a particular MOOC that's been coming up in my feeds.

i'm curious to see how this particular course on mathematical philosophy unfolds. at first glance it reads like a discrete mathematics course that i took for a computer science major^* but with more "relevant" and "edgy" material ..

such as the mοnty hall problem and ) arrow's voting paradox (also called an "impossibility theorem").^**

another interesting aspect is the topics in game theory, such as the von Neumann-Morgenstern axioms.

looking through the syllabus, i would rather call this a "mathematics literacy" course. the title of 'mathematical philosophy' suggests it as part of philosophy of mathematics, from whose recurrent themes strikes me as a wholly different thing!


* not that i ever completed the major. it was my final semester for my undergraduate degree, i had a 3-credit course left called "operating systems," but by all accounts i would have had to spend 20 hours per week, just coding. i decided to take a topics in ΡDEs course which, in the long run, was a wiser and more useful choice. (-:

** it's funny how we become aware of certain facts. i actually learned of the latter result from an acquaintance, while couchsurfing .. about 9 years ago? similarly, the vNM axioms came up when one of my students from a proofs class wanted to write his term paper on economics and game theory.

on writing: when you can no longer trust yourself ..?

// from sunday, 28 july 2013 @12:19 EEST:

currently i'm editing this one manuscript for the .. who knows: 27^th? 41^st?.. $n$^th time and it's getting to the point where it's hard to spot errors, however significant or small. i blame this on the general principle that the more exposure i have to this thing, the more familiar and not-out-of-the-ordinary it seems. (in other words, i've lived with this thing long enough that maybe, unconsciously, i believe it to be true.)

so i've decided on a different tack, which hardly seems efficient. i don't know any other way, though [1].

---

each time i think about proofreading the manuscript again, (1) i pick a random lemma, (2) i take out a few blank pages and try to prove it on my own, and (3) becoming aware of what is tricky about it, i focus on those parts in the writeup ..

.. and inevitably find gaps. they're getting smaller and less significant, though, so maybe it's some measure of progress?

at any rate, i had in mind (perhaps too optimistically) to send the manuscript to a big shot in the field today; let's see if i can actually keep to my word!

epilogue (29 july 2013): i didn't keep my word.
on the other hand, i found another gap and patched it.


[1] i've been given the advice before to read every line and check if it is true. in principle it's the best way to guarantee that a manuscript is as correct and consistent as possible, but i always fall short (and by a big margin): i guess i'm just that impatient .. \-:

a template .. of a slightly unusual kind.

disclaimer: as usual, this has nothing to do with actual maths.

recently, every work email i write begins with:
".. thanks for writing, and apologies for the late reply .."

i don't know why, but it always takes me a week to reply to anything ..

ARR! point & counterpoint.

so i've been reading about MOOCS again. here are a few excerpts i found:

---

"What you can do over the Internet this way is deliver information, but that's not education. Education, as any real teacher will tell you, involves more than just transmitting facts. It means teaching students what to do with those facts, as well as the skills they need to go out and learn new information themselves."

(as pointed out in "The MOOC Racket" @slate)

".. are we sure the only way to teach people what to do with facts is face-to-face? This seems like something that could at least conceivably be taught to more than one person at once. I can remember lots of professors teaching me what to do with facts via lectures in extremely large auditoriums, which is not that different than a lecture you watch online."

(a counterpoint via College Professors Are About to Get Really Mad at President Οbama @nymag)

---

ye gods, this issue is confusing, especially when one accounts for the perspectives of the given pundits. for one thing ..

