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Posts tagged with infinity

The dual V* of a vector space V  over ℝ matches lists of reals to linear functionals.

What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb quintuple.

image
linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!

pictures of lines through the origin with various slopes

Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is ƒ₅(x)=5•x. That ƒ₅ does things like

  • ƒ₅(■■■)=5+5+5 and
  • ƒ₅(■■■■■■)=5+5+5+5+5+5.

Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like _ / ∕ ...|. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say id, double, triple, quadruple, quintuple, ….

(Why did I jump so suddenly 0,1,… from _ flat to ⁄  45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).
image
To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T
image
hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)

 

These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same kind of thing. One is nouns and one is a verb.

But still, for any real number that I “counted” I could match up a function, just like I did with

  • "5 that I counted" and
  • "function ƒ₅ = quintuple

So these two qualitatively different things are in bijection. (One can hope for insights by viewing things through one lens or the other, noun or verb version.)

This one-dimensional story can be upgraded to a multi-dimensional one where

  • lists of reals (3.1, √2, −2.1852, ..., 6) 

match to

  • many-to-one functions ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].

Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.




A matrix is a box filled with numbers

\mathbf{A} =    \begin{bmatrix}   a_{11} & a_{12} & \cdots & a_{1n} \\   a_{21} & a_{22} & \cdots & a_{2n} \\   \vdots & \vdots & \ddots & \vdots \\   a_{m1} & a_{m2} & \cdots & a_{mn}   \end{bmatrix}.

with the context understood to be that they will be multiplying something in an inner-product sense.

image

i.e. “matrix on the left” is read with an arrow → going right across rows. “matrix on right” is read with an arrow ↓ going down columns.

If you read about spectral theorems or bounded linear operators or even just abstract vector spaces you might come across, as I did, mention of “infinite-dimensional spaces”. What could that even mean? How do the dimensions fit together? How can I picture an infinite-dimensional thing?

http://latex.codecogs.com/gif.latex?\dpi{200}%20\bg_white%20\large%20\!\!\!\!\!\!\!%20-\%2C\text{replace}-%20\\%20\text{\!\!\!\!\!\!sequence%20}%20\fbox{a_1%2C%20\ldots%2C%20\mbox{a}_N}%20\\%20\\%20\%3A%20-\text{%20\%20\%20by%20\%20\%20}-%20\\%20\text{\!\!\!\!\!\!function%20}%20\fbox{a(i)}%2C%20\\%20\\%20\quad%20i%20\in%20[1%2CN]%20\cong%20[0%2C1]%20\\%20\\%20\\%20\\%20\\%20\\%20\text{e.g.%2C%20replace%20}\\%20\fbox{1%2C4%2C9%2C16%2C25%2C\ldots}%20\\%20\text{%20by%20}%20\fbox{i^2}

I recently learned the answer and it’s not nearly as hard as I thought; I’ll share my new perspective with you.

  • Normally we talk about an entry a_{i,j} in the matrix. It’s indexed by {row,column} where i,j ∈ {1,…,N}.
  • The “infinite-dimensional vector space” idea uses the same a_{i,j} but i,j ∈ [0,1], the continuous line segment which bijects to [1,N] (another continuous line segment—just shift back by one and divide to biject it)
  • So the matrix entries function the same way, they’re just now to be thought of as “continuous rows”
  • …and the eigenvectors (discrete entries) become eigenfunctions (attaining values from continuous scale)
  • —if you have the mental machinery to envisage a probability distribution—even better a 2-D joint distribution—then you have what’s required to “picture” this thing.
    image

    gnuplot heatmap
    Example of the readout from the Zooplankton Acoustic Profiler (ZAP).

    enter image description here
    YlGnBu color map from colorbrewer


If you picture each of the matrix “blocks” as corresponding to a light/darkness value to represent the quantity inside

image

then the “infinite-dimensional” linear operator would just be “more subsquares in the grid”. If you want to allow complex values then Elias Wegert’s pictures (using colour as a “circular” value (complex argument) rather than brightness as a “straight” value)
image
then his pullbacks on a complex square Rect(Z)→arg(f) (I used 1000×1000 resolution) look fairly continuous—like an infinite-dimensional linear operator taking complex values a_{i,j}∈ℂ, i,j ∈ [0,1]

That’s the formal aspects taken care of. What kinds of things might an infinite-dimensional space be needed to represent? Here are some ideas:




The tropical semiring is arithmetic piped through a log with base →.

