Archive for the ‘Mathematics’ category

It Came From Planet Nine

January 26, 2016

There be dragons out there, waaaaay out there, in the dark, off the edge of the map.

Or rather, a virtuoso combination of observation and mathematical modeling has led to an exciting, in some ways joyously old-school prediction. Orbital oddities identified in a handful of distant Kuiper Belt Objects (KBOs) were subjected to the same kind of inquiry that allowed 19th century astronomers to infer Neptune from Uranus’s behavior, in what was widely understood to be a triumph of Isaac Newton’s “System of the World.”


The new analysis, by two Caltech astronomers, theoretician Konstantin Batygin and the observer and Slayer-of-Pluto Michael Brown, has led to a broad outline of what to expect — a ~10 Earth mass planet travelling a very eccentric orbit that never comes closer to the sun than ~250 Earth-Sun distances, a unit of measure known as the Astronomical Unit.

I’m sure many of you saw the news about this last week.  Alexandra Witze in Nature had a  good write-up, as did Alan Burdick in The New Yorker.  (For those (quite a few) on the blog with the urge to read the original Batygin-Brown paper — go here.)

I couldn’t be more excited by the news.  I sometimes forget what an extraordinary run of solar system exploration I’ve been privileged to witness.  The variety we’ve found exists in our near-environment has leapt unbelievably, just in the last two or  three decades, and the richness and complexity of our own solar system is allowing us to make more sense of the process of planet and planetary system formation as more and more data emerges about exo-systems.

But for all that excitement, there’s something special about a new major planet.  As I write in The Hunt for Vulcan [Shameless Plug Here], the idea of a whole new world joining the neighborhood had enormous romantic power in the eighteenth and nineteenth century.  Arguably, given our present immersion in the imagined reality of multiple worlds, that romance cuts deeper still today.

But. ButButButButBut….it’s important to remember that a prediction, no matter how well supported, how seemingly necessary, isn’t the same thing as proof, as the discovery itself.  That’s what I tried to say in this essay on the subject.  A sample:

In 1846, the discovery of Neptune turned Le Verrier into a celebrity; for a time, he was the most famous man of science in the world. He went on an international tour and seized the moment to rise to the top of power in the highly contentious and hierarchical world of French astronomy. Batygin and Brown are taking a much more measured tack with Planet Nineand for good reason.  “We felt quite cautious about making the statement we made,” Batygin says.  Why such concern? Because, he says, “immediately after the detection of Neptune spurious claims of planets in outer solar system began to surface. We didn’t want to be another red herring.”

It wasn’t just the distant reaches of the solar system that tripped people up: 


The only problem being, of course, that Vulcan was never there.

I’m much more hopeful for Batygin and Brown’s Planet Nine, but hopeful don’t pay the rent — or, as Batygin told me:

“If Newton is right, then I think we’re in pretty good shape,” says Batyagin. “We’re after a real physical effect that needs explanation. The dynamics of our model are persuasive.” And yet, he adds, that’s not enough. “Until Planet Nine is caught on camera it does not count as being real. All we have now is an echo.”

There’s a surfeit of terrestrial crazy to weigh us down.  It’s a relief, I find, to look up and out, and contemplate the ordered mysteries that so thoroughly dwarf Comrade Trump’s Yuuuuuggggge self conceit.

Images:  William Blake, Isaac Newton1795

Johannes Vermeer, The Astronomerc. 1668

Quote of the Day: Cosma Shalizi, or why mathematicians are delightfully different from other people

August 27, 2010

Presented without comment, from a stray announcement on Cosma’s invaluable, slow-cooked blog, Three Toed Sloth.

Every human relationship is a unique and precious snowflake, but do we treat them that way when we model them mathematically? No.

Image:  Hiroshige, “Kambara,” number 16 from The Fifty Three Stations of the Tokaido, 1831-4 (Hoeido edition). And yes, sometimes I do post so I can put up (and look at) pix I love.

