Framing mathematics

No, not framing it for a crime. I just read the article A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today by Frank Quinn (math professor). It establishes a surrounding explanation for how modern mathematics came to be structured as it is. In the process, it contrasts two ways of thinking about mathematics: the "old" style, and mathematical sciences, wherein math relates to observable facts and is intuitive (but often yields incorrect results), and the "new" style, which he calls the "core" of mathematical research, wherein math is the study of abstract rules which do not relate to reality (but whose results are provably, rigorously correct).

It is fascinating.

I had never paused to consider the evolution of my discipline. Yet Prof. Quinn highlights and summarizes my experience of grade-school math education: everything seemed much clearer and more reasonable when -- finally! -- worked as abstract symbols according to rules. This is how I learned geometry (my first proofs!), trigonometry, and calculus. I cannot imagine attempting to learn calculus through intuition. What terror! (Does this infinite series feel like it converges? What's your hunch about the derivative of f(x)?) And of course now in retrospect I think of the math I learned earlier -- multiplication, fractions, arithmetic -- in the more advanced terms I learned later.

The article was summarized for me by this: "the old dysfunction was invisible, whereas the new opacity is obvious." Yes, math is opaque; I've studied for years and this is the first thing I'd admit. And my topics are squarely in the "new/core" section: I've done research in precise definitions, logical proofs, completeness. Carefully justifying each step is a technique that I use in my dreams. Math for me has always been its own arena of knowledge, one of three (the humanities, sciences, and math), each with its own methods. Even though we use the science word "discovery," a mathematical discovery is nothing of the sort. And as a grad student, I am amazed at what other grad students do as "research."


This post's theme word is anemometer, "an instrument for measuring the speed of wind." This magical anemometer predicts the trends in sociology research!

Little sigh of relief

I just finished writing a paper. It's part of my qualification for candidacy so it has some import; on the other hand, professors and students alike tell me it doesn't really matter and so it might have less import. On the third hand -- the gripping hand -- or whatever noodly appendage is available -- this paper is only one part of my qualification: the rest is a presentation-and-grilling session, for which I now prepare. To quote Parade, "This is not over yet."

The writing felt like pulling teeth. There is something about academic writing that I truly dislike. It does not merely lack poetry; the poetry is bruised, abused, and forcibly expurgated. Sentences are dry and declarative. Words have one particular meaning and that meaning is sharply constrained to the topics under discussion. Having finished this piece of writing (at least for now), I want to write escapist fantasy. I want to be a novelist. I want to be a sculptor. I want to leave my keyboard to cool on my desk, and shortly I will.

But before I go, here's what I plan to do. WHEREAS academia is dessicating my delight in words, phrases, sentences, and constructions, I PROPOSE to begin a project, to be EXECUTED in my leisure moments, CONSISTING of one story, or many stories, using the sentences from the Bulwer-Lytton contests (EITHER as opening sentences OR as sentences appearing elsewhere), THEREFORE reviving my enjoyment of life.


This post's theme word is skueomorph, "a derivative object which retains ornamental design cues to a structure that was necessary in the original." Like copper-coated zinc pennies. That story is only a skueomorph of the inspirational first sentence.

Stupidity in scientific research

I just read Martin Schwartz's note "The importance of stupidity in scientific research." In simple language, it explains how graduate school is different from all the preceding schooling. Anyone in graduate school would do well to go read it right now. Highlights:

Doing significant research is intrinsically hard... We can't be sure whether we're asking the right question... if we don't feel stupid it means we're not really trying. ... Science involves confronting our `absolute stupidity'. That kind of stupidity is an existential fact, inherent in our efforts to push our way into the unknown.

At the end, he notes that "reasonable levels of confidence and emotional resilience help" which I think is a vast understatement. What else will keep you eagerly stupid, if not confidence and emotional resilience? -- especially since there is a whole world out there where the ex-graduate student can feel quite intelligent on a daily basis.

Part of what I've puzzled out in my graduate studies so far has been this: how do the professors do it? They seem to know which questions to ask in order to achieve meaningful answers and progress in research. Some of it is the buckshot approach: they have had the time to ask a hundred little questions, and the odds are in their favor: some of them hit a research target and became published papers. But also, they have an ineffable sense of which sorts of problems are good for research: this is summarized in the adjective "interesting," as in "that is an interesting research question" or "that is an interesting approach to this problem." I've learned that "interesting" is a key word, in bright flashing red letters, that indicates I'm doing Something Right with my research and should Keep It Up.

So while I curate my sense of academic stupidity, I'll continue to rely on my advisors' subtle (subconscious?) nudges and explicit advice.


This post's theme word is nescient, "lacking knowledge or awareness." The experts in the field are those most aware of their own nescience.