Also if you or someone you know  is first encountering a squeeze theorem or other a≤x≤A type reasoning, remark 2.1 might be a relatively painless calisthenic to warm you/them up to a≤x≤A type arguments.

by Gregory Mikhalkin










the sine of the reciprocal of [some angle between −1/π and 1/π]

at increasing resolution

s <- function(x) sin( 1/x )
    plot( s, xlim=c(-1/pi, 1/pi), col=rgb(0,0,0,.7), type = "l", ylab="output", xlab="input", main="compose [multiplicative inverse] with [vertical rect of a circle]" )

(Source: amzn.to)










The Cauchy distribution (?dcauchy in R) nails a flashlight over the number line

http://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/1000px-Number-line.svg.png

and swings it at a constant speed from 9 o’clock down to 6 o’clock over to 3 o’clock. (Or the other direction, from 3→6→9.) Then counts how much light shone on each number.

*

In other words we want to map evenly from the circle (minus the top point) onto the line. Two of the most basic, yet topologically distinct shapes related together.

image

You’ve probably heard of a mapping that does something close enough to this: it’s called tan.

http://www.calculatorsoup.com/images/trig_plots/graph_tan_pi.gif
Since tan is so familiar it’s implemented in Excel, which means you can simulate draws from a Cauchy distribution in a spreadsheet. Make a column of =RAND()'s (say column A) and then pipe them through tan. For example B1=TAN(A1). You could even do =TAN(RAND()) as your only column. That’s not quite it; you need to stretch and shift the [0,1] domain of =RAND() so it matches [−π,+π] like the circle. So really the long formula (if you didn’t break it into separate columns) would be =TAN( PI() * (RAND()−.5) ). A stretch and a shift and you’ve matched the domains up. There’s your Cauchy draw.

In R one could draw three Cauchy’s with rcauchy(3) or with tan(2*(runif(3).5)).

*

http://www.calculatorsoup.com/images/trig_plots/graph_tan_pi.gif

What’s happening at tan(−3π/2) and tan(π/2)? The tan function is putting out to ±∞.

I saw this in school and didn’t know what to make of it—I don’t think I had any further interest than finishing my problem set.

File:Hyperbola one over x.svg

I saw as well the ±∞ in the output of flip[x]= 1/x.

  • 1/−.0000...001 → −∞ whereas 1/.0000...0001 → +∞.

It’s not immediately clear in the flip[x] example but in tan[x/2] what’s definitely going on is that the angle is circling around the top of the circle (the hole in the top) and the flashlight of the Cauchy distribution could be pointing to the right or to the left at a parallel above the line.

Why not just call this ±∞ the same thing? “Projective infinity”, or, the hole in the top of the circle.

http://upload.wikimedia.org/wikipedia/commons/8/85/Stereographic_projection_in_3D.png




Readers of isomorphismes, you might enjoy powers of two tumblr.

powersoftwo:

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.




For those not in the know, here’s what mathematicians mean by the word “measurable”:

  1. The problem of measure is to assign a ℝ size ≥ 0 to a set. (The points not necessarily contiguous.) In other words, to answer the question:
    How big is that?
  2. Why is this hard? Well just think about the problem of sizing up a contiguous ℝ subinterval between 0 and 1.
    image
    • It’s obvious that [.4, .6] is .2 long and that
    • [0, .8] has a length of .8.
    • I don’t know what the length of √2√π/3] is but … it should be easy enough to figure out.
    • But real numbers can go on forever: .2816209287162381682365...1828361...1984...77280278254....
    • Most of them (the transcendentals) we don’t even have words or notation for.
      most of the numbers are black = transcendental
    • So there are a potentially infinite number of digits in each of these real numbers — which is essentially why the real numbers are so f#cked up — and therefore ∃ an infinitely infinite number of numbers just between 0% and 100%.


    Yeah, I said infinitely infinite, and I meant that. More real numbers exist in-between .999999999999999999999999 and 1 than there are atoms in the universe. There are more real numbers just in that teensy sub-interval than there are integers (and there are integers).

    In other words, if you filled a set with all of the things between .99999999999999999999 and 1, there would be infinity things inside. And not a nice, tame infinity either. This infinity is an infinity that just snorted a football helmet filled with coke, punched a stripper, and is now running around in the streets wearing her golden sparkly thong and brandishing a chainsaw:
    I think the analogy of 5_1 to Patrick Bateman is a solid and indisputable one.


