## Thursday, August 19, 2010

### Analysis vs Algebra predicts eating corn?

I like learning about odd connections between disparate things. This probably is the oddest example that I know.

Broadly speaking, mathematicians can be divided into those who like analysis, and those who like algebra. The distinction between the two types runs throughout math. Even those who work in areas that are far from analysis or algebra are very aware of the difference between them, and usually are very clear on which their preference is. I'll delve into this in more depth soon, but for now let's just take it for granted that this is a well-known distinction, and it has meaning for mathematicians.

Back when I was in grad school there was a department lunch with corn on the cob. Partway through the meal one of the analysts looked around the room and remarked, "That's odd, all of the analysts are eating corn one way and the algebraists are eating corn another!" Everyone looked around. In fact everyone was eating the corn in one of two ways. One way was to munch over the length of the corn in a straight line, back up, turn slightly, and do another row across. Kind of like how an old typewriter goes. The other way was to go around in a spiral. All of the analysts were eating in spirals, and the algebraists in rows.

There were a number of mathematicians present whose fields of study didn't make it clear whether they were on the analysis or algebra side of things. We went around and asked, and in every case the way they ate corn matched their preference. Since then I've made a point of amusing myself by asking mathematicians I meet whether they prefer algebra or analysis, and then predicting which way they will eat corn. I'm probably up to 40 or so by now, and in every case but one I've been able to correctly predict how they eat corn. The one exception was a logician who claimed to be exactly on the fence between the two. When I explained the corn thing to him he looked surprised, and said that he had an unusual way of eating corn. He went in loose spirals! In other words he truly was a perfect combination of algebra and analysis!

If you have even a passing familiarity of probability, it is clear that despite how unbelievable it initially is that the type of mathematics you prefer is connected to how you eat corn, it is pretty much certain that there actually is a very strong connection. If you believe, as I do, that this difference is connected to how we think about other things, then there must be some odd connection between how we like to understand the world and how we eat corn. Why is another matter.

How do I explain the distinction between algebra and analysis? Well the best way to understand it is to ask you to study advanced mathematics. You will have to take many courses with the word "algebra" in the name, and others with "analysis" in the name. By the time you're done you'll have experienced the difference, and you'll be clear on which you prefer. Odds are you won't do that, but that is the most reliable way to come to understand it.

If I have to wave my hands and explain it, I would explain it like this. In algebra there are sequences of operations which have proven to be important and effective in one circumstance. Algebraists try to reuse these operations in different contexts in the hopes that what proved effective in one situation will be effective again. By contrast an analyst is likely to form an idiosyncratic mental model of specific problems. Based on that mental model you have intuitions that let you carry out long chains of calculations that are, in principle, obviously going to lead to the right thing. Typically your intuition is correct to within a constant factor, and you're only interested in some sort of limiting behavior so that is fine.

If you don't know any advanced math, the odds are about equal that my explanation is going to mislead you as to give you an idea what I am talking about. You'd be better off figuring out your preference by looking at how you eat corn. That said, the distinction carries through into other subjects that I've learned about. But not in a clear and obvious way.

For instance I've noticed the difference cropping up in programming. The distinction is often hard to explain. There are a wide variety of programming techniques, and most programmers have only really learned a few. Some of those techniques appeal to analysts, and others to algebraists. But if you've only been exposed to techniques that are a good fit for one, then how do you know which you'd prefer? Worse yet, when two programmers talk and have different experience bases, how can they tell whether their natural intellectual tastes are similar or different?

Let me give some examples. Upon my first encounter it was clear to me that object oriented programming is something that appeals to algebraists. So if you're a programmer and found Design Patterns: Elements of Reusable Object-Oriented Software to be a revelation, it is highly likely that you lean towards algebra and eat your corn in neat rows. Going the other way, if the techniques described in On Lisp appeal, then you might be on the analytic side of the fence and eat your corn in spirals. This is particularly true if you found yourself agreeing with Paul Graham's thoughts in Why Arc Isn't Especially Object-Oriented. There was a period that I thought that the programming division might be as simple as functional versus object oriented. Then I encountered monads, and I learned that there were functional programmers who clearly were algebraists. (I know someone who got his PhD studying Haskell's type system. My prediction that he ate corn in rows was correct.) Going the other way I wouldn't be surprised that people who love what they can do with template metaprogramming in C++ lean towards analysis and eating corn in spirals. (I haven't tested the last guess at all, so take it with a grain of salt.)

