### Past Prologue
-[prologue might get broken out into a separate post?]]
+[prologue might get broken out into a separate post?—or maybe not; I want to include detail, but it doesn't feel important enough to be a separate post, so I'll just try to pack in as much detail as I can in a reasonable wordcount]
[TODO—
* I would say I had a pretty normal school experience—public elementary school, then Jewish day school from 5th–8th grades after my parents thought I was sad in normal school, then public high school.
The textbook was _An Introduction to Analysis_ (4th edition) by William R. Wade, the same book that had been used for "Real I" in Spring 2013. It felt in bad taste for reasons that are hard to precisely articulate. I want to say the tone is patronizing, but don't feel like I could defend that judgement in debate against someone who doesn't share it. What I love about Schröder is how it tries to simultaneously be friendly to the novice (the early chapters sprinkling analysis tips and tricks as numbered "Standard Proof Techniques" among the numbered theorems and definitions) while also showcasing the fearsome technicality of the topic in excruciatingly detailed estimates (proofs involving chains of inequalities, typically ending on "< ε"). In contrast, Wade often feels like it's hiding something from children who are now in fact teenagers.
-The assignments were a lot of work, but that was good. It was what I was there for—to prove that I could do the work. I could do most of the proofs with some effort. At SFSU in 2012–2013, I remembered submitting paper homework, but now, everything was uploaded to Canvas. I did all my writeups in LyX, a GUI editor (I know) for LaTeX.
+The assignments were a lot of work, but that was good. It was what I was there for—to prove that I could do the work. I could do most of the proofs with some effort. At SFSU in 2012–2013, I remembered submitting paper homework, but now, everything was uploaded to Canvas. I did all my writeups in LyX, a GUI editor (yeah, I know) for LaTeX.
One thing that had changed very recently, not about SFSU, but about the world, was the availability of large language models, which had in the GPT-4 era become good enough to be useful tutors on standard undergrad material. They definitely weren't totally reliable, but human tutors aren't always reliable, either. I adopted the policy that I was allowed to consult LLMs for a hint when I got stuck on homework assignments, citing the fact that I had gotten help in my writeup. Prof. Schuster didn't object when I inquired about the propriety of this at office hours. (I also cited office-hours hints in my writeups.)
I got along fine with the other students, but do not seem to have formed any lasting friendships. The culture of school didn't feel quite as bad as I remembered. It's unclear to me how much of this is due to my memory having stored a hostile caricature, and how much is due to my being less sensitive to it this time. When I was at SFSU a dozen years ago, I remember seething with hatred at how everyone talked about their studies in terms of classes and teachers and grades, rather than about the subject matter in itself. There was still a lot of that—bad enough that I complained about it at every opportunity—but I wasn't seething with hatred anymore, as if I had come to terms with it as mere dysfunction and not sacrilege. I only cried while complaining about it a couple times.
-One of my signature gripes was about the way people in the department habitually refered to courses by number rather than title, which felt [like something out of a dystopian YA novel](https://tvtropes.org/pmwiki/pmwiki.php/Main/YouAreNumberSix). A course title like "Real Analysis II" at least communicates that the students are working on real analysis, even if the opaque "II" doesn't expose which real-analytic topics are covered. In contrast, a course number like "MATH 770" _doesn't mean anything_ outside of SFSU's bureaucracy. It isn't how people would talk if they believed there was a subject matter worth knowing about except insofar as the customs of bureaucratic servitude demanded it.
+One of my signature gripes was about the way people in the department habitually refered to courses by number rather than title, which felt [like something out of a dystopian YA novel](https://tvtropes.org/pmwiki/pmwiki.php/Main/YouAreNumberSix). A course title like "Real Analysis II" at least communicates that the students are working on real analysis, even if the opaque "II" doesn't expose which real-analytic topics are covered. In contrast, a course number like "MATH 770" doesn't mean anything outside of SFSU's bureaucracy. It isn't how people would talk if they believed there was a subject matter worth knowing about except insofar as the customs of bureaucratic servitude demanded it.
