Posts tagged with Descartes

We want to take theories and turn them over and over in our hands, turn the pants inside out and look at the sewing; hold them upside down; see things from every angle; and sometimes, to quotient or equivalence-class over some property to either consider a subset of cases for which a conclusion can be drawn (e.g., “all fair economic transactions” (non-exploitive?) or “all supply-demand curveses such that how much you get paid is in proportion to how much you contributed” (how to define it? vary the S or the D and get a local proportionality of PS:TS? how to vary them?)

Consider abstractly a set like {a, b, c, d}. 4! ways to rearrange the letters. Since sets are unordered we could call it as well the quotient of all rearangements of quadruples of once-and-yes-used letters (b,d,c,a). /p>

Descartes’ concept of a mapping is “to assign” (although it’s not specified who is doing the assigning; just some categorical/universal ellipsis of agency) members of one set to members of another set.
image

  • For example the Hash Map of programming.
    {
     '_why' => 'famous programmer',
     'North Dakota' => 'cold place',
     ... }
  • Or to round up ⌈num⌉: not injective because many decimals are written onto the same integer.


    http://www.tigerlogic.com/tigerlogic/omnis/developers/images/technews/fnobj11ceilingfngraph.jpg

    http://mathworld.wolfram.com/images/interactive/CeilingReImAbs.gif
  • Or to “multiply by zero” i.e. “erase” or “throw everything away”:

In this sense a bijection from the same domain to itself is simply a different—but equivalent—way of looking at the same thing. I could rename A=1,B=2,C=3,D=4 or rename A='Elsa',B='Baobab',C=√5,D=Hypathia and end with the same conclusion or “same structure”. For example. But beyond renamings we are also interested in different ways of fitting the puzzle pieces together. The green triangle of the wooden block puzzle could fit in three rotations (or is it six rotations? or infinity right-or-left-rotations?) into the same hole.

image

By considering all such mappings, dividing them up, focussing on the easier classes; classifying the types at all; finding (or imposing) order|pattern on what seems too chaotic or hard to predict (viz, economics) more clarity or at least less stupidity might be found.

The hope isn’t completely without support either: Quine explained what is a number with an equivalence class of sets; Tymoczko described the space of musical chords with a quotient of a manifold; PDE’s (read: practical engineering application) solved or better geometrically understood with bijections; Gauss added 1+2+3+...+99+100 in two easy steps rather than ninety-nine with a bijection; ….

 

It’s hard for me to speak to why we want groups and what they are both at once. Today I felt more capable of writing what they are.

So this is the concept of sameness, let’s discuss just linear planes (or, hyperplanes) and countable sets of individual things.

Leave it up to you or for me later, to enumerate the things from life or the physical world that “look like” these pure mathematical things, and are therefore amenable by metaphor and application of proved results, to the group theory.

But just as one motivating example: it doesn’t matter whether I call my coordinates in the mechanical world of physics (x,y,z) or (y,x,z). This is just a renaming or bijection from {1,2,3} onto itself.

Even more, I could orient the axis any way that I want. As long as the three are mutually perpendicular each to the other, the origin can be anywhere (invariance under an affine mapping — we can equivalence-class those together) and the rotation of the 3-D system can be anything. Stand in front of the class as the teacher, upside down, oriented so that one of the dimensions helpfully disappears as you fly straight forward (or two dimensions disappear as you run straight forward on a flat road). Which is an observation taken for granted by my 8th grade physics teacher. But in the language of group theory means we can equivalence-class over the special linear group of 3-by-3 matrices that leave volume the same. Any rotation in 3-D

Sameness-preserving Groups partition into:

  • permutation groups, or rearrangements of countable things, and
  • linear groups, or “trivial” “unimportant” “invariant” changes to continua (such as rescaling—if we added a “0” to the end of all your currency nothing would change)
  • conjunctions of smaller groups

The linear groups—get ready for it—can all be represented as matrices! This is why matrices are considered mathematically “important”. Because we have already conceived this huge logical primitive that (in part) explains the Universe (groups) — or at least allows us to quotient away large classes of phenomena — and it’s reducible to something that’s completely understood! Namely, matrices with entries coming from corpora (fields).

So if you can classify (bonus if human beings can understand the classification in intuitive ways) all the qualitatively different types of Matrices,

image

then you not only know where your engineering numerical computation is going, but you have understood something fundamental about the logical primitives of the Universe!

Aaaaaand, matrices can be computed on this fantastic invention called a computer!

