/sci/ - Science & Math

>>6690359
It works in math-land, due to the axiom of choice. In physical reality, it does not.

it's just dat zorn's lemma son

not possible unless the pieces are infinitly small, which is impossible

>>6690372
it's more that the pieces are made up of nonmeasurable sense so they don't really make sense at all. you can do it with just 5 total pieces

>>6690374
*nonmeasurable sets

>>6690359
>Have an infinitely full hotel
>Everyone moves to room #*2
>Have a half empty hotel
>Same tenants

being in bijection to a proper subset of yourself is the definition of infinity.

I'm sorry but that's not what mathematicians believe. You said you start with a sphere, a 3 dimensional shape and cut it in half. You then use rotation to turn each half. You still only have two half spheres any mathematician would tell you.

By choosing a 3D object you have to work in a 3D plane. Even to use rotation to get your "two new spheres" would require a 3D plane.

What you're saying is just stupid. Mathematicians don't believe that. Cut a tennis ball in half and look at both halves face on.

Stop being dumb.

>>6690380
true but not really related. banach tarski doesn't just work because there are infinitely many points, it's a lot more complicated than that. the paradoxical part is that the two balls have the same measure (volume) as the original ball

>>6690391
i like how you decided to comment even though you have no idea what we're talking about

>>6690359
They fucked up the axioms of their geometry so operations that are supposed to preserve volume don't.

This is why rigor is overrated. You can be as rigorous as you like with your derivations, and find out centuries down the road that your axioms were wrong from the outset.

>>6690380
I don't get why it has to be times two. What prevents everyone from simply shifting up one?

>>6690405

Then you only get one extra room, but if everybody moves to room #*2, then all of the evenly numbered rooms will be occupied and all of the odd numbered rooms will be empty as even numbers are all integer multiples of 2.

>>6690402
>rigor is overrated
not sure how you came to that conclusion. modern mathematics is more rigorous thanks to examples like this; we use lebesgue integration instead of riemann and deal with measurable sets and functions precisely to avoid this type of pathological example. "operations that preserve volume" are still perfectly valid, but we realized that they only make sense on certain types of sets that are "well behaved." same reason we don't use terminology like the "set of all sets" thanks to russell's paradox.

>>6690401
To be fair to the guy, so did the OP.

>>6690429
but at least OP was asking a question not attempting to answer one

>>6690410
>>rigor is overrated
>modern mathematics is more rigorous thanks to examples like this
But since rigor is overrated, this doesn't really matter.

>>6690372
that is pretty much like saying 1*0=0=2*0 therefore 1=2

>>6690440
You fail to grasp that just stating that rigour is overrated does not actually make it overrated.

I think it's underrated. First defence from cranks on /sci/.

>>6690457
But if xy = xz, then it does imply that y = z.

>>6690457
Infinity is not a number. You cannot use it as such.

Doing so leads to retarded proofs, like that one numberphile video where he says that the summation of all positive integers is -1/12.

>>6690402
I don't agree with the sentiment of wrong or right of axioms to begin with.
If it werks, search for an application for engineers. If it doesn't werk, write a grant proposal for a better idea.
A nice related read is

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

That's worth it for the non-infinite quiz alone. The infinity-case bonus is troublesome, but also only if you take the scenario too seriously. IRL you can't realize it, just like taking a ball apart via it's set theoretic model. Nobody new to that stuff should confuse balls in real life with
<span class="math">\{(x_1,x_2,x_3)|x_1^2+x_2^2+x_3^3\le R\}[/spoiler].
PS: I wanted to discuss intuitionistic models for physics here >>6689795

>>6690402
I don't agree with the sentiment of wrong or right of axioms to begin with.
If it werks, search for an application for engineers. If it doesn't werk, write a grant proposal for a better idea.
A nice related read is

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

That's worth it for the non-infinite quiz alone. The infinity-case bonus is troublesome, but also only if you take the scenario too seriously. IRL you can't realize it, just like taking a ball apart via it's set theoretic model. Nobody new to that stuff should confuse balls in real life with
<span class="math">\{(x_1,x_2,x_3)|x_1^2+x_2^2+x_3^3<= R\}[/spoiler].
PS: I wanted to discuss intuitionistic models for physics here >>6689795

>>6690458
Lighten up. You responded to a statement that rigor was overrated because X, by saying, "No it's not, because of X we were able to make things more rigorous." That deserves at least a little mockery.