Going out on a limb, I wouldn't be surprised to find out that where people fall in the emacs/vi debate is correlated with how they eat corn. I wouldn't predict a very strong correlation, but I'd expect that emacs is likely to appeal to people who would like algebra, and vi to people who like analysis.

And now to wrap up, why would how we eat corn say something how we think? Here is what I think.

When you pick up a piece of corn on the cob, you have two cues for how to eat it. The first is that the corn is laid out in very nice rows. How can you not follow the lines that are laid out for you? The other is that as you eat, your teeth scrape down the corn. If you twist your wrist, you'll eat more efficiently. Why would someone want to eat inefficiently?

My best guess is that the cue you notice and follow reflects a natural tendency about how you tend to think in general. And this tendency is tied to such things as what kind of math you prefer or what programming techniques would prove interesting for you.

Jim Bower said...

This is terrific! My wife and I both have Ph.D.s in analytical chemistry. We frequently argue about approaches to problems. We eat our corn in opposing ways. I relayed the story to her. She told me that I should never doubt her again, as the corn was telling her what to do. ;-}

Anonymous said...

I'm most definitely on the Algebra side of things (master's thesis was on semigroups). However! I eat corn by going down the columns, not rows, which means I twist my hands (is that a mixture of the two methods?) And I'm definitely more on the Lisp side of things than OO. Does this mean I have some inner conflict I don't know about?

PS. My academic career has been entirely about interdisciplinary subjects, so maybe that explains it.

btilly said...

@forwearemany: Huh. That's very odd.

I wonder whether more math would have caused you to switch interests. If so, you may be similar to me. I initially started off in algebra (my first paper was in Ring theory) but after a couple of years found myself clearly in the analysis camp, even if I was using it in unexpected ways. The PhD thesis I never defended (long story) used real analytic techniques to show that a particular classification technique would not work on a class of combinatorics problems. (Yes, I eat corn in spirals, and I'm competent but not enthusiastic when it comes to OO programming.

Nicolás Brailovsky said...

Well, I like Vim, love templates and eat corn in rows. That probably makes me crazy, right?

btilly said...

@Nicolás Brailovsky: The template metaprogramming and vim comments were ones that I threw out there. I've never actually tested those guesses. So your experience is simply evidence that my guesses were wrong.

I have asked enough people who are strongly into things like design patterns and Haskell monads that I'm reasonably confident of that correlation with eating across in straight lines. And I know enough people who have learned design patterns and come away unimpressed to know that is correlated with eating in spirals. (Note that the opinions of people who have not really learned those things is irrelevant to any conclusions. Until you've learned to think in a particular way, you can't make an educated statement about how well that way of thinking suits you.)

Neither conclusion is as confident as the one about algebra vs analysis. But both are on much better grounds than my untested (and likely incorrect) guesses about editors and template metaprogramming in C++.

Spherical square said...

I just stumbled across your blog and read this post.I am just an undergrad student and have not studied much mathematics, so I cannot tell about whether my preference is for algebra or analysis.
However, the elementary algebra I liked, and complex analysis didn't particularly interest me.
But one of your predictions is perfectly true. After taking my beginner course in C++, I have started thinking that far too much emphasis is laid on OOP.And I love function programming. Haskell is so nice to learn!
And yeah, I always eat my corn in spirals:D

Dave Smith said...

Until giving up on corn, I would eat one row across on opposite sides, then finish the ear off by eating half-spirals. That drove my Dad, who must have been an Algebraist in a prior lifetime, nuts.

AlephNull said...
This comment has been removed by the author.
AlephNull said...

My field is theoretical physics - specifically, funamental physics - e.g. quantum field theory, particle physics, quantum gravity, string theory, etc.

As anyone familiar with these fields will know, they require a large arsenal of mathematical tools, from variational calculus and differential geometry to topology and group theory, i.e. a fairly balanced combination of 'analysis' and 'algebra'.