.. the first point comes from a university professor, who has probably developed an expertise in little-known fields (or at least poorly popularised) over years of study in academia. for him, relevant professional information typically arrives through academic channels and processing the information is a careful, length process of some depth. (think of the peer review process: ouch!)

the second point comes from a editor/journalist who has developed a different expertise in a widely-recognised occupation, probably by way of on the job training and less formal study [1]. relevant professional information probably comes through many diverse channels and rapidly so; the process of response probably requires similar speed (in order to remain relevant).

there are also tacit yet important questions here:

for a young adult, is college necessary for a future successful career?
if so, then what should (s)he learn at university?

honestly, i have no idea. there are too many types of careers out there for a simple answer. the issue gets even murkier when you account for advances in technology, even at the scale of a generation or two.

for example, it seems that there is a lack of available workers in the skilled trades, and the current infrastructure of civilization relies crucially on the fruits of their labor.

on the other hand, what if 3-D printing becomes robust enough, and available through a sufficiently diverse selection of materials, so that plumbing, welding, and soldering no longer require the work of human hands?

this sounds like science fiction, of course. i'll not discuss the likelihoods of certain events occurring .. mostly because i cannot even guess, much less quantify the time-dependent sample space of modern civilisation.

on the other hand, i would like to point out that they are real possibilities: take, for example, the history of the Luddites or how human computers were replaced by digital ones. now that i think about it, i wonder how many more travel agents there are nowadays, with the popularity of flight search engines and all ..? [2]

at any rate, the main problem is that we don't know what "workers of the future" need to know how to do, because many of those future jobs don't exist yet. (explain, for example, the notion of a web developer to someone in the 1970s.) at best we can only make decisions about how to help young adults now, with well-defined criteria ..

.. such as economic ones, i.e. whether they should be obligated to put themselves into tens of thousands of dollars in debt before the age of 30?



[1] this is not to say that the second writer knows any less than the first, nor is he any worse at his job. honestly, you cannot compare such experiences. if the second writer is a success, then i would guess that has to do with a lot of deliberate and systematic effort on his part. he may even have studied many journalists he has admired, read very carefully their work and took notes, which is clearly a kind of study, but not the formal kind you see in universities.

if this is his approach, then i applaud the guy. deliberate practice of this kind, regardless of the circumstance, is often necessary to succeed in many areas in life.

[2] actually, there may in fact be more travel agents than ever before. travel for pleasure has become more and more accessible; on the other hand, there is still a large population out there who cannot (or will not) deal with a computer .. or even afford a computer or high-speed internet. \-:

well, how big is $n$ again?

as found off a friend's feed:

courtesy of phdcomics (so don't sue).

(deceptive) cadence ..♫

you know, there is something very soothing about coding in $\LaTeX$ .. provided that you already know what you're putting into code, that is:

it's like being in the zone, letting the fingers go on autopilot. (i guess i mean "flow" in the sense of psychology.)

---

on a related note, maybe i was judging too quickly or harshly earlier.

so far, this and last afternoon have been rather productive. on the other hand, my typing on a computer doesn't necessarily imply that i am getting any work done.

from experience, $\LaTeX$ is as aesthetically pleasing as it is deceptive!


♫ the title was stolen from an NΡR classical music blog of the same name, which i recommend. (-:

quick post: round&round, and other visualisations (*UPDATED*)

apparently, this is π;
(courtesy of mkweb).


a realisation of a strange attractor
(courtesy of chaotic atmospheres)



a somewhat self-similar sculpture by carl jara (via thisiscolossal)



i immediately wondered what the π₁ of this sculpture was ..(-:
(art by henrique oliveira, via thisiscolossal)


lastly, something more "real" ..?


this is apparently a visualisation of what wifi networks would look like.
(photos by nickolay lamm @mydeals)

"it is easy to be logical; it is almost impossible to be logical to the bitter end" ..▲

today i drew a paper calendar of the next four weeks,
filled in various plans and obligations, both professional and personal,
and came to a single conclusion;

if everything is to work out,
then this week has to be mathematically unproductive. )-:



^▲ this quote is taken from A. Camus's the myth of sisyphus, an excellent book.

ARR! ... YEAH! (or: spot on!)

finally!

"Alan Turing, the Enigma codebreaker who took his own life after being convicted of gross indecency under anti-homosexuality legislation, is to be given a posthumous pardon.