There were two examinations: a midterm, and the final. Each involved stating some definitions, identifying some propositions as true or false with a brief justification, and writing two or three proofs. A reference sheet was allowed, which made the definitions portion somewhat farcical as a test of anything more than having bothered to prepare a reference sheet. (I objected to Prof. Schuster calling it a "cheat sheet." Since he was allowing it, it's wasn't "cheating"!)
I can only speculate that the occurrence of a student pointing out something about mathematical reality that wasn't on the test or syllabus was so unexpected, so beyond what everyone had been conditioned to think school was about, that no one had any context to make sense of it. A graduate statistics class at San Francisco State University just wasn't that kind of space. I did get an A.
-#### The Putnam Exam
+#### The 85th William Lowell Putnam Mathematical Competition
I also organized a team for the Putnam Competition, SFSU's first in institutional memory. (I'm really proud of [my recruitment advertisements](http://zackmdavis.net/blog/2025/01/recruitment-advertisements-for-the-2024-putnam-competition-at-san-francisco-state-university/) to the math majors' mailing list.) The story of the Putnam effort has been recounted in a separate post, ["The End of the Movie: SF State's 2024 Putnam Competition Team, A Retrospective"](http://zackmdavis.net/blog/2025/01/the-end-of-the-movie-sf-state-2024-putnam-competition-team-a-retrospective/).
#### "Measure and Integration" (Spring 2025)
-I had a busy semester planned for Spring 2025, with two graduate-level (true graduate-level, not cross-listed) analysis courses plus three gen-ed courses that I needed to graduate. (Following Prof. Schuster, I'm humorously counting "Modern Algebra I" as a gen-ed course.) I only needed one upper-division undergrad math course other than "Modern Algebra I" to graduate, but while I was at the University for one more semester, I was intent on getting my money's worth. (The registration webapp initially rebuffed me with "Must be graduate students [_sic_] to enroll in this course", but Prof. Schuster was happy to give me a permission code, and Prof. Lai offered no resistance when I asked for one at the end of the first class, pointing out that I had gotten A in "Real II" the previous semester.) I aspired to get a head start (ideally on all three math courses) over winter break and checked out a complex analysis book with exercise solutions from the library, but only ended up getting much traction on measure theory, doing some exercises from chapter 14 of Schröder, "Integration on Measure Spaces".
-
-[TODO: temporal sequencing of the registration anecdote; readers haven't been introduced to Prof. Lai yet]
+I had a busy semester planned for Spring 2025, with two graduate-level (true graduate-level, not cross-listed) analysis courses plus three gen-ed courses that I needed to graduate. (Following Prof. Schuster, I'm humorously counting "Modern Algebra I" as a gen-ed course.) I only needed one upper-division undergrad math course other than "Modern Algebra I" to graduate, but while I was at the University for one more semester, I was intent on getting my money's worth. I aspired to get a head start (ideally on all three math courses) over winter break and checked out a complex analysis book with exercise solutions from the library, but only ended up getting much traction on measure theory, doing some exercises from chapter 14 of Schröder, "Integration on Measure Spaces".
Prof. Schuster was teaching "Measure and Integration" ("MATH 710"). It was less intimate than "Real II" the previous semester, with a number of students in the teens. The class met at 9:30 a.m. on Tuesdays and Thursdays, which I found inconveniently early in the morning given my hour-and-twenty-minute BART-and-bus commute. I was late the first day. After running into to the room, I put the printout of my exercises from Schröder on the instructor's desk and said, "Homework." Prof. Schuster looked surprised for a moment, then accepted it without a word.
I would frequently be a few minutes late throughout the semester. One day, the BART had trouble while my train was in downtown San Francisco, and it wasn't clear when it would move again. I got off and summoned a Waymo driverless taxi to take me the rest of the way to the University. We were covering the Cantor set that day, and I rushed in with more than half the class period over. "Sorry, someone deleted the middle third of the train," I said.
