Quantcast

Posts tagged with generalised number

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).

http://www.cs.bham.ac.uk/~sjv/GeoFuzzy.pdf :

image

image

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




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!




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




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.




Vectors, concretely, are arrows, with a head and a tail. If two arrows share a tail, then you can measure the angle between them. The length of the arrow represents the magnitude of the vector.

The modern abstract view is much more interesting but let’s start at the beginning.

Force vectors

Originally vectors were conceived as a force applied at a point.

As in, “That lawn ain’t mowing itself, boy. Now you git over there and apply a continuous stream of vectors to that lawnmower, before I apply a high-magnitude vector to your bee-hind!”

Thanks Galileo, totally gonna get you back, man

The Galilean idea of splitting a point into its x-coordinate, y-coordinate and z-coordinate works with vectors as well. “Apply a force that totals 5 foot-pounds / second² in the x direction and 2 foot-pounds / second² in the y direction”, for instance.

Therefore, both points and vectors benefit from adding more dimensions to Galileo’s “coordinate system”. Add a w dimension, a q dimension, a ξ dimension — and it’s up to you to determine what those things can mean.

If a vector can be described as (5, 2, 0), then why not make a vector that’s (5, 2, 0, 1.1, 2.2, 19, 0, 0, 0, 3)? And so on.

4th Dimension Plus

So that’s how you get to 4-D vectors, 13-D vectors, and 11,929-D vectors. But the really interesting stuff comes from considering -dimensional vectors. That opens up functional space, and sooooo many things in life are functions.

(Interesting stuff also happens when you make vectors out of things that are not traditionally conceived to be “numbers”. Another post.)

Abstractions

In the most general sense, vectors are things that can be added together. The modern, abstract view includes as vectors:

Things you can do with vectors

Given two vectors, you should be able to take their outer product or their inner product.

The inner product allows you to measure the angle between two vectors. If the inner product makes sense, then the space you are playing in has geometry. (Not all spaces have geometry — some just have topology.)

And — this is weird — if the concept of angle applies, then the concept of length applies as well. Don’t ask me why; the symbols just work that way.

Magnitude

But the “length” of a song (one of my for-instances above) would not be something like 2:43. The magnitude of a song vector would be the total amount of energy in the sound wave | compression wave.

\| \text{song} \| = \int \text{compression wave}

What is the angle between two songs, two spike-trains, two security prices? What is the angle between two heartbeats? It’s the correlation between them.

Linear Algebra

Also, you can do linear algebra on vectors — provided they’re coming out of the same point. Some might say that the ability to do linear algebra on something is what makes a vector.

That can mean different things in different spaces — like maybe you’re superposing wave-forms, or maybe you’re converting bitmap images to JPEG. Or maybe you’re Photoshopping an existing JPEG. Oh, man, Photoshop is so math-y.

shearing the mona lisa

Shearing the mona lisa (linear algebra on an image — from the Wikipedia page on eigenvectors, one of which is the red arrow)




Pythagorean Theorem
This is how I first really understood the Pythagorean Theorem.
The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.
Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.
Ideas involved here:
scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)
circumcircle &amp; incircle
√2
This is also how I first really understood √2, now my favourite number.

Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

  • scaling properties of squared quantities
    (gravitational force, skin, paint, loudness, brightness)
  • circumcircle & incircle
  • √2

This is also how I first really understood √2, now my favourite number.


hi-res




The hallmark of a calculus course is epsilon-delta proofs. As one moves closer and closer to a point of interest (reducing δ, the distance from the point-of-interest), the phenomenon's measure is bounded by something times ε, a linear error term. The bound comes from the continuity of the function, also defined in epsilon–delta terms.

So everything moves gradually, in a sense. There are no sudden jumps. But human affairs are characterized by jumping, leaping, gapping, sparking, snapping, exploding processes.

One studied example is stock prices. If terrible news hits about a public company, you won’t be able to sell your shares for the previous price minus epsilon. You’ll have to unload a gap or a yawn lower. Not that it was ever possible to trade in arbitrarily small δ intervals anyway. The smallest increment on the NYSE is $.01 (it used to be  of a dollar), which by infinitesimal standards is huge.

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.

decomposition of the real numbers

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.)

the algebraic numbers in the complex plane, coloured by degree

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.

Economic Activity

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.

Innovation




Tits group

thank you Jacques Tits




That&#8217;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.

So say you&#8217;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&#8217;s brain [short vector], or measuring several points on an athlete&#8217;s body to see how they sync up [short vector].  That is not point data.  It&#8217;s a time series, or a &#8220;space series&#8221;, 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&#8217;re always trying to solve is the &#8220;big p, small n problem&#8221;.  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.

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.

f: {space} \to ’ {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