So, by your logic, I should have no preference for eating corn either in rows or in spirals, and you would be right; my preferred way of eating corn is to scrape the corn kernels from the cob using a spoon or knife, and then eat the kernels with a fork. :o)

Maybe you think that's just perverse, but I have perfectly sound reasons for eating corn this way; it is less messy (no melted butter on your hands, etc.) and it avoids getting those annoying bits of corn stuck between your teeth, which I find always happens when I chew the corn straight off the cob.

AlephNull said...

Oh, and btw, I HATE vi and C++, so that would tend to indicate a bias towards alegbra, according to your model, though I am not aware of such a bias when I think about which types of mathematics I prefer using; themindset of theoretical physicists is 'If it works, use it.' And if there is more than one tool that works on a problem, choose the simplest, most elegant one.

As an aside, would you say that the division between analysts and algebraists coincides with the division between those who prefer discrete mathematics (e.g. combinatorics, graph theory) to those who prefer continuous mathematics (e.g. calculus)?

thomas bushnell, bsg said...

I'm a functional programmer who is definitely on the algebra side. Never had any taste for analysis. Oh, and I eat my corn across, as all right-thinking people do.

Boxo said...

But Haskell programmers *hate* Java!

Erik said...

Im a physics student. I been studying in the last couple of years about error analysis and computational physics. My mathematical background is mostly calculus, statistics/probability and a little bit of algebra(linear algebra mostly).

And i do it in the same way as AlephNull do. I use a knife to scrape the corn kernel's from the cob.

So if there are more physicist that do it like that, we probable have the physicist way to eat corn.

pd:sorry about my english, im a native spanish speaker.

Sam DeHority said...

Math guy here - an algebraic topologist. I use haskell, am obviously an algebra guy, and eat my corn in rows, but I use vim. This is a surprising discovery.

Jonathan said...

What I've seen:
Mbti Ni-oop ; mbti Ne-lisp

Rishi said...

I dont think its so black and white .. Eating corn is instinct , people dont put any thoughts to it .. and nor while they eat , they think of " how to effectively finish it off completely" .

finally spirials and straight shifts are too elementary .. to say that one of it belongs to aljebra and the other to analysis.

however , when one is vacuuming a room, he /she often thinks deliberatly- how to effectively do it , so that its quick and the entire area is combed .

most people so in patches .. so can it be said that they are from group theory / or chaos theory . and people who finish it off in straight lines are linear algebra :)

sprocketeer said...

As Jonathan points out (somewhat cryptically), the axis you discovered seems to correspond to one axis of the Myers-Briggs Type Indicator (MBTI).

Specifically, it's the "sensing / intuition" axis. To quote from wiki:

Sensing and intuition are the information-gathering (perceiving) functions. They describe how new information is understood and interpreted. Individuals who prefer sensing are more likely to trust information that is in the present, tangible, and concrete: that is, information that can be understood by the five senses. They tend to distrust hunches, which seem to come "out of nowhere". They prefer to look for details and facts. For them, the meaning is in the data. On the other hand, those who prefer intuition tend to trust information that is more abstract or theoretical, that can be associated with other information (either remembered or discovered by seeking a wider context or pattern). They may be more interested in future possibilities. For them, the meaning is in the underlying theory and principles which are manifested in the data.

btilly said...

@sprocketeer: I am not sure what parallel you are trying to draw, but you are definitely drawing an incorrect one. I have known plenty of mathematicians in both algebra and analysis who are very, very solidly on the N site of the MBTI scale.

asdfghjkl said...

Shouldn't it remind you more of the judging/perceiving axis?

"INTPs value keeping their options open and find deadlines restrictive; it’s cutting them off from bringing in more possibilities. INTJs value coming to a conclusion so they can put their ideas in action and efficiently move on to the next task."

asdfghjkl said...

And I would conjecture that it's actually the other way around with Emacs/vi (consider Emacs with its endless extension capabilities and the fact that its basically a bunch of functions written in Lisp vs the simpler but extremely effective editor that is vi)

Tony Hursh said...

Possibly preferring recursion versus iteration might be a better test than OO versus functional?

Anonymous said...