-Measure theory was a test of faith which I'm not sure I passed. Everyone who reads _Wikipedia_ knows about the notorious axiom of choice. This was the part of the curriculum in which the axiom of choice becomes relevant. It impressed upon me that as much as I like analysis as an intellectual activity, I ... don't necessarily believe in this stuff? We go to all this work to define sigma-algebras in order to rule out pathological sets whose elements _cannot be written down because they're defined using the axiom of choice_. You could argue that it's not worse than uncountable sets, and that alternatives to classical mathematics just end up biting different bullets. (In computable analysis, equality ends up being uncomputable, because there's no limit on how many decimal places you would need to check for a tiny difference between two almost-equal numbers. For related reasons, all computable functions are continuous.) But I'm not necessarily happy about the situation.
+Measure theory was a test of faith which I'm not sure I passed. Everyone who reads _Wikipedia_ knows about the notorious axiom of choice. This was the part of the school curriculum in which the axiom of choice becomes relevant. It impressed upon me that as much as I like analysis as an intellectual activity, I ... don't necessarily believe in this stuff? We go to all this work to define sigma-algebras in order to rule out pathological sets whose elements _cannot be written down because they're defined using the axiom of choice_. You could argue that it's not worse than uncountable sets, and that alternatives to classical mathematics just end up biting different bullets. (In computable analysis, equality ends up being uncomputable, because there's no limit on how many decimal places you would need to check for a tiny difference between two almost-equal numbers. For related reasons, all computable functions are continuous.) But I'm not necessarily happy about the situation.
I did okay. I was late on some of the assignments (and didn't entirely finish assignments #9 and #10), but the TA was late in grading them, too. I posted a 31/40 (77.5%) on the midterm. I was expecting to get around 80% on the final based on my previous performance on Prof. Schuster's examinations, but I ended up posting a 48/50 (96%), locking in an A for the course.
Prof. Lai didn't object to my LLM hint usage policy when I inquired about it at office hours. I still felt bad about how much external help I needed just to get through the assignments. The fact that I footnoted everything meant that I wasn't being dishonest. (In his feedback on assignment #7, Prof. Lai wrote to me, "I like your footnote. Very genuine and is a modern way of learning math.") It still felt humiliating to turn in work with _so many_ footnotes: "Thanks to OpenAI o3-mini-high for hints", "Thanks to Claude Sonnet 3.7 for guidance", "Thanks to [classmate's name] for this insight", "Thanks to the "Harmonic Conjugate" _Wikipedia_ article", "This is pointed out in Tristan Needham's _Visual Complex Analysis_, p. [...]", _&c._
-If I were just trying to learn, it wouldn't have been an issue. I look things up all the time when I'm working on something I care about, but the institutional context of submitting an assignment for a grade seemed to introduce the kind of moral ambiguity that had made school so unbearable to me, in a way that didn't feel fully mitigated by the transparent footnotes.
-
-I told myself not to worry about it. Prof. Lai had said in class and in office hours that he trusted us, that he trusted me. The purpose of the "assignment" was to help us to learn about the theory of functions of a complex variable, and I was doing that. If I had wanted to avoid this particular source of moral ambiguity at all costs, but still wanted a Bachelor's degree, I could have taken easier classes for which I wouldn't need so much external assistance. But that would be insane. The thing I was doing now, of jointly trying to maximize math knowledge while also participating in the standard system to help with that, made sense. Minimizing perceived moral ambiguity (which was all in my head) would have just been a really stupid goal. Now, so late in life at age 37, I wanted to give myself fully over to not being stupid, even unto the cost of perceived moral ambiguity which was all in my head.