 

unf




Noncommutative & irreducible

  • a−a+b−b=0
  • a+b−a−b≠0

An organon of economic theory, contra Foucault, is that—just as a gas is nothing more than a composite of molecules—so is “a society” nothing more than a composite of individuals. (Although individuals vary considerably more than do atoms; a gas molecule can be characterised with only a handful of numbers.) “‘Society’ does not exist”if I cared to google some more, I could I think find utterances by Prime Minister Thatcher, Alan Greenspan, Russ Roberts, and Ayn Rand, to this effect.

Is that rubbish? I only specialise in stage 17 of the industrial-steel treatment process because other people have specialised in stages 16 and 18. But if we tossed out Cartesian decomposability, we’d be tossing out science (=reductionistic experimental method), and mathematics, and logic itself … right?

http://upload.wikimedia.org/wikipedia/commons/9/9c/Sacrificial_scene_on_Hammars_-_Valknut.png

Here with the Borromean rings, as with cohomology elements, we get an example of a global property which is lost at the local level. Nothing is special about any of the individual rings. It’s the way they combine that’s special—not an independent Cartesian product, but a thoroughly intermeshed interlinking. The whole is more than the sum of the parts.

Not saying that the world is Borromean or something as simplistic as that. But just like Cantor’s laughably small example of a three-part system (ω,ω²,π∙ω) disproved Nietzsche’s unfounded assertion (in The Eternal Return) that “Any complex system must return to its original state” (naïve conception of infinity) without suggesting that Society is a three-part system,—we might take the existence of a tiny primitive with global-non-local properties as at least evidence against the Great Organon, and possibly pointing the way toward a theory in which “society” can have a meaning beyond “set-like collection of individuals”.

 

Echoes of financial crisis. If you removed 30% of the banksters (and attorneys) from the problem centres (wherever they were!) in the instigation of the financial crisis, would we have averted an O($10 trillion) destruction of wealth? Or is it the incentives (echoes of Richie/Rosen’s “entailment structures”)? Or the “culture” (and can we give this meaning?) in which Gordon Gekko-worshipping


acolytes of capitalism buy and resell a Panglossian view of price-as-value-added where success comes only to those who serve the most? vision of capitalism—if indeed that replaced some earlier less casino-like culture to investment banking (http://www.tubechop.com/watch/923989).

But this sounds too vague and hand-wavy. A dystopian “system” that controls free-willed individuals? Constituted of “military-industrial-lobbying-banking complexes” and shadowy networks of faceless vice-presidents—but when one asks those who propound this woolly claptrap to point to specifics or give an atomistic description of what they think is going on, they can’t! Proving of course that their accusations are baseless.

http://3.bp.blogspot.com/_j1PMRiUP1iY/TU40j-3vIKI/AAAAAAAABlE/NMP3fCwksyw/s1600/penroseTriangle3.png

Where could anyone ever hope to find the tools to make a rigorous theory out of it?

(Source: math.cornell.edu)










Saying derivative is “slope” is a nice pedant’s lie, like the Bohr atom

image

which misses out on a deeper and more interesting later viewpoint:

|6,4,1> Orbital Animation|3,2,1>+|3,1,-1> Orbital Animation

 

The “slope” viewpoint—and what underlies it: the “charts” or “plots” view of functions as ƒ(x)–vs–x—like training wheels, eventually need to come off. The “slope” metaphor fails

  • for pushforwards,
  • on surfaces,
    image 
  • on curves γ that double back on themselves
    image 
  • my vignettes about integrals,
  • and, in my opinion, it’s harder to “see” derivatives or calculus in a statistical or business application, if you think of “derivative = slope”. Since you’re presented with reams of numbers rather than pictures of ƒ(x)–vs–x, where is the “slope” there?

"Really" it’s all about diff’s. Derivatives are differences (just zoomed in…this is what lim ∆x↓0 was for) and that viewpoint works, I think, everywhere.

I half-heartedly tried making the following illustrations in R with the barcode package but they came out ugly. Even uglier than my handwriting—so now enjoy the treat of my ugly handwriting.

 

Step back to Descartes definition of a function. It’s an association between two sets.

image

And the language we use sounds “backwards” to that of English. If I say “associate a temperature number to every point over the USA”

US temperatures

then that should be written as a function ƒ: surface → temp.,

(or we could say ƒ: ℝ²→ℝ with ℝ²=(lat,long) )

The \to arrow and the "maps to" phrasing are backwards of the way we speak.

  • "Assign a temperature to the surface" —versus— "Map each surface point to a temperature element from the set of possible temperatures”.

a function is an association between sets

{elf, book, Kraken, 4^π^e} … no, I’m not sure where that came from either. But I think we can agree that such a set is unstructured.

