## Sunday, December 01, 2013

### ARR!.. imagine computers as research collaborators?

sometimes i wonder if i should be a mathematician at all .. lately i've been reminiscing about the dreams my younger self had: among them was to be a successful novelist, perhaps in science fiction.

so when i read news articles like this one ..
"Some might argue that computers will never be able to match human ingenuity but it is difficult these days to argue they can't at least mimic many of our skills.

Take the eDavid painting robot. The computer-controlled arm - adapted from a welding machine - chooses from five brushes and 24 colours to create impressive artworks on canvas.

It works by snapping a photo of its subject matter and then making the necessary calculations to turn the image into a drawing or painting in a wide variety of styles.

Its creators admit that it has no awareness of what it is doing. But it is able to make decisions about things like shading and brushstrokes as it goes, tweaking its moves based on how the picture is evolving, rather than just creating a pre-determined image.
"

~ from "The quest to turn computers into creative artists" @bbc_tech

.. then i immediately begin to imagine the possibilities:

if the strength of a computer lies in being very efficient with a finite, fixed set of tools, then imagine if we could get computers to prove simple lemmata for us, just by giving them suitable hypotheses and a fixed set of axioms and existing lemmata ..

yes, it is hard enough to build a robust proof-checking program .. and some researchers have spent years of their lives focusing on a single, specific verification .. but i'm not talking about a universal engine:

i liken it to writing programs that play go or chess well. it's not that the computer can think on its own, but rather that it can traverse through the decision tree of possible games very efficiently. in fact, a competitive program mightn't even run through all the possibilities, but simply the games that grand masters have played before.

so imagine coding in all the basic, rigorous proofs from mathematics (e.g. the proof of the triangle inequality in Euclιdean space) and adding shortcuts into the space of all proofs. i wonder what lemmas a computer could tell me ..?

being a mathematician, sometimes i have mathematical daydreams.

## Monday, November 11, 2013

### a shot in the dark (but no updates yet: stay tuned)

(yes, it's been a while .. and no, i don't have time yet to write about what's happened since the last two posts ..)

[sighs]

i'm almost convinced that there is no good way to teach a first course in proofs to undergraduates. sometimes i even wonder if it's something that can be "taught" .. in the sense that if the student really wants to learn and to understand, then (s)he has to commit to a minimum amount of time for self-study and development.

it's like teaching someone how to be paranoid: as a skill, it only develops with time, experience, and stimuli ..

## Tuesday, September 24, 2013

### *sighs*

ye gods, i hate asking for money.

it's clearer to me now that there is a "rat race" to academia in general and to the sciences in particular. more and more i envision a future where i'll never stop writing grants and there will always be another meeting to sit in, another memorandum that i should have read (but have skimmed over, at best).

for a while i've wondered if i was cut out to be a mathematician, but i've made my peace with it now. it's been long enough that i wasn't going to cut it, then i would probably be doing something else by now.

i'm starting to wonder, though, if i'm cut out to be a professional mathematician.

the research is fine and the teaching, though time-consuming, is also fine and often enough fulfilling (if not enjoyable). as for the grants .. and the applications .. and the meetings, and so on;

i can see why many faculty "give up" upon earning tenure.

these professional aspects of the job were never advertised to me, as a ph.d. student; maybe the advisor was deliberately putting it in the background, if only so that we could have a greater focus, when working together. as a postdoc there seemed more and more of it, when discussing the nature of work with my colleagues.

who knows? maybe i've just always been naive;

my colleagues, near and far, seem quite able to maintain research as their primary focus and if anything, shape their other duties to complement this one singular priority. more and more i find this admirable.

maybe i'm just too new to this position, that these are all just growing pains, and that these shall pass with time and enough patience and a little humor. i don't know and it's hard to say.

i'm not giving up. it's just that i can see why others do.

## Friday, September 20, 2013

### ANH: from end to start, for now.

so it feels like ages since i last thought about a blog post of any kind. it seems like there's so much to saybut at the same time, none of it is really worth mentioning. that's always the difficulty of beginning a story at the beginning ..

