shapes, figures, and forms.isomorphismesTumblr (3.0; @isomorphismes)http://isomorphism.es/A sudoku of linear functionals<p>Today’s glossary provoked by <a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=0914098705&linkCode=as2&tag=isomorphismes-20&linkId=IKTOL4CO2QN7SDH4" title="chapter 4, paragraph 1" target="_blank">Michael Spivak</a>.</p>
<p><a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=0914098705&linkCode=as2&tag=isomorphismes-20&linkId=IKTOL4CO2QN7SDH4" title="chapter 4, paragraph 1" target="_blank"><img alt="tensors spivak diff geom vol 1 ch 4" src="https://31.media.tumblr.com/2d24b30ef5d27f1644943f86513cfd30/tumblr_inline_naowyvNmyg1qbdydv.png"/></a></p>
<p>Below I’ve drawn two <a href="http://isomorphism.es/post/1396693746/vectors" title="remember, vectors can represent tons of stuff, which is why they're interesting" target="_blank">vector</a> spac<a href="http://isomorphism.es/post/8696066280/semigroups" title="..but they can't represent everything. No rules against stopping time, for instance. (necessity of identity element)" target="_blank">e</a>s connected by a linear <a href="http://isomorphism.es/post/4187773646/rotations-are-linear-transformations" target="_blank">homomorphism</a> ƒ, plus a <a href="http://isomorphism.es/post/85638008449/vector-duals-and-musical-isomorphism-in-1-d" title="Functionals always map onto 1-D, like the «reduce» of «map-reduce». Coming-from-one-dimension linear functionals would be, like: double; triple; quadruple; quintuple; dectuple; reverse; reverse-and-double; throw-it-all-in-the-trash." target="_blank">linear functional</a> λ going to ℝ. After seeing these pictures I hope it’s easier to understand how the pullback ƒ* works.</p>
<p>Here’s one of the main pictures for flavour:</p>
<p><img alt="dual space linear functional" src="https://31.media.tumblr.com/b90eefeb36006099949e40d27a52b988/tumblr_inline_nape5eGDie1qbdydv.png"/></p>
<p>Also, you can probably just skim the pictures and get the point <small>(especially the Number Field one and the final one)</small>. That’s the fastest way to read this post.<br/><br/></p>
<hr><p><br/>Start with an abstract 𝓥ector space.</p>
<p><img alt="abstract vector space" src="https://31.media.tumblr.com/0f5677189fc35ca88c3bafabe303e552/tumblr_inline_napx6dPrL21qbdydv.png"/></p>
<p>I’ll do some <a href="https://twitter.com/isomorphisms/status/212400801975500801" title="by introducing an arbitrary coordinate system and therefore spending gauge, but now I can do calculations." target="_blank">violence</a> because I’ll need coordinates in a minute.</p>
<p><img alt="invent linear functional" src="https://31.media.tumblr.com/daa6d3f7a99d6bcca70d235d472947a2/tumblr_inline_napx6qTY3x1qbdydv.png"/></p>
<p><!-- more --></p>
<p>The simplest way to think about a linear function, in <a href="http://isomorphism.es/post/4810875961/determinant" title="if it were in multiple dimensions we'd be talking about a matrix. see ~/tagged/matrices or ~/tagged/matrix." target="_blank">1-D</a>, is as continued addition of some constant. Say the constant were <code>4</code>, then <a href="http://isomorphism.es/tagged/linear" title='I wrote this description down somewhere else under the "linear" tag....' target="_blank">linear</a> would be <code>+4+4+4+4+4+4+4+...</code>, “x” times, to use the jargon. If the constant were <code>m</code> then linear would be <code>+m+m+m+m+m+m+m+m+...</code> “x” times, or <code>y=<a href="http://isomorphism.es/tagged/matrix" title="conveniently we could use a capital M for the Matrix needed in multidimensional setting." target="_blank">m</a>x</code> in the usual parlance.</p>
<p><img alt="linear functional buildup" src="https://31.media.tumblr.com/268e9a9cc756e30b09ebec39e5839bc3/tumblr_inline_napx7ar8Lf1qbdydv.png"/></p>
<p></p>
<p>Linearity is a strong enough condition that I can infer everything that happens on my “horizontal number line” <small>(the first dimension I violently imposed)</small>. It has to be <code>+4+4+4+...,</code> adding that same constant every tick right.</p>
<p><img alt="linear functional buildup" src="https://31.media.tumblr.com/53cba5dcef814d6d8021d0815f6c9e05/tumblr_inline_napx7uDo2V1qbdydv.png"/></p>
<p>Linearity<small>—and weaker conditions like homomorphism—</small>fix zero. The zero of the domain has to map onto the zero of the codomain. So you always know linear subspaces are “centred”, they have to contain zero <a href="http://isomorphism.es/post/80742382617/rank-nullity-theorem" title="This is part of the context I think about with the rank-nullity theorem. You don't just have some "overlapping Venn diagrams" picture for linear subspaces -- all the Venn diagrams are centred the same, encasing each other. That doesn't quite work because linear subspaces are going to drop dimensions so like go from hyperspace→plane→line→zero. There are many paths to do this, which generates a poset structure. And the bottom of that poset is just one thing, {0}." target="_blank">at least</a>.</p>
<p><img alt="linear functional buildup" src="https://31.media.tumblr.com/e6c664dd7dcea8b806dfeeb1a180936c/tumblr_inline_napx8gfSGN1qbdydv.png"/></p>
<p></p>
<p>Since I’m doing 2-D, I need to make up another constant for what the λ <a href="http://projecteuclid.org/euclid.bams/1183500620" title="Functionals just mean functions onto →ℝ." target="_blank">functional</a> does in my “vertical direction”. To be really interesting I chose <code>+1+1+1+1+1+1+...</code>. Because of various freedoms I have, <small>like due to the possibility Gram-Schmidt orthogonalisation and such</small>, I can make the arithmetic really easy on myself. That is, I don’t have to work in some skewy, wonky basis. Instead, it’s necessarily possible to choose directions that will “work like” independent horizontal and vertical, even coming from an abstract vector space like the parameters for <a href="http://isomorphism.es/post/904006029/splines" target="_blank">spline</a>-based functions <a href="http://isomorphism.es/tagged/basis+functions" title="or something less tangible..." target="_blank">modelling</a> heart arrhythmia. So I’m making all my constants positive and aligning the axes <a href="http://isomorphism.es/post/20541715959/categorial-decomposition-galileo" title="so the arithmetic in one direction is INDEPENDENT of the arithmetic in the other. «as simple as possible, but no simpler.»" target="_blank">perpendicularly</a>.</p>