This is uncanny.

Ian Wilson said...

I performed better at algebra but I always felt like there was a curious mystery to analysis. Basic algebra overall seems much more programmatic. Sort of like analog versus digital. As I remember, mathematicians at my school divided the world into abstract algebra and real/complex analysis. Here is my contribution to your ongoing model:

I performed better at Algebra(Abstract/Linear versus Complex/Real Analysis).

I eat my corn in rows, left to right, like a civilized being.

I prefer iteration over recursion.

I prefer emacs over vi.

I prefer objects, not sure about deep OO, over purely functional programming.

Anonymous said...

ok the first part is uncanny but I have a different explanation. What attracts me about algebra is the breadth of the application whereas analysis seems more like "drilling down" into specific problems. So I see the analysts "killing" everything that's directly in front of their path, very focused-like, and the algebraists meandering about on the corn (perhaps "horizontal corn eating" and "breadth of application" are related for more than just linguistic happenstance)

Arvind Narayanan said...

This is a fantastic thesis, but sadly I must offer a counterexample. I have done only a little higher math, but enough to know that I come down on the analysis side of things, enough that your explanation about following intuition and being correct to within a constant factor makes sense to me.

I do a lot of programming, and I'm *definitely* on the functional side. And I use vi rather than emacs, having tried both.

I eat corn in rows :-(

rshewmaker said...

Knife first...I can't stand corn splatter when eating off the cob.

Alyx Clarke said...

I dont eat corn on the cob in any of these ways.
I hold it so the end is flat on a plate, I get a knife and cut the corn down in 4 strips so when its finished the cob looks like a perfect cuboid with 4 striped sided.
I guess that makes me a control freak or possibly crazy lol

Shaun G said...

Fascinating! I've made this the daily poll question on Correlated (http://www.correlated.org). Let's see what other surprising correlations we can make from this!

Tieros said...

My corn eating approach is spiral.
My MBTI is INTP.
I hate emacs, love vi.
I hate OOP, especially when forced (AKA Java), but tolerate object-aware languages. Prefer flexible scripting languages like PHP, LUA, and Perl.
No preference over iteration vs. recursion.
My job is Enterprise Architect, with a focus on human process optimization, and an obsession with code, database, and other technical optimizations.
Never liked algebra, calculus, or deadlines :-)

My wife alternates between spirals and the typewriter approach to eating corn. I always suspected she had multiple personalities :-)

Justin Steele said...

I think there may be some confusion. In other's comments I read that they have a degree focused more on one yet they don't follow the corn eating technique. I don't think this has anything to do with what you "have", is it not more correlated with what you LIKE?

Jason said...

question is, if children eat corn before they learn math can you predict the kinds of math they will be drawn to by the way they eat corn?

John Haugeland said...

You can tell a real programmer/gamer in how they state their corn strategy, rather than measure a group's approach and try to infer strategy from it.

Superior corn strategy involves maximizing group z-perpendicular surface area for efficient butter and salt distribution with minimal dripping.

If you're one-lining it, you're pessimizing your corn experience in order to not think. All experiences are combat ladders.

FIGHT HARDER. YOU CAN LEVEL AT CORN.

msutherl said...

"I wouldn't be surprised that people [...] lean towards analysis and eating corn in spirals. (I haven't tested the last guess at all, so take it with a grain of salt.)"

ba-dum-tsh

Tim Ottinger said...

Vim, OO, and rows. If you pry the kernels out instead of biting them off, you leave no sloppy residue behind and consume the corn entirely. You can't do that in spirals.

Tristen said...

physics...I hate eating corn on the cob, but when I do anyways b/c of nonoptional socials obligations I tend to do so in wavy lines, but would much rather have it loose on my plate :)

Scott Norble said...

It sounds like you can use this to determine if someone is a mapper or a packer.

http://c2.com/cgi/wiki?MappersVsPackers

Flavio Martins said...

One of my biggest pet peeves is the sound people make when they eat corn on the cob, disgusting!

I cut the corn off of the cob. I can't stand getting it stuck in my teeth, then eat it with a fork.

IT Executive Manager...not a mathematician.

Nick Kitten said...