>>6690461
Unless X = 0.

So is this sphere business just another 0.9999…. = 1 type of a paradox?

>>6690369
>Zorn's Lemma
Get a look at this pleb

>>6690391
>being this dumb and not knowing you're dumb

topkek

>>6690368
There are no spheres in physical reality.

As far as I know matter is discrete, so this wouldn't work in real life.

>>6690359
>This is what mathematicians actually believe

I'm using this.

Is this like the thing where you can turn a 3d sphere inside out without creasing its edges?

>>6690673
No, not at all.

>>6690678
ohh ok I get it. Wasted my time, not even cool math trick. 2/10 for effort

>>6690627
friend, is u srs

>>6690372
So, they use subsets of the sphere to make a second one?

>>6690692
Doesn't that involve stretching?

>>6690359
Is this paradox finally the proof that abstract maths is basically bullshit since it's based on a bunch of axioms that don't make any sense and create crazy results such as this one?

>>6690673
>Is this like the thing where you can turn a 3d sphere inside out without creasing its edges?

That sphere turning works if you allow surfaces to pass through each other. Banach-Tarski is a whole other level of insanity.

>>6690772
yep, you don't need to worry about math anymore!

>>6690374
But not with only 4 if I remember correctly. Surprising, isn't it?

>>6690802
Yeah I think that's true. And you can't do it with a disk in 2D, anyone know about higher dimensions? Is there minimum number of pieces as a function of dimension?

>>6690359
There's no analog to 'conversation of mass' is this mathematical space. There's an imfinite number of points in the continuum of math. Not so in the real world.

>>6690809
I just checked the wiki and it says that it works for all spheres of Rn. But I didn't see anything about a minimal number of part... Thats an interesting question.

Hey OP! What's an anagram for "Banach-Tarski"?

>>6690359
rigor and dogmatism lead to theory being accepted as fact and allow misunderstandings to blossom.
>inb4 theory is fact
Thats why they get superseded right?
Einstein was correct! Quantum physics is a silly concept!
Fuck your modern gravity, we impetus!
Muh -1/12..the proof is here against all intuition.

>>6691003
It has the word trashcan in it.

>>6691047
Banach-Tarski Banach-Tarski!

>>6691003
>>6691059
Back In a Trash

>>6691061

>>6691059
A Cranks Habit

>>6690359

What? Mathematicians never said any such thing at all.
Stop spewing bullshit.

>>6691072
So you've never heard of Banach-Tarski? How's high school?

>>6691073
>using paradoxical decomposition
>implying mathematicians state this is possible

Sounds like you're the one who needs to go back to high school. You should go finish your calculus before pretending to talk like you're one of the big boys.

>>6690489
that one you just replied to was my only post, you're talking to multiple people.

>>6690395
>the two balls have the same measure (volume)
No, volume is not a "measure" preserved by these operations. Lrn2topology

>>6690359
Best explanation I've seen:
http://www.irregularwebcomic.net/2339.html

>>6691145
very nice

>>6691003
hat rack basin
arab ski chant
bark at chains
chairs at bank
anti hack bras
bitch's ankara

>>6691125
volume is the lebesgue measure and is preserved by isometry. The paradoxical decomposition is possible because the pieces are nonmeasurable. The axiom of choice is fundamental to prove their existence, but without it, it's not possible to prove that all sets are measurable.

>>6691145
Could it be similar to the pi=4 circle and square meme?
The way I understand that is, length =/= shape.
Even when the square's perimeter remains at 4, can be chopped up so it finally is circle-shaped.