<p><img alt="linear functional buildup" src="https://31.media.tumblr.com/5d7135f8b8afb1a3c88f4a061a04f479/tumblr_inline_napx8wL44q1qbdydv.png"/></p>
<p><span>Now, again because of linearity, λ is very limited in what it can do. Remember homomorphisms ƒ always do ƒ(→+↑) = ƒ(→)+ƒ(↑). </span><small>(Here → and ↑ stand for “vertical” and “horizontal” movement—as I violently defined them—in the initial space 𝓥.)</small><span> So I can just maze through or Sudoku through all the points in 𝓥 and figure out what λ has to do there, now that I’ve chosen vertical constant to be </span><code>+1+1+1+1+1+...</code><span>. and horizontal constant to be </span><code>+4+4+4+4+4+...</code><span>..</span></p>
<p><img alt="linear functional buildup" src="https://31.media.tumblr.com/704f00b1c61da6c31fa2e12cfba7ff92/tumblr_inline_napx9jPk2m1qbdydv.png"/></p>
<p>So if over → is +4 and up ↑ is +1, then over & up →↑ = +5 = up & over ↑→. By homomorphic-ness of ƒ.</p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/71b574c9c1ef7ff7a1f72dc57fa6c441/tumblr_inline_napyocCw2K1qbdydv.png"/></p>
<p><span>I can continue to maze my way around, figuring out what λ has to do since I already chose the only two constants there are to choose for λ.</span></p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/34fe508ff656cdbca052bb0775809961/tumblr_inline_napynqWFfT1qbdydv.png"/></p>
<p>This is just like a sudoku! Except not painfully boring. I line up two thingies and they make a thingy.</p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/d8f04eeb49387aa44b1a81bf2a388c4e/tumblr_inline_napyn9RiRA1qbdydv.png"/></p>
<p>Yes! Take that, tube puzzles.</p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/2eba1195546d40c8d4ec39e6a9372a95/tumblr_inline_nap06228Aj1qbdydv.png"/></p>
<p>I’ll just <code>eval</code>uate those….</p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/a1012314bf054433ce73369e61253d31/tumblr_inline_napyljmbHg1qbdydv.png"/></p>
<p><span>…and remove the ornamentation now that you get the point…</span></p>
<p><img alt="linear functional sudoku" src="https://31.media.tumblr.com/4b078a1ec488be7d10ba79b20a33e46b/tumblr_inline_napyksEIzT1qbdydv.png"/></p>
<p>You can see it’s a proper number field! <small>(The most familiar n<a href="http://isomorphism.es/post/38251878380/differential-pushforward" title='eh ... "scalar field" ... who knows the word "scalar" outside of the classroom?!' target="_blank">umber fields</a>, if you watch TV, are the weather maps</small></p>
<p><a href="http://www.met.ie/climate-ireland/surface-temperature.asp" target="_blank"><img alt="average temperature" src="http://www.met.ie/images/climate/temp01.gif"/></a></p>
<p>but of course nowadays we do “heatmaps” of various things that are not <a href="http://isomorphism.es/tagged/temperature" title="∃ more interesting maths around temperature than you might think..." target="_blank">heat</a>:</p>
<p><a href="http://en.wikipedia.org/wiki/Winter_of_2010%E2%80%9311_in_Great_Britain_and_Ireland#Deaths" target="_blank"><img alt="deaths due to the winter of 2010" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/The_British_Isles_winter_weather_event_deaths_of_2010.png/800px-The_British_Isles_winter_weather_event_deaths_of_2010.png"/></a></p>
<p>By “number field” I just mean to each region <a href="http://isomorphism.es/post/5488198245/clouds" title="funny how there's a transition between the two..." target="_blank">or</a> <a href="http://isomorphism.es/post/14471242946/is-there-really-such-a-thing-as-a-point-well-not" title="points of infinitesimal size? I don't believe they actually exist." target="_blank">point</a>, <a href="http://isomorphism.es/post/38251878380/differential-pushforward" title="notice the phrasing is backwards of the symbology. Map *to* ℝ = assign *from* ℝ. Pesky" target="_blank">is</a> assigned a number ∈<a href="http://isomorphism.es/post/6796129033/banach-tarski-paradox" title="We don't like ℝ ... but that's what Spivak used, so we'll just wave our hands whilst saying "ℝ" and mimeo-tap with our feet the Morse code for "The concept of continuity has not been sufficiently elaborated"." target="_blank">ℝ</a>.</p>
<p><img alt="scalar field" src="https://31.media.tumblr.com/e976c93c7122fcf2c99e2b92a166bf19/tumblr_inline_napyk02Nf61qbdydv.png"/></p>
<p>This is a linear 2-D number <a href="http://isomorphism.es/tagged/scalar+field" title='Fine, "scalar field"!' target="_blank">field</a>, whereas Ghrist drew us a nice picture of a <a href="http://isomorphism.es/tagged/smooth" title="smooth ⊃ linear" target="_blank">smooth</a> 1-D number field (“differential one-form”), interpreted with relevance to <a href="http://isomorphism.es/tagged/dynamical+systems" target="_blank">dynamical systems</a> and their equilibria.</p>
<p><a href="http://isomorphism.es/post/9592554225/ghrist-one-form" target="_blank"><img alt="differential one-form" src="http://media.tumblr.com/tumblr_lqtrg2On8V1qbdydv.png"/></a></p>
<p>If you cared to, you could compute some stuff about this number field. Like the distance you have to go, in what direction, before the <a href="http://isomorphism.es/tagged/isoclines" title="isocline, isotherm, isobar, isoutility curve..." target="_blank">level set</a> increments up. Or the slope of the level sets. For instance look at one of the 11’s and try to find the other 11 by looking at the pattern of the other equi-coloured pairs.</p>
<p><img alt="level sets of a scalar field" src="https://31.media.tumblr.com/12927a43b9092a0dce689eb136f0856b/tumblr_inline_napyjmsx451qbdydv.png"/></p>
