"fortunately, mathematics is useless."

if i've ever said that i'm good at explaining to others the mathematics i do, then i retract those statements.

i'm terrible at explaining what i do to people. while on a shuttle from the airport, a biology postdoc asked me what sort of math do i do. it was then that i froze.

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from now on, i will not call them applied mathematicians but mathematical scientists.

i'm convinced that theoretical mathematicians cannot explain themselves, even to scientists. we are like the O-negative blood type: our work can be used by non-mathematicians, but we cannot use the work of non-mathematicians. hence applied folks, if capable of speaking to scientists, must be scientists.

that doesn't mean that they aren't mathematicians. far from it: they may be excellent mathematicians, capable of great things and fine work. they just happen to be able to present themselves also as scientists. we theorists have no such flexibility.


also, i will hereby use the default answer of:

"oh, what i study is absolutely useless"

whenever someone asks me about the applications of my work. it's true, from my viewpoint. there are no immediate, non-mathematical applications which come to mind.

if they want to know how it is related to science, say how the field began, then sure, i'll answer that. but they have to ask explicitly that, because it's not worth wasting time and proffering illusions of mathematics as some science, or something very real.

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mathematics is not real, or at least, not fully so. it is abstraction. for some inexplicable reason, when polished to something computational, it just happens to be useful .. in frighteningly many settings, and it will probably continue being useful.

but i still cannot say why, why it is useful. a mathematicial scientist would know better; i leave such things to such people. i'm happy to be useless, contribute nothing to today's society except sate its sense of progress. we are as useless as artists and poets and philosophers ..

.. but isn't it funny, how it is art and literature and philosophy that we remember from older civilizations, and not their technology, their science, and their engineering? we don't fully know how the egyptians built their pyramids, and except for reasons of historical posterity, i don't think we much care.

after all, with bulldozers and cargo planes and laser cutters and explosives, we could build better pyramids, and more quickly and efficiently.

we have abandoned steam engines and vacuum tubes. if we retain our computers, then still, their form will change. we already smirk at 386's, right?


but we remember culture; it persists in memory, despite not contributing directly to the progress of its society, in its time. so how is that, for "useless?"

lies that a mathematician friend might tell you, because i do.

so today i told several lies .. i think,
but these are very common lies for mathematicians.

twice today, i was asked about the things that i study: once by my sister and once by a friend from high school, and both of them are neither mathematicians nor know much mathematics.

oh well; nobody's perfect. q:
then again, they dared ask and that is something!

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perhaps "lie" is the wrong word, and "simplification" is more correct. as a general rule, i never explain actual mathematics to non-mathematicians. the best i can hope for is to suggest an idea, or give a rough explanation of one key idea which is key in the work i do.

have i written about this before?
i must be getting redundant.

so to err on the side of caution, let me not rant excessively. spot the 'lies' if you find any, but

1. to my sister i described rectifiable sets as surfaces which look like terribly crumpled aluminium foil, because i couldn't think of a way to describe to a general audience the currents of federer and fleming, much less the notion of a current on a metric space.
   
   happily, she remembered what a tangent 2-plane was .. intuitively, at least.
   
   i then suggested the minimal surface (plateau) problem as an application, and to motivate the study of general metric spaces, i speculated that there are plenty of strange geometries which are interesting, even in real-world terms, like the parameter space for a bicycle or connectivity graphs from a computer network.
   
   
2. to my friend i tried to explain why limits aren't as intuitive as the terminology suggests. he fancies the example of achilles and the tortoise, and i suggested the pathology of porosity:
   
   what if the racetrack was full of holes, at all scales? so say that you want to sample various times to determine how achilles will finish; how do you avoid choosing bad sample times over the race?
   
   i then suggested to him that as a sequence, an orthonormal basis on an infinite-dimensional hilbert space [1] has no limit, even though no point is that much farther from another. as a concrete example of why infinitely many dimensions is reasonable, i suggested possible states of an electron ..
   
   .. even though i don't know any quantum mechanics. i wonder if that remains true.
   