The B-T would be a more complicated 3D variation, but in both cases weirdness would be based on the fact that real numbers can't fit in a hilbert hotel.

>>6691070
9/10
That post is /s4s/ in a nutshell.

"counter-intuitive" is a euphemism, in this case, for: you can prove stupid things if you are clever enough because math is broken.

after -1/12 fiasco, I have my doubts about a lot of what mathematicians say.

also, did you guys know that most math problems do not have a unique right answer?

YES, let that sink in for a second.

one of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.

>>6691571
>one of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.
<span class="math">5+9=sqrt{196}[/spoiler]

>>6691571
I hate you so much.
-1/12 is completely viable, understood and correct. The answer is not within the real numbers only.
/sci/ is so fucking dumb.

>>6691571
>-1/12 fiasco
>what is Riemann and Dini's theorem
>>6691079
>paradoxical composition
There's nothing paradoxical about it, it's just counterintuitive

>>6691593
Quoted the wrong comment; also
>*decomposition

>>6691593
>*decomposition

>>6691571
>-1/12 fiasco
kek

>>6691597
>>6691593
>>6691584

Do you three really believe that if you sum up ALL POSITIVE NUMBERS that you end up with a negative rational?!?!?!??!?!

>>6690359
yeah its possible you just start with a thicker sphere then make two thinner ones retard, try it with chocolate

>>6691626
they're solids.

>>6691632
solids have thickness, or they would crumble and wouldn't be solid retard

>>6691619
>implying it's the result of a standard summation
>implying you know what you're talking about
>>6691626
They are not spheres, they are balls. Balls don't have a thickness since they are "full".

>>6691619
Confirmed for not knowing what Riemann's theorem is

Cutting a sphere up and reassembling the pieces into two spheres of equal volume to the original isn't normal.

But on math it is.

Math: not even once.

>>6691635
>solids have thickness, or they would crumble and wouldn't be solid retard

I can't believe I'm replying to someone this retarded. Solids are different from shells. Go back to that short bus, kid.

>>6690395
It is a different situation, but the people who are amazed by banach tarski probably don't know about Hilbert's hotel. Like if you look at OP's greentext, his amazement would be applied just as well to "wow you can split infinity into two infinities"

>>6691655
>Confirmed for not knowing what Riemann's theorem is

Uh oh, who cares what it's called when it makes NO SENSE.

>>6691652
>implying it's the result of a standard summation

LOL! Yes, it's a standard summation, dummy!

Some of these mathtards say that if you SUM UP ALL OF THE POSITIVE INTEGERS, you end up with a NEGATIVE RATIONAL as the result.

Think about that for a second.

If you count and sum up everything in the universe, you will end up with less than with what you started with.

If you think this makes sense, just because it has some fancy name, you're a brainwashed moron.

Fact is, you summation of DIVERGENT series makes NO SENSE! It's like dividing by zero!

>>6691816
I'd love to make rational elaborations on this topic, but I don't want to end up having responded to such a troll.

>>6690359
okay OP, i hope you're still reading this thread because i don't want to explain banach-tarski's paradox for a bunch of troll...

i won't go in a lot of details, in fact i'll only explain the idea of how it works.

but first some explanations on why it's an interesting result. Take the segment [0;1], take each point x of this segment and put it at the position 2x, you now have a segment [0;2] which is twice as long as the previous one, only by shifting some parts of the first segment. the tricks here is that there is infinitely many (it's even uncountable) parts. One other example would be to take some sub set of the plane, to stretch it and tada! you end up with a bigger piece (now that's just obvious).

The interesting part is that here you take a finite number of pieces and the transformations of the pieces are isometric (only rotations and translations). The trick here is that the pieces are so horrible, so complicated that you can't even define a volume/area for them. If you can't define the areas of the pieces there is no reason the area of the result is the same that the area of the begenning.

cont.

>>6691816
>I don't understand it, so it makes NO SENSE!