.. so, being lazy at the moment, i'll not. i'll begin at the ending instead, which is today.

so today i gave a lecture about metric spaces to my students. it's a first course in analysis and the textbook [1] happens to cover the topic, which to me sounds like a license to expound on it for 75 minutes.

so i showed them the discrete metric on any set, and how the unit circle would look if the set were the euclidean plane. i showed them the L-infinity norm, how the unit circle looks like the usual unit square, and how short the proof is for its triangle inequality. this is in contrast to how the proof of the triangle inequality goes for the usual L-2 distance, which uses Cauchy-Schwarz and in turn, a nod to Pythagoreas's theorem.

i thought it was cool. it would be the kind of lecture that would have inspired me as a student .. but i don't know. i'm getting to know the students in my class, but i'm still learning all the time.

[1] we're using baby Rudin.

## Thursday, September 19, 2013

### ARR! more machine now, than man .. twisted and evil.

"On the one hand, today’s computers feature programming and writing tools more powerful than anything available in the twentieth century. But, in a different way, each of these tasks would be much harder: on a modern machine, each man would face a more challenging battle with distraction ... Kafka, Kerouac, and Wozniak had one advantage over us: they worked on machines that did not readily do more than one thing at a time, easily yielding to our conflicting desires. And, while distraction was surely available—say, by reading the newspaper, or chatting with friends—there was a crucial difference. Today’s machines don’t just allow distraction; they promote it. The Web calls us constantly, like a carnival barker, and the machines, instead of keeping us on task, make it easy to get drawn in—and even add their own distractions to the mix. In short: we have built a generation of “distraction machines” that make great feats of concentrated effort harder instead of easier."

~ from "HOW TODAY'S COMPUTERS WEAKEN OUR BRAIN @newyorker "

## Sunday, September 15, 2013

### on how our choices can haunt us later.

if i sit and think about it, then it feels i have a lot to say about the last two weeks, of this new job, at this university.

i don't know where to begin, though;
if i start now, then it will all come out as chaos.

maybe i've been writing too many lectures lately, and habit urges me to put some order or narrative into it. after all, life is simply a sequence of events; any additional order or structure on it is an inherently human contribution.

my guess is that it will take months for me to make sense of it all: these experiences, mistakes, small joys, and frequent setbacks. (i don't know.)

as for something small to share ..
.. during the first lecture of multivariable calculus, on a whim i decided to pronounce the letter z as zed, just like how they seem to do in europe and the u.k.

as a result, now i feel compelled to be consistent and remember, from now on, to refer to the vertical axis (in 3 dimensions) as the 'zed-axis" .. or else risk being caught as a pretentious snob!

## Saturday, September 14, 2013

### ARR!.. apparently i still have more trigοnometry to learn.

well, i learned something new today:
It sounds cumbersome now, but doing multiplication by hand requires a lot more operations than addition does. When each operation takes a nontrivial amount of time (and is prone to a nontrivial amount of error), a procedure that lets you convert multiplication into addition is a real time-saver, and it can help increase accuracy.

The secret trig functions, like logarithms, made computations easier. Versine and haversine [1] were used the most often. Near the angle $\theta = 0$, $\cos(\theta)$ is very close to $1$. If you were doing a computation that had $1-\cos(\theta)$ in it, your computation might be ruined if your cosine table didn’t have enough significant figures. To illustrate, the cosine of $5$ degrees is $0.996194698$, and the cosine of $1$ degree is $0.999847695$. The difference $\cos(1^o)-\cos(5^o)$ is $0.003652997$. If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. And a table with only three significant figures of precision would not be able to distinguish between 0 degree and 1 degree angles. In many cases, this wouldn’t matter, but it could be a problem if the errors built up over the course of a computation.

~ from "10 Secret Trig Functions Your Math Teachers Never Taught You" @sciam
in other news: it's been more than two weeks into this new job, and i still feel disoriented. often i feel exhausted, too.

on the bright side: i finally found an expensive apartment and signed a lease .. after a month of searching (and simultaneously teaching, for the last 2 1/2 weeks).

[1] these are defined, respectively, as $\textrm{versin}(\theta) = 1-\cos(\theta)$ and $\textrm{haversin}(\theta) = \frac{1}{2}\textrm{versin}(\theta)$. suggestively, "ha" mean half.

## Saturday, August 31, 2013

### ANH: first day unease.

today [0] i taught two classes, each of which were 75 minutes long. in each i felt like i was saying obvious things [1] and wondered if i was boring the students into a desperation of some kind, that i just stop talking and dismiss class early.

i don't know why, but i hate being boring ... well actually, i do know:
first, i don't want to enforce the stereotype that maths is hard and boring;
on the other hand, it takes a while for me to get anywhere interesting.

a lot of times i struggle with writing lectures because i can't convince myself of really, is this it? come on! there has to be something interesting in this whole topic..!

last night, after deliberating on and off, i finally put some lecture notes down on paper .. at about 1am. then i promptly fell asleep, glad to be rid of the task.
i don't think i succeeded. i don't think i "get" the students yet, and i don't think they "get" me.

