

Explain xkcd : It's cause you're dumb - aufreak3
http://www.explainxkcd.com/

======
lmkg
Here's what's going on with the Banach-Tarski paradox:

First, in case this wasn't clear from the explanation, you have to carve the
pumpkin into some very specific pieces in order to be able to re-assemble them
into two pumpkins. In particular, you have to carve them into some pieces that
are _non-measurable sets_ , which basically means that the carving is so
exotic that it is not possible to define a consistent concept of volume for
these pieces. This is sort of the heart of the paradox: you get volume-
preserving transformations to disobey preservation of volume by using
intermediate steps with undefined volume.

Non-measurable sets are not 100% legit in mathematics. Constructing them
relies on the Axiom of Choice. I won't get into the debate about the Axiom of
Choice, but suffice it to say that if you reject the Axiom of Choice, then the
Banach-Tarski Paradox need not be true, and you can self-consistently reject
the BTP if you are OK losing the AOC.

The biblical reference from King Solomon is probably referring to the parable
of splitting the baby. Two women came to King Solomon, each claiming to be the
true mother of the baby. King Solomon suggested cutting it in half, in order
to observe the claimants' reactions. The fake mother was ok with this, and the
real mother would rather give up her child than see it killed, . XKCD suggests
that King Solomon was actually attempting to use the Banach-Tarski paradox to
create two babies.

~~~
Dove
There's a bit more to the subtext than that. The Axiom of Choice sounds quite
innocuous: If you have a bunch of sets, you can construct a new set by picking
one element out of each of them.

Turns out you can reason your way from that to "I can make any solid 3d object
into any other solid 3d object just by cutting and rearranging the parts".
(Banach-Tarski is often explained as the ability to turn one sphere into two,
which is true, but the full implication is more along the lines of being able
to rearrange a pea into the earth.) This is just one of many scary dragons
that lurk in set theory.

Historically speaking, Banach-Tarski is _why_ the Axiom of Choice is
controversial. The other (I forget, nine?) axioms in ZFC, garner a tepid
response--along the lines of, "Hooray, someone axiomatized set theory." But AC
remained controversial for a long, long time. I cannot find the quote, but
there was a mathematician who refused to accept the Axiom of Choice, and when
asked about it, commented, "It is because I do not believe one and one make
three." He was referencing Banach-Tarski.

That is the dilemma that has faced mathematicians historically: either reject
something so innocuous as the Axiom of Choice or embrace something so
nonsensical as the Banach-Tarski paradox. "Controversial" is putting it
lightly; that's a hard decision. One still sees textbooks that flag all
theorems depending on the axiom of choice, so you can reject them if you want
to. I would say the field as a whole has come to embrace AC and accept Banach-
Tarski; one of my professors one commented, "If we need the axiom of choice in
this proof, we'll use it; why do mathematics with one arm tied behind your
back?" I think this reflects the common attitude. But it wasn't an easy
choice.

And that's the joke. If the guy didn't want one pumpkin to equal two, he
should have gone the other way and rejected AC. It is not unfair to
characterize Banach-Tarski as the sorry consequence of something
mathematicians really wanted to do -- collateral damage, the lesser evil. One
could regret it.

~~~
Dylan16807
To be more specific about what exactly the Axiom of Choice is:

If you have a finite number of sets, you don't need the Axiom of Choice. You
can enumerate them and pick out of each.

If you have a rule to choose items, like picking the smallest element in a set
of natural numbers, you don't need the Axiom of Choice. There is no choice to
be made, you just follow the rule.