>>6691816
>Some of these mathtards say that if you SUM UP ALL OF THE POSITIVE INTEGERS, you end up with a NEGATIVE RATIONAL as the result.
Yes, using certain summation methods that works. But it is NOT a "normal" sum. It's not like you just add one number after the other and you get that result. A way to get that result is using a Ramanujan summation, which assigns a "sum" to divergent series too, even if it doesn't follow the classical definition of sum.

>>6691860
Adding to this answer, this blog post by Terry Tao should be enlightening: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

>>6690359
>>6691856


and now let's get to banach-tarski.

Let's take a sphere and two axes, the north/south axe and one other, perpendicular to the first. Now let's chose an angle Y (in radian) such that Y/2pi isn't rational. Now we call A the rotation of angle Y around the first axe and B the rotation of angle Y around the second axe. Accordingly the rotation of angle -Y around the first axe will be called A^-1 and the rotation of angle -Y around the second axe will be called B^-1.

We can do rotations one after the other, for example first +Y around N-S and then -Y around O-E, this rotation would be called A*B^-1. Now you have to see that this rotations are NOT commutative, it means that AB is not equal to BA (you can try that yourself with a book and a rotation of pi/2). If we take one point x of the sphere and we rotate the sphere with the rotation A the point x we'll also be rotated and be called Ax. In that way A and B are doing exaclty like matrix and A*A^-1=A^-1*A is like no rotations at all and therefore will be called Id. And of course A*A will be called A^2 A*A^2=A^2*A=A^3 and so on... You can just considere that A and B are matrix, in fact you can write rotations in terms of matrix.

An other thing to notice is that, since Y/2pi is irrational, if you take any point x on the sphere but the two poles then (A^n)x=(A^m)x if and only iff n=m (for two positive or negative integer n and m).

Now let's take a point x of the sphere which is neither on the NS-axe nor on the OE-axe. Then take a string of caracter only constitued with A, A^-1, B and B^-1 (this will be a rotation of the sphere). If in this string contains something like A^-1*A or B*B^-1 we'll erase it because A^-1*A=Id. So the string ABA*A^-1B^-1 would become ABB^-1 and then A.

cont.

>>6691584
>>6691593
>>6691597
You are right about the -1/12 thing. It is possible to receive that value from that summation.

However, saying that "1+2+3+4+5+6..." is equivalent to "-1/12" in the same way that "1+1" is equivalent to "2" is absolutely incorrect. You need to use zeta function regularization or Ramanujan summation to get this answer.

If solved as a standard infinite sum, the series is divergent.

>>6691879
Oh, and the way Numberphile did it is totally bullshit and can be used to show that 1=0, which in turn means that all numbers are equivalent to all other numbers, which in turn breaks math. There's a reason series don't work that way.

>>6691843
>being this dumb

>>6691885
>Oh, and the way Numberphile did it is totally bullshit and can be used to show that 1=0, which in turn means that all numbers are equivalent to all other numbers, which in turn breaks math. There's a reason series don't work that way.

Numberphile is full of idiots and that Brady guy's a jerk off.

>>6691874
Awesome! Please continue!

>>6690359
>>6691874


So, because Y/2pi is irrational and the two axes are perpendicular if you take two different string U and V only made of A,A^-1, B and B^-1, then Vx is not equal to Ux.

Now we call S the set of all the finite string only made of A, A^-1,B and B^-1. So Sx is a set of points of the sphere, precisely the set of all the points you reach with x by rotating the sphere with A, A^-1, B and B^-1. Let's split S in 4 parts. The first part is Sa+, it contain all the strings of S who begin with the rotation A. The second is Sa-, it contain all the strings of S who begin with the rotation A^-1. The same goes for Sb+ with the rotation B and Sb- with the rotation B^-1.

now if we take a string from Sa+, since the first rotation is A and we said that A*A^-1 was automatically erased the second rotation must be B or B^-1. So if we take a string U of Sa+ and we multiplicate it by A^-1 by the left (so we look at A^-1*U) we get a string of either Sb+ or Sb-. Therefore A^-1*Sa+ is a sub set of Sb+ union Sb-. In fact you can easyli see that A^-1*Sa+=Sb+ union Sb-.