[0] that is, on friday: i finished this post later that night.

[1] which, of course, they were .. to me. that's not a statement of arrogance; any seasoned calculus instructor would probably tell you the same.

## Monday, August 26, 2013

### ANH*: life comes first, then blogging.

.. yes, it's been pretty quiet on this blog. i guess you could say that i've been busy:
i'm teaching two courses this fall,
with plenty of preparations to do;

i'm starting a new job at a new department,
which involves figuring out how things work here;

i'm still trying to find an apartment, which is incredibly frustrating ..!
*sighs*
maybe things will calm down soon.

* this is the pre-amble that will probably lead to a year-long series, where i'll comment (read: complain) about starting a new academic life in a new department .. this time, as an assistant professor on the tenure track.

as longtime readers of this blog may recall: i get weirded out by the term "professor." in fact, during my postdoc i told my student not to call me that, due to inaccuracies. now that it's part of the job title, i suppose i can't really escape it anymore ..

.. anyway, expect this to read like a "how-NOT-to" type of guide ..!

lastly, for star wars fans it's probably clear that ANH is short for a new hope; for the longest time i thought i'd fade out from academia like obi-wan kenobi, but apparently that didn't work out .. or rather, that did work out.

## Thursday, August 15, 2013

### not quite ARR! numbers and symbols, as viewed by a man of letters.

these are some excerpts from the pleasures and sorrows of work by a. de botton, one of my favorite authors. he has a way of revealing the sublime aspects about everyday life.

anyway, this is his take on science ..
"Gone were the days of geniuses in their observatories and workshops, single-handedly rerouting scientific history. We had entered the sober era of the collaborative laboratory, where astrophysicists and aeronautical engineers banded themselves together for decade-long assaults on minor mysteries, resisting the media's attempts to raise any one of their number into a contemporary Galileo. A company might limit itself to perfecting the performance of silver-zinc batteries in zero-gravity conditions, rightly sensing the foolishness of expanding to address further puzzles in satellite electrics. A scientist might spend a lifetime examining the properties of titanium at high temperatures or the behaviour of hydrogen at the moment of ignition. The sum total of one's contributions to mankind might end up in an issue of the Journal of Advanced Propulsion Methods."
.. and this is his take on maths, viewed from the non-technical viewpoint:
"Noting my puzzlement, Ian told me that he was calculating the force of gravity at work on the cable, and that in his equation $l$ stood for the length of the span, $w$ for the effective weight per unit of length, and $T_H$ for the constant along the line. He explained that transmission engineers were unusually blessed in having at their fingertips a highly precise, efficient and universal vocabulary with which to convey even the most labyrinthine electrical scenarios, so that from Iran to Chile, $\psi$ referred to electric flux, $\mu$ to permeability, $\mathcal{P}$ to pereance, and $\alpha$ to the temperature coefficient of resistance.

I was struck by how impoverished ordinary language can be by contrast, requiring its user to arrange inordinate numbers of words in tottering and unstable piles in order to communicate meanings infinitely more basic than anything related to an electrical network. I found myself wishing that the rest of mankind would follow the engineers' example and agree on a series of symbols which could point incontrovertibly to certain elusive, vaporous, ad often painful psychological states -- a code which might help us to feel less tongue-tied and less lonely, and enable us to resolve arguments with swift and silent exchanges of equations.

There seemed to be no shortage of feelings to which the engineers' brevity might be profitably applied. If only a letter could have been identified, for example, with which elegantly to allude the strange desire one occasionally has to elicit love from people one does not even particularly like ($\beta$, say); or the irritation evoked when acquaintances seem to be more worried about one's illnesses than one is oneself ($\omega$); or the still vaguer sense one can sometimes have that different periods of one's life are in coexistence, so that one would have only to return to one's childhood home to find everything the same as it once was, with no one having died and nothing having changed ($\xi$). Possessed of such a notational system, one would be able to compress the free-floating nostalgia and anxiety fo a typical Sunday afternoon into a single pellucid and unambiguous sequence ($\beta + \omega | \xi \times 2$)" and attract sympathy and compassion from the friends around whom one would otherwise have grunted unhelpfully."