Cartesian function from non-space to weird space

Great. I drew above a set “without other structure" as the source (domain) and a branched, partially ordered weirdy thing as the target (codomain). Now it’s possible with some work to come up with a calculus like the infinitesimal one on ℝ→ℝ functions that’s taught to many 19-year-olds, but that takes more work. But for right now my point is to make that look ridiculous and impossible. Newton’s calculus is something we do only with a specific kind of Cartesian mapping: where both the from and the to have Euclidean concepts of straight-line-ness and distance has the usual meaning from maths class. In other words the Newtonian derivative applies only to smooth mappings from ℝ to ℝ.

 

Let’s stop there and think about examples of mappings.

(Not from the real world—I’ll do another post on examples of functions from the real world. For now just accept that numbers describe the world and let’s consider abstractly some mappings that associate, not arbitrarily but in a describable pattern, some numbers to other numbers.)

successor function and square function

sin function

(I didn’t have a calculator at the time but the circle values for [1,2,3,4,5,6,7] are [57°,114°,172°,229°,286°,344°,401°=41°].)

I want to contrast the “map upwards” pictures to both the Cartesian pictures for structure-less sets

image

and to the normal graphical picture of a “chart” or “plot”.

image

image

Notice what’s obscured and what’s emphasised in each of the picture types. The plots certainly look better—but we lose the Cartesian sense that the “vertical” axis is no more vertical than is the horizontal. Both ℝ’s in ƒ: ℝ→ℝ are just the same as the other.

And if I want to compose mappings? As in the parabola picture above (first the square function, then an affine recentering). I can only show the end result of g∘ƒ rather than the intermediate result.

image

Whereas I could line up a long vertical of successive transformations (like one might do in Excel except that would be column-wise to the right) and see the results of each “input-output program”.

(Además, I have a languishing draft post called “How I Got to Gobbledegook” which shows how much simpler a sequence of transforms can be rather than “a forbidding formula from a textbook”.)

Another weakness of the “charts” approach is that whereas "Stay the same" command ought to be the simplest one (it’s a null command), it gets mapped to the 45˚ line:

image

Here’s the familiar parabola / plot “my way”: with the numbers written out so as to equalise the target space and the source space.

Parabola with the domain and codomain on the same footing.

 

Now the “new” tool is in hand let’s go back to the calculus. Now I’m going to say "derivative=pulse" and that’s the main point of this essay.

linear approximations (differentials) of a parabola (x²)

Considering both the source ℝ→ and the target →ℝ on the same footing, I’ll call the length of the arrows the “mapping strength”. In a convex mapping like square the diffs are going to increase as you go to the right.

image

OK now in the middle of the piece, here is the main point I want to make about derivatives and calculus and how looking at numbers written on the paper rather than plots makes understanding a push forward possible. And, in my opinion, since in business the gigantic databases of numbers are commoner than charts making themselves, and in life we just experience stimuli rather than someone making a chart to explain it to us, this perspective is the more practical one.

differences on a scalar field (California)

I’m deliberately alliding the concepts of diff as

  • difference
  • R's diff function
  • differential (as in differential calculus or as in linear approximation)
because they’re all related.
differentials on a surface (Where is the Slope?)
a U-neighbourhood of Los Angeles
In my example of an open set around Los Angeles, a surface diff could be you measure the temperature on your rooftop in Los Feliz, and then measure the temperature down the block. Or across the city. Or, if you want to be infinitesimal and truly calculus-ish about it, the difference between the temperature of one fraction of an atom in your room and its nearby neighbour. (How could that be coherent? There are ways, but let’s just stick with the cross-city differential and pretend you could zoom in for more detail if you liked.)
 

Linear

I’m still not quite done with the “my style of pictures” because there’s another insight you can get from writing these mappings as a bar code rather than as a “chart”. Indeed, this is exactly what a rug plot does when it shows histograms.

a rug plot or carpet plot is like a barcode on the bottom of your plot to show the marginal (one-dimension only) distribution of data

Here are some strip plots = rug plots = carpet plots = barcode plots of nonlinear functions for comparison.

 image

image

The main conclusion of calculus is that nonlinear functions can be approximated by linear functions. The approximation only works “locally” at small scales, but still if you’re engineering the screws holding a plane together, it’s nice to know that you can just use a multiple (linear function) rather than some complicated nonlineary thingie to estimate how much the screws are going to shake and come loose.