<p>If you can see through the <a href="http://isomorphism.es/tagged/colour" target="_blank">colour</a> barf, then you’ll notice the level sets are all straight and all march with <a href="http://isomorphism.es/tagged/curvature" title="uneven pace would be either concavity or convexity... some kind of nonlinearity" target="_blank">even</a> pace.</p>
<p><img alt="convexity and concavity, vs linearity" src="http://media.tumblr.com/a4114f2289011ea2aa05153ea7c19cfa/tumblr_inline_mf8x96llxR1qbdydv.png"/></p>
<p>This derives from the linearity and homomorphic-ness of λ.</p>
<p><img alt="level sets of a linear scalar field" src="https://31.media.tumblr.com/bdbd4b2ff6a53299a71488b08df0d55f/tumblr_inline_napyiyCqk41qbdydv.png"/></p>
<p>Now that’s λ sorted. What was the original question? Oh yeah, <a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;camp=1789&amp;creative=9325&amp;creativeASIN=0914098705&amp;linkCode=as2&amp;tag=isomorphismes-20&amp;linkId=IKTOL4CO2QN7SDH4" target="_blank">Spivvy</a>.</p>
<p><img alt="image" src="https://31.media.tumblr.com/19f8db934cf8020475388046bff1c506/tumblr_inline_napyhw8Pgs1qbdydv.png"/></p>
<p>He started us off with three things: 𝓥, ƒ, and λ.</p>
<p><img alt="image" src="https://31.media.tumblr.com/e6fc9a4f13e5b66ed2adb78f1746a4f5/tumblr_inline_napyh6uxEG1qbdydv.png"/></p>
<p><span>I assumed you </span><a href="http://isomorphism.es/post/21037878642/emotional-zero-vector-size-distance-addition" title="abstract vector spaces 'n' 'at" target="_blank">knew</a><span> 𝓥 and only talked about λ, some linear N</span><a href="http://isomorphism.es/tagged/map-reduce" title='the "reduce" of "map-reduce" is N→1 dimensional.' target="_blank">→</a><span>1-dimensional function. I’ll do a bit about ƒ, the 3rd ingredient—a homomorphism of </span><a href="http://isomorphism.es/post/12848094184/space" title="linear spaces = abstract vector spaces, in this case" target="_blank">spaces</a><span>—as well.</span></p>
<p><img alt="image" src="https://31.media.tumblr.com/26ebe94993ff80b7470b7a9ced1b442f/tumblr_inline_napygiFzI91qbdydv.png"/></p>
<p><a href="http://isomorphism.es/post/4187773646/rotations-are-linear-transformations" title="Like, a rotation would be an example of a homomorphism." target="_blank">Homomorphisms</a> work for “<a href="http://isomorphism.es/post/39608848450/go-around-it" target="_blank">going the long way</a>" because they <a href="http://isomorphism.es/post/33909584057/the-same-like-how" title='keep "the same", meaning "the same as far as vector-space properties go"' target="_blank">preserve</a> algebra.</p>
<p><img alt="image" src="https://31.media.tumblr.com/e0bf2a9b8efa1aa9e2169083e15dfbb9/tumblr_inline_napyfwoGx31qbdydv.png"/></p>
<p><span>All of </span><a href="http://isomorphism.es/post/11407834056/matrices-linear-algebra" target="_blank">the usual matrices</a><span> preserve linear structure. Meaning we can do convenient things like pretend (as I did above) that I can </span><a href="http://isomorphism.es/tagged/decomposition" target="_blank">decompose</a><span> an arbitrary abstract vector-space into </span><a href="http://isomorphism.es/post/20541715959/categorial-decomposition-galileo" target="_blank">independent</a><span> pieces: because that goal is just a matrix transform away. Since matrix transforms preserve all of the properties I’m using (linearity), I just need to keep a spare gauge in my pocket, knowing I can </span><a href="https://terrytao.files.wordpress.com/2012/11/blog-book.pdf" target="_blank">spend</a><span> it if necessary to enter the annoying world of actual numbers to work something out. But, knowing that I </span><em>could</em><span> if I</span><em> wanted to</em><span>, I’ll dispense with the actual work whenever possible.</span></p>
<p><img alt="image" src="https://31.media.tumblr.com/028dd371a6a4c1140c2de9defc563f9b/tumblr_inline_napyf4GfR61qbdydv.png"/></p>
<p><span>There’s </span><a href="http://stanford.edu/class/ee364a/videos.html" title='S. Boyd used the "programming" distinction in his convex-optimisation lectures (¿I think lecture 3?) and I really liked that.' target="_blank">something</a><span> about “data types” here. In the abstract vector spaces 𝓥 and ƒ(𝓥) = 𝓦, we allowed ourselves quite a bit of freedom. In fact that’s why this is interesting: because 𝓥 and 𝓦 are general enough to represent something </span><a href="http://isomorphism.es/post/1396693746/vectors" title="like a face, or the history of sea levels at some particular buoy station, or handwriting styles, or ¿personality types? ¿feelings?" target="_blank">interesting</a><span>, not just the triangles and squares and circles and numbers of grade-school mathematics.</span></p>
<p><img alt="image" src="https://31.media.tumblr.com/2158d6792ef55bbc3b8ddc3e16f4b659/tumblr_inline_napyebVplS1qbdydv.png"/></p>
<p>What that means is that I allowed myself to call 𝓥 and 𝓦 vector spaces, so long as I could come up with <em>some</em> properly commuting, distributing, zero-and-one-having pair of <a href="http://isomorphism.es/tagged/bilinear" title=""binary" meaning "taking two arguments". Eg 125+8712, g⁂q, ƒ(a,b). Bilinear being a subset of binary but I've written more about it. If you can find the Wikipedia article with asterism ⁂ and some other crazy symbol representing the two binary operations in the example, please leave a comment pointing me to it! It was the most abstract and wacky symbology I've seen for "two binary operations which interact [however]"." target="_blank">binary</a> operations. But that means that 𝓥’s “plus” and 𝓦’s “plus” might not be the same! Since both are relatively free concepts on their own. But as a homomorphism, ƒ still needs to match them up.</p>
<p><span>So I’ve tried to indicate the situation with colour barf above. Does that work for you??? 𝓥’s stuff gets purplish/bluish colours and 𝓦’s stuff gets an equally tasteful, yet distinctive, pink/green/brown palette.</span></p>
