isomorphismes

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 :

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”.

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.

• ∞ + 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:

• 1 + ∞ + 3   =   ∞ + 3
• 418 + ∞ + 3   =   ∞ + 3
• 1729 × ∞ + 3   =   ∞ + 3
• 43252003274489856000 × 287 × 1.4142135623730954 + 3→3→64→2  ×  ∞ + 3   =   ∞ + 3

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:

It’s called ε₀, short for “epsilon nought gonna understand what you are talking about anymore”. More comes after ε₀ 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,  ε₀ + 1, …

Man, infinity just got a lot bigger.

PS Hey Josh: Cobra Kai sucks. Can’t catch me!

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

That’s all.

($i++ for programmers)

Which is why ∞ seems very small to the mind of a mathematician.

With projective geometry you can map ℝ 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.

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:

• lists
• 1-D arrays
• bitstreams
• linear functionals
• force vectors
• personal preferences
• decisions
• desires
• the flow of heat
• dinosaur tracks
• tying a knot
• economic transactions
• a story or article, in any language
• a poem
• water flowing
• time series
• one instant during an argument
• a curve
• probabilities associated with various outcomes
• marketing data
• statistical observations
• bids and asks
• the quantum numbers of an atom
• waveforms
• signals
• songs
• heartbeats
• one solution of a differential equation
• a polynomial
• a jpeg or bitmap of the Mona Lisa
• a set of instructions
• a dance move
• someone’s signature
• a secret message
• a Taylor series
• a Fourier decomposition
• turning your mattress
  (which you’re apparently supposed to do once a season)
• intentions
• electromagnetic flux
• part of a trajectory
• one wisp of the wind
• states of affairs
• logical propositions
• distortions in a crystal lattice
• a rotation of Rubik’s cube
• neuronal spike-trains — so, thoughts? perceptions?
• colour

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.

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