Quickie Post, to let you know that David Brooks has finally revealed the secrets of conservative math.

March 23, 2010

I’m on the road again in yet one more Newtonpalooza, so no substantive posting is going to happen, but I saw in my morning check of Balloon Juice (the only source for news you can really use) that David Brooks has produced another of his considered analyses to explain the real meaning of critical events.

The whole thing is yet one more sample of the unique combination of credulousness and really dangerous hackery-in-defense-of-establishment-power that characterizes Mr. Brooks’ work, and I’m going to try to go blog-medieval on it in the near future.   But here I just want to point out the implications of the delicious sentence that Mr. Brooks writes one truly revealing sentence, the one quoted by DougJ in his BJ snark:

Nobody knows how this bill will work out. It is an undertaking exponentially more complex than the Iraq war, for example.

The overt dumbness has already been dealt with at Balloon Juice.

As the commenters there point out, the only honest response is “Uh…..noes.”  It is also all you need to explain why the GOP so badly botched everything about that war.  They thought and think this is true, that destroying a country and rebuilding would be simpler and cheaper than regulating insurance companies.

(On that note — about two years into the Iraq war I had the chance to talk, completely informally with Madeleine Albright.  Among much else interesting, she told me that in the briefing she and other former high-ranking Clinton and Bush I officials received in the run-up to the war, the Rumsfeld DOD had made essentially no after-conflict plans, which we know now to be exactly right.  She told me she raised the thought that this was wrong, that real post-conflict planning had to be done to deal with all kinds of things, from the vacuum in civil power to economic matters.  She was, she said, brushed off by the Bush version of the Best and the Brightest — the So-So and the Not-Quite-Set-To-Be-Watered-Twice-A-Day).

But enough of ancient history.  I’m still wallowing in the mess of trying to understand  Mr. Brooks tortured diction.  Just what the hell could he mean by “exponentially” in this context?

Well — math jokes are not for amateurs, and I certainly don’t want to dive into xkcd territory (No! No! No!  Quantum leaps are really smalllllll), but it occurs to me that Mr. Brooks’ statement is more than usually meaningless if you don’t know what exponent he’s thinking about.

And then it became clear.  The only way any of Mr. Brooks’ attempts to assert some connection between his thought and that fundamental tool of science, mathematical reasoning actually makes sense, given the gap between reality and his accounts of it, is if that exponent contains the factor “i.”

That is all.

Image:  Nicolas Neufchâtel, “Nürnberger Schoolmaster Johann Neudörffer and a Student,” 1561.

Quickie math (arithmetic)/journalism peeve: I don’t think “party lines” means what you think it means/NY Times edition.

August 7, 2009

Update: a couple of changes below to try to make the point I thought I was making.

From today’s NY Times article on the Senate vote to extend the Cash for Clunkers program:

In the end, the bill passed largely along party lines, with 51 Democrats, 2 independents and 7 Republicans voting for it.

What’s wrong with that?

This phrase:  “largely along party lines.”

Let’s do the math, shall we?

7 Republican senators voted with the majority to pass the bill.  4 Democrats joined the minority opposing the bill.  So the Democrats did come close to holding party lines on the bill; only 6.7 % of the Democratic caucus defected.

But 7 Republicans?   That’s 17.5% of the rump opposition party.  That’s a substantial defection, a significant breach in party discipline, especially given the record of GOP monolithic “no-ness” on most major votes this year.  Even more so when you consider that this is in essence a small supplement to the economic stimulus bill, and as such would naturally be a target for a Republican party that has pushed a lot of chips onto the table betting against the success of Obama’s economic policies.

Numeracy matters — and it does so not just at the level in which an understanding of statistics in any kind of deep way might be helpful, but even at this simplest level of fifth grade math.

I’ve written before on the importance of distinguishing between raw numbers and some representation of the data that permits comparisons to be made across different circumstances or events.  See this post for a much sadder example.  But the point keeps getting made by the way journalists handle numbers.  Simply the act of deriving a percentage from the tally sheets is way too often a bridge too far.