<p><span>The harmless-looking algebra of ƒ(v+w) = ƒ(v) + ƒ(w) is not trivial at all! So I try to substitute something like ƒ(v+w) = ƒ(v) ⊕ ƒ(w) to emphasise that these are not the same plus signs. They’re not operating on the same <a href="http://qchu.wordpress.com/2013/05/28/the-type-system-of-mathematics/" target="_blank">datatype</a>: + operates on 𝓥’s things and ⊕ operates on 𝓦’s things. It could be something like + operating on <a href="http://isomorphism.es/tagged/fourier+transform" target="_blank">frequency</a> bands and ⊕ operating on circuit <a href="http://isomorphism.es/post/5045646882/superposition-of-emotions" target="_blank">superpositions</a>—freq-bands and circuits aren’t even the same <em><a href="http://isomorphism.es/post/34359201729/inheritance" title="Yeah, yeah, you could have different data-types that inherit operations from each other, like $USD datatype inherits addition from Number datatype ... let's not complicate the issue." target="_blank">kind</a></em> of thing.</span></p>
<p><span>(Also 𝓥’s “zero” and 𝓦’s “zero” aren’t the same. Like a null circuit ≠ a null set ≠ a null operator ≠ a null <a href="http://isomorphism.es/tagged/trading" title="Could just be an order that gets routed and comes back from the broker with a signal confirming it was received but no action taken. Like a ping." target="_blank">trade</a> ≠ a null wave … they all do nothing, but they do it to different kinds of things.)</span></p>
<p><img alt="image" src="https://31.media.tumblr.com/6caa7537332c343ae298b6997192ec1a/tumblr_inline_napycxF2YK1qbdydv.png"/></p>
<p><span>So </span><a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;camp=1789&amp;creative=9325&amp;creativeASIN=0914098705&amp;linkCode=as2&amp;tag=isomorphismes-20&amp;linkId=IKTOL4CO2QN7SDH4" target="_blank">Spivak</a><span> started us with essentially a diagram chase:</span></p>
<p><img alt="image" src="https://31.media.tumblr.com/6f2b9515ab24758c5fc327fa4f2385d9/tumblr_inline_napycgDB9G1qbdydv.png"/></p>
<p></p>
<p>The simplest homomorphism ƒ I can think of would be <a href="http://isomorphism.es/tagged/rotation" title="and this is significantly easier in 2-D than in 3-D. Rotations in 3-D are a bit more like the complex numbers--I'm alluding to a more surprising homomorphism between SO(3) and SU(2)--in that there aren't quite 2 independent directions for rotation in 3-D. If I rotate 180 degrees in one of the directions, the other direction gets orientated backwards. That's like the sesquilinear ("1.5 dimensional") property of complex numbers, which happens because e^( √−180 degrees) = −1. Did a post about that way back." target="_blank">rotating</a> the basis,</p>
<p><ins><a href="http://arxiv.org/abs/gr-qc/9712019" title="chapter 1" target="_blank"><img alt="image" src="http://media.tumblr.com/a4f7694308aa17338141db3ad0e0ce10/tumblr_inline_n9nt9zgvBR1qbdydv.png"/></a></ins></p>
<p><br/>so let’s do that one.</p>
<p><img alt="image" src="https://31.media.tumblr.com/5cfd4917e44c931daaaf53fcb9f3484a/tumblr_inline_napyc3hbBF1qbdydv.png"/></p>
<p>In that case there’s an “obvious” way to put λ and the rotation ƒ together: just rotate the number field.</p>
<p>You could transform the number field by any old <a href="http://isomorphism.es/post/11407834056/matrices-linear-algebra" target="_blank">matrix</a> as well – I just didn’t want to draw those ones. But if you were plotting a linear heatmap in <a href="http://r.isomorphism.es" target="_blank">R</a> then matrix-transformations of that heatmap would be easy enough to do and you could see the precomposition of λ with ƒ that way.</p>
<p>Now, that’s not explaining the pullback ƒ*, but the “precomposition” idea is what I guess you need to understand that passage in <a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&amp;camp=1789&amp;creative=9325&amp;creativeASIN=0914098705&amp;linkCode=as2&amp;tag=isomorphismes-20&amp;linkId=IKTOL4CO2QN7SDH4" target="_blank">Spivak</a>. Here it is again.</p>
<p><a href="http://smile.amazon.com/gp/product/0914098705/ref=as_li_qf_sp_asin_il_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=0914098705&linkCode=as2&tag=isomorphismes-20&linkId=IKTOL4CO2QN7SDH4" title="chapter 4, paragraph 1" target="_blank"><img alt="spivak dg ch 4" src="https://31.media.tumblr.com/2d24b30ef5d27f1644943f86513cfd30/tumblr_inline_naowyvNmyg1qbdydv.png"/></a></p>http://isomorphism.es/post/95450363264http://isomorphism.es/post/95450363264Fri, 22 Aug 2014 04:23:00 -0600linearfunctionalsone-formsdual spacepushforwardMichael SpivakGram-Schmidt orthogonalisationevalmapreducemap-reduceGram-Schmidt orthogonalizationorthogonalisationorthogonalizationsudokuscalar fieldsgauge theorypullbackprecompositionbinary operationsfunctionsmathsmathematicsmathdifferential geometrytensorsdimensionsdiagram chasingdata typestype theorywaterpaperink:
Misty Mountains
An old painting I found while...<img src="http://38.media.tumblr.com/c75a234457aa15b11d60d23f4bd8dbbf/tumblr_n67mmtyoXj1r6u0tdo1_500.png"/><br/><br/><p><a class="tumblr_blog" href="http://www.waterpaperink.com/post/87103036909/misty-mountains-an-old-painting-i-found-while" target="_blank">waterpaperink</a>:</p>
<blockquote>
<p>Misty Mountains</p>
<p>An old painting I found while doing some spring cleaning</p>
</blockquote>http://isomorphism.es/post/95433633407http://isomorphism.es/post/95433633407Thu, 21 Aug 2014 22:30:00 -0600artnaturecolour"For what is the theory of determinants? It is an algebra upon algebra; a calculus which enables us..."“For what is the theory of <a href="http://tmblr.co/ZdCxIy4Um2Gv" target="_blank">determinants</a>? It is <b>an algebra upon algebra</b>; a calculus which enables us to combine and foretell the results of algebraical operations, in the same way as algebra itself enables us to <b>dispense with the performance</b> of the special operations of arithmetic. All analysis must ultimately clothe itself under this form.