Happy Thanksgiving!

Today we celebrate our gluttonous excesses and the (late) harvest. It may interest you to know that one deadly sin (gluttony) leads to another (lust). Science proves it! (Via slashdot.) In particular, these scientists had men smell various food odors and measured the subsequent arousal.

They found that "the number one odor that enhanced penile blood flow was a combination of lavender and pumpkin pie," which "increased penile blood flow by an average of 40 percent."

Temper your excitement, baker women! Because "every odor we tested aroused the participants... nothing turns a man off."

So, nevermind. Science simply found that men are aroused by smelling things and having their penile blood flow measured. Whoopee. (A summary of their paper is available here. Black licorice is surprisingly sexy-smelling.)


This post's theme word is hyaloid, "transparent." This post's bonus theme word is plethysmograph.
This post written like: Dan Brown.

Acceptance

Another acceptance, at a big conference this time. Whoo!


This post's theme word: convoke, "to call together for a meeting."

Git r done

This month feels like an aggregation of small and irksome tasks -- emails to act upon, articles to read, travel to arrange, deeds to follow-up, essays to write. I keep making ε-progress on the small tasks, context-switching too rapidly to accomplish something meaningful, juggling these many things while also trying to make large progress on my large task (research!). It takes a lot of brain space to keep so many things loaded in my RAM.

In the spirit of Merlin Mann's Inbox Zero, my rallying cry for the remainder of the month is thus: git r done! I aim to finish these tasks and clear all the irritation out of my mental cache.


This post's theme comic is from SMBC:

First acceptance

It's just a workshop, but still: first acceptance. Yay.


This post's theme word: legerdemain, "a slight of hand" or "a display of skill." Very apropros for imposter syndrome.

VC-dimension?

Do you know anything about computing VC-dimension? Or about the Sauer-Shelah lemma?

Both Wikipedia and the CC blog state the lemma in a way that I do not understand. Namely, they define the VC-dimension as, at some point, determined by the cardinality of the intersection of the set {x₁,…,x_k} with some other set. And both sources upper bound this number by 2^k, which seems absurd: how can it possibly be more than k? We have indexed the set; it has exactly k elements; set intersection is a strictly nonincreasing function in terms of set size.

It also seems unlikely to me that both these sources would have typos. So I must be misunderstanding something.

If you know something about VC-dimension, the Sauer-Shelah lemma, or related combinatorics, please send me an email and I shall grill you with my questions (seasoned with confusion... and paprika, if you like).

Update: I figured it out with the help of these lecture notes. It was an issue of misunderstood notation. (My bad, as usual.)


This post's theme word: curtilage, "an area of land encompassing a dwelling and its surrounding yard, considered as enclosed whether fenced or not." In my present frame of mind, it relates to set theory.

First rejection

I just received my first paper rejection. Ah, well. We pick ourselves up and move along.


This post's theme word: moirologist, "a hired mourner."

Seminar skills

I gave a seminar today, entitled "An introduction to Kolmogorov complexity." The abstract I provided:

This week, I'll cover some introductory Kolmogorov complexity (including how to pronounce it!), definitions and applications to complexity theory,
including the relation of Kolmogorov complexity to the halting problem, and defining resource-bounded computational hierarchies from Kolmogorov
complexity. No background knowledge of Kolmogorov complexity is required; this seminar will be self-contained.

General consensus? It went very well. There was a lot of audience participation. Perhaps too much, since it got a bit derailed with people trying to explain each others' questions and answer them. Afterwards, more senior grad students offered me this advice:

• Never admit you are wrong. Never erase and edit what you've written on the board. I wrote one thing wrong and then 10 minutes were wasted fiddling with it. Relatedly,
• Don't answer all the questions. Make sure everyone has a basic understanding. If the question is about details that won't improve a basic understanding, postpone it until after the seminar is over.
• Proof by assertion. Related to the above two points. If there are too many details, or you don't quite remember how to prove it, or it's too hard, or whatever, then just say, "Obviously, ..." and move on before anyone derails you.
  
• Don't let audience members talk amongst themselves.
  

Obviously some of these should be applied judiciously. I'll have to work on these points, now that I seem to have the basics down.


This post's theme word: expatiate, "to speak or write at length" or "to move about freely."

Swamped

I was sick last week, and now I feel swamped with the tasks before me.


This post's theme word: coeval, "having the same age or duration."

Award-winning students

At CCC (currently happening!), two of Steve Cook's students won awards. Former PhD Mark Braverman won best paper, and current PhD Aktioshi Kawamura won best student paper. That's great! I am re-intimidated by them. (Hat tip: Lance Fortnow.)


This post's theme word: hagiography, "a biography that idealizes or idolizes the person (especially a person who is a saint)."