The government signalled on Friday that it is prepared to support a backbench bill that would pardon Turing, who died from cyanide poisoning at the age of 41 in 1954 after he was subjected to "chemical castration".

~ from "Enigma codebreaker Alan Turing to be given posthumous pardon" @guardian

then again, i don't know how sure of a bet this is. anyone know what a third reading is?

"Lord Ahmad of Wimbledon, a government whip, told peers that the government would table the third reading of the Alan Turing (statutory pardon) bill at the end of October if no amendments are made. "If nobody tables an amendment to this bill, its supporters can be assured that it will have speedy passage to the House of Commons," Ahmad said."

---

on a (barely) related note, i've been posting a blitz of ARR! posts in the last week and a half. this blog was never meant to be an aggregate maths news website, so i apologise if the content lately has seemed rather .. commercial(?) in nature.

related to this: i've recently experienced a few notable changes in my life ..

.. with more to come, like moving across the ocean ..

.. so my focus isn't 100% on maths, at the moment. to be honest, it makes me feel like i'm not a "real mathematician," even though that sounds ridiculous. (it just goes to show you how insidious "impostor syndrome" can be.)

somewhat oppositely:

ten years ago i entered a ph.d. program. five years ago, i defended a dissertation. it's been a long enough time and i've dodged enough bullets that, perhaps, it's safe for me to consider myself a mathematician and that i'll be around for the long haul, after all.

my point is that research life has settled down and i know that, if i work hard enough, i am capable of good work. as a result, the level of "mathematical drama" in my life has been toned down.

i guess this is a round-about way of saying:

yes, my life is boring now, compared to my student days ..
but i like it that way! (-:

irrelevant, but ..

odd: i cannot seem to spell square correctly, today. every time i start typing it out, my fingers move to the keys

..s ..e ..q ..u ..e ..?

maybe my unconscious is trying to tell me something, via psychography?..

.. that i should take a sequence, perhaps a bounded one,
and choose a convergent subsequence ..? (-;

ARR! the good fight.

first, the student side: this is so cool. the idealist in me wishes for all education to be free, and in some places it's still possible;

it's nice to see that part of that might still be kept alive.

"A unique aspect of the Cooper Union case is that several of the students fighting for the cause have already graduated, and remaining undergraduates still won’t have to pay tuition while they’re students. The reason the students have been sleeping in the president’s office for the past two months is because they fear what will happen to the school after they leave, how a decision to charge tuition might affect the character of the incoming classes and the direction of their alma mater."

~from "Can Cooper Union Find A Way To Continue Free Tuition And Its Social Mission?" @fastco

now, the faculty: when i was in the middle of my ph.d. i recalled a fellow student say the worse the teacher, the more students have to work and the more they have to learn. this was in response to seeing how neurotically i was writing my lectures, so that my students could learn in the most effective way possible.

thinking about it now, i must have been the calculus version of a helicopter parent ..! \-:

anyway, that came to mind when i read the following excerpt. apparently some students do react that way (in extreme cases).

"I had a teacher in college whose lectures were so incredibly clear that it made me think physics was the easiest thing in the world. Until I went home and tried to do the problem set. He was truly amazing, but sometimes I think he was TOO good. I didn't struggle to understand his lectures--but maybe I should have."

~ from "Do the Best Professors Get the Worst Ratings?" @psytoday

there's more:

"When you measure performance in the courses the professors taught (i.e., how intro students did in intro), the less experienced and less qualified professors produced the best performance. They also got the highest student evaluation scores. But more experienced and qualified professors' students did best in follow-on courses (i.e., their intro students did best in advanced classes).
..
To summarize the findings: because they didn't teach to the test, the professors who instilled the deepest learning in their students came out looking the worst in terms of student evaluations and initial exam performance."

on a related note: over the course of my career thus far, i've learned from various occasions to be wary of those instructors whose evaluation scores are too high, almost perfect.

lauding your first-year mathematics instructors is a little like thinking that your parents did a perfect job raising you while you are being raised through childhood. i don't deny that students have good, discerning taste .. but it's hard to accurately judge how someone is conveying lessons to you when you don't completely understand the lessons in question!

not quite ARR!.. but a logarithmic spiral, embedded in spacetime.

photo by arvin rahimzadeh (@ thisiscolossal)

ARR! muscles in polar coordinates ..