<br/><br/>
I have in previous papers defined a ‘<a href="http://isomorphism.es/tagged/matrx" target="_blank">Matrix</a>' as a rectangular array of terms, out of which different systems of determinants may be engendered, <b>as from the womb of a common parent</b>; these cognate determinants being by no means isolated in their relations to one another , but subject to certain simple laws of mutual dependence and simultaneous deperition.”<br/><br/> - <em><p>James Joseph Sylvester, 1851</p>
<pre> </pre>
<p><small>(a relevant fact is <a href="https://en.wikipedia.org/wiki/Determinant#Cramer.27s_rule" target="_blank">Cramer’s rule</a>: knowing only the <a href="http://isomorphismes.tumblr.com/post/4810875961/determinant" title="What is a determinant?" target="_blank">determinants</a> of submatrices you can find the <a href="http://isomorphismes.tumblr.com/post/4261903646/eigenvector" title="What is an eigenvector?" target="_blank">eigenvectors</a>, which are the stable fixed-points under the matrix operation)</small></p>
<p>from the same source, quoting Sylvester’s <em>Apotheosis of Algebraical Quantity</em> (1884):</p>
<blockquote>
<p>A matrix … regarded apart from the determinant … becomes an empty schema of operation, … only for a moment<strong> looses the attribute of quantity to emerge again as quantity</strong>, … of a higher and unthought-of kind, … in a glorified shape-as an organism composed of discrete parts, but having an essential and undivisible unity as a whole of its own .… The conception of multiple quantity thus rises on the field of vision.</p>
</blockquote>
<p>The romantic view expressed there doesn’t sound very different, to me, of <a href="https://twitter.com/isomorphisms/timelines/476964783414333440" target="_blank">Deleuze on calculus</a></p></em>http://isomorphism.es/post/95142718130http://isomorphism.es/post/95142718130Mon, 18 Aug 2014 18:30:00 -0600matricesromancealgebradeterminantsJames Joseph SylvesterCramer's rulematrixdeterminantmadewithisometric:
Made with Isometric
<img src="http://31.media.tumblr.com/e9e172a74e4be0612f92944767bc9d55/tumblr_mpvle8D8bh1sxg2xko1_500.jpg"/><br/><br/><p><a class="tumblr_blog" href="http://madewithisometric.com/post/55378176301/made-with-isometric" target="_blank">madewithisometric</a>:</p>
<blockquote>
<p>Made with <a href="http://isometricapp.com" target="_blank">Isometric</a></p>
</blockquote>http://isomorphism.es/post/95048342138http://isomorphism.es/post/95048342138Sun, 17 Aug 2014 18:30:09 -0600art"adults, unlike children, rarely cry in public. They wait until they’re in the privacy of their..."“<p>adults, unlike children, rarely cry in public. They wait until they’re in the privacy of their homes—when they are alone or, at most, in the company of one other adult. On the face of it, the “crying-as-communication” hypothesis does not fully hold up, and it certainly doesn’t explain why we cry when we’re alone, or in an airplane surrounded by strangers we have no connection to…</p>
<p>In the same 2000 study, Vingerhoet’s team also discovered that, in adults, crying is most likely to follow a few specific antecedents. When asked to choose from a wide range of reasons for recent spells of crying, participants in the study chose “separation” or “rejection” far more often than other options, which included things like “pain and injury” and “criticism.”</p>”<br/><br/> - <em><p>about a paper by <a href="https://pure.uvt.nl/portal/files/388957/cryrev.PDF" title="I'm not sure if this is actually the correct paper but at least Vingerhoets is an author on the paper being discussed." target="_blank">Vingerhoets, Cornelius, Heft, Beck</a></p>
<p><em>Towards a Model of Adult Crying</em></p></em>http://isomorphism.es/post/94952955033http://isomorphism.es/post/94952955033Sat, 16 Aug 2014 18:30:04 -0600utility theorywealthpowerrejectionseparationcryinghurtfeelingsevolutionary psychologyevo psychpurposemoneyeconomicsbiologytears"yes trees opening up like a scream yes the wolves moving their bodies before their own shadows yes..."“yes trees opening up like a scream yes the wolves moving their bodies before their own shadows yes claws falling in love with the air yes smoke always moving in the direction opposite of our bodies yes down yes a tearing in the ground yes a dream of a throne of birds yes throwing our bones upward saying yes that is my shape yes that is my dream yes stepping out of the fire with a mouth full of snow yes angels building houses in the screams yes a goodness yes oh my god yes my spine yes i think my spine is more of a spine and less of a shiver yes less of a fire more of a flowering yes less of a knife more of a knife yes raising a body like a sharp deliberate thing yes towards the sky yes delighting in whatever falls from it like a rain yes like a fire yes like a frost yes whatever falls it will not be me yes it will not be the trees yes it will not be the fur yes it will not be you yes i am more of an opening more of a deliberate thing”<br/><br/> - <em>aheartlikea-socket, via <a class="tumblr_blog" href="http://myshoesuntied.tumblr.com/post/93343175038/aheartlikea-socket-jesus-fuck-goddamn" target="_blank">tiny ghost hands</a> </em>http://isomorphism.es/post/94862552473http://isomorphism.es/post/94862552473Fri, 15 Aug 2014 18:30:24 -0600artlaaaaaaandsat:
