Archive for September, 2011

September 30, 2011

Shivers by The Boys Next Door

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September 29, 2011

LHC sequence (23) [26238FlowerCircles_05_autoLIN_2]

by holgerlippmann

What Comes After Infinity?

September 28, 2011

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.

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”.

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!

September 27, 2011

291.2768-100.jpg (via jim soliven, via wowgreat)

September 26, 2011

Human psychology is such that we all like to imagine ourselves at the center of a dramatically coherent and meaningful story, whether we work in a Dilbert-style cubicle zoo or commute to weekly Executive Committee meetings on the company G550.

“The Epicurean Dealmaker”

September 25, 2011

Amos Tversky

September 24, 2011

None of [the attendees at Davos 2007] seemed to understand that when you create a derivative you don’t add to the sum of total risk in the financial world; you merely create a means for redistributing that risk. They have no evidence that financial risk is being redistributed in ways we should all worry about.

Michael Lewis

Uh-oh … this isn’t what he says in The Big Short: it’s actually the opposite.

September 24, 2011

Melanie Authier – Portfolio (via nullscapes)

September 23, 2011

Paul Ricard, créateur

September 23, 2011

Briefly: the linear regression model. We suppose we can explain or predict y using a vector of variables x. As in Gauß’ estimation theory, y is supposed to be unobservable, and thus has to be estimated. The assumption that y depends on x is expressed this way: the posterior distribution Prob{ Y | X } is different from the prior distribution Prob{ Y }.

The minimization of variance of the difference between [our estimation of Y given X] and [Y] leads to a unique solution: the conditional expectation.

The linear hypothesis says that the estimated value should be an affine expression of X. Moreover, the affine parameters which minimise the variance of the error are given by:

The above linear model coincides with the optimal conditional expectation model when X,Y are Gaussian.

Michel Grabisch, in Modeling Data by the Choquet Integral
(liberally edited)