What makes us happy?

I just finished reading Joshua Wolf Shenk's article "What Makes Us Happy?" from the Atlantic. It was interesting, for all science is interesting, and it focused on Harvard and psychology (and, tangentially, the challenges of an academic career and obtaining funding). But I think that what makes the article most interesting is a self-interested desire for the reader (me!) to compare my life with those described, and see how I measure up. With a title like "What Makes Us Happy?" I would expect that the article answers. What I'm looking for is a rubric for happiness, so that I can check off some boxes and -- bam! -- be happy.

Of course, there's not an answer, though several are presented. The now-lead researcher Valliant named seven "major factors that predict healthy aging:"

Employing mature adaptations was one. The others were education, stable marriage, not smoking, not abusing alcohol, some exercise, and healthy weight.

These are all forehead-smackingly obvious, and I do pretty well by this metric (though my education is not yet complete and I'm unmarried). But later the article reports that good adjustment in youth/early adulthood (right where I am) is no predictor for health and happiness in old(er) age.

In an interview in the March 2008 newsletter to the Grant Study subjects, Vaillant was asked, “What have you learned from the Grant Study men?” Vaillant’s response: “That the only thing that really matters in life are your relationships to other people.”

Again, this seems obvious. (I have to work on this one.)

Shenk's unacknowledged assumption in writing the article is that everyone wants to live to a healthy, happy, bad-memory-obliterating old age. It seems like one of Valliant's goals for the study is to determine some guidelines, or at least predictive factors, for achieving healthy, happy old age. We'll never find out what the self-evaluations are of the study participants who died young. If self-evaluations are what really matters, then those who lived and lived happily are successful, merely by the fact of their happiness.

Bah, humbug. The article was interesting -- I'd recommend that you read it -- but questionable in its romantic portrayal of the way that valiant Valliant kept the study alive. And the "poetry" of the very subjective parts of the study. The ideas of a scientific study is divorced from this researcher-dependent qualitative study, but this makes for such fun reading. (Hat tip: postpostpre via A. via C.)


This post's theme word: kismet, "ineluctable destiny, fate."

Good progress

I have of late, but wherefore I know not, lost all my mirth. And by "mirth," I mean "enthusiasm for my current research project." And it is a slight lie to claim that wherefore I know not. I have some musings on "wherefore."

My advisor tells me I am making "good progress," but I am doubtful. (Note: advisor's word = pronouncement of truth, and can be accepted as true axiomatically in every model. I AM MAKING GOOD PROGRESS.)

My project is insubstantial, as so much theoretical computer science is. I'm not building anything real: I'm not gluing or sawing or constructing anything that can be held in the hands, passed around in a circle, plugged into a socket, or brought in for show-and-tell. The only sense in which I'm building is a metaphorical one: I'm building mathematical tools. I'm holding theories in my hands and passing them around for comment. I plug one logical system into another. I write on blackboards and tell people about this mathematical project I'm constructing. It's exceedingly abstract, even in metaphor.

If I were to wink out of existence, little of my research would endure. Some papers, some files. But most of my project is in my brain: the motivations, the way each little mathematical tendril wraps around another thing to root my project in the context of significance to computer science.

I think I need to get some Lego robotics kits and spend some time outside. Maybe find a pottery studio -- that kinetic therapy improved my abstracted, mathematical senior[thesis!] year. This frigidly air-conditioned cinderblock cell where I work is responsible for this funk. It's beautiful outside. Maybe next summer I'll get a job as a bike messenger. I'll deliver your messages promptly, with a quick side-dish of context-appropriate Shakespeare. ("News from Verona! How now, Balthasar? How doth my lady?")


This post's theme word: recondite, "abstruse." (That is, "difficult to understand.") Abstruse is the 13th most-looked-up word on the New York Times website (via MetaFilter).

Victoria Day!

I had no idea that today was a holiday. Lo! It is. My calendar told me so. It also told me that I had a meeting with my advisor this morning. We were the only people in the (locked) building. (More international students showed up later.) We were both rather vague on the purpose and scheduling of Victoria Day, facts which I could look up now. But I leave that task to the reader.

The meeting went well. Then I worked, typing up my results of last week and reading about becoming a permanent resident. It will make me eligible for more scholarships and lower tuition.

Later today I walked around downtown and enjoyed the sun and cool weather. And resolved to write here more frequently. I have a few ideas to write up this week.


This post's theme word: zephyr, "a gentle breeze from the west."