.. or, should i say, cγlindrical cοordinates?

"One of the major discoveries that David Williams brought to light is that force is generated in multiple directions, not just along the long axis of muscle as everyone thinks, but also in the radial direction.
..
The basics of how a muscle generates power remain the same: Filaments of myosin tugging on filaments of actin shorten, or contract, the muscle – but the power doesn’t just come from what’s happening straight up and down the length of the muscle, as has been assumed for 50 years. Instead, University of Washington-led research shows that as muscles bulge, the filaments are drawn apart from each other, the myosin tugs at sharper angles over greater distances, and it’s that action that deserves credit for half the change in muscle force scientists have been measuring.

~ from Biceps bulge, calves curve, 50-year-old assumptions muscled aside @u-dub

courtesy of the university of washington (so don't sue)

despite this being a mechanical and biological process, the funny thing is that the totals are apparently hard to measure:

"“The ability to model in three dimensions and separate the effects of changes in lattice spacing from changes in muscle length wouldn’t even have been possible without the advent of cloud computing in the last 10 years, because it takes ridiculous amounts of computational resources,” Williams said."

it makes me wonder: what basic but subtle aspects of nature have we been missing, all this time?

thoughts, while idle: clarκe's descriptions of geometries.

so i took this past weekend off, joining friends and a special someone to idle away the days in a beach resort town. it was good fun, though i believe now that the idea of a beach is often more attractive than the actual experience.

during those sun-soaked days, i couldn't really concentrate on anything. i tried a little, but having friends around all the time turned even those peaceful periods an hour or two into unusable blocks.

then again, it's a holiday; the point is not to be working!

i did start rendezvοus with rama by arτhur c. clarκe, which made for good episodic reading; the story stretches its telling ..

.. at least until page 100, where i stopped last night..

.. and the plot isn't too intricate that you have to pay attention all the time.

i make it sound simple, but it isn't. the beauty of this book is how it points out the subtlety of human perception towards mathematical possibility.

to give you an idea, the back cover of this book reads:

Rama is a vast alien spacecraft which enters our Solar System. A perfect cylinder some fifty kilometres long, spinning rapidly, Rama is a technological marvel, a mysterious and deeply enigmatic alien artifact. It is Mankind's first visitor from the stars and must be investigated ...

yes, we all know what a cylinder is and we have all (probably) driven as far as 50km on the highway. combining those two bits of information together, however, must make for a rather unique sight. even the large hadrοn collider at CERN is only 27km in circumference, and i don't know if anyone has ever caught the entire thing within a single frame of vision ..!

a view of the LHC, courtesy of alienshift

anyway, here is an excerpt from p. 74:

"Because they were now standing on the edge of a fifty-metre cliff, it was possible for the first time to appreciate the curvature of Rama. But no one had ever seen a frozen lake bent upwards into a cylindrical surface; that was distinctly unsettling, and the eye did its best to find some other interpretation. It seemed to Dr Ernst, who had once made a study of visual illusions, that half the time she was really looking at a horizontally curving bay, not a surface that soared up into the sky. It required a deliberate effort of will to accept the fantastic truth."

moreover, clarκe does an excellent job of suggesting that, though space and its contents may be absolute, our perspectives of it can be rather relative:

from p. 42:

"And now, Karl Mercer told himself, I have to make my first decision. Am I going up that ladder, or down it?

The question was not a trivial one. They were still essentially in zero gravity, and the brain could select any reference system it pleased. By a simple effort of will, Mercer could convince himself that he was looking out across a horizontal plain [1], or up the face of a vertical wall, or over the edge of a sheer cliff. Not a few astronauts had experienced grave psychological problems by choosing the wrong coordinates when they started on a complicated job."