The problems come out when you have an infinite number of sets and you pick
items arbitrarily. This is what the Axiom of Choice allows you to do. Infinity
is dangerous.

~~~
Dove
Actually, I think countably infinite sets of sets are okay, too. It's when
you're allowed to choose an arbitrary element from an _uncountably_ infinite
number of sets that weird things start happening.

(Not that I can call pedantery given the sheer number of who-cares-about-the-
details glosses in my original comment ;)

And I don't know that I'd call infinity dangerous so much as I'd call it
sanity-stretchingly _counterintuitive_.

Which makes sense, really. Where would we have developed intuitions for
dealing with the infinite?

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dctoedt
At the end of the post, the author says he doesn't get the King Solomon
reference. It's to the story of the king's offering to resolve a dispute about
which of two women was the mother of the child by cutting the baby in two. See
<http://en.wikipedia.org/wiki/Judgment_of_Solomon>;
[http://www.biblegateway.com/passage/?search=1Kings%203:16-28...](http://www.biblegateway.com/passage/?search=1Kings%203:16-28&version=NIV)

------
bediger
I thought the first panel was the funniest. "I carved a pumpkin" is true in at
least two ways for the character, and untrue in the "This is not a pipe" Rene
Magritte way.

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jcdreads
There is also <http://xkcdexplained.com/>

~~~
rogerclark
XKCD Explained is much better -- because it has the right perspective; it's
explaining the comics to people who already understand them, shitting on the
author and his idiot audience.

XKCD is a condescending comic whose audience revels in its own perceived
superiority, while any self-respecting nerd with a real sense of humor knows
that putting a science or technology reference in a cartoon is not a joke in
itself.

~~~
dasil003
You're very wrong about XKCD. There's a whole slew of un-funny techy strips,
"User Friendly" being the canonical example. XKCD by comparison is often
hilarious. The fact that it requires inside knowledge and targets a specific
audience does not automatically make it unfunny, nor does it make anyone who
enjoys it "revel in their superiority".

~~~
JoeAltmaier
perhaps rogerclark is referring to this comic alone; it indeed seems to offer
a technical reference as the sole point of humor, at least in the final frame.

Although I do enjoy frame 3 on its own merits.

~~~
dasil003
Grammatically that is clearly not what he meant unless he is not a native
English speaker and thus not aware of the finer points of article usage, but I
doubt that.

~~~
lwhi
Oh my god. Irony is beautiful.

~~~
dasil003
I don't think you know what irony means.

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Natsu
I can't say that I've ever needed another website to explain this. If there's
something I don't get, a quick visit to Google or Wikipedia usually fills in
the rest and people in the comic thread even supply links to the relevant
parts rather quickly.

I have to say, though, that I'm surprised by how many people know Banach-
Tarski, but who never heard the story about King Solomon threatening to cut
the baby in two in order to discern who cared about its life the most (in
other words, to find the real mother).

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xyzzyz
I do not understand it. "If you want to be a hipster, you need to read xkcd,
and if you do not get it, it is no problem, because we can explain it to you"?

Come to think of it, maybe recent xkcd panels would amuse me more if I did not
understand them.

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chris_l
From wikipedia: "Unlike most theorems in geometry, this result depends in a
critical way on the axiom of choice". By "taking" the AoC the figure ended up
making two pumpkins. Not really funny at all, I'm afraid.

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ck2
If you can cut something into infinite pieces and then reassemble it into two
of the same object - you obviously have a rounding error where you are
"rounding up" each fraction when it should not be!

No?

~~~
cynicalkane
[https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%...](https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%93Tarski_paradox)

It works because, if you assume the axiom of choice, you can postulate
"pathological" sets that don't have a mathematically well-defined volume. You
can use this to backdoor volume changes, by cutting something with well-
defined volume into a buncha pieces without a well-defined volume, then
reassembling this into something with a different volume than the original.

You can't get around this trick by redefining what it means to measure
something: If you pick any reasonable definition, it must admit the Banach-
Tarski paradox. [https://secure.wikimedia.org/wikipedia/en/wiki/Non-
measurabl...](https://secure.wikimedia.org/wikipedia/en/wiki/Non-
measurable_set)

These pathological sets are too weird to exist in physical reality.