    Talking still of that particular infinity: in a set-theoretic continuum sense, ∃ infinite number of points between Barcelona and Vladivostok, but also an infinite number of points between my toe and my nose. Well, now the simple and obvious has become not very clear at all!
    Europe
Data set:&gt; eurodist                 Athens Barcelona Brussels Calais Cherbourg Cologne CopenhagenBarcelona         3313                                                       Brussels          2963      1318                                             Calais            3175      1326      204                                    Cherbourg         3339      1294      583    460                             Cologne           2762      1498      206    409       785                   Copenhagen        3276      2218      966   1136      1545     760           Geneva            2610       803      677    747       853    1662       1418Gibraltar         4485      1172     2256   2224      2047    2436       3196Hamburg           2977      2018      597    714      1115     460        460Hook of Holland   3030      1490      172    330       731     269        269Lisbon            4532      1305     2084   2052      1827    2290       2971Lyons             2753       645      690    739       789     714       1458Madrid            3949       636     1558   1550      1347    1764       2498Marseilles        2865       521     1011   1059      1101    1035       1778Milan             2282      1014      925   1077      1209     911       1537Munich            2179      1365      747    977      1160     583       1104Paris             3000      1033      285    280       340     465       1176Rome               817      1460     1511   1662      1794    1497       2050Stockholm         3927      2868     1616   1786      2196    1403        650Vienna            1991      1802     1175   1381      1588     937       1455                Geneva Gibraltar Hamburg Hook of Holland Lisbon Lyons MadridBarcelona                                                                   Brussels                                                                    Calais                                                                      Cherbourg                                                                   Cologne                                                                     Copenhagen                                                                  Geneva                                                                      Gibraltar         1975                                                      Hamburg           1118      2897                                            Hook of Holland    895      2428     550                                    Lisbon            1936       676    2671            2280                    Lyons              158      1817    1159             863   1178             Madrid            1439       698    2198            1730    668  1281       Marseilles         425      1693    1479            1183   1762   320   1157Milan              328      2185    1238            1098   2250   328   1724Munich             591      2565     805             851   2507   724   2010Paris              513      1971     877             457   1799   471   1273Rome               995      2631    1751            1683   2700  1048   2097Stockholm         2068      3886     949            1500   3231  2108   3188Vienna            1019      2974    1155            1205   2937  1157   2409                Marseilles Milan Munich Paris Rome StockholmBarcelona                                                   Brussels                                                    Calais                                                      Cherbourg                                                   Cologne                                                     Copenhagen                                                  Geneva                                                      Gibraltar                                                   Hamburg                                                     Hook of Holland                                             Lisbon                                                      Lyons                                                       Madrid                                                      Marseilles                                                  Milan                  618                                  Munich                1109   331                            Paris                  792   856    821                     Rome                  1011   586    946  1476               Stockholm             2428  2187   1754  1827 2707          Vienna                1363   898    428  1249 1209      2105
Multi-dimensional scaling of the distances:
&gt; cmdscale(eurodist)                        [,1]        [,2]Athens           2290.274680  1798.80293Barcelona        -825.382790   546.81148Brussels           59.183341  -367.08135Calais            -82.845973  -429.91466Cherbourg        -352.499435  -290.90843Cologne           293.689633  -405.31194Copenhagen        681.931545 -1108.64478Geneva             -9.423364   240.40600Gibraltar       -2048.449113   642.45854Hamburg           561.108970  -773.36929Hook of Holland   164.921799  -549.36704Lisbon          -1935.040811    49.12514Lyons            -226.423236   187.08779Madrid          -1423.353697   305.87513Marseilles       -299.498710   388.80726Milan             260.878046   416.67381Munich            587.675679    81.18224Paris            -156.836257  -211.13911Rome              709.413282  1109.36665Stockholm         839.445911 -1836.79055Vienna            911.230500   205.93020
Plot
     require(stats)     loc &lt;- cmdscale(eurodist)     rx &lt;- range(x &lt;- loc[,1])     ry &lt;- range(y &lt;- -loc[,2])     plot(x, y, type="n", asp=1, xlab="", ylab="")     abline(h = pretty(rx, 10), v = pretty(ry, 10), col = "light gray")     text(x, y, labels(eurodist), cex=0.8)
    So it’s a problem of infinities, a problem of sets, and a problem of the continuum being such an infernal taskmaster that it took until the 20th century for mathematicians to whip-crack the real numbers into shape.
  3. If you can define “size” on the [0,1] interval, you can define it on the [−535,19^19] interval as well, by extension.

