?

Log in

No account? Create an account

Previous Entry | Next Entry

Math Man's Burden

Wow: runny nose, stuffed-up sound, sniffling and nose-blowing... this body is malfunctioning.

Seriously, I get sick only once every 12d4 months or so. I've gone well over three years without illness before (once when I lived in chambana, and nearly the past three years here). If you searched back though my LiveJournal archives, you'd find nary another entry in which my mood is genuinely "sick". That'll teach me to go gallivanting off on damfool idealisticanti-Sith crusades without my jacket.

Today's whimsical title is a result of a weekend's convalescent rumination over the difference between a mathematician and a normal scientist or engineer. Now, we who are engineers do not like to admit to any inherent intellectual inferiority relative to science, and we who are scientists do not always like to suppose that there is anything magical in the so-called universal language that elevates itself above our mere mortal tongues. But let me just come right out and say it plainly: if you talk to a pure mathematician, the probability is high that, somewhere in the course of your conversation, you will recognize that he or she takes it for granted that there is something loftier about mathematics, the realm of abstract structures and patterns of change.

Maria Zamfir-Bleyberg, a professor emeritus of my department who was on the committee that recruited me, once called computer science a "selfish discipline", one that brooks no rivals of attention or preference. By this she meant that personal time, particularly that devoted to other creative and nurturing activities (yes, even parenthood!) sometimes had to be balanced against the demands of the discipline. What makes CS different from any other field, you ask? Well, first of all, Maria didn't say "balanced against", she said "given up to", and of course many of our graduate students - single or married, with children or childfree - might understandably balk at that characterization. But I digress. I came here today to talk abut math, and what is it that makes mathematics not only a selfish discipline, but a lonely and antisocial one. Herewith, my theories.1


Arrogation. Mathematicians are better. No, they don't just think they are: they have to be. Olympians walk alone. Which is as much to say: when a tradition - be it a religious sect, a secular order, a profession, or even an ethnicity - gets so caught up in its own legend, its peculiar perfectionism - that legend becomes as much a burden and a crushing responsibilty as it is an incentive. Carl Friedrich Gauss is said to have encouraged his sons to study law rather than mathematics, rather than taint the name of Gauss with mediocrity. How do you like them apples, zengeneral? Lawyers. Lawyers. (Fermat was a lawyer too, was he not?)

The terror of being second. Sometimes math appears to brook no seconds, either in ability or in temporal order. Gauss famously tantalized and taunted Jacobi, who would go to him with months of work at new theorems he had proven, new properties he had discovered - only to have Gauss reach into his cabinets and pull out unpublished notes from fifteen years before, independently tracing the same steps. In reading about the brilliant and promising career of Theodore Kaczynski, the convicted Unabomber, one gets the distinct impression that this kind of environment at least exacerbated his anti-technological singularity-phobia. I wonder if the same pressures that some withstand and cause others to snap is not a catalyst in some people's retreat from society. I don't think I am exaggerating when I say it is sometimes more than we can reasonably expect from human beings.

The urge to ineffability. Mathematics delights in the ineffable, the inscrutable. A true mathematician will grin (if inwardly) when you scratch your head and look duly intimidated by the massive sprawl of symbols he or she has neatly arranged on the board. An intelligent systems or programming language researcher is merely pleased that you find the derivation original, meaningful, and edifying; the grin is at your respect for the impressive display of sequent rules, denotational semantics, or algebra. A pure mathematician is glad you don't get it. This peculiar pathology is so common among mathematicians that my father used to regale me humorously with stories of his old chemical engineering classmates, who proposed to take handguns to their Ph.D. defenses. To shoot their committees if they failed? No - to shoot themselves if the committee understood the work, rendering it so tawdry and commonplace as to be readily comprehensible, applicable, et cetera.

