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IntroductionPreparation

Research ProjectTransition to Graduate School

Conclusions

My logic experience as an undergraduate

Christopher Hardin

Smith College

March 10, 2007, Gainesville, FL

Christopher Hardin Undergrad Logic Exp.



Conclusions

Overview

Amherst College, 1994–1998 (BA, Math and Comp. Sci.)Honors thesis in logic, advised by Daniel VellemanResulting paper in JSL

(Job as Unix admin, 1998–1999)

Cornell University, 1999–2005 (PhD, Math; MS, Comp. Sci.)

Smith College, 2005–present (visiting professor)




Conclusions

Caveat

This is not intended as a template for how undergraduate researchshould be conducted. It’s just one way that worked for at least oneperson.




Conclusions

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




Conclusions

Logic courses at Amherst

Introduction to Logic (Philosophy 13)Natural deduction, elements of FOL

Mathematical Logic (Math 34)FOL, Godel’s completeness and incompleteness theorems

Philosophy of Mathematics (Math/Philosophy 50)Foundations and controversies

(Other logic courses are available at UMass/Amherst and othernearby schools.)




Conclusions

Further preparation

(Did an REU, but it was in astronomy and didn’t involvemuch research.)

Although not a logic course, Amherst’s Moore methodtopology course was tremendously valuable preparation forresearch.




Conclusions

Motivation

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.




Conclusions

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.Christopher Hardin Undergrad Logic Exp.



Conclusions

Meetings

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 clearingup misunderstandings; choosing a research problem;feedback).




Conclusions

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 greattopic for me.

We eventually opted for reverse mathematics. A significantfactor: Steve Simpson’s writing style was clear andself-contained.




Conclusions

Learning about reverse math

To get up to speed on the subject, I worked through chapters 1–4of a draft of Simpson’s Subsystems of Second Order Arithmetic,doing any exercises, filling in details of proofs, finding typos orminor errors.




Conclusions

Reverse Mathematics

Goal: Classify the strength of known theorems in terms of the setcomprehension axioms required.

...

Π11-CA0 Π1

1 comprehension Cantor-Bendixson

ATR0 arithmetical transfinite recursion Perfect set thm.

ACA0 arithmetical comprehension Bolzano-Weierstrass

WKL0 RCA0 + Weak Konig’s Lemma maximum principle

RCA0 ∆01-comprehension, Σ0

1-induction IVT




Conclusions

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 WKL0 over RCA0.

Perhaps one could obtain a similar reversal for the MVT?




Conclusions

Research process

Lots of work.

Trying to show reversal RCA0 |= MVT → WKL0 by adaptingthe existing proof of RCA0 |= (maximum principle) → WKL0.

Letting the difficulties in proving RCA0 |= MVT → WKL0

shed light on possible proof of RCA0 |= MVT.

Letting the difficulties in proving RCA0 |= MVT shed light onpossible proof of RCA0 |= MVT → WKL0.

Ping-ponging, as above.

Finally reaching a solution.

Role of advisor here: pithy observations and questions.




Conclusions

Reaching a result

Theorem

RCA0 |= MVT

Proof outline.

(Show Rolle’s theorem; MVT follows as a corollary.) Take asequence of progressively narrower “peaks” (x , y , z) withx < y < z with f (x) < f (y), f (y) > f (z). Avoid pathologicalcases by putting a niceness condition on the peaks. Express theproof in terms of approximations, so that it can be carried out inRCA0.




Conclusions

Writing it up

Started writing in December, before finished with actualresearch. (In an undergraduate research project, much of theproduct is exposition on known material. The writing on thiscan begin while research is ongoing.)

Turned in and defended thesis (50 pages) in mid-April.




Conclusions

Publishing

After I had completed the thesis, Dan adapted the main resultinto a journal article, which we submitted to the Journal ofSymbolic Logic in May 1998.

“The Mean Value Theorem in Second Order Arithmetic”appeared in the JSL, September 2001.




Conclusions

Deciding to go to graduate school

At the beginning of my senior year, graduate school was aserious option, but I was not yet committed to it. (So, I tookthe math GRE, but did not apply to schools that year.)

By the end of my senior year, I was committed to going tograduate school. This was based on many factors other thanmy thesis: encouragement from Duane Bailey; job interviews.

I was still uncertain about whether to study math or computerscience.

I got a job as a Unix admin after graduating.




Conclusions

Applying to graduate school

Learning about CMU’s program in pure and applied logic iswhat led me to apply to graduate school. I also applied toseveral other schools for good measure.

I met my future thesis advisor while visiting Cornell, thoughthis research was very interesting, and ultimately decided onCornell.




Conclusions

Recruiting

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




Conclusions

Preparation

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




Conclusions

Choosing a topic and getting up to speed

Instead of just jumping into a single topic, it can be helpful totry multiple topics, and ultimately stick with the one that isworking out best.

The accessibility of the material can be as much about howwell the writing fits the student as it is about the materialitself (up to a certain point).

An issue specific to logic: Students might have a thinnerbackground than in a subject like algebra, for curricularreasons.




Conclusions

Frequent meetings

Frequent meetings prevent misunderstandings from persistingtoo long. (E.g., ∃x1 · · · ∃xn can really mean ∃x1∀x2 · · · ∃xn.)

Undergrads have been conditioned to think in terms ofdeadlines. I.e., they’re almost event-driven.

Students want their advisor’s respect, so there may be no needfor the advisor to say “Do such and such by Monday.” Rather,just having to report on progress will keep things moving.




Conclusions

The ignorance of undergraduates

Undergraduates often do not understand the nature of academicsand grad school. Most particularly:

A number of Smith students I’ve spoken with were surprisedto learn that, financially, getting a PhD in math is more like alow-paying job than it is like med school. (This is cleared upwhen students look into the possibility of grad school, butsome may never look into the possibility.)




Conclusions

(End)


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