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/sci/ - Science & Math

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997048 No.997048 [Reply] [Original]

"The Banach-Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball."

what the fuck, science. are you even trying anymore?

>> No.997071

wait, what?
saw this posted some time ago, instantly assumed it was a troll
seriously?

>> No.997085

OK, if the parts are assembled in such a way that two balls can be created out of one, is it gay?

>> No.997093

>>997071
Yes, it's a theorem. This is the kind of shit you get when you allow infinitely divisible objects.

>> No.997094

>>997048

banach-tarski is only a problem is we assume that all sets are measurable...if there exists non-measurable sets, then banach-tarski is not a problem

>> No.997092

>>997071
yep, according to "science", you can chop anything into finite number of objects and then reassemble it to get the same thing twice.

makes sense.

>> No.997099

>>997085
only if they touch

>> No.997102

>>997092
This is math, not science, and is based on assumptions. One of them is that the ball may be cut infinitely accurately, which is an obvious impossibility. There's no science here.

>> No.997113

>>997102
>This is math, not science
I know how you mean it, but it made me lol

>> No.997115

The "pieces" are really sets of points that need not be solid as we intuitively imagine it. They are non-measurable. If you think this theorem is stupid or obviously false, you are simply narrowminded and ignorant.

>> No.997133

>>997115
well, it surely isn't intuitive

>> No.997141

>>997133
indeed, that much is certainly true

>> No.997159

mathematicians is just a circle jerk, the 'proofs' are mostly fake nonsense.. ever wondere why it's "peer review"? Because they just publish their friends articles. Taking this math stuff seriously is the most stupid mistake you can make.

>> No.997183

>>997159
just because you don't understand it doesn't mean it's all fake

>> No.997192
File: 94 KB, 480x480, 1266820825549.png [View same] [iqdb] [saucenao] [google]
997192

>>997159
0/10

>> No.997199

>>997159

even if it is rubbish, at least their rubbish doesn't cost the taxpayers 70 million dollars to conduct one experiment, and publish a paper that will be outdated in 10 years.

>> No.997210

>>997048
How is it a paradox?

>> No.997249

>>997199
that physics not science

>> No.997277

>>997210
Volume is not conserved through a series of operations which conserve volume.

>> No.997294

The Banach Tarski paradox was created to show what kind of strange things happen when you allow the Axiom of Choice.

>> No.997305
File: 3 KB, 409x259, 1218919210073.png [View same] [iqdb] [saucenao] [google]
997305

>>997199

you really dont get how science works.

weather or not something becomes outdated (an generally things don't; they become superseeded or are shown to be limiting cases/statistical approximations) it is irrelevant. To make later progress, you need the original progress, new discoveries are built on the old ones. The 70 million dollar investment isn't just in one experiment or finding but in all the potential research that could come from it.

Also, gaining knowledge is intrinsically valuable (as a mathfag I figured you would know this)

4/10, would rage again.

>> No.997308

>>997294

i think the axiom of choice being false is even stranger

>> No.997327

>>997277
10/10

The "pieces" cannot be given a "volume" because they are nonmeasurable sets. You can't speak of "conservation of volume" in a setting where you can't even speak intelligibly of volume.

>> No.997337

>>997092
This is not science. Mathematicians create systems based on very very precise sets of rules, then determine what must be true about the system. The paradox discussed here is a demonstration of what can happen if you allow the Axiom of Choice into your math.

Still most mathematicians will allow the axiom of choice because it is so useful. Nobody is going to claim that you can turn one basketball into two. Banach Tarski requires you to split the "basketball" into an infinite number of pieces then reassemble them. Clearly this will not happen in any real life problem.

>> No.997344

>>997308
Nah, there are just some things you'll never be allowed to 'choose', it's just not allowed by the universe that we exist in.

>> No.997345

>>997199
Fuck taxpayer dollars. Progress is more important than letting the consumer blowing the money on donuts and guns. When the government funds science, it helps save the lives of those fat idiots when they have a heart attack while watching American Idol.