+It's been said that the real-world usefulness of LLM agents has been limited by low reliability impeding the [horizon length of tasks](https://metr.org/blog/2025-03-19-measuring-ai-ability-to-complete-long-tasks/): if the agent can only successfully complete a single step with probability 0.9, then its probability of succeeding on a task that requires ten correct steps in sequence is only 0.9<sup>10</sup> ≈ 0.35.
-Prof. Lai eschewed in-person exams in favor of take-homes for both the midterm and the final. He said reasonable internet reference usage was allowed, as with the assignments. I didn't ask for further clarification, but resolved to myself that for the take-homes, I would allow myself static websites but obviously no LLMs. (I had already neurotically asked for clarification about the assignments once more than was necessary. I suspect he would have allowed LLMs if I had asked—I didn't get the sense that he yet understood the edge that the latest models offered over mere books and websites—but even asking would have been undignified.)
+That was about how I felt with math. Schuster was assigning short horizon-length problems from Axler, which I could mostly do independently; Lai was assigning longer horizon-length problems from Stein and Shakarchi, which I mostly couldn't. All the individual steps made sense once explained, but I could only generate so many steps before getting stuck.
+
+If I were just trying to learn, the external help wouldn't have seemed like a moral issue. I look things up all the time when I'm working on something I care about, but the institutional context of submitting an assignment for a grade seemed to introduce the kind of moral ambiguity that had made school so unbearable to me, in a way that didn't feel fully mitigated by the transparent footnotes.
-I got a 73 on the midterm. [TODO: describe errors, my takehome/midterm split looks bad]
+I told myself not to worry about it. The purpose of the "assignment" was to help us to learn about the theory of functions of a complex variable, and I was doing that. Prof. Lai had said in class and in office hours that he trusted us, that he trusted me. If I had wanted to avoid this particular source of moral ambiguity at all costs, but still wanted a Bachelor's degree, I could have taken easier classes for which I wouldn't need so much external assistance. But that would be insane. The thing I was doing now, of jointly trying to maximize math knowledge while also participating in the standard system to help with that, made sense. Minimizing perceived moral ambiguity (which was all in my head) would have just been a really stupid goal. Now, so late in life at age 37, I wanted to give myself fully over to not being stupid, even unto the cost of perceived moral ambiguity which was all in my head.
-There were only 9 homework assignments during the semester (contrasted to 12 in "Measure and Integration") to give us time to work on an expository paper and presentation on one of the Gamma function, the Reimann zeta function, the prime number theorem, or elliptic functions. I wrote four pages on "Pinpointing the Generalized Factorial", explaining the significance of the Gamma function, except that I'm not fond of how the definition is shifted by one from what anyone would expect, so I wrote about the unshifted Pi function instead. I wish I had allocated more time to it.
+Prof. Lai eschewed in-person exams in favor of take-homes for both the midterm and the final. He said reasonable internet reference usage was allowed, as with the assignments. I didn't ask for further clarification, but resolved to myself that for the take-homes, I would allow myself static websites but obviously no LLMs. I had already neurotically asked for clarification about the policy for the assignments once more than was necessary. I wasn't a grade-grubber; I would give myself the authentic 2010s take-home exam experience and accept the outcome.
-This was my one opporunity to "write a paper" and not merely "complete an assignment"
+(I suspect Prof. Lai would have allowed LLMs on the midterm if I had asked—I didn't get the sense that he yet understood the edge that the latest models offered over mere books and websites. On 29 April, a friend told me that instructors will increasingly just assume students are cheating with LLMs anyway; anything that showed I put thought in would be refreshing. I said that for this particular class and professor, I thought I was a semester or two early for that. In fact, I was two weeks early: on 13 May, Prof. Lai remarked before class and in the conference room during Prof. Schuster's office hours that he had given a bunch of analysis problem to Gemini the previous night, and it got them all right.)
-[TODO: 95/100 final, math.SE use, "don't worry, everyone will get a good grade"; getting an A; I could have used the inverse function theorem
+I got a 73 on the midterm. Even with the (static) internet, sometimes I would hit a spot where I got stuck and couldn't get unstuck in a reasonable amount of time.