[1] more true to words, "suppose you had two coordinate axes, and now three. now imagine four, which could be, so why not five .. and so on: imagine that there is no end, and there are an infinite number of independent directions .."

thus far (update from new york)

my visit to new york is half over, and as a vacation (from mathematics) it's been mostly successful.

there was one morning where i was the first one awake (my brother and a sister are also visiting this weekend) and i wrote as i tried to remember how things stand with my research.

on another day, i was killing time for some reason and tried to make rigorous an example that the advisor and i sketched out. the key word is "tried" and it's still not quite rigorous.

i'm sure it's true, or at the very least, contains something true and purposeful in the computation. maybe i'll find time for it on monday -- during a train ride into manhattan -- or on wednesday -- while waiting at airports.

in the meanwhile, i'm trying to be a person again, and relearn how to do normal things.

pondering, on whether or not to ponder ..

tomorrow i visit my parents and the house i called "home" in my adolescent years. except for a new pair of running shoes i bought last new years, i can't remember what i left in that dusty chamber that used to be my bedroom, and that makes it hard to pack.

no matter: it's summertime.
packing clothes is easy.

packing mathematics is much harder. i'm still thinking of taking the week off, but that feels wrong. as a friend of mine suggested, i can always split the difference and just read: read the papers that i've been meaning to read ..

.. because when will i ever have the time to sit and read them, anyway? it will definitely be more relaxing than .. say, research. after all, thinking can be quite tiring.

maybe i should take it as a vacation.

besides, what's the worst that could happen? if indeed i get so obsessed with a good idea that i break stride, and prove something great, then it was meant to be. in the meanwhile, why not enjoy myself?

then again, why does this sound like excusing myself, and being lazy?

bad news.

yesterday a fellow grad student stopped by my office and asked me about one of my proofs. he thought there was a flaw in it and i asked him to explain.

sure enough, there was. he apologized for not telling me sooner (he knew on monday) but i don't think it really mattered.

so i lost a handful of theorems .. mostly corollaries from that one "theorem" .. and there's not much of my collected work left.

it's been a bad 24 hours.

i met with the advisor today, and we came up with a few ideas. none of mine worked, so i'm trying one of his.

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the thing which stings the most is that i didn't catch my own error, and it's not for any reason of vanity or embarrassment. the fact that i didn't spot the error means that i'm not as good of a mathematician that i thought i was.

just when things were going right, that i felt like i was making progress .. and now? now i feel like i'm starting over, pillaging the salvage.

not surrender, but a retreat.

i think i've lost another few days. last night and this morning i was tempted to throw away the notes and scratch work i've jotted since last saturday.

to clarify that: i'm a packrat. i never throw anything away, least of all written ideas .. that is, if they contain any value at all.

blame it on paranoia.

the last thing i'd want to happen is realising that an old idea of mine will work, only i can't remember the argument and i threw away the paper. i don't think i'd forgive myself.

for similar reasons, when i think about a problem, i start with the most obvious, 'stupid' ideas first [1]. it's the safest course of action, i think. if the 'stupid' ideas don't work, then fine: usually there is some lesson in why they don't work.

now what if the 'stupid' idea works, and you never tried it?

anyway, i've been thinking about a problem lastly and nothing's working. i feel like i've been arguing myself in circles to the same impasse.

i can never reach one particular step, and coincidentally, i can't seem to apply one particular hypothesis in that step of the argument.

i've concluded that i don't know enough to solve the problem. there's something i need to know, or discover, before i can tell whether this argument will succeed or fail.

but as of now, i'm not ready for this problem,
and that is a VERY frustrating thing to say.

it's not quite a surrender, but it's an order of retreat. it still feels like giving up .. and i HATE giving up.

more than that, i hate being unproductive. so now it's a matter of switching gears, and starting work on something else: picking battles, so to speak.

it still feels .. wrong, but i can't afford to lose any more days without learning anything. picking battles, and living to fight another day.