Path 93, Row 66 2013-12-04 at...<img src="http://31.media.tumblr.com/f9bd77a90562a4e99b35bcc3a5023386/tumblr_mx9nh17TqZ1swt5zko1_500.jpg"/><br/><br/><p><a class="tumblr_blog" href="http://laaaaaaandsat.tumblr.com/post/68951939029/path-93-row-66-2013-12-04-at" target="_blank">laaaaaaandsat</a>:</p>
<blockquote>
<p><a href="http://ift.tt/IDspqC" target="_blank">Path 93, Row 66 2013-12-04 at 2013-12-04 00:03:00.959321 GMT</a></p>
</blockquote>http://isomorphism.es/post/94771624751http://isomorphism.es/post/94771624751Thu, 14 Aug 2014 18:30:05 -0600"If your vector space is a shopping cart full of groceries, then the checkout clerk is a linear..."“If your vector space is a shopping cart full of groceries, then the checkout clerk is a <a href="http://isomorphism.es/tagged/linear" target="_blank">linear</a> <a href="http://isomorphism.es/tagged/operator+theory" target="_blank">operator</a> on that space.”<br/><br/> - <em>Bill Burke, <a href="http://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf" target="_blank"><em>Div, Grad, and Curl are Dead</em></a></em>http://isomorphism.es/post/94581062011http://isomorphism.es/post/94581062011Tue, 12 Aug 2014 18:30:00 -0600lineardual spacedifferential geometryrow vectorsoperator theoryvector calculusdivgradcurlgeometric algebramusical isomorphismfunctionalsMoises Mahiques<img src="http://37.media.tumblr.com/4828d602a55ce70dcece3f5205679933/tumblr_n8zqdva6C11qc38e9o1_500.jpg"/><br/><br/><p>Moises Mahiques</p>http://isomorphism.es/post/94580510249http://isomorphism.es/post/94580510249Tue, 12 Aug 2014 18:23:07 -0600art"[F]or leaders, it’s important that people know you are consistent and fair in how you think about..."“<p>[F]or leaders, it’s important that people know you are consistent and fair in how you think about making decisions and that there’s an element of predictability. If a leader is consistent, people on their teams experience tremendous freedom, because then they know that within certain parameters, they can do whatever they want. If your manager is all over the place, you’re never going to know what you can do, and you’re going to experience it as very restrictive.</p>
<p>[Employees should be saying that] the manager treats me with respect, the manager gives me clear goals, the manager shares information, the manager treats the entire team fairly.</p>”<br/><br/> - <em><em><strong><a href="http://www.linkedin.com/pub/laszlo-bock/8/234/114" target="_blank">Laszlo Bock</a></strong></em></em>http://isomorphism.es/post/94503532506http://isomorphism.es/post/94503532506Mon, 11 Aug 2014 22:30:00 -0600managementfeedbackhappinessmeasurementSolid, Liquid, Gas<ul><li><strong>solid</strong> — the category <code>FinSet</code> <img alt="http://upload.wikimedia.org/math/4/b/0/4b01e1d7f710de6818f24f140d5528cb.png" class="decoded" src="http://upload.wikimedia.org/math/4/b/0/4b01e1d7f710de6818f24f140d5528cb.png"/>, a sack of wheat <img alt="http://cloud.graphicleftovers.com/23704/516160/the-scattered-bag-with-wheat-of-a-grain.jpg" class="decoded" src="http://cloud.graphicleftovers.com/23704/516160/the-scattered-bag-with-wheat-of-a-grain.jpg"/>, a bag of marbles; atoms; axiom of choice; individuation. The urelemente or wheat-kernels are <a href="http://isomorphism.es/post/33909584057/the-same-like-how" target="_blank">interchangeable</a> although they’re technically distinct. Yet I can pick out just one and it has a mass.</li>
<li><strong>liquid</strong> — continuity; probability mass; Lewis’ gunky line; <a href="http://www.tc.umn.edu/~hellm001/SelectedPublications.html" target="_blank">Geoff Hellman</a>; the pre-modern, “continuous” idea of water; <a href="http://ncatlab.org/nlab/show/motivation+for+cohesive+toposes" target="_blank">Urs Schreiber</a>; <a href="http://researchmap.jp/ymaruyama/?lang=english" target="_blank">Yoshihiro Maruyama</a>; <a href="http://philpapers.org/rec/BELWAP" target="_blank">John L Bell</a></li>
<li><strong>gas</strong> — Lebesgue <a href="http://isomorphism.es/tagged/measure+theory" target="_blank">measure theory</a>; sizing <a href="http://isomorphism.es/post/9923467729/wiener-processes" target="_blank">Wiener</a> processes <img alt="image" src="http://media.tumblr.com/tumblr_lr44rfH8uV1qbdydv.png"/> or other things in other “smooth” categories; here I mean again the pre-atomic vision of gas: in some sense it has constant mass, <em>but</em> it might be so de-pressurised that there’s not much in some sub-chamber, and the mass might even be so dispersed not only can you not pick out atoms and expect them to have a size <small>(so each point of probability density has “zero” chance of happening)</small>, but you might need a “significant pocket” of gas before you get the volume—and unlike liquid, the gas’ volume might confuse you without some “pressure”-like concept “squeezing” the stuff to constrain the notion of volume.</li>
</ul>http://isomorphism.es/post/94386212869http://isomorphism.es/post/94386212869Sun, 10 Aug 2014 18:44:00 -0600sensesideationmathematicssizestuffcategory theoryPV=nRTconceptsSchopenhauerindividuationcountingJohn L BellUrs SchreiberGeoffrey HellmanYoshihiro Maruyamamereologyimagination"Every year, wealthy countries spend billions of dollars to help the world’s poor, paying for cows,..."“Every year, wealthy countries spend billions of dollars to help the world’s poor, paying for cows, goats, seeds, beans, textbooks, business training, microloans, and much more…. Much of this aid … works. But [such aid is] expensive….
