Famous Fluid Equations Are Incomplete

July 29, 2015 12:34 AM

The Singular Mind of Terry Tao - "Imagine, he said, that someone awfully clever could construct a machine out of pure water. It would be built not of rods and gears but from a pattern of interacting currents." (via)

Tao has emerged as one of the field's great bridge-­builders. At the time of his Fields Medal, he had already made discoveries with more than 30 different collaborators. Since then, he has also become a prolific math blogger with a decidedly non-­Gaussian ebullience: He celebrates the work of others, shares favorite tricks, documents his progress and delights at any corrections that follow in the comments. He has organized cooperative online efforts to work on problems. "Terry is what a great 21st-­century mathematician looks like," [mefi's own] Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison, who has collaborated with Tao, told me. He is "part of a network, always communicating, always connecting what he is doing with what other people are doing."

also btw...
A 115-year effort to bridge the particle and fluid descriptions of nature has led mathematicians to an unexpected answer - "The evidence suggests that truer equations of fluid dynamics can be found in a little-known, relatively unheralded theory developed by the Dutch mathematician and physicist Diederik Korteweg in the early 1900s. And yet, for some gases, even the Korteweg equations fall short, and there is no fluid picture at all."

posted by kliuless (17 comments total)

Excellent read.
posted by flippant at 2:08 AM on July 29, 2015

Good read. I like how it stressed that math research requires creativity, not computation. I'm finding that creativity is really what makes a good researcher, not pure smarts or technica skills.
posted by MisantropicPainforest at 4:56 AM on July 29, 2015 [3 favorites]

Good read. I like how it stressed that math research requires creativity, not computation

Agreed. Those of us who are good enough at math to be dangerous (I majored in Physics in college) know this to be true. When you see the work of a great mathematician you see a mind that can make imaginative leaps in abstract realms.

Mathematics is really the highest form of thinking, the brain's deepest forays into purer and purer abstraction. It is a shame that people who don't get this tend to portray the art of mathematics as the juggling of algorithms or tricks of memorization. (Although I blame the education system for this reductive view.)

Relevant xkcd
posted by vacapinta at 5:32 AM on July 29, 2015 [10 favorites]

I'm finding that creativity is really what makes a good researcher, not pure smarts or technica skills

Or typing/spelling skills, for that matter.
posted by MisantropicPainforest at 5:40 AM on July 29, 2015 [3 favorites]

I remember Manuel Delanda talking about water machines in Mondo 2000 in precisely the same way. Not a bad thing, but surely prior art.

The thing is that there are already mind boggingly amazing water based machines. Like the sea and animals in general. Just saying.
posted by n9 at 5:44 AM on July 29, 2015 [1 favorite]

Good read. I like how it stressed that math research requires creativity, not computation

Agreed. Those of us who are good enough at math to be dangerous (I majored in Physics in college) know this to be true. When you see the work of a great mathematician you see a mind that can make imaginative leaps in abstract realms.


You need all of it. Certainly the most impressive mathematical thinkers I've known have been good at "doing the math." ("Computation" sounds too much like "arithmetic." Math people often protest exaggerated infacility with arithmetic. But there is plenty of reasoning that is not arithmetic but which it is at least incomplete to call "creative.") This seems to be born out by historical lore. E.g. Fermi problems require good intuitive facility with quantities.

But I like this discourse because it's an interesting test case for what we mean by creativity. An old classmate of mine (and a very good mathematician) now studies math education, and he emphasizes that the models of creativity used in psychological research tend to be extremely narrow. In most experimental paradigms, what's measured as "creativity" is basically "divergent thinking" -- to caricature a little, giving different answers from other people. But divergent thinking doesn't seem sufficient to explain what's creative in creative mathematical work.
posted by grobstein at 5:52 AM on July 29, 2015 [3 favorites]

On my blog, I expanded a little on my quote.
posted by escabeche at 6:01 AM on July 29, 2015 [2 favorites]

Back in the day I did fluid dynamics / engineering as part of my degree... there was a hell of lot of 'yeah, we really don't know what is actually happening here so we're just gonna add this made up constant / factor to make the equations balance.'
posted by fearfulsymmetry at 6:07 AM on July 29, 2015 [1 favorite]

'yeah, we really don't know what is actually happening here so we're just gonna add this made up constant / factor to make the equations balance.'
posted by fearfulsymmetry

Eponysterical.

And seconded.
posted by RolandOfEld at 7:16 AM on July 29, 2015 [3 favorites]

An old classmate of mine (and a very good mathematician) now studies math education, and he emphasizes that the models of creativity used in psychological research tend to be extremely narrow. In most experimental paradigms, what's measured as "creativity" is basically "divergent thinking" -- to caricature a little, giving different answers from other people. But divergent thinking doesn't seem sufficient to explain what's creative in creative mathematical work.

grobstein, not to derail, but this sounds super interesting. Do you have any citations/references from your classmate about better/fuller models/paradigms of creativity?
posted by suedehead at 7:50 AM on July 29, 2015 [1 favorite]

I don't see how you could be referring to creativity without relying centrally on novelty. Maybe Grobstein you are thinking of insight or profundity instead?
posted by leibniz at 7:53 AM on July 29, 2015

Well generally speaking you can have exact and accurate equations, or useful and solvable equations, but not both. So it's more productive to experiment and figure out how much your approximate, but useful equation is off by, use that, and let scientists figure out all the complex minutiae of why over the next century or two.
posted by Zalzidrax at 7:53 AM on July 29, 2015

I don't see how you could be referring to creativity without relying centrally on novelty. Maybe Grobstein you are thinking of insight or profundity instead?

