In a college calculus class I once correctly solved a bullshit test question by taking the average of two infinities.
Hell, for all I know I invented a new branch of mathematics.
“It’s a very common teaching method to introduce a counter-intuitive concept…”
I am not convinced that any teaching happened. I suspect that those who did not understand, or believe, the concept before the example still don’t understand the concept (or believe the assertion) after the example was provided.
In looking at the comments it seems to be divided between the people who say “I’ve known this and it is really cool” and those who say “That’s a bunch of BS”. There are no commenters saying, “Oh, now I get it.” This data set is consistent with no learning.
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But the rule set wasn’t “changed”, what it did was introduce two rules (the special cases of each n where infinity wasn’t reached) as well as the stipulation that n went to infinity. At infinity, the simpler construction (all the numbers in the set being considered can be squared) makes the infinite number of special cases relatively irrelevant. It’s not a rule change, it’s a case where the algorithmic grind produces a trend that isn’t borne out when infinity is considered.
The original rule still operates. It wasn’t removed. To say that it was would be like describing a higher-degree polynomial, where the higher degree is fractional, as “cheating” because its value declines past x=1.
No, a procedure isn’t a series sum; and being a person who can see the future, you would quickly decide what you want to be the last cohort of large numbers and whether it’s a fun furnishing as ‘the last numbers,’ or overflow related to picking the last set you’d tolerate stacking in bin (once log(n) was 8k and your shorthand was getting dicey.) Maybe the Long Now Society is getting pretty crazy and you want to have limits. So the last thing in the bin is The Ramones. Ask Mathematica and it might say the Mighty Mighty Boztones x Voivod.
Wait, it could be precision large integer libraries’ verbose compiler errors for people who ask for infinite schlub. (Plus a residue consistent with the default limit of the library.) That would make the thing PrecisionLibsplaining.
Or morning notions from someone who thought this was a lay book lay book and now sees that it’s a key party for people who start their cars with a Classical Greek Election.
“real” and “imaginary” seem to be almost completely meaningless when you talk about math. It is pretty unclear what being real would even mean, let alone how we would ever determine if it something were real. At best, we know that many of these things are “real enough” in that they are part of models that describe observable reality to a high degree of precision.
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Infinity. brrr. My biggest boggle came when I realized the “arms” of a parabola get closer and closer and closer to parallel as they get farther, farther, farther from their focal point - but they still get infinitely far apart!
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Are Chuck Norris jokes played out already? 'Cause I’ve got one.
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I think he means, in the instance of the ball with “5” on it, you are not adding five of something, you are adding one ball, identified with the label “5”.
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Oh. Ohhhhhh. Yeah, of course, duh. Don’t know how I missed that. I’ll have to give the whole thing another look when it’s not wee hours of the morning.
Which balls will be placed in the drawer?
All of them, eventually. But you will never have an empty box because you can never stop adding new balls.
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Thank you all for reminding me why I love boingboing. Also:
NEEEEEEEEEEEEERRRRRRDDDSSSS!
…That is all.
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I’m guessing that you’ve never had the pleasure of actually teaching someone a concept they had never considered before, and that was interesting, pleasing, and surprising to them when they got it.
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Edward de Bono shaves with Occam’s razor.
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What the article is saying comes down to this:
That’s not a very fun thing say and not particularly accessible. So it’s introduced and explored with the balls and drawers and boxes analogy.
One of the parts of this that is hardest to “get” is that the cardinality of a set is NOT the same as the set.
The cardinality of the set of all of my surviving siblings is 3. But the number 3 is no substitute for a family reunion 
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There is no “last ball” because all the balls are placed in the box simultaneously, not one after the other.
Yes, it would be a difficult task for you and me, but perhaps we could use an assistant:
It’s strange how many BB threads end up being about the right to bear arms.
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I like this one:
What is 1-1+1-1+1-1 …?
Is it (1-1)+(1-1)+(1-1)+…=0+0+0+…=0?
or …
Is it 1-(1-1)-(1-1)-(1-1)-…=1-0-0-0-…=1?
(note: The negative signs distribute through the parentheses, so it is the same.)
Answer: Just average the answers and get 1/2.
Not hairy enough.
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I prefer:
1-1+1-1+1-1+…
= 1+x+x^2+x^3+x^4+x^5+… at x=-1
but
1+x+x^2+x^3+x^4+x^5+… = 1/(1-x)
( a standard calculus trick which is valid when |x|<1 )
so, at x=-1,
1-1+1-1+1-1+… = 1/(1-(-1)) = 1/2
No, that does not apply here. (Among other things, the infinity he gets as the limit is not any kind of cardinal — it is a statement of convergence to one of the ends End (topology) - Wikipedia of the real line.) He is (subject to also taking the ceiling or floor function) measuring the finite cardinality of the quantity of numbers in the box at step n in a perfectly valid way.
It’s just that this bigger and bigger set (in the natural numbers) also has a lower bound which is shooting off to infinity even though the upper bound is getting there faster.
Since Parker’s book also seems to be about 4-dimensional geometry, I’ll just leave this 4-dimensional rotation of a horse here:
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