To me, analyst vs algebraist sounds like the classic Aristotelian vs Platonic philosophies, and isn't all that difficult to explain: the one starts with data, and the other starts with logical primitives. Of course, most of the people posting are the ones who think they buck the trend (everyone likes to feel special and difficult to categorize), but it makes sense that a trait which would probably reveal itself in a personality test would also come out in more subtle ways. Now I'm not going to be able to eat corn without thinking about this!

Elizabeth Marston said...

I'm into cryptography. At some point I forced myself to switch from eating my corn in rows to eating corn in as random a pattern as I could generate.

I can see now my intuitions were correct: I *was* closing a dangerous side channel.

Mikko said...

I've been a programmer since 1983. I eat carrots.

Singletoned said...

Eating in rows creates a series of small steps that will eventually solve the problem.

Eating in a spiral creates a single graceful solution that will solve the problem.

I think that if you ask people if they think there is a right way to eat corn, the people who answer yes will be more likely to fit your theory, with the no's being more likely to be outliers.

Andrew L Coleman said...

I am an algebraist, eat my corn in neat rows, and greatly prefer object orientation. Although my favorite language is powershell because it has even more organization than Java if you let it. I'd say, your theory applies to me.

fsdafsdbbbbbb said...
This comment has been removed by the author.

I really do not agree with the claim that algebra has no intuition. Actually, I believe it is a lot more intuitive that analysis. It is all about the set of integers, roots of polynomials, etc. You can explain a seventh grader the field extensions, the ring of integers of an algebraic number field, the Galois group of an integer, or the complex field using algebra. Try to do that through analysis and it is hell on earth.

The problem is that for the first few semesters of algebra we learn facts on groups and rings without making much sense of them. By the time when we try to explain what these things really mean we have lost most of the people.

A Breaking Change said...

reminds me of the subjectivity of the word "simple." I often as interviewees "which of the following two expressions is simpler? (A) 1 + 2 * 3, (B) 123timesplus

I've given this interview question dozens if not hundreds of times, and the answers are evenly split.

Epsilon Given said...

I'm an Algebraist who loves anything geometric. I eat my corn in rows. And while I appreciate Haskell, I'm attracted to the techniques in On Lisp, and understand where Paul Graham is coming from in why he doesn't like object orientation. On the other hand, I also like how Haskell's type system is basically object oriented, even if certain people don't see it that way.

Thus, I'm a case where I match the mathematician corncob hypothesis, but not the mathematician programming-style hypothesis--but then, I'm not nearly as conversant in programming as I am in math, so my preferences might change as I learn even more. Although, for what its worth, when I first made the transition from BASIC to C++ in high school, I really liked OOP!

wordsmith said...

I have always eaten corn in rows, and the first time I ever saw anyone eat corn in spirals/circles, I was shocked. Algebra? Perhaps ~ I thought that discrete math and number theory were fun, but on the other hand I also enjoyed real analysis. I had to work to get an A in topology, and would have liked to taken more (only got my MS, but I had flirted with the idea of going on, and so had begun taking some of the coursework necessary for the qualifiers). The one area I absolutely couldn't stand was PDEs; however, this aversion could very well have been due to the instructor's presentation of the subject.

But what really clinches it for me is emacs :D

Aretae said...

Algebraist. Rows. Fascinating.

S.P. Zeidler said...

Physicist, prefer Algebra, eating corn on the cob row-wise but not typewriter-style but weaving shuttle-style, and preferring vi. Mostly preferring procedural languages for the simplicity.

Curt Sampson said...

I don't think that the emacs versus vi distinction will work out. I'm clearly an algebraist (I like algebraic structures, hate calculus, liked GoF back when I was still an OO programmer, and switched to Haskell), but I invariably use vi rather than Emacs when I need to edit a file.

However, I don't particularly admire vi, while I do admire Emacs. I use vi in part because it's what I happened to start on thirty years ago, and in part because emacs, while elegant in so many ways, is significantly less efficient for the most common editing operations.

It gets more complex: from the viewpoint of a Unix user, I don't find the monolithicity of Emacs to be elegant at all. If I frequently had to use Windows as well, I would probably think differently about this.