That is no country for old men. Like chess, musical composition, gymnastics, and many other prodigious acts of youth, math demands too much, too soon - the early display of precocity, the sustained output during a "period of trial". When Evariste Galois lay mortally wounded from a pistol duel, he reputedly said to his brother, "do not cry; I need all my courage to die at twenty". Any aspiring mathematician I know, myself included, will tell you that it takes something of a greater level of courage to survive. Math, the Discipline, does not mind if we burn out: how few of the shining lights of the Elder Days did not? There is a romantic ideal of mathematics, to die prematurely as Abel, Galois, Ada Lovelace, or Ramanujan did, if you accomplished anything of worth, is to not have to worry about that messy rest of your life. To live to a ripe old age as Euler, Gauss, Newton, and Erdős did is to look over your shoulder every day, wondering if your moment of glory is past - and often knowing it has passed. Note that the four worthies I named were anomalously prolific throughout their long lives. You still wonder why some aspiring mathematicians plan on dying at 30? lao3 er2 bu1 si3 wei3 shi3 zei2, goes the Chinese aphorism: "the old who do not die are scoundrels" (that is, they rob the young of their birthright).

From mountaintop to mountaintop. Finally, mathematicians delight not in low obfuscation, but in astoundingly virtuous leaps. Like Asimov's "Eureka effect" of science, wherein scientists who happen upon a discovery by plodding methodicality feel obligated to "make up a story" about how "something wondrous happened", mathematicians also delight in jumping bodily from mountaintop to mountaintop without climbing down to earth. Abel admired Gauss with the remark, "He is like the fox, who effaces his tracks in the sand with his tail." Make no mistake - that is a compliment. I remember Uday Reddy, a type theorist whose group I worked in as a first-year graduate student, remarking of Gaisi Takeuti: "He is Sherlock Holmes! Everything is elementary to him." In 1994, at 75 years of age, Professor Takeuti was hailed as the Grandfather of Modern Proof Theory. I took his linear logic course, and was baffled week after week by the effortlessness with which he would produce proofs that took him seconds to our hours. He had an uncanny instinct: an automated deductive theorem prover would take days to come up with the same assertions, if it did at all. And he would do this with a good-natured but backhanded humility, calling every proof "kindergarten stuff". My point, though, is that Dr. Takeuti, like many who survived their twenties, thirties, and forties, knew what he was about. He knew exactly the width of the chasm he could leap, and by knowing it, could comfortably spend his fifties, sixties, and seventies leaping safely. It was a lot more than we could do, yet perhaps a young prodigy might do better. Dr. Takeuti acknowledged this: when faced with a very difficult proof, he would pause and say, "maybe this is first grade". I can only imagine what kind of near-impossible proofs would elevate a theorem to second grade status.

Method: eschew or embrace? Many computer scientists I know, some of them instructors of courses taken by students of higher mathematics, lament that the mathematician mindset (to be distinguished from the mathematical mindset) seems to shun any received methodology. Whether it be nomenclature, proof procedure, structure of communication (lemmas and intermediate steps), or just simply "using all the tools given", we really do seem to like to do things Our WayTM. In light of the above points, this is not so hard to explain: virtuosity is the one true virtue; methodicality is praised and starves, to paraphrase Juvenal. However, my point (and I do have one) is that method is a foundation of CS, of science, and yes, of engineering. And guess what, sports fans? It isn't a strict hierarchy, however much the numerati would like us to believe.

I wrote this for two reasons: First, because I have been sometimes thrilled, sometimes dismayed at the attitudes prevalent in my "home court", the CS-Math double major, and I wanted to sort out why things might be the way they are. Second, because I wanted to convey to the nonmathematician a little of the mindset: there really should be a field called Psychology of Mathematics (psych applied to math, not the other way around). Third, because I hope the above will shed a tiny bit of light for the bears in the mist: know yourself, and know it doesn't have to be any way but how you restrict or free yourself.

Oh, was that three reasons? Heh.

1 At some points in my discourse I will use "they" to refer to mathematicians, and at others I will use "I". This apparent schizophrenia is explained thus: sometimes I feel like a mathematician; sometimes I don't. Almond Joy's got nuts; Mounds don't.

--
Banazir

Comments

( 22 comments — Leave a comment )
digby_tantrum
Apr. 4th, 2005 11:38 am (UTC)
Fermat was indeed a lawyer.
In my experience the real difference between mathematicians and 'normals' tend to be that of dress: mathmos are scruffy buggers, as a rule. I recall one frighteningly clever FRS who could seemlessly generate proofs as if it were an automatic process involving no conscious thought, but could not master the basics of buttoning up his shirt properly.

