I was once described as a "beautiful, intelligent iguana".
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…nobody, it seems, is prepared to dump the system that writes 7,654,321 in place of seven million, six hundred fifty- thousand, three hundred twenty-four; the unwieldy prolixity here is too obvious to ignore. But why stop at numbers?
Brian Rotman - Thinking Dia-Grams
Seriously, this essay is so good. Holy crap.
- I cheated on you. ∄ way to restore the original pure trust of our relationship.
- The broken glass. Even with glue we couldn’t put it back to its unblemished state.
- I got old. ∄ potion to restore my lost youth.
- Adam & Eve ate from the tree of the knowledge of good & evil. They could not unlearn what they learned.
- “Be … carefulwhat you putin that head because you will never, ever get it out.” ― Thomas Cardinal Wolsey
- We polluted the lake with our sewage runoff. The algal blooms choked off the fish. ∄ way to restore it.
- Phase change. And the phase boundary can only be traversed one direction (or the backwards direction costs vastly more energy). The marble rolls off the table, the leg poisoned by gangrene. The father dies at war. The unkind words can’t be unsaid.
…[I]t takes years of dedicated study before scientific truth in its truest, mathematical and symbolic forms can be understood. The rest of us rely on experts to explain it, someone who has seen and understood the truth and can dumb it down for us….
The Last Psychiatrist (via isomorphismes)
Yes! Video games are the best way to explain basic topology.
- You know those special levels in Super Mario Bros. where the screen doesn’t move with you — you’re just in a “room” and if you go off to the right you arrive back on the left?
- That was just a convenience for the game programmers.
- But think about this: what’s the difference between a straight line that meets back to its other end and you loop over and over and over the same spot while running forward — and a circle?
- (Answer: there is no difference. If we wanted to imagine Mario being a 3-D person, we could — he’d be running around a cylinder (this room he’s jumping to get the coins in would just be maybe 2-3 shoulders wide and dug in a circle underneath the ground. Or the brick platform he’s running on equally wide and it’s built in a cylinder above off the ground. We just don’t see that he’s constantly adjusting to bear a few degrees left as he’s running.
- How about Star Fox battle mode?
- If you drive off the north of the screen you end up at the south, and if you fly off the east you end up on the west.
- At first I thought this just meant we were flying around an entire planet Like The Little Prince’s moon or so. (The non-map visuals—the main flying visuals—could go along with this story, since there’s suelo below and cielo above.)
- But on further consideration this can’t be the case. Think about a globe of the Earth: east and west are connected contiguously ::globe pic:: — but the North Pole is as far away from the South Pole as you can get. ::mathworld pic::
- Think about running away from your enemy to the northeast corner and disappearing very quickly off the north to the south, then disappearing just as quickly from east to west. What allows you to do this quick of a dodge?
- Think about if the North Pole anhttp://arxiv.org/pdf/1205.6044v1.pdfd the South Pole WERE equivalent. Picture the Earth and start “sucking the two towards each other”.
- You would end up with….
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”.
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 coloured 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 quantity to infinity, you are still left with infinity.
- 3 + ∞ = ∞
- 555 + ∞ = ∞
- 2 × ∞ = ∞
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.
- ∞ + 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 digit, the hundreds digit, the thousands digit from third grade. Well after infinity there’s a ∞ digit, a ∞2 digit, a ∞3 digit, 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!