-> #4 Yes. The key is all closed loops either wrap both poles and does not wrap any pole. Therefore, primitive exists. I could put a point off here, but this is a subtle point, so let's make it OK.
-]
+There were only 9 homework assignments during the semester (contrasted to 12 in "Measure and Integration") to give us time to work on an expository paper and presentation on one of either the Gamma function, the Reimann zeta function, the prime number theorem, or elliptic functions. I wrote four pages on "Pinpointing the Generalized Factorial", explaining the motivation of the Gamma function, except that I'm not fond of how the definition is shifted by one from what anyone would expect, so I wrote about the unshifted Pi function instead.
-[TODO: Discord log of Prof. Lai's reaction to frontier LLMs
+I wish I had allocated more time to it. This was my one opportunity in my institutionalized math career to "write a paper" and not merely "complete an assignment"; it would have been vindicating to go over and above knocking this one out of the park. (Expository work had been the lifeblood of my non-institutionalized math life.) There was so more I could have said about the generalized factorial, and applications (like the fractional calculus), but it was a busy semester and I didn't get to it all. It's hardly an excuse that Prof. Lai wrote an approving comment and gave me full credit for those four pages.
-4/29 discussion
-Fyodorov — 4/29/25, 9:13 PM
-Nah it's fine
-That's exactly the kind of thing I might do.
-After all, instructors will increasingly just assume you cheated with LLMs anyway, anything that shows you put actual thought in will be refreshing.
-"He actually bothered to ask the LLM to explain it rather than just cheating? Wow."
-Zack M. Davis — 4/29/25, 9:14 PM
-for this particular class/prof, I think I'm a semester or two early
+I was resolved to do better on the take-home final than the take-home midterm, but it was a struggle. I eventually got everything, but what I submitted ended up having five footnotes to various _math.stackexchange.com_ answers. (I was very transparent about my reasoning process; no one could accuse me of dishonesty.) For one problem, I ended up using formulas for the modulus of the derivative of a Bashke factor at 0 and the preimage of zero which I found in David C. Ulrich's _Complex Made Simple_ from the University library. It wasn't until after I submitted my work that I realized that the explicit formulas had been unnecessary; the fact that they were inverses followed from the inverse function theorem.
-5/13
-... two weeks early (before class and in the conference room, this prof was remarking with surprise to people that he tried giving a bunch of analysis problems to Gemini last night and it got them right)
+Prof. Lai gave me 95/100 on the final, and an A in the course. I think he was being lenient with the points. Looking over the work I had submitted throughout the semester, I don't think it would have been an A at Berkeley (or Princeton).
-]
+I guess that's okay because grades aren't real, but the work was real. If Prof. Lai had faced a dilemma between watering down either the grading scale or the course content in order to accomodate SFSU students being retarded, I'm glad he chose to preserve the integrity of the content.
#### "Modern Algebra I" (Spring 2025)
-One of the quirks of being an autodidact is that it's easy to end up with an "unbalanced" skill profile relative to what school authorities expect. As a student of mathematics, I consider myself more of an analyst than an algebraist and had not previously prioritized learning abstract algebra nor (what the school authorities cared about) "taking" an algebra "class", neither last semester nor in Fall 2012/Spring 2013. (Over the years, I had taken a few [desultory swings at Dummit & Foote](http://zackmdavis.net/blog/2019/05/group-theory-for-wellness-i/), but never got very far.)
+One of the quirks of being an autodidact is that it's easy to end up with an "unbalanced" skill profile relative to what school authorities expect. As a student of mathematics, I consider myself more of an analyst than an algebraist and had not previously prioritized learning abstract algebra nor (what the school authorities cared about) "taking" an algebra "class", neither the previous semester nor in Fall 2012/Spring 2013. (Over the years, I had taken a few [desultory swings at Dummit & Foote](http://zackmdavis.net/blog/2019/05/group-theory-for-wellness-i/), but had never gotten very far.) I thus found myself in Prof. Dusty Ross's "Modern Algebra I" ("MATH 335"), the last "core" course I needed to graduate.
