Posts tagged with generalised number

## Wiggly Numbers

http://en.wikipedia.org/wiki/Local_ring#Ring_of_germs:

we consider real-valued continuous functions defined on some open interval … of ℝ. We are only interested in the local behaviour of these functions … ∴ identify two functions if they agree on some (possibly very small) open interval…. This identification defines an equivalence relation, and the equivalence classes are the “germs of real-valued continuous functions at 0”. These germs can be added and multiplied and form a commutative ring.

(Of course 0 is just an arbitrary centre—but hey, why not 0?)

I feel like this is something I imagined, certainly not in this level of specificity, but at least wished for at some point in the past. Why should I be multiplying numbers like 13 and 27,714 when things in life are so less precise?

But we can still get the general idea of 13 and 27,714 without being so sticklerish about it. And no, the germ doesn’t need to be defined on the frazzwangled continuum, it could be done on other topologies as well. (Wikipedia gets into it.)

Here’s the only picture I could find of a germ online (the crayon splotches are my addition).

Yes, it’s no wonder this Grothendieck stuff is called an impressionistic mathematics.

## 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!

Paul Finsler believed that sets could be viewed as generalised numbers. Generalised numbers, like numbers, have finitely many predecessors. Numbers having the same predecessors are identical.

We can obtain a directed graph for each generalised number by taking the generalised numbers as points and directing an edge from a generalised number toward each of its immediate predecessors.

It has been shown that these generalised numbers can be “added” and “multiplied” in a natural way by combining the associated graphs. The sum a+b is obtained by “hanging” the diagram of b onto that of a so the bottom point of a coincides with the top point of b. The product a·b is obtained by replacing each edge of the graph of a with the graph of b where the graphs are similarly oriented.

Paul Finsler, David Booth, Renatus Ziegler in Finsler set theory: platonism and circularity

## Maths Infinity

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

That’s all.

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

## The Continuum

Speaking of infinitesimal standards, I need to digress for a few paragraphs so my point will make sense to all readers. Real numbers ℝ — any number you can construct with infinity decimal places, so essentially any number that most people consider a number at all — are thick, dense, an uncountable thicket. They are complete.

"The Reals" ℝ are made up of rational ℚ and irrational ℚᶜ numbers.

Rational numbers ℚ are ratios of regular counting numbers, 1, 2, 3, ℕ, etc., and their negatives −ℕ. However the rational part ℚ of the reals ℝ — the part that’s easy to conceive and talk about and imagine — is a negligible part of the real number line.

The irrational part ℚᶜ is further divided into algebraic 𝓐 and transcendental 𝓐 parts. Again the algebraic part 𝓐 is easier to explain and is, literally, negligible in size compared to the transcendental part.

$\dpi{200} \bg_white \begin{matrix} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mathbb{Q} \ \ = \ \pm \ ^\mathbb{N} \! / _\mathbb{N} \quad \longleftarrow \scriptsize{\text{negligible}} \\ \qquad \qquad \qquad \qquad \nearrow \\ \qquad \qquad \qquad \mathbb{R} \\ \qquad \qquad \qquad \qquad \searrow \\ & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mathbb{Q}^c \rightarrow \mathbb{A} \qquad \quad \longleftarrow \scriptsize{\text{negligible}} \\ & \searrow \qquad \qquad \; \qquad \qquad \ \ \qquad \\ & \qquad \qquad \mathbb{A}^c \qquad \longleftarrow \Normal{\text{ almost everything}} \end{matrix}$

Algebraic numbers 𝓐𝓐 are the x's that solve various algebraic equations, like x²=2.

Whatever number x you square to get 2, is an algebraic number 𝓐. We invent a symbol  and put it in front of the 2 symbol to express the number we’re talking about. Although there is no such symbol to express the number x that solves x² + x = 2 — square this number, then add itself to the result, and you get two — that is also an algebraic number.

Now add in all other finite-length equations with integer or fraction coefficients. That’s a lot of equations. Their solutions constitute the algebraic numbers 𝓐. But like I said above, 99% of the real numbers — those simple things you learned about in 3rd grade when they taught you the decimal system — are NOT IN THERE.

(99% of infinity, what am I talking about?  It doesn’t make sense, I know, just work with me here.)

## Transcendental

OK so now I have gotten to these hard-to-describe numbers called transcendental 𝓐ᶜ. The black part in the picture above. It took me a few paragraphs just to sloppily say what they are. If you have never thought about this issue before it might take you hours to wrap your head around them.

But it’s these transcendental numbers 𝓐ᶜ— can’t be assembled without an infinitely long equation — which essentially make calculus work. Calculus depends upon the real numbers ℝ and continuity therein, and without this thick, dense, impenetrable subset 𝓐ᶜ called transcendentals, its theorems would be unprovable and illegitimate.

I don’t know about you, but I haven’t seen any transcendental numbers around lately! Other than e and π, I mean. Despite transcendental numbers being the most numerous, only a few are known, most based around e and π. That’s right, these are the largest exclusive subset of the real numbers, and we don’t really know that many of them. We use them in proofs but not by name. Just knowing that they’re there ensures that calculus works.

But in the real world, you can’t buy e/3 eggs.  That, among other reasons, means you can’t optimize — even in principle — a purchasing decision at the grocery using calculus. (Maybe you don’t think you would be using calculus anyway, but the economic theorists treat you like a gas particle dispersing in the room — and while the particle doesn’t think it’s using calculus to decide where to move, it obeys those laws. So they are relevant somewhere, contrary to the title of this post.)

## Spiky

So calculus works in gas diffusion and solving various states of atoms / molecules via Schrödinger equations. But what about us people?

Here is a topographical picture of where people live. Notice that there is a lot of spikiness. Sudden jumps.

For a long time there’s no people because you’re in the middle of Nevada, and then — all of a sudden — Vegas! Holy cow there are people EVERYWHERE. Flowing in and out at a phenomenal rate. But there is zero flow and zero inhabitants just a few miles away. Molecules don’t behave like that.

Here is another picture — got it out of the same book, which my girlfriend is reading — of economic output by region.

Again, much spikiness.  Not much calculus. Discontinuous outputs. Maybe that is how we are. Maybe calculus doesn’t work on us.

$\dpi{300} \bg_white a^2 = b^3 = (ab)^{13} \\ = (a^{-1}b^{-1}ab)^5 \\ =(a^{-1}(bab)^{-1}a \; bab)^4 \\ = (ababababab^{-1})^6 \\ = 1$

thank you Jacques Tits

That’s functional in the sense that the data of interest forms a mathematical function or curve, not in the sense that flats are functional and high heels are not.

$\dpi{300} \bg_white f: \{ \rm{space} \} \to \{\rm{another\ space} \}$

So say you’re dealing with like a bit of handwriting, or a dinosaur footprint [x(h), y(h)], or a financial time series \$(t), or a weather time series [long vector], or a bunch of electrodes all over someone’s brain [short vector], or measuring several points on an athlete’s body to see how they sync up [short vector].  That is not point data.  It’s a time series, or a “space series”, or both.

Techniques include:

• principal components analysis on the Fourier components
• landmark registration
• using derivatives or differences
• fitting splines
• smoothing and penalties for over-smoothing

The problem you’re always trying to solve is the “big p, small n problem”.  Lots of causes (p) and not enough data (n) to resolve them precisely.

You can see all of their examples, with code, at http://www.springerlink.com/content/978-0-387-95414-1.

hi-res