Posts tagged with the continuum

## One

Counting generates from the programmer’s successor function `++` and the number one. (You might argue that to get out to infinity requires also repetition. Well every category comes with composition by default, which includes composition of ƒ∘ƒ∘ƒ∘….)

But getting to one is nontrivial. Besides the mystical implications of 1, it’s not always easy to draw a boundary around “one thing”. Looking at snow (without the advantage of modern optical science) I couldn’t find “one snow”. Even where it is cut off by a plowed street it’s still from the same snowfall.

And if you got around on skis a lot of your life you wouldn’t care about one snow-flake (a reductive way to define “one” snow), at least not for transport, because one flake amounts to zero ability to travel anywhere. Could we talk about one inch of snow? One hour of snow? One night of snow?

Speaking of the cold, how about temperature? It has no inherent units; all of our human scales pick endpoints and define a continuum in between. That’s the same as in measure theory which gave (along with martingales) at least an illusion of technical respectability to the science of chances. If you use Kolmogorov’s axioms then the difficult (impossible?) questions—what the “likelihood” of a one-shot event (like a US presidential election) actually means or how you could measure it—can be swept under the rug whilst one computes random walks on trees or Gaussian copulæ. Meanwhile the sum-total of everything that could possibly happen `Ω` is called 1.

With water or other liquids as well. Or gases. You can have one grain of powder or grain (granular solids can flow like a fluid) but you can’t have one gas or one water. (Well, again you can with modern science—but with even more moderner science you can’t, because you just find a QCD dynamical field balancing (see video) and anyway none of the “one” things are strictly local.)

And in my more favourite realm, the realm of ideas. I have a really hard time figuring out where I can break off one idea for a blogpost. These paragraphs were a stalactite growth off a blobular self-rant that keeps jackhammering away inside my head on the topic of mathematical modelling and equivalence classes. I’ve been trying to write something called “To equivalence class” and I’ve also been trying to write something called “Statistics for People Who Program Computers” and as I was talking this out to myself, another rant squeezed out between my fingers and I knew if I dropped the other two I could pull One off it could be sculpted into a readable microtract. Leaving “To Equivalence Class”, like so many of the harder-to-write things, in the refrigerator—to marinate or to mould, I don’t know which.

But notice that I couldn’t fully disconnect this one from other shared-or-not-shared referents. (Shared being English language and maybe a lot of unspoken assumptions we both hold. Unshared being my own personal jargon—some of which I’ve tried to share in this space—and rants that continually obsess me such as the fallaciousness of probabilistic statements and of certain economic debates.) This is why I like writing on the Web: I can plug in a picture from Wikipedia or point back to somewhere else I’ve talked on the other tangent so I don’t ride off on the connecting track and end up away from where I tried to head.

The difficulty of drawing a firm boundary of "one" to begin the process of counting may be an inverse of the "full" paradox or it may be that certain things (like liquid) don’t lend themselves to counting in an obvious way—in jargon, they don’t map nicely onto the natural numbers (the simplest kind of number). If that’s a motivation to move from discrete things to continuous when necessary, then I feel a similar motivation to move from Euclidean to Hausdorff, or from line to poset. Not that the simpler things don’t deserve as well a place at the table.

We thinkers are fairly free to look at things in different ways—to quotient and equivalence-class creatively or at varying scales. And that’s also a truth of mathematical modelling. Even if maths seems one-right-answer from the classroom, the same piece of reality can bear multiple models—some refining each other, some partially overlapping, some mutually disjoint.

## 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.

## Nonterminating decimals do not make sense.

The Banach-Tarski paradox proves how f#cked up the real numbers are. Logical peculiarities confuse our intuitions about “length”, “density”, “volume”, etc. within the continuum (ℝ) of nonterminating decimals. Which is why Measure Theory is a graduate-level mathematics course. These peculiarities were noticed around the turn of the 20th century and perhaps never satisfactorily resolved. (Hence I disagree with the use of real numbers in economic theory: they aren’t what you think they are.)

Axiom of Choice → Garbage

The paradox states that if you assumed the axiom of choice (or Zorn’s Lemma or the well-ordering of ℝ or the trichotomy law), then you could take one ball and make two balls out of it. It follows that you could make seven balls or thirty-seven out of just one. That doesn’t sound like real matter (it’s not; it’s the infinitely infinite mathematical continuum).

I can’t think of anything in real life that that does sound like. Conservation-of-mass-type constraints hold in economics (finite budget), probability (∑pᵢ=1), text mining, and in all the phase and state spaces I can think of as well. Generally you don’t make something out of nothing.

If it’s broke, throw it out.

The logical rule-of-inference Modus Tollens says that if A→B and ¬B, then ¬A. For example if leaving the fridge open overnight leads to rotten food, and the food is not rotten, I conclude that the fridge was not open overnight. Let A = Axiom of Choice and B = Banach-Tarski Paradox. Axiom of Choice leads to Banach Tarski paradox; said paradox is false; so why don’t we reject the Axiom of Choice? I have never gotten a satisfactory answer about that. ℝ is still used as a base corpus in dynamical systems, economics, fuzzy logic, finance, fluid dynamics, and as far as I can tell, everywhere.

How does the proof of paradox work?

The proof gives instructions of how to:

1. Partition a solid ball into five unmeasurable disjoint subsets.
2. Move them around (rigidly, without adding mass).
3. Get a new solid ball, whilst leaving the first ball intact.

The internet has several readable, detailed explanations of the above. You’ll end up reading about Fuchsian groups, Henri Lebesgue’s measure, and hyperbolic geometry (& the Poincaré disk) along the way.

Stan Wagon has also written a Mathematica script to display the subsets in a hyperbolic geometry (whence these pictures come). Thanks, Stan!