That aside, I think the unconscious superiority prevalent in mathematicians' attitudes to lower beings can be traced to the fact that 'doing maths' involves taking part in a tradition of philosophical thought that has been around since BC. Computer scientists tend to be more useful, but it's hard to be lofty when you're subject's still in the kindergarten stage.

(Oh, and the willful obscurity thing as well. There's something great about being able to talk about Riesz and Borel measures in the knowledge that you're the only one in the room who knows what they are.)
digby_tantrum
Apr. 4th, 2005 03:33 pm (UTC)
Re: Fermat was indeed a lawyer.
'... you're subject's'? Oh dear.
banazir
Apr. 23rd, 2005 06:29 am (UTC)
Re: Fermat was indeed a lawyer.
In my experience the real difference between mathematicians and 'normals' tend to be that of dress: mathmos are scruffy buggers, as a rule.
Now that I am paying attention, this is indeed a severe distinction, but as the old adage goes: correlation is necessary but not sufficient for causality. I hypothesize a latent (cultural) cause.

I recall one frighteningly clever FRS who could seemlessly generate proofs as if it were an automatic process involving no conscious thought, but could not master the basics of buttoning up his shirt properly.
:nod:

That aside, I think the unconscious superiority prevalent in mathematicians' attitudes to lower beings can be traced to the fact that 'doing maths' involves taking part in a tradition of philosophical thought that has been around since BC.
Yes, but then, you have to include symbolic logic (to which AI and semantics, overlapping subareas of CS, trace their roots) in that. On the one hand, you can delineate proof theory as its own subfield of mathematics and as a substrate of mathematics; OTOH, it is a substate of CS by the same token, and theoretical CS would have as much "claim" to it as any single aspect of mathematics such as combinatorics or algebra.

Computer scientists tend to be more useful, but it's hard to be lofty when [your] subject's still in the kindergarten stage.
Yes and no. CS has indeed evolved as:

1. a theoretical field with its own independent life from mathematics
2. an applied mathematical and scientific discipline, cf. aspects of statistics, operations research, and numerical analysis
3. a technical engineering outgrowth of computation, which is sometimes as far from computer science as chemical engineering is from chemistry

(Oh, and the willful obscurity thing as well. There's something great about being able to talk about Riesz and Borel measures in the knowledge that you're the only one in the room who knows what they are.)

... and I thank you for demonstrating my point. ;-)

--
Banazir
casecob
Apr. 4th, 2005 01:51 pm (UTC)
This was a really interesting read - I'll have to send over ruakh to read this post and to see what his take is.

It's nice to have time to read other people's posts once in a bit...
banazir
Apr. 5th, 2005 03:47 am (UTC)
Hello there!
This was a really interesting read - I'll have to send over ruakh to read this post and to see what his take is.
Cool. I see he's an ECE - the more comments, the merrier.

It's nice to have time to read other people's posts once in a bit...
Welcome to my friends list, if I haven't bid you welcome before!

--
Banazir
yahvah
Apr. 4th, 2005 02:13 pm (UTC)
I agree with the second commenter. This was a read worth reading more than once. I always tell people who say that they don't like math or think they can't do well in math that they should take a moment, sit back and look at how quickly and how accurately they can type. If they can type fast with great accuracy, chances are that they will be good at mathematics, but the distinction you make between the mathematician mindset and the mathematical mindset is interesting. It's interesting because now I should say that people who type quickly and accurately will have a mathematical mindset. I say this to people as an attempt to build them up because the parts of the parietal lobe that give us the ability to type also gives us the ability to make music and do mathematical things. Of course, that's extremely simple, and there's more than just the parietal lobe, but it's the parietal lobe that gives us our number sense and that connects to our frontal lobe and allows us to express our number sense.
prolog
Apr. 4th, 2005 05:49 pm (UTC)
I've always heard a lot of about the math-music connection, but I've always wondered exactly how it works. I'm a decent musician (I play several instruments in my university's ensembles), but I've never been able to do as well at math - not even close.
yahvah
Apr. 4th, 2005 05:56 pm (UTC)
Never underestimate the power of your 'level of interest' with regards to how easily you'll learn something. That makes a big difference as well.
banazir
Apr. 23rd, 2005 06:48 am (UTC)
The limitless power of interest
Absolutely; I was just remarking as much to oikade here.