Novelty (or originality) is certainly part of our conceptions of creativity. But it leaves out a lot. Creativity can't be reduced to novelty. We celebrate the creativity of Terry Tao, not Timecube guy (whose contributions may be stipulated to be completely novel).

My friend's gloss, to put it roughly, is that psychological measures of creativity pick out the Timecube guy. So we have studies that say you are more creative if you are in a distracting setting. Well, what does that mean? You get more focused answers if you let people focus; you get more scattered answers if you don't let people focus. "Novel" or "divergent" kinda just means: dispersed or non-central in the space of answers. So if you break up people's searches, you'll get that more often. And if you're not giving them any tough or interesting challenges, if your instruments are designed to measure creativity via scatteredness, then you will find that distracted people are more creative.

But I doubt you would make Terry Tao more creative by forcing him to do all his proofs in a coffee shop. (And Paul Erdos famously took benzedrine in his coffee, not cough syrup in his daiquiris.)

But, again, this isn't my research. Maybe I don't quite have the point. My friend is Ben Dickman. Here's his faculty page. His dissertation contains a useful review of creativity literature, making the point I am making (possibly butchering) above. He's also done a paper on Paul Cohen (forcing) as a case study of mathematical creativity, which sounds relevant but which I haven't read. I'll send him this thread in case he wants to clarify anything.
posted by grobstein at 8:23 AM on July 29, 2015 [7 favorites]

Thanks, grobstein. That sounds fascinating, and somewhat related to Paul Feyerabend's idea of epistemological anarchy, which I'll badly summarize as a rejection of any sort of meta-rules concerning scientific methodology - consistency, universally applicable rules, etc.

One question for escabeche and any mathematicians out there:

I find that disciplines with a lot of discussion and name/paper-referencing have a shared foundation and a structure of knowledge that is cumulative or easy to build upon. Computer science, for example, has named algorithms, processes, acronyms, jargon, because it's incredibly convenient to communicate with such methods. And such communication is only possible because, among many reasons, if person A knows loops and B knows loops, they can verify that they know the same thing. On top of that, if B learns bubble sort, then B can take advantage of their prior knowledge in a way that doesn't invalidate that prior knowledge but just builds upon it. Or functional programming building upon a knowledge of imperative programming, etc.

Other disciplines/practices have different structures; therapy/coaching, for example, in which techniques are highly discussed and named, yet progress in the field doesn't appear like this 'stacking' of techniques, but the refinement and creation of new ones. Or architecture/design, which mixes problem-solving, engineering, and aesthetics, and can have a maddeningly bespoke / custom-fit stance to solving one-off problems, largely because large-scale problem solving seems hard to generalize, let alone 'stack'. In these realms, references and acronyms and names play a large role, but they're a bit more social/referential, as they operate as pointers to a wide variety of examples in the world, rather than as units of knowledge that build upon each other cumulatively.

Reading the kind of social discussion and references in Tao's blog, it seems like it's a mix of the two. The presence of these references means that there is a clear shared language built upon prior work. At the same time, it seems that a lot of the discussion is within a given community, perhaps a little bit like certain philosophical blogs, which leads me to wonder if it's not also a lot of speculative discussion around examples and references that don't necessarily contribute on top of each other.

I suppose my question is: at the highest level of math, what does that discussion feel like? Philosophical debate, of facts mixed with speculation? Legalistic banter, where references are always given to prior work deemed 'correct'? Something totally different?
posted by suedehead at 9:58 PM on July 29, 2015

speaking of delanda and computation (by way of deleuze!) i saw this expressed recently: "I heard once that geeks come in two flavours: those who read A Thousand Plateaus; those who read Godel, Escher, Bach."

here's delanda on assemblage theory fwiw :P

at the highest level of math, what does that discussion feel like?

tao took a crack at it!

-What is good mathematics? [1,2]
-There's more to mathematics than rigour and proofs [1,2]

oh and john baez posted a "great description of what it's like to learn math" :P

The thing is that there are already mind boggingly amazing water based machines. Like the sea and animals in general. Just saying.

i guess i've been thinking about combinatorial, uh, explosion lately re: the 'growth of information' wrt the rise of computer-aided explanation, price theory and prigogine: "The equations of motion we've found to work very nicely in most applications --- whether Newtonian or quantum-mechanical --- are reversible, but led to irreversible phenomena in aggregate."
posted by kliuless at 10:29 AM on July 30, 2015 [2 favorites]

this thread is brilliant, and one of the reasons I read metafilter. I'm going to lose a week of work digging into all the links. Thanks kluiless and everyone for the great discussion.
posted by herda05 at 11:54 AM on July 30, 2015

Tim Gowers, in "The Two Cultures of Mathematics", writes about the tension between "stacking" and "non-stacking" in mathematics.
posted by escabeche at 4:23 PM on July 30, 2015 [2 favorites]