Comments on: Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning/
Comments on MetaFilter post Proofs and Pictures: The Role of Visualization in Mathematical and Scientific ReasoningSun, 20 Aug 2006 12:08:32 -0800Sun, 20 Aug 2006 12:08:32 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning
<a href="http://www.mediasite.com/Player/?p=218">Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning</a> <small>[video]</small> "The picture is a telescope for looking into Plato's heaven." -- <a href="http://www.chass.utoronto.ca/~jrbrown/">James Brown</a> <small>[<a href="http://64.233.167.104/search?q=cache:b-b60hPfOmUJ:www.chass.utoronto.ca/~jrbrown/+james+brown+toronto&hl=en&ct=clnk&cd=1">cached</a>]</small>post:www.metafilter.com,2006:site.54065Sun, 20 Aug 2006 12:08:20 -0800ChucklesphysicsproofmathematicsvisualizationsciencereasoningperimeterinstitutelectureBy: Chuckles
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407847
Many other great <a href="http://streamer.perimeterinstitute.ca:81/mediasite/viewer/FrontEnd/Front.aspx">Perimeter Institute public lectures</a> are available.
<img src="http://streamer.perimeterinstitute.ca/live_images/8cee2b41-842b-41d7-a478-9581fc52c7c3/Slide_0008.jpg">
QED MFcomment:www.metafilter.com,2006:site.54065-1407847Sun, 20 Aug 2006 12:08:32 -0800ChucklesBy: MonkeySaltedNuts
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407861
yeah. Pictures can be proofs for simple cases. But induction is a general method. I would like to see someone come of with a general method of making picture proofs for complicated answers you can prove by induction.comment:www.metafilter.com,2006:site.54065-1407861Sun, 20 Aug 2006 12:38:43 -0800MonkeySaltedNutsBy: orthogonality
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407864
Oh, fuck, that's a beautiful picture, Chuckles.comment:www.metafilter.com,2006:site.54065-1407864Sun, 20 Aug 2006 12:39:40 -0800orthogonalityBy: pracowity
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407871
Yay. So... can anyone show a picture that illustrates <a href="http://ask.metafilter.com/mefi/36264">this problem</a>? <blockquote>If there's a 20 percent chance of rain on Monday, a 20 percent chance on Tuesday, and a 30 percent chance on Wednesday, what's the overall chance that it'll rain during this three-day period? How do you calculate this?</blockquote>comment:www.metafilter.com,2006:site.54065-1407871Sun, 20 Aug 2006 12:51:14 -0800pracowityBy: TedW
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407882
Wow: who knew <a href="http://en.wikipedia.org/wiki/James_Brown_(musician)">Soul Brother #1</a> was such a math geek?comment:www.metafilter.com,2006:site.54065-1407882Sun, 20 Aug 2006 13:00:55 -0800TedWBy: MonkeySaltedNuts
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407889
<i>So... can anyone show a picture that illustrates this problem?</i>
For n days the picture is a n-dimensonal cube. Looking at it in 2 dimensions will give you the general pictuire.comment:www.metafilter.com,2006:site.54065-1407889Sun, 20 Aug 2006 13:07:26 -0800MonkeySaltedNutsBy: number9dream
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407911
I was at that talk. While the guy was interesting, he wasn't much of a mathematician. The "picture proof" of the Continuum Hypothesis was the most laughable thing I've ever seen somebody try to present as serious mathematics.comment:www.metafilter.com,2006:site.54065-1407911Sun, 20 Aug 2006 13:55:20 -0800number9dreamBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407915
<img src="http://img.photobucket.com/albums/v606/Oewd/weather.png">comment:www.metafilter.com,2006:site.54065-1407915Sun, 20 Aug 2006 14:05:22 -0800WolfdogBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407917
More clearly, the uncolored squares give (7*16)/250=44.8%, the chance of no rain.comment:www.metafilter.com,2006:site.54065-1407917Sun, 20 Aug 2006 14:07:55 -0800WolfdogBy: nickyskye
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407932
Nice work Wolfdog! And artistically attractive too.
Then there's the quirky <a href="http://indexed.blogspot.com/">probability version of things</a>.