[1] the first two times i read this paragraph, i thought that the word was 'plane' and not 'plain.' (-:

the how is done, but what about the why ..?

so i've been working systematically on the same problem all this week, and i think i've come up with a good proof. in that sense, it's been a satisfying week.

this morning, though, i looked at the result that the proof implies .. and debated whether it is worth publishing.

to its credit, the topic is mainly about fractals but not exactly the self-similar kind. nevertheless the diagrams should be pretty to look at.

it's not too technical either. most of the work lies in building the right lιpschitz functions, actually.

then again, it's about these objects called (metrιc) derivatiοns that come up in analysιs and geοmetry of metrιc-measure spaces. i've been working with these things for a while, but my feeling is that few people care about them .. or about metrιc spaces in general.

maybe i should advertise it as a gmτ result, of some kind. one corollary is that certain kinds of fractals cannot arise as flat chaιns (in the sense of whitηey), yet their weak tangeηts are flat.

i don't know how interesting that is in gmτ, though;

maybe i should just shelve the result for now, and think about something else. for one thing, i promised one newly-met colleague that i'll think about systems of ρde's, next week!

à la pοincaré: the 4-hour day.

tuesday, 9 july 2013 @ 11:44 EEST:

my new plan: i'll concentrate on being as productive as i can, but specifically during the hours that i know that i am usually productive (even if it means that i am working far fewer hours than i usually do).

this includes first thing in the morning ..

today was already productive:
i constructed a whole class of counter-examples!

.. and probably late afternoon into early evening, just before i usually go running or join friends for climbing, actually.

odd: why would the "bookend" times be the most productive?

maybe i should also keep a "journal" of what i did during those hours, and how long it took? this is in reaction to how ineffective my to-do lists are: they are always overloaded and i can never check everything off, which eventually gets depressing.

the tricky thing is how to be productive in the hours in between. one possibility is that if i'm going to waste time, then i may as well enjoy myself, like reading novels or blogging or taking photos of the city, but another possibility is to fill it with drudgery, such as writing a review for this article that is 4 months late .. etc.

on the other hand, maybe a few hours of real work is enough; i mean if it was good enough for poincare .. [0]

---

wednesday, 10 july 2013 @ 08:38 EEST:

ups and downs again:

two days ago i knew what worked .. or at least, i thought i did;

yesterday i was convinced that a colleague's "conjecture" [1] is false and even had a rough proof in mind,

but just now i found a gap in my argument,
so i'll spend today working on a patch.

despite having been privately wrong [2] this has been a pretty exciting week, mathematically speaking. the uncertainty of it all is .. not intoxicating, but like a high. not knowing what exactly is true is all the more motivation to figuring out what it could be .. especially if it keeps on switching, due to increased effort.

so work is fun again and i can obsess about it with new abandon.



[0] to be fair, i wonder what poincare would consider "work." my guess is that the man had high standards, so a lot of things he wouldn't consider work, i would!

[1] i like to reserve the word 'conjecture' for those unproven claims that are particularly nontrivial .. and no, i don't have a definition of nontrivial, but just a sufficient condition: if, by telling someone related to your field of the basic notions, by stating the problem, and by showing why it is hard or what it would imply, that that someone become interested, then it's nontrivial.

[2] 'privately wrong' is doing something stupid in the privacy of your own workspace. 'publicly wrong' is having many people know about it. i've been both, and the former is very, very much preferable to the latter.

ARR! strange occurrences in science

// initially written: last week (or before)

ok: so what exactly is a measurement, then?

"Just like position and momentum, quantum theory predicts that the polarization along two different axes cannot simultaneously be known with certainty (see Nature). The team adopted a strategy in which the polarization is initially probed using a series of ‘weak’ measurements — detections that barely disturb the system but must be repeated several times to record the same information that a single ‘strong’ measurement can detect. They found that, on average, the polarization measurements disturbed the system by only about half as much as Heisenberg’s original formulation of the uncertainty principle dictates."