    If you can’t even define “size” on the [0,1] interval — how do you think you’re going to define it on all of ℝ? Punk.
  4. A reasonable definition of “size” (measure) should work for non-contiguous subsets of ℝ such as “just the rational numbers” or “all solutions to cos² x = 0(they’re not next to each other) as well.

    Just another problem to add to the heap.
  5. Nevertheless, the monstrosity has more-or-less been tamed. Epsilons, deltas, open sets, Dedekind cuts, Cauchy sequences, well-orderings, and metric spaces had to be invented in order to bazooka the beast into submission, but mostly-satisfactory answers have now been obtained.

    It just takes a sequence of 4-5 university-level maths classes to get to those mostly-satisfactory answers.
    image
    One is reminded of the hypermathematicians from The Hitchhiker’s Guide to the Galaxy who time-warp themselves through several lives of study before they begin their real work.

imageimage

For a readable summary of the reasoning & results of Henri Lebesgue's measure theory, I recommend this 4-page PDF by G.H. Meisters. (NB: His weird ∁ symbol means complement.)

That doesn’t cover the measurement of probability spaces, functional spaces, or even more abstract spaces. But I don’t have an equally great reference for those.

Oh, I forgot to say: why does anyone care about measurability? Measure theory is just a highly technical prerequisite to true understanding of a lot of cool subjects — like complexity, signal processing, functional analysis, Wiener processes, dynamical systems, Sobolev spaces, and other interesting and relevant such stuff.

It’s hard to do very much mathematics with those sorts of things if you can’t even say how big they are.




When I was in kindergarten, we would argue about whose dad made the most money. I can’t fathom the reason. I guess it’s like arguing about who’s taller? Or who’s older? Or who has a later bedtime. I don’t know why we did it.

 

  • Josh Lenaigne: My Dad makes one million dollars a year.
  • Me: Oh yeah? Well, my Dad makes two million dollars a year.
  • Josh Lenaigne: Oh yeah?! Well My Dad makes five, hundred, BILLION dollars a year!! He makes a jillion dollars a year.
    (um, nevermind that we were obviously lying by this point, having already claimed a much lower figure … the rhetoric continued …)
  • Me: Nut-uh! Well, my Dad makes, um, Infinity Dollars per year!
    (I seriously thought I had won the argument by this tactic. You know what they say: Go Ugly Early.)
  • Josh Lenaigne: Well, my Dad makes Infinity Plus One dollars a year.




I felt so out-gunned. It was like I had pulled out a bazooka during a kickball game and then my opponent said “Oh, I got one-a those too”.

image

Sigh.

Now many years later, I find out that transfinite arithmetic actually justifies Josh Lenaigne’s cheap shot. Josh, if you’re reading this, I was always a bit afraid of you because you wore a camouflage T-shirt and talked about wrestling moves.

Georg Cantor took the idea of ∞ + 1 and developed a logically sound way of actually doing that infinitary arithmetic.




 

¿¿¿¿¿ INFINITY PLUS ?????

You might object that if you add a finite amount to infinity, you are still left with infinity.

  • 3 + ∞   =   ∞
  • 555 + ∞   =   ∞
  • 3^3^3^3^3 + ∞   =   ∞

and Georg Cantor would agree with you. But he was so clever — he came up with a way to preserve that intuition (finite + infinite = infinite) while at the same time giving force to 5-year-old Josh Lenaigne’s idea of infinity, plus one.

to infinity, and beyond

Nearly a century before C++, Cantor overloaded the plus operator. Plus on the left means something different than plus on the right.

1 + \infty \ \ = \ \ \infty\ \ < \ \ \infty + 1

  • ∞ + 1
  • ∞ + 2
  • ∞ + 3
  • ∞ + 936

That’s his way of counting "to infinity, then one more." If you define the + symbol noncommutatively, the maths logically work out just fine. So transfinite arithmetic works like this:

All those big numbers on the left don’t matter a tad. But ∞+3 on the right still holds … because we ”went to infinity, then counted three more”.

image

image

By the way, Josh Lenaigne, if you’re still reading: you’ve got something on your shirt. No, over there. Yeah, look down. Now, flick yourself in the nose. That’s from me. Special delivery.