<br/><br/>
Part of [the expense] is due to overhead, but overhead [gets too much] attention…. [Worse] is <b>the [cost] of procuring and giving away</b> goats, textbooks, sacks of beans, and the like.
<br/><br/>
…the nonprofit Bandhan spends $331 to get $166 worth of local livestock and other assets to the [recipients]”<br/><br/> - <em><div class="article-field-authors"><a href="http://www.foreignaffairs.com/author/christopher-blattman" target="_blank">Christopher Blattman</a> and <a href="http://www.foreignaffairs.com/author/paul-niehaus" target="_blank">Paul Niehaus</a></div></em>http://isomorphism.es/post/94382847104http://isomorphism.es/post/94382847104Sun, 10 Aug 2014 18:00:00 -0600povertyaidcharity sectornonprofitsefficiencyPeter DruckermeasurementThoth<iframe width="400" height="299" src="https://www.youtube.com/embed/Sjnd__isN0Y?feature=oembed" frameborder="0" allowfullscreen></iframe><br/><br/><p><em>Thoth</em></p>http://isomorphism.es/post/94309229504http://isomorphism.es/post/94309229504Sat, 09 Aug 2014 22:30:00 -0600Thothperformance artSan Franciscorentspiritualityhomelessliving with your parentscharitydonationsartracecommunismloveracial equalityimaginationcreativitycompassiondancingself-loveshyfree expressionself-doubtconfidencemoving in with your parentsindividualityvoyeurismmaking ends meetmusicNYCsuccessMexican Aftershow Party by Kevin Drew<iframe width="400" height="224" src="https://www.youtube.com/embed/vtxcfqeE-gQ?feature=oembed" frameborder="0" allowfullscreen></iframe><br/><br/><p><em>Mexican Aftershow Party</em> by Kevin Drew</p>http://isomorphism.es/post/94211066592http://isomorphism.es/post/94211066592Fri, 08 Aug 2014 20:30:12 -0600musicKevin DrewMexican Aftershow PartymakeoutsSpirit If...Broken Social Scene"For as well as complaining about the “piracy” of mechanical music, Sousa also complained about the..."“For as well as complaining about the “piracy” of mechanical music, Sousa also complained about the cultural emptiness that mechanical music would create. As he testified: When I was a boy … in front of every house in the summer evenings you would find young people together singing the songs of the day or the old songs. Today you hear these infernal machines going night and day. We will not have a vocal cord left. The vocal cords will be eliminated by a process of evolution, as was the tail of man when he came from the ape.”<br/><br/> - <em><a class="tumblr_blog" href="http://worxe.tumblr.com/post/63115430875" target="_blank">Worx: </a> </em>http://isomorphism.es/post/94120334294http://isomorphism.es/post/94120334294Thu, 07 Aug 2014 20:30:00 -0600what instrument do you play? iPodTV rots the brainidiot boxJohn Philip SousacreativityRoger Penrose, The Road to Reality, §12.8
(part of the Amazon...<img src="http://37.media.tumblr.com/6362c49860cc7b0654c83d3e3d0ba02a/tumblr_n96mvzNC941qc38e9o2_500.png"/><br/> <br/><img src="http://38.media.tumblr.com/d334b0caa6c5630f65d69c4cc34c9e6a/tumblr_n96mvzNC941qc38e9o3_500.png"/><br/> <br/><img src="http://38.media.tumblr.com/fec351fb0fed233d2dc34c3b028678e9/tumblr_n96mvzNC941qc38e9o1_500.png"/><br/> <br/><p>Roger Penrose, <a href="http://amzn.to/1rBYB2i" target="_blank"><em>The Road to Reality</em></a>, §12.8</p>
<p><small>(part of the Amazon Preview)</small></p>http://isomorphism.es/post/94026454633http://isomorphism.es/post/94026454633Wed, 06 Aug 2014 20:30:11 -0600Roger Penrosetensorsstring diagramsalgebra
87-year-old Cristina Calderón is the last speaker of the Yamana...<img src="http://33.media.tumblr.com/7e5be183a6403be16bf322236232e339/tumblr_n79m7elWnw1qc38e9o1_500.jpg"/><br/><br/><blockquote>
<p>87-year-old <strong>Cristina Calderón</strong> is the last speaker of the Yamana language. Three of her ancestors met Charles Darwin and the King and Queen of England. She lives in Puerto Williams, Navarino Island, near the southern tip of Patagonia</p>
</blockquote>
<p><img alt="http://www.kimmacquarrie.com/wp-content/uploads/2011/10/language-families-sa.GIF" class="decoded" src="http://www.kimmacquarrie.com/wp-content/uploads/2011/10/language-families-sa.GIF"/></p>
<p><img alt="http://www.kimmacquarrie.com/wp-content/uploads/2011/10/language-families-the-americas.gif" class="decoded" src="http://www.kimmacquarrie.com/wp-content/uploads/2011/10/language-families-the-americas.gif"/></p>http://isomorphism.es/post/93930849324http://isomorphism.es/post/93930849324Tue, 05 Aug 2014 20:30:22 -0600PatagoniaYaghanlanguagessuraméricaNoise scape by Holger Lippmann<img src="http://31.media.tumblr.com/07c72040edffd98e966cef9fa4cff092/tumblr_mvi7jasZly1qzg58bo1_500.jpg"/><br/><br/><p><a href="http://www.lumicon.de/wp3/wp-content/gallery/noise_scape/" target="_blank">Noise scape</a> by <a class="tumblr_blog" href="http://wowgreat.tumblr.com/post/65560483451/http-www-lumicon-de-wp3-wp-content-gallery-noise" target="_blank">Holger Lippmann</a></p>http://isomorphism.es/post/93734925521http://isomorphism.es/post/93734925521Sun, 03 Aug 2014 20:00:12 -0600artThis Much I Care by Skaters
(skip to 27”)<iframe width="400" height="224" src="https://www.youtube.com/embed/VRDNvSsbLW8?feature=oembed" frameborder="0" allowfullscreen></iframe><br/><br/><p><em>This Much I Care</em> by Skaters</p>