For me, at least, way too many years of solving y=mx+b obscured the fact that linear functions are just multiples. You take the space and stretch or shrink it by a constant multiple. Like converting a currency: take pesos, divide by 8, get dollars. The multiple doesn’t change if you have 10,000 pesos or 10,000,000 pesos, it’s still the same conversion rate.

image

image

linear maps as multiplication

linear mappings -- notice they're ALL straight lines through the origin!

the flip function

So in a neighborhood or locality a linear approximation is enough. That means that a collection of linear functions can approximate a nonlinear one to arbitrary precision.

building up a nonlinear function from linear parts

That means we can use computers!

Calculus says Smooth functions can be approximatedaround a local neighborhood of a pointwith straight lines

 

Square

I can’t use the example of self times self so many times without exploring the concept a bit. Squares to me seem so limited and boring. No squizzles, no funky shapes, just boring chalkboard and rulers.

But that’s probably too judgmental.

image

recursive "Square" function

After all there’s something self-referential and almost recursive about repeated applications of the square function. And it serves as the basis for Euclidean distance (and standard deviation formula) via the Pythagorean theorem.

How those two are connected is a mystery I still haven’t wrapped my head around. But a cool connection I have come to understand is that between:

  • a variety of inverse square laws in Nature
  • a curve that is equidistant from a point and a line
  • and the area of a rectangle which has both sides equal.

inverse square laws

what does self times self have to do with the geometric figure of a parabola?

parabola

I guess first of all one has to appreciate that “parabola” shouldn’t necessarily have anything to do with x•x. Hopefully that’s become more obvious if you read the sections above where I point out that the target ℝ isn’t any more “vertical” than is the source ℝ.

image

The inverse-square laws show up everywhere because our universe is 3-dimensional. The surface of a 3-dimensional ball (like an expanding wave of gravitons, or an expanding wave of photons, or an expanding wave of sound waves) is 2-dimensional, which means that whatever “force” or “energy” is “painted on” the surface, will drop off as the square rate (surface area) when the radius increases at a constant rate. Oh. Thanks, Universe, for being 3-dimensional.

inverse square laws  why, why, why, WHY?!?!

What’s most amazing about the parabola—gravity connection is that it’s a metaphor that spans across both space and time. The curvature that looks like a-plane-figure-equidistant-to-a-line-and-a-point is curving in time.




The origins of mass & the feebleness of gravity by Frank Wilczek

tl,dr:

  • dark matter & dark energy
  • "Even though protons, neutrons, and electrons comprise only 3% of the universe’s mass as a whole, I hope you’ll agree that it’s a particularly significant part of the mass." lol
  • "Just because you can say words and they make sense grammatically doesn’t mean they make sense conceptually. What does it mean to talk about ‘the origin of mass’?”
  • "Origin of mass" is meaningless in Newtonian mechanics. It was a primitive, primary, irreducible concept.
  • Conservation is the zeroth law of classical mechanics.
  • F=MA relates the dynamical concept of force to a kinematic quantity and a conversion factor (mass).
  • rewriting equations and they “say” something different
  • the US Army field guide for radio engineers describes “Ohm’s three laws”: V=IR, I=V/R, and a third one which I’ll leave it as an exercise for you to deduce”
  • m=E/c²
  • Einstein’s original paper Does the inertia of a body depend on its energy content? uses this ^ form
  • You could go back and think through Einstein’s problem (knowing the solution) in terms of free variables. In order to unite systems of equations with uncommon terms, you need a conversion factor converting a ∈ Sys_1 to b ∈ Sys_2.
  • Min 13:30 “the body and soul of QCD
    http://www.lightnessofbeingbook.com/LOBColorPlates/images/Plate01_ThreeJet.jpg
    http://www.fnal.gov/pub/today/images11/CMSResult042211figure1.jpg
    image
    img_lrg/jet.jpg not found

  • Protons and neutrons are built up from quarks that are moving around in circles, continuously being deflected by small amounts. (chaotic initial value problem)
  • supercomputer development spurred forward by desire to do QCD computations
  • Min 25:30 “The error bounds were quite optimistic, but the pattern was correct”
  • A model with two parameters that runs for years on a teraflop machine.
  • Min 27:20 The origin of mass is this (N≡nucleon in the diagram): QCD predicts that energetic-but-massless quarks & gluons should find stable equilibria around .9 GeV:
    Full-size image (27 K)

