# 80-110: The Nature of Mathematical Reasoning

## Carnegie Mellon University

## Spring 2015

## Possible essay topics

### Due Friday, May 15

The following are some ideas for extra-credit essay topics. You're welcome to write about something that's not on this list, but send me an email first to check that it's an appropriate topic! I've linked some potential resources, but you should find your own as well.

These papers should be a minimum of 1500 words. Be sure to cite whatever resources you use (Wikipedia is not a good resource). If you are writing on something historical, carefully explain the timeline and progression of how the topic developed. If you are writing on something philosophical, explain different viewpoints that are found in the literature, and justify your own opinion. Please write in clear, grammatical prose: the fact that we're talking about math does not excuse sloppy English! Essays will be graded on both style and content, and will be worth a maximum 10% increase to your course grade.

The final deadline for these papers is Friday, May 15. I will not grade anything received after that. If you are a graduating senior, your grades are due to the registrar at 4:00 pm on May 14; to meet this deadline, you need to get your papers to me earlier, by Sunday, May 10. I will be on planes to Australia starting the morning of May 12, and don't know what internet availablility will be like when I arrive; if you are late submitting your paper, you are tempting fate.
- Geometric constructions via origami. Link
- Arithmetic reasoning with primitive tools: abacus, slide rule, etc.
- In general, ancient cultures had varying approaches to arithmetic: pick one and describe their methods.
- Pythagorean number mysticism.
- Women in mathematics (see here for a starting point).
- Connections between projective (noneuclidean) geometry and perspective in art.
- The debate between formalism and platonism in the philosophy of mathematics.
- The debate about Godel's Theorem applied to the philosophy of mind (see, e.g., John Lucas,
*Minds, Machines, and Godel*, and the response by Paul Benacerraf, *God, the Devil, and Godel*). (Note: this could be a difficult topic!)
- The controversy between Newton and Leibniz about the discovery of calculus.
- Intuitionistic logic, and the school of intuitionism founded by Brouwer and Heyting.
- Multi-valued logics: for example, relevance logic or the work of Graham Priest. In these logics, statements are not just "true" or "false."
- Applications of mathematics in music. ( is a technical resource on this. But there are lots of directions to take this topic.)