I like his poetry on <a href="http://streamer.perimeterinstitute.ca/live_images/8cee2b41-842b-41d7-a478-9581fc52c7c3/Slide_0021.jpg">this image</a>:"The picture is a telescope for the mind's eye looking into Plato's heaven."comment:www.metafilter.com,2006:site.54065-1407932Sun, 20 Aug 2006 14:29:23 -0800nickyskyeBy: OverlappingElvis
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407950
I got really excited about this post when I saw a quote from James Brown (!) talking about math... until I realized it wasn't the right James Brown...comment:www.metafilter.com,2006:site.54065-1407950Sun, 20 Aug 2006 15:11:31 -0800OverlappingElvisBy: Earl the Polliwog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1407983
I entered this topic thinking that it was perhaps another <a href="http://www-personal.umd.umich.edu/~jwbrown/">James Brown</a>, whom I have heard many good things about.comment:www.metafilter.com,2006:site.54065-1407983Sun, 20 Aug 2006 16:13:19 -0800Earl the PolliwogBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408024
<small>(Just to be sure, the picture is constructed as if the three events are independent, which models the statement of the problem, but is probably not a realistic weather model. In the absence of any other information, if you don't like the independence assumption, you can still use a block-counting picture to show that if the given information in that problem is accurate, then there's at least a 30% chance and no more than a 70% chance of getting some rain.</small>)comment:www.metafilter.com,2006:site.54065-1408024Sun, 20 Aug 2006 16:53:11 -0800WolfdogBy: fleacircus
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408030
How would Wolfdog's method work for this nearly identical problem:<blockquote>If there's a cos(1) chance of rain on Monday, a 1/π chance on Tuesday, and a 43.2387315 percent chance on Wednesday, what's the overall percent chance that it'll rain during this three-day period (to 7 decimal places)?</blockquote>comment:www.metafilter.com,2006:site.54065-1408030Sun, 20 Aug 2006 17:13:07 -0800fleacircusBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408034
You go to the store, you buy a sheet of transcendentally ruled graph paper, and proceed as before.comment:www.metafilter.com,2006:site.54065-1408034Sun, 20 Aug 2006 17:19:55 -0800WolfdogBy: fleacircus
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408038
That could work, though knowing the area of those squares might get a bit tricky.comment:www.metafilter.com,2006:site.54065-1408038Sun, 20 Aug 2006 17:27:50 -0800fleacircusBy: spacediver
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408042
I took a course on thought experiments last semester with James Brown.
Very interesting course.comment:www.metafilter.com,2006:site.54065-1408042Sun, 20 Aug 2006 17:37:24 -0800spacediverBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408048
To be a little less glib, I'm a big advocate for the use of figures to help find an argument, support an argument, or be the entire argument - I'm perfectly happy with the picture versions of 1+2+...+n = (n^2+n)/2 and 1+3+5+...+(2n-1) = n^2 as essentially complete proofs that moreover contain some insight that an induction version might be lacking.
But probably the main reason math teachers spend more time teaching induction and algebra and counting techniques and epsilons and delta, and so on, is pointed up precisely by fleacircus - the cases where you can make a beautiful, insightful picture are very special ones. They're gems when they turn up, but a lot of problems just don't have nice pictures, or the picture is so hard to get to & interpret that it's not worth the effort. His pictures for the series (1/2)+(1/4)+(1/8)+... and (1/4)+(1/16)+(1/64)+... are nice - the latter one's beautiful - and they're even important when you're first trying to get the very idea that an infinite number of regions can be contained within a finite amount of area; but if that's your only technique and I do like fleacircus and tell you "Now, do the same thing to figure out what 1+(1/π)+(1/π<sup>2</sup>)+... is", (let alone Σ_{k=2}^∞ [C(n+k-2,k)*k*(k-1)/n<sup>k</sup>] )... well, good luck.