--
Banazir
banazir
Apr. 23rd, 2005 06:49 am (UTC)
And I'm just the complement
... so what gives? ;-)

--
Banazir
prolog
Apr. 23rd, 2005 03:42 pm (UTC)
Re: And I'm just the complement
I assume we act as some sort of balance within the universe. ;)
banazir
Apr. 23rd, 2005 06:45 am (UTC)
Correlation and causality
the distinction you make between the mathematician mindset and the mathematical mindset is interesting.

I'm glad you find the distinction interesting. What I was getting at, in retrospect, was a distinction between the basic technical aptitude at the vocation and the "high intellectual endeavor". I'd liken it to the ability to appreciate music, art, or theatre and perform or reenact basic works (mathematical) versus hearing a calling to become a musician, artist, actor, or playwright (mathematician). These are actually orthogonal: One can, in fact, be a perfectly competent number-cruncher (and here I mean numerical analyst, not actuary) but be happy to work as a software engineer or data analyst. One can also be a mathematician who struggles for years to get one elusive proof and is ultimately disappointed. Like all reality TV, math has its horde at the bottom of the pyramid.

It's interesting because now I should say that people who type quickly and accurately will have a mathematical mindset.
I'll have to think more about the neuroscientific angle. I'm not saying those connections aren't there, but as I just quoted to digby_tantrum, correlation is necessary but not sufficient for causality.

--
Banazir
yahvah
Apr. 23rd, 2005 03:21 pm (UTC)
Re: Correlation and causality
correlation is necessary but not sufficient for causality.

I actually knew that. ;)

Even though there are always critics who want every book bought off the shelf to be a work of genius, I enjoyed Keith Devin's The Math Gene.
prolog
Apr. 4th, 2005 05:47 pm (UTC)
I really enjoyed reading this, and especially agree with your "from mountaintop to mountaintop" argument. Mere mortals like myself, who have been only average at mathematics at every step of life, can't help but feel frustrated when the mountain-hoppers don't understand how the earth-dwellers can't visualize the proof, or how it's not intuitive to them.

I remember a first year calculus course: I was in awe of Jacek Szmigielski, a genuinely nice and helpful mathematician, and how easily he could hop from step to step of the proof, while the gears in my mind turned madly trying to comprehend.
banazir
Apr. 5th, 2005 05:41 am (UTC)
The gaps between fields, like the valley between mountains
I really enjoyed reading this, and especially agree with your "from mountaintop to mountaintop" argument. Mere mortals like myself, who have been only average at mathematics at every step of life, can't help but feel frustrated when the mountain-hoppers don't understand how the earth-dwellers can't visualize the proof, or how it's not intuitive to them.
Thank you. Part of my point was that very good scientists can be but average mathematicians: these are semi-orthogonal aptitudes, and should be seen so IMHO. Somebody's got to be mediocre, or it'd be Lake Wobegon Day's all the way: "98% of the children are above average".

We don't think twice of saying that a bioinformatician is a lousy bench biologist but a great algorithmist, or vice versa, nor that a computer scientist is a good theorist but an average software engineer, nor that a molecular biologist is a top-notch lab biologist but a so-so biochemist. Why does every science acknowledge that ge2 ge2 ru2 ge2 shan1 ("The divisions between fields are as the divisions between mountains," one of my dad's favorite proverbs)?

I remember a first year calculus course: I was in awe of Jacek Szmigielski, a genuinely nice and helpful mathematician, and how easily he could hop from step to step of the proof, while the gears in my mind turned madly trying to comprehend.
I know exactly how you feel. We all (well, all but a very few) have our moments of being the Salieri to a Mozart. But who's to say who is personally fortunate. In the long run, achievement comes rarely on the edge of a mighty leap. This is perhaps recently true, as zengeneral implies here, but it is true nevertheless.

--
Banazir
sze
Apr. 4th, 2005 07:35 pm (UTC)
what is this dark science you speak of called Math?!? ;)
banazir
Apr. 5th, 2005 03:23 am (UTC)
Math, the dark science
LOL, you tell me!

--
Banazir
sui_degeneris
Apr. 4th, 2005 07:45 pm (UTC)
Yabbut... They both have chewy coconut...
...EWWWWWW!