+
+Prof. Ross is an outstanding schoolteacher, the best I encountered at SFSU. I choose my words here very carefully. I don't mean he was my favorite professor. I mean that he was good at his job as understood in Society: his lectures were clear and well-prepared, and puncutated with group work on well-designed worksheets (pedogogically superior to the whole class just being lecture). The assignments and tests were fair, and so on.
-But "Modern Algebra I" was a graduation requirement,
+"Modern Algebra I" met on Monday, Wednesday, and Friday. All of my other classes met Tuesdays and Thursdays. I had wondered whether I could save myself a lot of commuting by ditching algebra most of the time, but started off the semester dutifully attending—and, as long as I was on campus that day anyway, also sitting in on Prof. Ross's "Topology" ("MATH 450") and doing the worksheet with an acquaintance in that class, even though I couldn't commit to a fourth math course for credit.
-so I found myself in Prof. Dusty Ross
+On Monday of the second week, Prof. Ross stopped me after class to express disapproval with how I had brought out my copy of Dummit & Foote and referred to Lagrange's theorem during the group worksheet discussion about subgroups of cyclic groups; we hadn't covered that yet. I asked whether he cared whether I attended class, and he said that the answer was already in the syllabus. (Attendance was worth 5% of the grade.) After that, I mostly stayed home on Mondays, Wednesdays, and Fridays unless there was a quiz, and didn't show up to topology again.
- * the course numbers are actually informative
+The course covered the basics of group theory, with a little bit about rings at the end of the semester. The textbook was Joseph A. Gallian's _Contemporary Abstract Algebra_, which I found to be in insultingly poor taste. The contrast between "Modern Algebra I" ("MATH 335") and "Theory of Functions of a Complex Variable" ("MATH 730") that semester did persuade me that the course numbers did have semantic content in their first digit (3xx = insulting, 4xx or cross-listed 4xx/7xx = requires effort, 7xx = potentially punishing).
-Prof. Ross is an outstanding schoolteacher, the best I encountered at SFSU. I choose my words here very carefully. _My_ favorite professor was Alex Schuster.
+I mostly treated the algebra algebra coursework as an afterthought to the analysis courses I was devoting most of my focus to. I tried to maintain a lead on the weekly algebra assignments (five problems hand-picked by Prof. Ross, not from Gallian), submitting them an average of 5.9 days early—in the spirit of getting it out of the way. One week I started working on the prequisite chapter on polynomial rings from the algebraic geometry book Prof. Ross had just written with his partner Prof. Clader, but that was just to show off to Prof. Ross at office hours that I had at least looked at his book; I didn't stick with it.
-Prof. Ross is a better teacher, but Prof. Schuster is a better person, because Dusty Ross _believes_ in San Francisco State University; Alex Schuster just works there.
* algebra tutoring and punch card
+
### Not Sweating the Fake Stuff (Non-Math)
#### Queer Literatures and Media (Fall 2024)
* being more aggressive about working the system
* Peter Verdone https://www.peterverdone.com/academia-math-trans-and-a-ton-of-other-stuff/
* it was chance that I ended up deciding to finish before moving; finishing at Reno would be harder
-
+ * I would have leaned on office hours more if LLMs didn't exist
+ * the struggle and external resources are probably normal
Afterwards, Prof. Schuster encouraged me via email to at least consider grad school, saying that I seemed comparable talent-wise to his peers in the University of Michigan Ph.D. program (which was ranked #10 in the U.S. at that time in the late '90s). I demurred: I said I would consider it if circumstances were otherwise, but in contrast to the last two semesters to finish undergrad, grad school didn't pass a cost-benefit analysis.
projects / An_Algorithmic_Lucidity.git / commitdiff