>> No.997347

since math is free, and science costs billions every year, I question the reasonableness of the latter investment more critically

>> No.997351

>>997345
So wait, they're stealing my money to help fatassery subsist longer? WHAT HAS SCIENCE DONE

>> No.997356

>>997347
Knowledge is power. No price is too great for it and no money has ever been wasted trying to gain it.

>> No.997359

>>997327
Obviously. I was merely stating why it's called a paradox; when it's also true.

>> No.997363

>>997356

if you believe that science is ever really "knowledge" or "fact", and not just ever-increasingly detailed accounts of how we see the universe around us.

>> No.997361

Dean, to the Physics Department. "Why do I always have to give you guys so much money, for laboratories and expensive equipment and stuff. Why couldn't you be like the Maths Department - all they need is money for pencils, paper and waste-paper baskets. Or even better, like the Philosophy Department. All they need are pencils and paper."

>> No.997360

>>997351
>stealing my money
If you don't want to contribute to society then go live in the government-free Utopia that is Somalia.

>> No.997367

>>997347
I don't know about US, but where I live, investments to science are much smaller than investments to, I donno, social system or healthcare

(downside of public healthcare, I guess)

>> No.997371

>>997363
What's the difference?

>> No.997372

>>997277
Implied by Axiom of choice. The sphere is splited in parts that are nonmeasurable (theres no way to define a volume to that parts). There are other models of ZF that imply that you cant split the sphere like in the banach tarski theorem. Axiom of determinacy implies that all subsets of reals are measurable, so you can't split the sphere in that way.

>> No.997373

>>997371

the difference between objective conclusions, and subjective observations

>> No.997376

>>997337
The axiom of choice is not required if you use hyperbolic representation.

>> No.997381

>>997373
That's philosophy.

>> No.997389

>>997361
why dont they just empty the trashcans out and use them again istead of buying new ones

>> No.997391

>>997376
I call shenanigans

>> No.997392

This is why I hate set theory

>> No.997409

>>997159

the funny thing is, the computer you are using works because some people came up with fake nonsense that can accurately describe its behaviour

>> No.997427

>>997409
Stop trying to make sense!

>> No.997442

>>997409
Computers are works of devil anyway.

>> No.997446

>>997159
Mathematicians think a curved line is still one-dimensional.

>> No.997464

>>997446
only if it curves counterclockwise (has positive curl) but not the other way

>> No.997468

whats everyones favorite anagrams? mines:
Banach-Tarski
=
Banach-Tarski Banach-Tarski

>> No.997470

>>997446
according to that logic there is no straight line.

>> No.997473

>>997468
Why did I laugh at this.

I hate you

>> No.997477

>>997470
Only because you're bent

>> No.997480

>>997159
Pretty much this. half the shit they do no one cars about, yet they just fucking keep taking funding for their experiments, and the funders are pressured into paying by the rest of the mathematicians taking the backs of their colleges. So fucking stupid, not like anyone cares about math IRL, it's all abstract shit.

>> No.997483

>>997468

that's an analogy, not an anagram

>> No.997484

>>997480
*cares
And I mean except basic algebra and trig.

>> No.997488

>>997480
>>997159
samefag

>> No.997492

>>997446

it takes exactly two pieces of information to identify location on a line...it has two dimensions

>> No.997497

>>997492

haha bad typo...I meant ONE instead of TWO

>> No.997503 [DELETED] 

>>997492
except you can parametrise it as
<span class="math"> \gamma(t) = (\gamma_1(t), \gamma_2(t),\ldots, \gamma_n(t)) [/spoiler] so it's actually one dimensional

>> No.997509 [DELETED] 

>>997503

i know know i know, i fixed my mistake immediately

>> No.997524
File: 107 KB, 815x275, irreg2339.jpg [View same] [iqdb] [saucenao] [google]
997524

It makes perfect sense, just read this:
http://www.irregularwebcomic.net/2339.html#annotation

>> No.997615

>>997524
thanks for that link, good sir.