Intriguing post. Makes me wonder about myself. I'm pretty sure I have a mathematical mind rather than a mathematician's mind, but I never really considered it before.

And I'm about to start babbling about something that's not really related to your topic, so I'll shut up now.
banazir
Apr. 7th, 2005 08:55 pm (UTC)
Re: Yabbut... They both have chewy coconut...
...EWWWWWW!
WHAT?! You blashpheme against the true god, Cowconut?
Kri jaffa!

Intriguing post. Makes me wonder about myself. I'm pretty sure I have a mathematical mind rather than a mathematician's mind, but I never really considered it before.
I think you have aome of both.

And I'm about to start babbling about something that's not really related to your topic, so I'll shut up now.
Huh? Everything's related to my tyopic!

--
Banazir
sui_degeneris
Apr. 10th, 2005 04:49 am (UTC)
Re: Yabbut... They both have chewy coconut...
WHAT?! You blashpheme against the true god, Cowconut?
Kri jaffa!

Fresh coconut is yummy. The dried stuff doesn't thrill me.

Stick that in your pipe and smoke it!

Intriguing post. Makes me wonder about myself. I'm pretty sure I have a mathematical mind rather than a mathematician's mind, but I never really considered it before.
I think you have aome of both.
Goodness, that's a frightening thought.

May I never get so caught up in virtuosity of any sort that I forget that not everyone is at the same level.

And I'm about to start babbling about something that's not really related to your topic, so I'll shut up now.
Huh? Everything's related to my tyopic!
Okay. On your head be it!

Text: http://www.livejournal.com/users/sui_degeneris/68618.html
Pictures: http://www.livejournal.com/users/sui_degeneris/70726.html
banazir
Apr. 23rd, 2005 06:34 am (UTC)
Lindenmayer systems and other fractals
Fresh coconut is yummy. The dried stuff doesn't thrill me.
But... but... eet can be grated on to CHOKLIT!

Stick that in your pipe and smoke it!
Ewww... smorking DRIED COCONUT?
I'd ruther have a stern loonk.

I think you have aome of both.
Goodness, that's a frightening thought.
Bha! I'm knot afraid!

May I never get so caught up in virtuosity of any sort that I forget that not everyone is at the same level.
Oh, amen, amen, Ms. Crochet qoeee... (J/K! *ducking*)

Huh? Everything's related to my tyopic!
Okay. On your head be it!

Text: http://www.livejournal.com/users/sui_degeneris/68618.html
Pictures: http://www.livejournal.com/users/sui_degeneris/70726.html

Well, I'll be. A Lindenmayer system!

I wuvs fractals, d'ye know?

--
Banazir
sui_degeneris
May. 1st, 2005 02:13 am (UTC)
Re: Lindenmayer systems and other fractals
But... but... eet (dried coconut) can be grated on to CHOKLIT!
Waste of good CHOKLIT, I call that.

Bha! I'm knot afraid!
But I AM!

May I never get so caught up in virtuosity of any sort that I forget that not everyone is at the same level.
Oh, amen, amen, Ms. Crochet qoeee... (J/K! *ducking*)

/me lobs freshly rolled ball of acrylic yarn at you

Actually, that's the sort of thing I'm talking about. I know I'm okay at crocheting. Better than some, worse than others. May I never forget (or worse, get annoyed when I'm reminded) that not everyone knows what a double crochet is. And, especially, may I never forget that some of the people who currently know less than I do about a topic will far surpass my knowledge before the end.

Thank you for the link the wikipedia article on Lindenmayer systems. I hadn't heard of them before. I also think that fractals are very cool, but hadn't considered that this beastie fell into that group.

Any idea for programs I can use to get the 'puter to do the crunching and plotting of these things, so I can just enjoy the pretty patterns?
( 22 comments — Leave a comment )

Latest Month

December 2008
S M T W T F S
 123456
78910111213
14151617181920
21222324252627
28293031   

KSU Genetic and Evolutionary Computation (GEC) Lab

Teunciness

Breakfast

Science, Technology, Engineering, Math (STEM) Communities

Fresh Pages

Tags

Powered by LiveJournal.com
Designed by Naoto Kishi