[1] i must be getting thin-skinned or something, but these days, words like 'obvious' or 'trivial' or 'stupid' bother me, when used mathematically.

a mathematician's disclaimer (cross-posted)

i originally meant to write about vacations on the other blog [0] but as do many things in my life, it turned mathematical.

so i've copied-and-pasted it below.

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how often do people take vacations, in this day and age?

i can't tell and even if i could, my opinion cannot be trusted: i'm an academic and apart from one afternoon per week [1], i set my own work hours. there aren't so many hours, either, but there are enough to make it feel like something close to work and definitely not play ..

.. well, not for most days of the week, anyway.

* * * * begin: waxing mathematical * * * *

not all academics can be so casual, i suppose. we mathematicians are a strange class of academic. our field is rife with technical jargon and the terminology must be both intuitive and precise in order to serve any use. but we are not quite scientists: we have no laboratories and we have no loyalty to any workbenches. a chalkboard helps, but paper and pen will do, and if there is neither then we can always think it out carefully.

according to legend, archimedes drew figures in the sand, and eventually he died amidst his figures.

nobody seems to understand us, because simply put, mathematics is a language and there are few fluent speakers. i tell people that practitioners of theoretical mathematics are more akin to philosophers than to scientists. we may refer to 'thought experiments,' but those are more thoughts than experiments.

scientists must bow to reality. despite how abstruse and abstract the theories of the physicists, there is the motivation to explain observable processes.

instead, mathematics begets more mathematics, and mathematics has nothing to do with reality [2]. if we are helpful to science, it's because science poses problems that are mathematically interesting; otherwise mathematicians would have stopped listening to scientists, long ago.

science entertains mathematics. if we produce something of value, then let the scientists have it, make their progress, and reinvent the world in their own image. they have offered us stimuli and now we can ponder new curiosities, and now the engineers can invoke this progress and harness it to whatever end will serve or sate our society.

for example, a mathematician doesn't care whether string theory is correct or not. to an algebraic topologist, string theory is another motivation for problems in cohomology, and to a complex analyst or hyperbolic geometer, string theory is another reason to revive the notions of conformal structures and teichmuller spaces.

but i digress: we will always be misunderstood. we cannot share our work to anyone but other mathematicians. we are like a type of nerd, who will bander on and on about something that is pointless to everyone else in the world, but to us, it's SO DAMNED COOL. think of WoW gamers or foodie cooks or linux hackers; we are like them.

i know some mathematicians who love to talk about their work, but amongst non-mathematical company they clam up and are nervous, in the way that people become nervous when you tell them not to imagine their best friend naked.

"it's so damned cool, but it won't make any sense if i tell you and you will be annoyed if i do tell you, so i should shut up .. but it's so .. damned .. cool .."

as non-mathematicians, you will never know us in the same way that you might never understand an artist and his art. artists must really, really care and think their art is wonderful and important; otherwise, why would they devote so much time to it, when they could be doing something more fun? why would they exile themselves in isolation, when they could spend more time with their friends and loved ones, or at least, work a less lonely occupation?

think of them, and think of us.

if you really want to understand what we do, we'll try to explain. but be warned: we mathematicians don't mean to offend if we say, "it's complicated," because it is complicated and hard and often frustrating. if it were easy, we'd have sorted it all out and be at the beach ..

.. and probably, we'd be drawing an occasional figure in the sand .. at least, when nobody's looking.

* * * * waxing ceases * * * *


on an unrelated but funny note: i'll bristle at being referred to as an 'intellectual' but i have no problem with being called an 'academic.'


[0] yes, i have another blog: two others, actually. one you can find on my profile (see sidebar) and it's about vegetarian cooking. the other blog is .. well, the other blog.

[1] the weekly research meeting with the advisor. today we ended up talking mathematics for 3 hours or so; it would have been shorter, were i better at math.