In order to see that take a string U of Sb+ or Sb-, then A*U is in Sa+ and so the string U will effectively be in the set A^-1*Sa+ because A^-1*A*U=U. Because the choice of U is abitrary this goes for every string of Sb+ and Sb- and we have the equality of A^-1*Sa+.

So (A^-1*Sa+) x=(Sb+)x union (Sb-)x. With one part of Sx and by rotating it we got two part of Sx. The same thing can be applied to the other parts :

(A*Sa-) x=(Sb+)x union (Sb-)x
(B^-1*Sb+) x=(Sa+)x union (Sa-)x
(B*Sb-) x=(Sa+)x union (Sa-)x

And so by rotating the four parts of Sx we get 2 exact copies of Sx.

That's the trick behind banach tarski theorem. (or at least this and the axom of choice...)

cont.

>>6691903
great. plz continue.

>>6691918

This is all on wikipedia, ya know.

>>6690359
>>6691903

last part about banach tarski.

There is a lot of problem left to take care of now. For example the set Sx isn't equal to the sphere, indeed Sx is countable while the sphere isn't. But it's no big deal, Just take an other point x_1 which isn't in Sx and do the same with S(x union x_1). But S(x union x_1) isn't equal to the sphere and you have to take another point and another and another. The problem is that you'll need infinitely many other points (uncountable infinity) so you need the axiom of choice to assure you that such a collection of points exists.

One other problem is that we only took x on the sphere but not on the axes, so there are 4 points that are problematic. That's why the real version of the banach tarski paradox is done with 5 pieces instead of four like we did. A fifth piece is needed to take care of thoses points. although it's not really a big problem, take a sphere withouth 4 points, chop it in 4 pieces, rotate/translate thoses 4 pieces : you get 2 copies of the exact same sphere without the 4 points, it's still looks paradoxal.


Of course it's not possible to chope those pieces in reality, because the components of the sphere is discrete (atoms and such), because the axiom of choice doesn't tell you what algorithm to follow to construct the pieces etc etc... But those are (in my opinion) stupid concern, there is no perfect sphere in the real world, you'll never walk into the number one in the street, (have one apple means that your set of apples can be put in bijections with the set 1={{}}, where {} is the empty set, so the number one is just a set, you really never see a one in real life) or the empty set. But this has much more to do with semantics and phylosophie than with math and real world applications.

What i believe that banach tarski paradox teach us is that there is indeed some really weird fucked up subset of R^3.

end.

>>6691921
>This is all on wikipedia, ya know.

Sure, but Wikipedia is absolutely _the worst_ for anything math related. Their explanations are complete shit. Totally unreadable.

>>6691923
worst I've seen on a popular subject:
https://en.wikipedia.org/wiki/Function_(mathematics)#Notation
links to
https://en.wikipedia.org/wiki/Functional_notation

>>6691928
How is that function notation one bad?

>>6691928

How is that unreadable? I mean to say, sure the link to more on function notation is not well coordinated and pointless, but the subsection isn't unreadable

>>6690641
That statement is unfalsifiable at first glance, isn't at second glance, and at third glance it is indubitably so.

>>6691937

Vast majority of math topic on Wikipedia are unreadable unless you have a PhD in math. Sure, everything in there might be 100% correct but the value of it as a tutorial on anything is extremely low. It's worse than the worst textbook that you can find on some topic.

>>6691940
That function section is an example of needing a PhD in math?

>>6691856
>>6691874
>>6691903
>>6691922
Great explanation. Capped for future generations (and for myself since I will undoubtedly forget it before tomorrow).

>>6691944

I didn't post that function link. But yes, pick anything that's not HS math on WIkipedia and you'll have really rough time trying to learn the subject.

