My logic experience as an undergraduate

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1 My logic experience as an undergraduate Smith College March 10, 2007, Gainesville, FL

2 Overview Introduction Amherst College, (BA, Math and Comp. Sci.) Honors thesis in logic, advised by Daniel Velleman Resulting paper in JSL (Job as Unix admin, ) Cornell University, (PhD, Math; MS, Comp. Sci.) Smith College, 2005 present (visiting professor)

3 Caveat Introduction This is not intended as a template for how undergraduate research should be conducted. It s just one way that worked for at least one person.

4 Amherst College Amherst College: Selective liberal arts college Undergraduate only 1600 students Math at Amherst College: 10 faculty 15 math majors graduate per year 3 honors theses in math per year

5 Logic courses at Amherst Introduction to Logic (Philosophy 13) Natural deduction, elements of FOL Mathematical Logic (Math 34) FOL, Gödel s completeness and incompleteness theorems Philosophy of Mathematics (Math/Philosophy 50) Foundations and controversies (Other logic courses are available at UMass/Amherst and other nearby schools.)

6 Further preparation (Did an REU, but it was in astronomy and didn t involve much research.) Although not a logic course, Amherst s Moore method topology course was tremendously valuable preparation for research.

7 Further preparation (Did an REU, but it was in astronomy and didn t involve much research.) Although not a logic course, Amherst s Moore method topology course was tremendously valuable preparation for research.

8 Motivation Introduction Motivation for doing an honors thesis: Extrinsic: academic recognition; expectations Intrinsic: It would be interesting and hard. Motivation for thesis in logic: Had gained interest in logic through logic courses.

9 Overview and timeline Summer 1997 Did some reading on automated theorem proving. Fall 1997 (senior year) Did further readings suggested by advisor. Settled on reverse mathematics, broadly, as topic. Got up to speed. Started research and writing. Spring 1998 Did more research and writing. Came up with a result. Completed and defended thesis. Submitted paper, co-authored with advisor, to JSL. September 2001 Paper appears in JSL.

10 Meetings Introduction We held weekly meetings, about an hour long. Just having the meetings at all helped maintain pace. The role of meetings changed to suit phase of project (choosing a topic, broadly; answering questions and clearing up misunderstandings; choosing a research problem; feedback).

11 Selecting a topic I was interested in automated theorem proving. Dan suggested readings in a few other topics. Automated theorem proving didn t turn out to be a great topic for me. We eventually opted for reverse mathematics. A significant factor: Steve Simpson s writing style was clear and self-contained.

12 Learning about reverse math To get up to speed on the subject, I worked through chapters 1 4 of a draft of Simpson s Subsystems of Second Order Arithmetic, doing any exercises, filling in details of proofs, finding typos or minor errors.

13 Reverse Mathematics Goal: Classify the strength of known theorems in terms of the set comprehension axioms required.. Π 1 1 -CA 0 Π 1 1 comprehension Cantor-Bendixson ATR 0 arithmetical transfinite recursion Perfect set thm. ACA 0 arithmetical comprehension Bolzano-Weierstrass WKL 0 RCA 0 + Weak König s Lemma maximum principle RCA comprehension, Σ0 1-induction IVT

14 Topic for individual research I focused on the mean value theorem, at Dan s suggestion. The standard proof of MVT uses maximum principle. The maximum principle is equivalent to WKL 0 over RCA 0. Perhaps one could obtain a similar reversal for the MVT?

15 Research process Lots of work. Trying to show reversal RCA 0 = MVT WKL 0 by adapting the existing proof of RCA 0 = (maximum principle) WKL 0. Letting the difficulties in proving RCA 0 = MVT WKL 0 shed light on possible proof of RCA 0 = MVT. Letting the difficulties in proving RCA 0 = MVT shed light on possible proof of RCA 0 = MVT WKL 0. Ping-ponging, as above. Finally reaching a solution. Role of advisor here: pithy observations and questions.

16 Reaching a result Theorem RCA 0 = MVT Proof outline. (Show Rolle s theorem; MVT follows as a corollary.) Take a sequence of progressively narrower peaks (x, y, z) with x < y < z with f (x) < f (y), f (y) > f (z). Avoid pathological cases by putting a niceness condition on the peaks. Express the proof in terms of approximations, so that it can be carried out in RCA 0.

17 Writing it up Started writing in December, before finished with actual research. (In an undergraduate research project, much of the product is exposition on known material. The writing on this can begin while research is ongoing.) Turned in and defended thesis (50 pages) in mid-april.

18 Publishing Introduction After I had completed the thesis, Dan adapted the main result into a journal article, which we submitted to the Journal of Symbolic Logic in May The Mean Value Theorem in Second Order Arithmetic appeared in the JSL, September 2001.

19 Deciding to go to graduate school At the beginning of my senior year, graduate school was a serious option, but I was not yet committed to it. (So, I took the math GRE, but did not apply to schools that year.) By the end of my senior year, I was committed to going to graduate school. This was based on many factors other than my thesis: encouragement from Duane Bailey; job interviews. I was still uncertain about whether to study math or computer science. I got a job as a Unix admin after graduating.

20 Applying to graduate school Learning about CMU s program in pure and applied logic is what led me to apply to graduate school. I also applied to several other schools for good measure. I met my future thesis advisor while visiting Cornell, thought his research was very interesting, and ultimately decided on Cornell.

21 Recruiting Introduction Getting students to do research: Research-like experiences in coursework (e.g., Moore method) Requirement for honors? Getting students to do research in logic: Presence and prominence of logic within curriculum

22 Introduction Coursework in logic obviously helped. The Moore method topology course was helpful in two ways: Psychological preparation High expectations Appreciation for the process of getting stuck, getting stuck, and finally solving a problem Technical preparation Ability to work independently Greater facility in proving theorems

23 Choosing a topic and getting up to speed Instead of just jumping into a single topic, it can be helpful to try multiple topics, and ultimately stick with the one that is working out best. The accessibility of the material can be as much about how well the writing fits the student as it is about the material itself (up to a certain point). An issue specific to logic: Students might have a thinner background than in a subject like algebra, for curricular reasons.

24 Frequent meetings Frequent meetings prevent misunderstandings from persisting too long. (E.g., x 1 x n can really mean x 1 x 2 x n.) Undergrads have been conditioned to think in terms of deadlines. I.e., they re almost event-driven. Students want their advisor s respect, so there may be no need for the advisor to say Do such and such by Monday. Rather, just having to report on progress will keep things moving.

25 The ignorance of undergraduates Undergraduates often do not understand the nature of academics and grad school. Most particularly: A number of Smith students I ve spoken with were surprised to learn that, financially, getting a PhD in math is more like a low-paying job than it is like med school. (This is cleared up when students look into the possibility of grad school, but some may never look into the possibility.)

26 (End)

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