Organizing papers

I am looking for a way to organize the academic papers that have, over the past year or so, accumulated in big snowdrifts across my desk and shelves. A digital way. I can store them - as my advisor does - in filing cabinets alphabetically by author, but this is less searchable than I'd like. What I really want is some kind of already-customized personal wiki that I can use to store pdf/dvi files of papers, along with some metadata, like a short, searchable summary: title, authors, year, publication name, and maybe my own summary of the results and usefulness of the paper. Enough to remind me of reading the paper, and in a format that is easily searchable.

A private wiki seems to reasonably satisfy my requirements. In a wiki, I could cross-link articles that cite each other, and even do fancier things like have a page where I list all the claims in my M. Sc. research paper and link to the pages for their citations.

Does such software exist? I am surely not the first academic to need and think of such a thing.

I looked at Bibsonomy, but it is not quite what I have in mind. I don't need a social site, or even an online service. Ideally, it would be private and I could maintain it on my own.


This post's theme comic:

Presenting research

I don't like presenting my research. Not because it's unpleasant, but because I psych myself out beforehand and then shake during the presentation. I thought that stage-fright was supposed to decrease with age.

I'm not shy. I think I just fear being judged on something that I really care about. I like my research and want it to be good, accessible, and accepted. I am a decent public speaker, too, I just clam up when I have to present math/CS -- it involves thinking hard while also making a coherent presentation. I just need more practice.

Next presentation? In 15 days.


This post's theme cartoon:

"You are nominated for a Ph. D."

That's the promising title of an email I found in my inbox this morning. Unfortunately, it was spam:

WHAT A GREAT IDEA! We provide a concept that will allow anyone with sufficient work experience to obtain a fully verifiable University Degree. Bachelors, Masters or even a Doctorate.

Maybe I'm going about this all wrong -- years of work, supervision, and learning. All wrong! I could be earning money at "work experience" and also inching closer to a doctorate!

Today I signed more paperwork and received an assignment from one of the department secretaries: assemble my Ph. D. committee. I have to have an oral exam soon. Yikes.


This post's theme word: maugre/mauger, "in spite of." It's a preposition, as in, "Mauger all the solicitous emails, I have decided to get a[nother] degree the hard way."

Master of Science

This morning, after a long writing and revision period, my advisor (S.) declared my M. Sc. thesis research paper officially dead done. Whoopee. I am almost two months late on the departmental deadline (although I have been granted preemptive clemency), and I was feeling pretty unenthusiastic about finishing.

In talking about the broader scope of the project, S. commented, "This project turned out to be much harder than I thought it would be." Oh! Good, now I feel alright that I was so slow to iron out the details in writing. Then he picked up the document -- nearly 30 pages! -- and said, "It's hefty."

And *poof* I felt much better. Whoopee!

A while back, closer to the "real" departmental deadline for M. Sc. research papers, C. tagged me for his "wordle your thesis" meme. Wordle is cute, but easily manipulated. Given a bunch of text, it creates a "word cloud" of the most frequently-used words, with the more frequent words indicated by larger size.

The wordle of my research paper (pdf):

Even though wordle just uses frequency count, it conveys some meaningful information (not a lot). I like that you can tell, just from this word cloud, that my paper is about strings, functions, matrices, numbers, definitions, axioms and theory. I grabbed this text from the pdf, so some symbols didn't convey properly -- like \SigmaB.


The wordle of my research paper (LaTeX):

This tells you that I used a lot of mathcal, Sigma, vec, and wedge. It's cute that some of my own defined tags/symbols got such usage (parityL). In comparing the two word clouds, one can see that I have a very LaTeX-heavy paper: the actual words of the paper are crowded out of the top-150-most-frequent word count when LaTeX commands are included. "String," "matrix," "function," and "theorem" still survive, but look how large "mathcal" and "begin" and "end" are!


This post's featured quote, from S. upon reading the paperwork he has to complete:

Is 'transitioning' even a word?

His real paper dictionary from the '50s says that "transition" is a noun, with possible adjective form "transitional." There was no verb "to transition" in the '50s, apparently.

Saturday break

I'm putting my intertubes embargo on hold for a few hours this afternoon, because (1) I am worried that something important might be going overlooked, and (2) I might give myself an aneurysm trying to magically foresee the overlooked thing without checking my email. Also, (3) good for me, I got a lot of thesis stuff, done, while slowly going into connectivity withdrawal.

Oh goodness there is more than a PAGE of emails in my account! I usually maintain inbox zero, or perhaps inbox 4 or 5. While I'm ruthlessly emptying my inbox, I'll also be posting some recent stuff here, tersely. My life goes on and after awhile, the fiddly details go stale and unpostable. I'd like to post them while they're relatively fresh in my mind.

Enjoy your Saturdays, everyone. I have allotted myself this brief respite, and some ice skating this afternoon, and then the nose is back to the LaTeX grindstone.


This post's theme word: abstemious, "not self-indulgent."