In short, I think it's too mixed an issue for a simple one-of-two answer to have any significance.

Minerva said...

I absolutely adore template metaprogramming, and I eat my corn in neat rows. I am also passing fond of OOP and algebra. So there's an answer to your last prediction there!

Jack said...

Interesting... I'm an algebraist by training who has jumped onto the "Big Data" bandwagon. I eat corn in rows and, yes, I tend to think in terms of OOP when coding (but not exclusively so).

However, I prever vim over emacs.

Harkonnen Jesuit said...
This comment has been removed by the author.
Dibyo Majumdar said...

I like eating my corn in spirals but I also like OOP quite a lot. Where does that place me? (I like to think of myself as an analyst).

Melanoman said...

I eat my corn in neat columns. Eat a give width rotating, move over and repeat. Is this what you meant by "spirals" or are people actually moving over a little bit each bite to make a spiral?

C.J. Bott said...

Holds for my wife and I as well. I'm an algebraist who eats corn in rows, and she's an analyst who eats in spirals. So great! :)

John Baez said...

Note also that the term "kernel" has different meanings in algebra and analysis. In algebra a kernel is the set where a homomorphism vanishes, while in analysis a kernel is a function of two variables used to define an integral operator.

Unfortunately this doesn't seem to explain the different styles of eating corn.

Siddhartha said...

Very interesting, but the programming language correspondence assumes the opposite of OOP is functional - and scala (which I use) shows this is not the case. I would venture that the analysts are those who choose to use C/C++ and hand-optimize, while algebraists (such as myself go for any abstraction available - OOP, Functional (Monads etc.), even both. And algebraists will not only prefer emacs to vi, but atom to both.

Paul Franklin said...

While I have never, to my best knowledge, taken an analytic course, I have much enjoyed algebraic math. From your description of analytic math, I doubt I would like it. And I eat typewriter style. But your analysis of why I might do so is off. When I'm done with a corn cob, every bit of kernel is gone, clipped off precisely at the root. Perhaps I have never observed an analyst eating corn, but everyone who eats around the cob (and I used to do so) leaves massive amount of kernel in his wake.

So does an analyst leave a clean cob?

Justin Barry said...

I have my bachelor's in math and computer science. I loved Abstract Algebra and had a tough time with Analysis. Def lean more toward OO programming and I love Haskell's type system! I eat corn in rows! whoa!

Dr Rick said...

I wrote object-orientated code in emacs when working on my postdoc in, roughly speaking, algebra. And yeah, corn in rows.

Unknown said...

I like to categorize analytics and algebraics as "calculation vs. theory." I find that most people, even the non-technically-minded, can grab that a bit better.

I am definitely a theory guy; it was my focus at university. I'm also a devout object-oriented programmer. Sure enough, rows is the only right way to eat corn from the cob.

MC said...

LOVE that this seven year old post is still getting comments. I just stumbled upon it via reddit and posted it to facebook, which ended up causing a mathematical existential crisis among some of my spiraling algebra-loving friends (although, most of the commenters fit nicely into either the typewriter/algebra or spiral/analysis box). Personally I am on (and feel strongly about) the rows/algebra side. This has made my week!

Charles Taylor said...

But, people drawn to Java probably like the orderliness that comes with static typing.
The type system doesn't go far enough for Haskellers and the language constrains you in other ways.
OO and FP programmers both care about high faluting abstractions that help them understand their code base. There's just serious disagreement about what makes a program orderly and comprehensible. (I'm strongly believe FP achieves it's stated goals better than OO. Yet OO still dominates...)

Unknown said...

No offense, but putting group theory in a bunch of analytical tools doesn't make you an algebraist! Most physicists are definitely more on the analytical side of thigs, and your description definitely fits that. There are symmetric things everywhere, but that doesn't make you an algebraist!

The algebraist feel is often described as trying to solve a problem by seeing some structure as part of a larger family of other structures and seeing how the members of this family behave with respect to one another (using morphisms). Repeating this step a few times, you go further and further away in terms of levels of abstraction, and the fun begins...!!!

Which you just don't do in physics. I don't see any physicist go "oh, this space-time has this problem, let's deform space-time as a family of space-times and... No. Haha