Why the failure to count on one’s fingers at such an elevated institution as the Times?

I don’t know for sure, certainly — I wasn’t in Matthew Wald’s head when he wrote the piece, nor the room in which his editors ran their eyes ove rhis copy.  But I do note that the running journalistic line for weeks now, significant especially in the context of the health care debate, has been that the Democrats have failed to meet promises of bipartisanship, and that hence, their legislative agenda and even their (our) legitimacy is in doubt.

If that’s the story line, then clearly each significant vote must be characterized as evidence of the ongoing division between the parties, and the inability of the Democrats to construct legislation attractive to a rumored center and the mostly mythical moderate wing of a party dominated by Limbaugh dead-enders.

And so, even in this case, when Democratic legislation on a high profile economic and environmental* program attracts a significant number of Republican votes, enough to release some Democratic senators to cast what may be either or both politically useful and or principled nay votes, the Times still feels compelled to assert, in essence, that this Democratic — and partly Republican — legislative victory was in fact yet another blow to bipartisanship.

The moral of this story:  this is how destructive tropes get made.  In the construction of myths epithets matter.  Little phrases, rote characterizations —  “the wily Oddysseus,” “loose-tongued Thersites” — they frame the reader’s interpretation of all the action surrounding the thus-labeled person or event.  They serve as received wisdom, a collectively agreed value judgment, to speed the plot.

And here, the phrase approximating an epithet, “party lines,” is the tell that the writer has substituted such prior qualitative interpretation for thought.  The cash for clunkers bill saw close to one fifth of the opposition party break ranks.

But to say so, to recognize that a Democratic policy initiative was sufficiently successful/popular that a significant subset of the GOP senate caucus chose to embrace it would have forced the writer (and editors) of the Times’ story  to break out of that sweet sleep in which there is no need to actually do the simplest bit of arithmetic.

Image:  Jan Breugel the Elder, “A Fantastic cave with Odysseus and Calypso,” c. 1616.

Sunday Links

June 7, 2009

Running late — deadlines on three (3!) pieces by tomorrow at eleven at the latest.

So far I have one in draft, one that I will conjure out of another piece of writing on the same subject for a different purpose, expanding by 60% (all good stuff, no groats for fillers) and a third that is but a gleam in my eye.

Plus it’s Sunday and my son wouldn’t mind having his dad share the same spacetime coordinates with him, and nor, emphatically would his father.

So not much today, except to thank all those who listened to my appearance on Ira Flatow’s NPR TOTN:  Science Friday to discuss Newton and the Counterfeiter— and especially those who mentioned the gig beforehand on Twitter or in their blogs.  (I haven’t managed to send my personal thanks to all those in the latter group yet, but I will.)

If y’all missed it, you can hear my dulcet tones here.*

So just to demonstrate that I’m only mostly about me, and not solely, I’d like to point folks to a running series  of cool over at Daily Kos, Orinoco’s “Fundamental Understanding of Mathematics” farrago, now up to its eighteenth installment.

As long term readers of this blog know, I’m a big proponent of evangelizing the idea that injecting the most basic and simple of mathematical ideas into one’s own reasoning and into public discourse pays enormous benefits.

I’ve given a few scattered examples over the 18 months or so I’ve been writing this blog — e.g. the conversion of raw numbers into abstractions like percentages** but Orinoco, a school teacher who has taught math to young kids, is doing a wonderful job in building a systematic appreciation for both the constructs and habits of mind of mathematics accessible to just about anyone.  Highly recommended — and especially if you read through the comments where a number of math and computer geeks expand on Orinoco’s themes.  This is what I love about the internet:  spontaeneous collective creativity.

Anyway, here’s the link to diary one, and you can find the rest by navigating through Orinoco’s diary page.

*And yes, on the higher traffic day of Monday, I’ll give this another plug — I enjoyed it; I think it went well; and I’d like people to hear it.