~ from "Proof mooted for quantum uncertainty" @nature

speaking of science, maybe this explanation of mercury's liquid state is well-known .. but admittedly, i was ignorant of it.

"Relativity states that objects get heavier the faster they move. In atoms, the velocity of the innermost electrons is related to the nuclear charge. The larger the nucleus gets the greater the electrostatic attraction and the faster the electrons have to move to avoid falling into it. So, as you go down the periodic table these 1s electrons get faster and faster, and therefore heavier, causing the radius of the atom to shrink. This stabilises some orbitals, which also have a relativistic nature of their own, while destabilising others. This interplay means that for heavy elements like mercury and gold, the outer electrons are stabilised. In mercury’s case, instead of forming bonds between neighbouring mercury atoms, the electrons stay associated with their own nuclei, and weaker interatomic forces such as van der Waals bonds hold the atoms together."

~ from "Relativity behind mercury's liquidity" @rsc

---

lastly .. judging from how i wrote this post, it's safe to assume that those monday roundups aren't returning anytime soon.

let me clarify: i still think that rounding up articles is a good idea, but a weekly time constraint feels slightly artificial to me. i'd much rather collect items with a common theme and let the ideas percolate into something coherent. (i don't know if i'll do this weekly; it really depends on how much interesting stuff appears on the blogosphere.)

that said, i'm open to taking requests;
if you want my opinion on a topical article, then send me the link.

that said, the MoAR label will become just ARR!, where AR refers to ARticle, R to Roundup, and ! to indicate that i have an opinion!

off the cuff: snippets before the weekend.

// snippets from thursday, 4 july 2013

i'm trying to unlock a door, figure out blueprints for objects that, in their full generality and obscurity, requires believing in statements like the axiοm of choice and their implications [1]. for the right skeleton i've recast someone else's conjecture for bones and soft cartiliage.

i have examples. i know that these things exist, at least in crude forms, but how sophisticated must the designs really be?

---

i don't know if unlock or build is the right word, mathematically speaking. i don't know what the right word could be.

all i know is that i'm stumped at the mathematics, and all i can do now is discuss the frustration of those mathematics.

[1] apparently "axiom of choice" is also the name of a band.

unfocused .. (updated)

maybe i should take a vacation: a short one, maybe a week.

i haven't been able to focus for a while. today i was in the office and rather than sticking to a list of things that i really should do, i spun the same ideas on the same problem that i've failed to solve for .. well, years.

..

it's one of those transitional times in my life again:

1. in six weeks i'll start a new job in the states, one that could be for life .. that is, if they like me enough;

2. i recently solved an open problem that's plagued me from the latter years of my ph.d. it was the kind of problem that if i had a spare day, then i'd just attack it with any idea i had .. even if it meant that the day would be lost to compulsion, folly, and frustration.

so i can't seem to get excited about any new projects. there's plenty of things to do, of course, but .. i can't convince myself to do anything.

sometimes i'll catch myself staring through the window, not thinking about anything in particular.

// added 2 july 2013 @ 03:46 EST

so perhaps i spoke too soon:

when i woke up today and made coffee, i decided not to attack the problem again but instead, work out a related formula that has stumped me before.

it turns out that it follows pretty easily from basic principles .. at least, easily after a good night's sleep (and/or subconscious hacking).

now that i think about it: it would make sense that i had proven it before, forgotten it, and just now re-discovered it. i don't know, i can't remember, and my notes are not archived well enough for me to check it readily.

[shrugs]

well, at least it's true. it doesn't solve that aforementioned problem, of course, but it does narrow the gap so that there are fewer things to try.

..

related to all of this, i keep forgetting about the happy clarity that mornings can provide. let me make this clear: though it might be nice to be part of the 5am club, i'm happy enough enjoying the evenings and getting a few hours in the morning before heading to the office. (-: