" as another, Zeno's paradox (a paradox being, as Quine defined it, an apparently absurd conclusion with an argument to sustain it) "exists," because Zeno stated that argument and conclusion and we can read what he said, regardless of what you conclude about its paradoxical nature.

(4.) Explanations are clear, and are understandable with no special background beyond this course.

(5.) Paper is well-organized, with a clear theme and a logical flow of ideas throughout.

(6.) Writing style is good, both clear and graceful.

Watch especially for using words, phrases, and sentence structures that are unnecessarily complex and pompous.

(7.) Technical details of spelling, grammar, citations, etc. are correct.

In connection with both this point and the previous one, I recommend The Elements of Style by Strunk and White.

Paper Two

In class on Friday, October 7, we worked together to solve some math problems using the method of infinitesimals.

For this paper, you are to present and explain a solution to one of these problems using infinitesimals. You may use the problem you were assigned, and which you and your team solved, or you may use any of the other problems on the list.

In order to explain you may want to use: A discussion of the method of infinitesimals. Pictures or diagrams. A description of a physical situation corresponding to the problem. A graph or collection of graphs. Examples of special cases (e.g., particular values of the variables), and discussion of the important features of those cases.

Your paper may have any format you want. You may write a straightforward math paper, a dialogue, an essay, or even a science fiction story, as long as the focus of your paper is solving a math problem.

Grading

The grade for your second paper will be based on how well it meets these criteria:

(1.) Problem is solved correctly, using the method of infinitesimals.

(2.) Explanations are clear and understandaable by a college student who has never studied calculus.

(3.) Other material (examples, graphs, anecdotes, etc.) helps in understanding the problem and its solution.

(4.) Paper is well-organized, with a clear theme and a logical flow of ideas throughout.

(5.) Writing style is good, both clear and graceful.

(6.) Technical details of spelling, grammar, citations, etc. are correct.

(7.) Paper is made as interesting as possible for the reader (without distracting from the central purpose of explaining the solution to a math problem.)

Paper Three

Your third paper is to be a research paper on a topic either connected with material we have covered in the course or connected in some other way with the topic of infinity. Recommended length, 10-15 pages.

If you are unsure of the suitability of your proposed topic, please talk to me about it.

The first assignment for this paper is not a draft of the paper, but a proposal. In your paper proposal, you should not only describe your proposed topic, but also say as much as you can about the projected content of your paper. For example, you can discuss the approach you plan to take, and you can say what particular aspects of your topic you will discuss. Your proposal should also include a list of several sources you plan to consult. It should describe each source briefly and say what you expect to get from each source.

In other words, you can expect to do a good deal of preliminary research before writing your paper proposal. Proposals, whether scholarly paper proposals like this one, scientific research grant proposals, or business project plans, are often extremely important pieces of writing. They should meet the same standards of clear and correct communication as any other piece of scholarly work.

Your project proposal will count toward your paper grade, but only on a credit or no credit basis. That is, you will get full credit for an acceptable (and timely) proposal, and if your proposal is not acceptable you will get a chance to rewrite it.

Grading

The grade for your third paper will be based on how well it meets these criteria:

(1.) Paper has a clear focus and substantive content.

(2.) Paper includes a synthesis of material drawn from a number of different, appropriate, and reputable sources.

(3.) Synthesis and analysis is well thought out and supported.

(4.) Paper is well-organized, with a clear theme and a logical flow of ideas throughout.

(5.) Writing style is good, both clear and graceful.

(6.) Technical details of spelling, grammar, citations, etc. are correct.

(7.) Paper is understandable and interesting for the general reader.

Paper Four

Your fourth paper is to be a scholarly philosophical essay on a topic that touches on both mathematical and philosophical concerns. To write a philosophical essay, as opposed to some other sort of scholarly paper, you must take a stand on some issue and support your position. Your paper should be explicitly set in the context of the existing literature; you should contribute to or engage with, rather than work independently of, current scholarship.

Your paper may be partly a research paper, like your third paper. In this case, your paper should be more than simply a presentation of material; your research should be in support of your position. Your paper may, like your second paper, present the solution to a mathematical problem. In this case, your paper should be more than simply a presentation of a mathematical result; your problem solving should be in support of your position.

Recommended length, 5-8 pages.

Here are some examples of suitable theses for this paper. This is far from an exhaustive list. It is fine to have a different thesis for your paper, although you may want to check your idea out with me first. You will notice that you can contradict any of the theses in this list and arrive at a reasonable thesis; the same should be true of your thesis.

Formalism is a critically flawed philosophy of mathematics.

Formalism is a better philosophy of mathematics than is realism.

The fact that infinitesimals can be used consistently does not justify belief in their existence.

Ultimately, the intuitionist movement had a positive effect on the practice of mathematics.

We should believe in the existence of numbers for some of the same reasons we should believe in the existence of God.

Grading

The grade for your fourth paper will be based on how well it meets those of the criteria set for the first three paper assignment that are relevant to your choice of topic and approach. For example, if your paper is in part a research paper, then the criteria regarding the selection and use of sources apply to your paper.

Other Assignments

Other short credit / no credit writing or math assignments may be assigned in class, and are due the next class.

Marcia J. Groszek
Last updated May 31, 2008 12:24:46 EDT