**And yes again, I know that numbers are themselves abstractions.

Image: William-Adolphe Bouguereau, La leçon difficule (The Difficult Lesson), 1884

Friday xkcd break: When Mathematicans Stray

December 5, 2008

With thanks, as always to xkcd, in whose head it must be passing strange, (though fun) to live.

Saturday Math/Funk Unification post

September 6, 2008

Because of this

…I get to post this:

What Does the Public Really Need To Know?: Science/Math edition.

July 14, 2008

So, last week I have the good fortune (a) to junket in LA (thanks, History Channel — look for their latest Einstein documentary sometime between October and the new year) and, thus geographically advantaged, the chance to raise a glass or two with Sean Carroll and Jennifer (new digs) Ouellette (familiar haunt) — two of the brightest lights among those who blog the physical sciences.

Among much other discussion (how to do good science on television, whether there is any useful algorithm available to help navigate LA traffic) we drifted into that hardy perennial: what, really, does the general public need to know about science. Not for the greater good of science, not to secure more complaisant support for big accelerators or stem cell research, but for them/ourselves?

There are lots of facts that I think would give people pleasure — I love knowing that Albert Einstein patented a hearing aid (with Rudolf Goldschmidt); that chimpanzees fashion tools in the wild; that the first reaction written down in something like the modern form of a chemical formula was that describing the fermentation of alcohol. There are ideas that are enormously powerful — and some of them are clearly of value as part of anyone’s mental apparatus in confronting daily life. (Natural selection, offers insights well beyond the history of life, for example, (though great care must be taken, as we know, to our sorrow) and as general a heuristic as Ockham’s Razor would help people deal with silly season stories like this one.)*

But while these and much more are part of what I think any education should provide, the question I asked over something-or-other in martini glasses last week,** and re-ask here, is what the minimal body of knowledge is that every adult should possess.

Regular readers of this blog will guess the answer I gave: the bare minimum is arithmetic, or more broadly, a grasp of quantitative reasoning and a set of simple rules to apply such reasoning in everyday life.

For example — these posts sought to illustrate of the value of remembering to do something as basic as converting a cardinal number into a percentage, to make it possible to compare different data points.

Another example: the habit in this country of focusing on miles-per-gallon as a measure of fuel efficiency leads systematically to bad decision making. If we instead looked at gallons-per-mile (or hundred miles), it would make it clear that replacing a 16 mile per gallon SUV with a 20 mpg station wagon is a much better choice than replacing a 34 mpg compact with a 50 mpg hybrid, assuming equal miles driven for each vehicle. No one reading this needs much help figuring out why — but for the details, listen to the NPR story from which this particular example came. (See — I had to say something nice about NPR after slagging them for their Shakespeare follies.)

In sum: I’ve been at the popular science game for a quarter of a century now. I’ve written about climate change and physics and cancer research and precision guided weapons and big telescopes and the origins of the pentatonic scale and I can’t remember it all now. I hope everything found some audience who got something out of it. But more and more now I look for stories that in their telling express some of the basic habits of scientific thinking — whatever the body of facts with which I may be dealing.

There is much more to such habits than a quantitative turn of mind — notions of observation, of framing answerable questions and lots besides . But more and more the starting point seems to me to be conveying how much mastery of the world one can get from astonishingly simple acts of counting and comparing.

What do y’all think?

Update: See Chad Orzel’s recent post on John Allen Paulos’ Innumeracy for another swipe at the same problem. (h/t Bora)

*For an antidote to the “Who wrote Shakespeare” tomfoolery, you can begin here with James Shapiro’s latest — one of the best of a spate of Shakespeare-as-window-on-the-birth-of-the-modern books that have appeared recenlyy.

**Fortunately, the waiter in the very chic bar in which the three of us chatted had never heard of what I tried to order, a French 75, which is the only reason I remained unfogged enough to have any kind of a conversation that night. Just the mention of it makes me feel a little shaky. Enjoy, but at your own risk.