    http://www.yorku.ca/lewisr/images/Yspectrum.png
    image
    Or said alternately, the origin of mass is the balance of quark/gluon dynamics. (and we may have to revise a bit if whatever succeeds QCD makes a different suggestion…but it shouldn’t be too different)
  • OK, that was QCD Lite. But the assumptions / simplifications / idealisations make only 5% difference so we’ll still explain 90% of the reason where mass comes from.
  • Computer ∋ 10^27 neutrons & protons
  • The supercomputer can calculate masses, but not decays or scattering. Fragile.
  • Minute 36. quantum Yang-Mills theory, Fourier transform, and an analogy from { a stormcloud discharging electrical charge into its surroundings } to { a "single quark" alone in empty space would generate a shower of quark-antiquark virtual pairs in order to keep a balanced strong charge }
  • Minute 37. but just like in QM, it “costs” (∃ a symplectic, conserved quantity that must be traded off against its complement) to localise a particle (against Heisenberg uncertainty of momentum). And here’s where the Fourier transform comes in. FT embeds a frequency=time/space=locality tradeoff at a given energy (“GDP" in economic theory). The “probability waves" or whatever—spread-out waveparticlequarkthings—couldn’t be exactly on top of each other, they’ll settle in some middle range of the Fourier tradeoff.
  • "quasi-stable compromises"
  • This is similar to how the hydrogen atom gets stable in quantum mechanics. Coulomb field would like to pull the electron on top of the proton, but the quantum keeps them apart.
    http://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bohr-atom-PAR.svg/500px-Bohr-atom-PAR.svg.png
    http://upload.wikimedia.org/wikipedia/commons/a/a4/HydrogenOrbitalsN6L0M0.png
  • "the highest form of musicality"
  • Quantum mechanics uses the mathematics of musical notes (vibrating harmonics).
  • Quantum chromodynamics uses the mathematics of chords, specifically triads since 3 colour forces act on each other at once.
    image
    image
  • Particles are nothing more than stable tradeoffs that can be made between localisation costs (per energy) from QM and colour forces.
  • (Aside to quote Wikipedia: “Mathematically, QCD is a non-Abeliangauge theory based on a local (gauge) symmetry group called SU(3).”)

  • Minute 40. Because the compromises can’t be evened out exactly due to quanta, there’s some leftover energy. It’s the same for a particular kind of quark-gluon interaction (again, because of the quanta). The .9 GeV overshoot | disbalance | asymmetry in some particular quark-gluon attempts to balance creates the neutrons and protons. And that’s the origin of mass.

Minute 42. Feebleness of gravity.

  • (first of all, gravity is weak—notice that a paperclip sticks to a magnet rather than falling to the floor)
  • (muscular forces are the result of a lot of ATP conversions and such. That just happens to be even weaker—but if you think of how far removed those biochemical electropulses and cell fibres are from the fundamental foundation, maybe that’s not so surprising.)
  • Gravity is 40 orders of magnitude weaker than the electrical force. Not forty times, forty orders of magnitude.
  • Planck’s vision; necessary conversion; a theory of the universe with only numbers.
  • The Planck distance, even for nuclear physicists, is about 20 orders of magnitude too small.
  • The clunkiness of Planck’s constants mocks dimensional analysis. “If you measure natural objects in natural units, you should get something of the order of unity”.
  • "If you agree that the proton is a natural object and the Planck scale is a natural unit, you’d be off by 18 orders of magnitude".
  • Suppose gravity is a primitive. Then the question becomes: “Why is the proton so light?” Which now we can answer. (see above)
    http://www.grin.com/object/external_document.243440/7414ac37090135af37b2813c60318981_LARGE.png
    http://www-cdf.fnal.gov/physics/new/qcd/diphoton_2012/diphoton_xsec_10fb/plots/TXSec_massQLSSV01_07Jun12.gif
    http://cms.web.cern.ch/sites/cms.web.cern.ch/files/styles/large/public/field/image/Fig3-MassFactSoBWeightedMass.png
  • Simple physics (local interactions, basic = atomic = fundamental = primitive behaviours) should occur at Planck scales. (More complex behaviours then should “emerge” out of this reduction.)
  • So that should be, in terms of energy & momentum, 10^18 proton masses, where the fundamental interactions happen.
  • The value of the quark-gluon interaction at the Planck scale. “Smart” dimensional analysis says the quantum level that makes protons from the gluon-quark interactions then gets us to ½, “which I hope you’ll agree is a lot closer to unity than 10^−18”.
  • Minute 57. “A lot of what we know about the deep structure of the Standard Model is summarised on this slide”
  • weak force causes beta decay
  • standard model not so great on neutrino masses
  • SO(10)’s spinor representation has all the standard model’s symmetries as subgroups
  • Minute 67. Trips my regression-analysis circuits. Slopes & intercepts. Affine!
  • Supersymmetry would have changed the clouds and made everything line up real nicely. (The talk was in 2004 and this week, in 2012, the BBC reported that SuSy was kneecapped by the latest LHC evidence.)
  • "If low-energy supersymmetry turns out to be false, I’ll be very disappointed and we’ll have to think of something else."

(Source: mit.tv)