Skill with figures is an essential part of the problem-solving toolkit; <a href="http://en.wikipedia.org/wiki/How_to_Solve_It">Polya</a> didn't just throw "Can you draw a figure?" in his list for the hell of it. But if it's all you've got, you're going to be out of luck; and the special cases where you can do a figure are often exactly the same neat and tidy cases that afford the best practice with induction or other symbolic techniques when you're getting acquainted with them.comment:www.metafilter.com,2006:site.54065-1408048Sun, 20 Aug 2006 17:50:41 -0800WolfdogBy: Chuckles
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408091
I think you're overstating Brown's claim, Wolfdog. You still need a hammer, even if you only have a couple of nails.comment:www.metafilter.com,2006:site.54065-1408091Sun, 20 Aug 2006 18:43:23 -0800ChucklesBy: Chuckles
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408095
What is the problem with disproving the continuum Hypothesis? It is intuitively obvious that any interval on the reals has an uncountable number of intermediate values. So, is it just very very hard to show that formally, or am I just missing something? Brown said something about no well-ordering of the reals is known - is that the problem?comment:www.metafilter.com,2006:site.54065-1408095Sun, 20 Aug 2006 18:49:06 -0800ChucklesBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408099
Well, I think in his talk he overplays the "I do proofs by picture, I'm a rogue who doesn't play by THE MAN's rules" kind of angle, as if there was some sort of conspiracy against pictorial arguments. Probably just to make the talk a little more exciting, because it's more fun to do something if it seems a little illicit.comment:www.metafilter.com,2006:site.54065-1408099Sun, 20 Aug 2006 18:53:20 -0800WolfdogBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408154
I've only just gotten to the punchline of his continuum hypothesis refutation. He definitely overstates the novelty of the style of argument - whether there are pictures of darts involved or not, it's a familiar sort of "paradox" that arises when you make agruments based on probability <i>and</i> you allow yourself the sort of weird sets that the axiom of choice can summon into existence. At the heart of the argument is the same sort of stuff that makes the <a href="http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">Banach-Tarski paradox</a> (<a href="http://mathworld.wolfram.com/Banach-TarskiParadox.html">also</a>) tick. The wiki article is pretty concise at the top: some people <i>do</i> view this sort of thing as evidence that AC should not be accepted as an axiom; likewise, philosophically, you may buy Freiling's argument as real evidence that CH is not acceptable as a model of "how things really work".
If you view throwing out AC and CH as a "conservative" point of view, just be aware that you can be too conservative. By analogy, the parallel postulate in geometry - which occupies a similar indecidable position based on the commonly accepted axioms - turns out to give rise to <i>useful models of real things</i> whether you accept it OR one of its negations. It's not unreasonable to think that CH and (not CH) both give rise to set theories that are each correct models - for different things; we're just not subtle enough yet to understand which the problems are that should be modeled with a CH version of set theory, and which shouldn't, and why.</a>
There's a reference for Freiling's argument in <a href="http://www.cs.uwaterloo.ca/~alopez-o/math-faq/node71.html#SECTION001030000000000000000 ">this</a> CH page.comment:www.metafilter.com,2006:site.54065-1408154Sun, 20 Aug 2006 19:48:37 -0800WolfdogBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408158
<i>It is intuitively obvious that any interval on the reals has an uncountable number of intermediate values...</i>
With the usual notion of order, yes; intuitively obvious and also correct. Remember, he's talking about some exotic way of ordering the reals. And his order doesn't have to be in any way compatible the usual topology (nearness relations) of the real line.comment:www.metafilter.com,2006:site.54065-1408158Sun, 20 Aug 2006 19:51:37 -0800WolfdogBy: Chuckles
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408233
<em>Remember, he's talking about some exotic way of ordering the reals.</em>
Okay. And the usual notion of ordering the reals is specifically not a well-ordering (an open interval doesn't have a well defined first element, for anyone who's curious - this fact probably should have been clear from Brown's talk, I guess). I really don't have the background, and I'm starting to exhaust my attention span for the evening..
I find the attempt to apply discrete thinking to a continuous field troubling. Is that ridiculously naive?comment:www.metafilter.com,2006:site.54065-1408233Sun, 20 Aug 2006 21:59:43 -0800ChucklesBy: nickyskye
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408677
Dang, *gush warning*, this conversation is light years above my head but it seems so cool you guys are talking about this stuff.comment:www.metafilter.com,2006:site.54065-1408677Mon, 21 Aug 2006 09:59:55 -0800nickyskyeBy: beatrice
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1408932
This is really neat, thanks.comment:www.metafilter.com,2006:site.54065-1408932Mon, 21 Aug 2006 12:28:56 -0800beatriceBy: Wolfdog
http://www.metafilter.com/54065/Proofs-and-Pictures-The-Role-of-Visualization-in-Mathematical-and-Scientific-Reasoning#1411276
<img src="http://img.photobucket.com/albums/v606/Oewd/skatulyaelv.jpg">comment:www.metafilter.com,2006:site.54065-1411276Wed, 23 Aug 2006 05:15:38 -0800Wolfdog