 

#### ORDINAL NUMBERS ####

W******ia's articles on ordinal arithmetic, ordinal numbers, and cardinality flesh out Cantor's transfinite arithmetic in more detail (at least at the time of this writing, they did). If you know what a “well-ordering” is, then you’ll be able to understand even the technical parts. They answer questions like:

  • What about ∞ × 2 ?
  • What about ∞ +  ? (They should be the same, right? And they are.)
  • Does the entire second infinity come after the first one? (Yes, it does. In a < sense.)
  • What’s the deal with parentheses, since we’re using that differently defined plus sign? Transfinite arithmetic is associative, but as stated above, not commutative. So (∞ + 19) + ∞   =   ∞ + (19 + ∞)
  • What about ∞ × ∞ × ∞ × ∞ × ∞ × ∞ × ? Cantor made sense of that, too.
  • What about ∞ ^ ? Yep. Also that.
  • OK, what about ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^  ? Push a little further.

I cease to comprehend the infinitary arithmetic when the ordinals reach up to the  limit of the above expression, i.e.  taken to the exponent of  times:

\lim_{i \to \infty} \ \underbrace{{{{{{{{ \infty ^ \infty  } ^ { ^ \infty} } ^ {^ \infty}} ^ {^ \infty}} ^ { ^ \infty  }} ^ {^ \infty }} ^ {^ \infty}   } ^ {^ \ldots  }   }_i

It’s called ε, short for “epsilon nought gonna understand what you are talking about anymore”. More comes after ε but Peano arithmetic ceases to function at that point. Or should I say, 1-arithmetic ceases to function and you have to move up to 2-arithmetic.





 

===== SO … WHAT COMES AFTER INFINITY? =====

You remember the tens place, the hundreds place, the thousands place from third grade. Well after infinity there’s a ∞ place, a ∞2 place, a ∞3 place, and so on. To keep counting after infinity you go:

  • 1, 2, 3, … 100, …, 10^99, … , 3→3→64→2  , … , ∞ + 1, ∞ + 2, …, ∞ 43252003274489856000   , ∞×2∞×2 + 1, ∞×2 + 2, … , ∞×84, ∞×84 + 1,  … , ∞^∞∞^∞ + 1, …, ∞^∞^∞^∞^∞^… , ε0,  ε+ 1, …




Man, infinity just got a lot bigger.

PS Hey Josh: Cobra Kai sucks. Can’t catch me!




Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop.

That’s all.

successor function
($i++ for programmers)

Which is why seems very small to the mind of a mathematician.

With projective geometry you can map to a circle, in which case there is a point-sized hole at the top where you can put ∞ (or −∞, or both).

Same thing with the Riemann Sphere.

stereographic projection of the Riemann sphere

So to them ∞ is very reachable. It’s just a tiny point.

Graham’s Number

It takes much more mental effort to conceive of Graham’s Number than ∞. It took me several hours just to begin to conceive Graham’s number the first time I tried.

\underbrace{     {{{{{{{{{{{{3^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^{\cdots}  }       }_{   3^{3^{3^{3^{\cdots}}}}  \text{ times}  }

Graham’s Number is basically a continuation of the above, recursed many times. Maybe I’ll do a write-up another time but really you can just look at Wikipedia or Mathworld. It’s absolutely mind-blowing.

Bigger

Here’s what’s weird. Infinity is obviously bigger than Graham’s Number. But Graham’s Number takes up more mental space. Weird, right?

EDIT: Maybe ∞ takes up less mental space than g64 because its minimal algorithmic description is shorter.




When I was young, I used to — as an exercise — try to conceive of ∞. We would hear in Sunday School that God is Infinite, that you can’t comprehend God’s Infinite-ness.

I would imagine myself in a spaceship flying out to the edge of the universe. I would imagine all of the stuff we had left behind us, flying at the speed of imagination.

Then I would zoom out the camera, seeing that in fact we had only gotten to the edge of a tiny speck. I would recurse this and try to recurse the recursions until my brain got tired. “Infinity is so big,” I would think. “The Universe is so big. God is so big.”

All of this was on purpose. I wanted ∞ to fill up my mind. I think there are lots of religious people who do this — meditate, in a way, on ∞.

Imagining ∞ as a mathematician is easy in comparison. Using the M.O. of stereographic projection, I can conceive of infinity in an Augenblick.

Nowadays it’s up to me, whether I want to view ∞ as large or small.