Regarding π Day

I've already mentioned in a previous post that I wouldn't be able to publish Chapter 2 of π on the 14th, also known as "π Day," but now that it's up, I can reveal that that was because I'd already published Chapter 1 on π Day --- Specifically, the Trixie-approved π Day, March 2nd.

Comments

[q97randomguy]

3.2???
For real?
You never know what an idiot will think of.

[PoweredByTea]

Humm... just a little note on the video. Obviously, you can square the circle. If you give me a circle of radius 1, then the problem is solved by a square of side-length root-pi. Pi (and by extension) is a perfectly respectable real number. In fact, it's a computable number. Any Universal Computer is capable of calculating approximations of it to any given precision, no problem.

What pi isn't, however, is algebraic. It's not the the solution of any polynomial with rational coefficients. In other words (almost), you can't write it down with the basic mathematical symbols like integers, "+", "-", "x", "/" and various roots that you learned about in school and anyone who tells you otherwise is wrong because a proof exists by Évariste Galois that says that you can't.

(Incidentally the story of Évariste Galois is rather interesting and tragic and involved him dying in a duel. Look it up if you're interested.)

Numbers like pi are call transcendental. There are even weirder classes of numbers, such as the uncomputables and even stranger, the unnamable number, which is a set of numbers whose members cannot be specified ("named") in any way imaginable.

Em, yes. Right. Now to read the new chapter.

[nemryn]

921873
1) 'Square the circle' usually refers to the specific problem of constructing a square with side root-pi using only a compass and a straightedge, which is impossible.

2) Can't you specify an unnameable number by pointing to it on a number line?

3) The funnest part of unnameable numbers is, if you pick a number at random between 0 and 1, you're almost certain to pick an unnameable number.

[PoweredByTea]

921960
1) I'm afraid I didn't feel like I was familiar enough with straight edge and compass constructions to comment.

2) If you try that, you'll find your finger is too fat. Even if you file a pointer's point down to a single atom, you're still pointing at an uncountable infinity of real numbers. But more importantly, using real, physical objects in a such a way isn't really playing by the rules of mathematics. As V from V for Vendetta would say, "A circle is an idea. And ideas are bulletproof not subject to physics", or, as Lockheart puts it better in his famous lament:

The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination. For one thing, the question of how much of the box the triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. That is, unless you want to talk about some sort of approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple, and consequently it is an ugly question which depends on all sorts of real-world details. Let’s leave that to the scientists. The mathematical question is about an imaginary triangle inside an imaginary box. The edges are perfect because I want them to be— that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.

So if you want to specify a unnameable real in the sense of an imaginary idea (that is, imaginary as in common usage, not imaginary as in complex numbers) you need to specify its unambiguously, somehow, with language.

Actually, this has got me thinking. We can specify a unnameable like this. Let a be an unnameable real between, say 0 and 1. We can then continue and prove that 2a must also be unnameable and so on. That seems like cheating, but we also "cheated" in 0 and 1 as we specified them as axioms. So there are probably more rules here in the formal definition than in the informal definition. Humm... I don't know. I may be overthinking this.

3) I always thought the funniest part was the way they make the real number line sound like it's full of Lovecraftian monstrosities.

[Jesin]

922075 921960 The formal definition for what it means to "name" a real number, in that sense, is this:
If you can write down a grammatically-correct statement (call it "P(x)") in which every variable except "x" is defined, and there exists some real number x0 such that "P(x0)" is true, but "P(y)" is false if "y" is any real number other than x0, then you have named "x0".

We usually assume that a "grammatically-correct statement" is a sequence of symbols taken from a finite alphabet, and each statement must terminate (that is, you can't have an infinitely-long sentence). Under this assumption, you can prove that there are simply a lot more real numbers than there are sentences, so most real numbers can't be unambiguously specified by a sentence.

[PoweredByTea]

923373
Ah, so which members of a set are unnameable is a property of your grammar. We choose the usual axioms that define the real numbers as acceptable statements to be made without proof (so we could name 2 as "true for x if x is equal to 1+1" with both the "1" and the "+" coming from the field axioms) and bamb, through the diagonal argument there is a countably infinite number of such statements but an uncountably infinite number of reals.

Okay I think I get this now. Thanks.

[Jesin]

921960 As for (2), only if your finger has zero width.

923684 That sounds pretty much right, yep.

[nemryn]

924019
It's an idealized, mathematical finger, so yeah it does.

[Jesin]

924028 Nice. If you can dip that into some idealized ink and make a perceptible zero-width marking in finite time, you'll be able to solve problems that even so-called "universal" Turing machines can't.

[nemryn]

924064
Sweet! Now I just need to go see if Twilight has any spells for that stuff.