[2] except in the platonic sense, i suppose, but i'll leave that to the philosophers.

something wrong, something right, and neither very promising.

it's been a frustrating 6 days.

i spent the first 4 days trying to prove something, aware of what shouldn't work in the proof .. due to known facts and non-examples .. and i couldn't prove it.

worse yet, it's because the argument is incomplete. as of 2 days ago, i had no more ideas or inspiration to fill the gap. so i don't know if the argument is wrong, either.

after these many [1] years in graduate school, i've learned not to feel that bad about being wrong or entertaining a stupid idea ..

more accurately, i would call it an emotional callous

.. but to be wrong and not to have learned anything .. ye gods, that's frustrating. it's like having a mild conspiracy theory stuck in your thoughts, and not knowing if you are indeed a crackpot or not.

i wonder, if only on a practical level, whether that explains why some conspiracy theorists are crazy .. that it's not the crackpot theory that drives them mad, but the uncertainty which does.

as for the other 2 days, i may have actually proved something, but the argument of proof doesn't look right.

you see, it's not my theorem. a few months ago i read a particular theorem in a paper from 1999 or 2000, and i couldn't understand the proof.

so i tried to cut up the author's argument into claims, and prove those. some i could prove and some i couldn't, so i thought of other claims which would make sense ..

.. but having let this take a life of its own, my current proof has noticeable difference from the author's proof.

so yes, i've had better weeks. \:

[1] and yes, i'm griping. i know that plenty of you have taken longer to finish your own ph.d.'s and that i have no basis to gripe about time.

for the record, i am griping about something else.

scattered and unproductive.

i don't think i'm being very productive. certain things i do by habit, like waking up and then thinking about mathematics for a few hours, over coffee .. taking notes and pondering problems.

lately i've been confused.

i think i've been able to coax all the 'easy' results from the fusion of these two topics that i've learned .. though i don't think i've learned them very well .. and now the work will become more difficult.

put one way, i would say that the theorems i've proven over the last two months have been pointed observations and casual conclusions (and not all of them mine).

i wouldn't call them all of them corollaries, but i've needed to use a nontrivial fact or two from the works of others. without them, these theorems would still be claims, dead in the water, without proof.

now it won't be so easy.

i have a claim in mind, but i can't prove it, and those facts from before are no longer applicable and not obviously relevant. i think i'll have to invent or discover a small something, but a new something, in order to solve this part of the problem ..

.. but i don't quite know what something it should be. i suppose that if i did, i'd probably settle the problem and have a laugh about it. for now, i'm not laughing.

most of the time, i'm frowning at those pieces of paper.

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my mind feels scattered.

i can't really encapsulate my research as a thesis problem anymore. then again, we've always kept this second "problem" open-ended, like an exploratory mission of sorts. but before, it was easier [1] to say what i was doing ..

.. but now this idea runs this way, and another idea runs the other way. often i don't know which to follow, which is more productive or fruitful, and often i just pick one, hope for the best, and get to work ..

.. and every so often, wondering if "the grass is greener."

i also find myself switching around these sub-projects. for example:

yesterday & today, i thought about a problem which has more to do with the currents of Federer & Fleming on euclidean spaces, rather than the new-fangled theory in the context of metric spaces.

the flat forms of whitney and wolfe also appeared briefly, as those in the know .. well, could probably have known.


two days ago, i was pondering the (first) heisenberg group and what could be said there, in my line of work. i conclude what i always conclude:

does anyone actually understand this space? it's a beautiful geometry, but do we know how it works, analytically?


the days before that i was working on metric spaces which are general enough to do what i'd like them to do. the advisor then asked me about something, which was a good idea.

i'll probably try and work it out tomorrow .. unless i keep with this Federer-Fleming thing until i reach an intractable dead-end, which might take a while, and then there's the rare chance that it could work out ..

but you get the idea: "scattered." (:

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[1] well, "easier" if you narrow down the world to a few dozen people, maybe even a hundred.

i'd like to think that i could explain the rough idea to a mathematician, but then come the "prerequisites," such as

* measure theory & functional analysis, for starters,
* a little differential geometry, for motivation,
* some awareness of the analysis on metric spaces, for relevance,
* some geometric measure theory, for intuition and analogy,

i might be able to gloss over certain things, but it would be hard-going.