Image: Codex Vigilanus, 976 C.E., in which Arabic numerals first appeared in a Western European manuscript. Source: Wikimedia Commons.

We Love Math, Electoral College Department.

June 12, 2008

Andrew Sullivan says that this question-and-answer is why he doesn’t do math.

That Sullivan is quantitatively challenged is, of course, no surprise to anyone who reads his blog on a regular basis.  But his excuse here is pretty lame.

The problem posed at was “How many unique ways are there to acquire at least 270 electoral votes without any excess?”

The solution to that did indeed turn on a sophisticated application of combinatorial methods.  According to the analysis by Isabel Lugo (posted at 5:22 p.m. on June 10), there are 51,199,463,116,367 different possible  ways to accumulate 270, 271, 272 or 273 electoral votes.

Lugo’s solution does indeed demand both smarts and training, and she received her just due of praise from the comment thread.  Certainly, though I can follow, gasping, the reasoning as she explained it, I can’t claim any greater chance of cracking such a problem than Sullivan could — which is to say, none.

But Sullivan’s surrender — “it’s just too hard, and look at the cute, big number” makes me crazy, and illustrates one of the persistant reasons why our discourse is so bad, why, as Brad DeLong keeps asking, we can’t get a better press corps.

That is: there is a difference between ignorance of advanced math (in which I take second place to no one), and an inability or unwillingness to master the basics of quantitative reasoning.

What’s remarkable, is how far you can get with not that much, just a basic disciplined approach to simple concepts — estimation, use of ratios and so on.

And with such simple tools it is possible to get a handle, if not always a precise result, even for such subtle, complex problems as the electoral vote question that so flummoxed Sullivan.

As Lugo pointed out, introducing her analysis — her exact number was anticipated by a much simpler simulation by commenter Brian at 4:43.  Even if you don’t follow Brian all the way through the simulation, his exercise begins with a simple piece of arithmetic that gives the first hint of the scale of a likely solution, the fact that with 51 jurisdictions there are 2 to the fifty first power, or 2.25 quadrillion possible win/lose outcomes.

That’s enough to tell you from the start that you are dealing with a big number. The next steps take you further, and show how the simulation produces a plausible argument that the number of outcomes where the electoral vote totals hit the desired range (270-273) is going to come in at just a bit under three percent of that huge total number of outcomes, or right in the range of the 51.2 trillion outcomes that Lugo derived.

And my point is that whether or not you can imagine performing this bit of computer-mediated approximation, even the very first step, one that comes from high school math, is enough to get you into the right neighborhood, the right scale in which any answer will have to land.

It’s a necessary skill for any reporter today, I think, really any citizen.  I won’t go here into the same riff I’ve blogged many times before.  I’ll outsource instead to my new blog humor BFF xkcd:

Welcome to Cosmic Variance folks — and a question

March 11, 2008

Welcome to all coming via Sean Carroll’s very kind shout out.

Come on in, look around, enjoy yourselves.

And if you have a moment, consider answering this prompt. In this post written a couple of weeks ago, I wrote a complaint about some lousy reporting on the housing crisis — but my larger point touched on one of the big themes of this blog, how applying even the simplest quantitative reasoning makes a huge difference to one’s ability to make sense of (detect the bullshit in) everyday experience. I argued that this was one of the foundations of what is often miscalled (IMHO) scientific literacy as it applies to the public. I pointed to a couple of examples, one from Freeman Dyson, and another by J.B.S. Haldane to show how such minimal math makes a difference in real science as well.

And then I made this request: Perhaps readers could be persuaded to post examples of what they think are elegant, simple insights about everyday experience such simple applications of math can give us?

Anyone want to belly up to the bar?

In any event — glad to have you all here.

Image: Hans Holbein, “Portrait of the Astronomer Nikolaus Kratzer (detail)” 1528. Image: Wikimedia Commons