Posts tagged with infinity

## Bounded linear operators on ∞-dimensional vector spaces

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.

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?

$\dpi{200} \bg_white \large \!\!\!\!\!\!\! -\,\text{replace}- \\ \text{\!\!\!\!\!\!sequence } \fbox{a_1, \ldots, \mbox{a}_N} \\ \\ \: -\text{ \ \ by \ \ }- \\ \text{\!\!\!\!\!\!function } \fbox{a(i)}, \\ \\ \quad i \in [1,N] \cong [0,1] \\ \\ \\ \\ \\ \\ \text{e.g., replace }\\ \fbox{1,4,9,16,25,\ldots} \\ \text{ by } \fbox{i^2}$

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.

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

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)

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.

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

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.

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

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)).

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.

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.

## 2¹⁰⁰

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

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.

## Measure: Sizing up the Continuum

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.

• 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 [¼√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.
• 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:

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!

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.

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.

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.

## What Comes After Infinity?

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”.

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.

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

$\large \dpi{200} \bg_white 1 + \infty \ \ = \ \ \infty\ \ < \ \ \infty + 1$

• ∞ + 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”.

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:

$\large \dpi{200} \bg_white \lim_{i \to \infty} \ \underbrace{{{{{{{{ \infty ^ \infty } ^ { ^ \infty} } ^ {^ \infty}} ^ {^ \infty}} ^ { ^ \infty }} ^ {^ \infty }} ^ {^ \infty} } ^ {^ \ldots } }_i$

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!

## Maths Infinity

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

That’s all.

$\dpi{300} \bg_white \begin{matrix} s: \mathbb{N} \to \mathbb{N} \\ s \mapsto s+1 \end{matrix}$
(\$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.

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.

$\dpi{300} \bg_white \underbrace{ {{{{{{{{{{{{3^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^{\cdots} } }_{ 3^{3^{3^{3^{\cdots}}}} \text{ times} }$

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.

## Religious Infinity

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.

This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

$\large \dpi{200} \bg_white f^0+f^1+f^2+f^3+f^4+f^5+f^6+ \ldots = {1 \over 1-f}$

Or, in fancy notation:

$\large \dpi{200} \bg_white \sum_{i=0}^\infty f^{\,i} = {1 \over 1-f} , \quad 0

Or, in C++:

long big = 9999999999;float frac = .70;double total = 0;for ( i = 0; i < big; i++){  total += frac∗∗i;  }cout << total;                 # in this case, prints 1 / .3 = 10/3cout << total - 1/(1-frac)     # prints 0 for any value of frac

Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

$\large \dpi{200} \bg_white {1 \over 2} + {1 \over 4} +{1 \over 8} + {1 \over 16 } + {1 \over 32} + {1 \over 64} + {1 \over 128} + {1 \over 256} + \ldots = {^1\! / \!_2 \over 1- {1 \over 2} } = 1$

Take that, Zeno!