<p><small>(skip to 27”)</small></p>http://isomorphism.es/post/93521086584http://isomorphism.es/post/93521086584Fri, 01 Aug 2014 13:30:00 -0600I'm not sure why this song is about gold-diggerswhateverit's Fridaypump-upmusicSkatersThis Much I Care"The entrepreneurial class, during its rule of scarce one hundred years, has created more massive and..."“<p><b>The entrepreneurial class</b>, during its rule of scarce one hundred years, <b>has created more massive and more colossal <a href="http://isomorphism.es/tagged/economic+growth" target="_blank">productive</a> forces than have all preceding generations together.</b> Subjection of Nature’s forces to man, machinery, application of chemistry to <a href="http://isomorphism.es/tagged/industrial" target="_blank">industry</a> and <a href="http://isomorphism.es/tagged/agriculture" target="_blank">agriculture</a>, … — what earlier century had even a presentiment [of] such productive forces…?</p>
<p>The need of a constantly <a href="http://www.entrepreneur.com/video/224710" title="from Entrepreneur Magazine's «expanding markets» tag" target="_blank">expanding</a> market for its products chases businesspeople over the entire surface of the globe. They must nestle everywhere, settle everywhere, establish connexions everywhere.</p>
<p>Business has subjected the country to the rule of the towns. It has created <a href="http://www.economist.com/node/9070726" target="_blank">enormous cities</a>, has greatly increased the urban population as compared with the rural, and has thus rescued a considerable part of the population from the idiocy of rural life. </p>
<p>Entrepreneurs, wherever they have got the upper hand, have put an end to all feudal, patriarchal, idyllic relations. They have pitilessly torn asunder the motley feudal ties that bound man to his “natural superiors”….</p>
<p><b>Entrepreneurs cannot exist without constantly <a href="https://www.hackerleague.org/hackathons/techcrunch-disrupt-sf-2013" title="#hackdisrupt" target="_blank">revolutionising</a> the instruments of production, and thereby … the whole relations of society.</b></p>”<br/><br/> - <em><p>Charles Marx, 1848</p>
<p><small>(I just changed all reference to “the bourgeoisie”, which has an archaic or leftist ring to it, to “entrepreneurs”, which sounds more contemporary.</small> Eat your heart out, Thomas Friedman.)</p>
<p><!-- more --></p>
<pre> </pre>
<p>If I wanted to be more adventurous in substituting words for “the bourgeoisie” I could obtain also these gems:</p>
<ul><li>[Economists] have torn away from the family its sentimental veil, and has reduced the family relation to a mere money relation.</li>
<li>[Economists] have drowned the most heavenly ecstasies of religious fervour, of chivalrous enthusiasm, of philistine sentimentalism, in the icy water of egotistical calculation. They have resolved personal worth into exchange value, and in place of … chartered freedoms, have set up … — Free Trade.</li>
<li>
<p>A similar <a href="/tagged/OWS" target="_blank">movement</a> is going on before our own eyes. Modern [finance], … [which] has conjured up such gigantic means of production and of exchange, is like the sorcerer who is no longer able to control the powers of the nether world whom he has called up by his spells.<br/><img alt="http://3.bp.blogspot.com/-RzAPgd3RKoM/T42p2CpGEPI/AAAAAAAAAEk/3Spbk-320Yk/s1600/fantasia+sorcerror.jpg" class="decoded" src="http://3.bp.blogspot.com/-RzAPgd3RKoM/T42p2CpGEPI/AAAAAAAAAEk/3Spbk-320Yk/s1600/fantasia+sorcerror.jpg"/><br/>For many decades … the history of industry and commerce ….. It is enough to mention the [financial] crises that by their periodical return put the existence of [modern finance] on … trial, each time more threateningly. In these crises, a great part not only of the existing [<a href="http://isomorphism.es/tagged/GDP" target="_blank">GDP</a>], but also of the previously created [GDP growth], are periodically destroyed.<br/><img alt="US quarter-to-quarter GDP growth, not seasonally adjusted, in percent, from 1947 to present" src="https://31.media.tumblr.com/3f3e370d1ce16b388c38bad27464f19e/tumblr_n9hj4peO3u1qc38e9o2_r1_1280.png"/><br/>In these crises, there breaks out an epidemic that, in all earlier epochs, would have seemed an absurdity — the epidemic of <a href="http://isomorphism.es/tagged/subprime+crisis" target="_blank">over-[eager</a> lending]. Society suddenly finds itself put back … as if a famine, a … war … industry and commerce seem to be destroyed; and why?<br/><br/>Because there is too much civilisation, too much means of subsistence, too much industry, too much commerce. The productive forces at the disposal of society no longer … <a href="http://davidaedwards.tumblr.com/post/93226309269/the-best-known-critic-of-financial-engineering-is" target="_blank">further</a> … development …; on the contrary, they have become too powerful …, … they are fettered, and [when] they overcome these fetters, they bring disorder into [#hackdisrupt with creative distruption] … the whole of [modern capitalist] society, endanger the existence of [financial goods]. … And how do [bankers] get over these crises? …[B]y the conquest of new markets, and by the more thorough [development] of the old ones. That is to say, by paving the way for more extensive and more destructive crises, and by diminishing the means whereby crises are prevented. <a id="035" name="035"></a></p>
</li>
</ul></em>http://isomorphism.es/post/93513335669http://isomorphism.es/post/93513335669Fri, 01 Aug 2014 11:42:00 -0600globalisation1848innovationentrepreneurshipcitiesurbanisationeconomic growthfree tradeOWScrisis of 2008subprime crisisfinancecapitalism