Whenever someone complains about Wikipedia, my response is that a) it's free and b) it's your job to make it better. It's an open encyclopedia, after all.
Also, I can guarantee you that the majority of math articles isn't even written by PhD holders, you don't need more than 1 year of studying any subject to be able to read a Wikipedia article on it.

>>6691954

But functions are HS math

>>6691940
>Wiki as a tutorial
No, it's a reference.

>>6691976
Who said they were not?

>>6691989
Sure, which means that it's fucking useless unless you already know the subject.

>>6692208
>it's fucking useless unless you already know something
FTFY

>>6691003
I hack ant bras.

>>6691940
That's because WP isn't meant to serve as a tutorial, only a repository of knowledge
>>6691951
Fukken saved

>>6691874

>That feel when I can't understand shit.

;_;

>>6691954
I can't read through an undergrad math or physics textbook but I jam through wiki articles.

Obviously supplemented by 102930289 papers and the most advanced textbook I can find.

>>6690402
>volume
What is volume?

>>6690461
...if <span class="math">x[/spoiler] is a left-invertible element with respect to the operation denoted by juxtaposition.

>>6691070
Why is there a picture of Voltaire when it says "Rene Descartes"?

Fuck your fake quotes.

>>6692761
>What is volume?
In those critical years I learned how to be alone. But even this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation (1945-1948),when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

>>6692794
Based Grothendieck.

>>6691619
>implying I'm not a mathematician

>>6692794
>>6692795
that was well said

>>6692795
this is great

>>6691879
people forget that -every- equality is only meaningful in a certain context. wtf does "1+1=2" mean? If you believe Russell, making this rigorous is not something to take lightly.

>>6693035
>people forget that -every- equality is only meaningful in a certain context. wtf does "1+1=2" mean? If you believe Russell, making this rigorous is not something to take lightly.

take one grain of wheat, add it to another grain of wheat and how many grains do you have? 2.

take one grain of wheat, add 2 grains to that, add 3 grains etc… and you end up with -1/12 grains of wheat.

sorry, but if that makes sense to you, you're an idiot.

FACT is that once you start performing operations such as the ones in that idiotic Numberphaggot video, you can just as easily prove that 1 = 0 because you're dealing with divergent series.

http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/

>>6693052
Just ignore the part about Russell. Pretend you really know how to rigorously define equality in a formal system. Go on. Think you know better.

>>6693052
Exactly. 1+2+3+4+5+6+... can, in some contexts, be represented by the value -1/12, but that's only when dealing with infinite series.

>>6693052
>take one grain of wheat, add 2 grains to that, add 3 grains etc… and you end up with -1/12 grains of wheat.
>sorry, but if that makes sense to you, you're an idiot.
Have you ever tried that? Since you clearly don't understand how math works and this is the science&math board, use the scientific method to disprove it. Perform an experiment. Show us that when adding one grain of wheat to two grains of wheat to three grains of wheat and so on you don't get -1/12 grains of wheat. Go on, I'll be waiting. And when you're finished after the end of time I'll be there ready to call you a moron because adding grains of wheat is totally unrelated to what we're talking about.

>>6693232
I think he was making more of a point about what the average person perceives the symbol "=" to mean.

A lot of people learn math by screwing around with marbles in elementary school, and continue to understand math in terms of physical quantities until their death. In THAT sense, no, of course 1+2+3+4+5... isn't -1/12.

However, -1/12 is a useful value to assign to that series.

>>6693052
>take one grain of wheat, add it to another grain of wheat and how many grains do you have? 2.
>take one grain of wheat, add 2 grains to that, add 3 grains etc… and you end up with -1/12 grains of wheat.
>sorry, but if that makes sense to you, you're an idiot.

What you say here isn't an argument against 1+2+3+...=-1/12.
Let me elaborate:
(the first part is gonna be basic)
Yes, most here will agree that 1+1=2. That's “obviously self-evident”. People have know how to add grains of wheat and written down the rules for addition accordingly.
So what is 1+3 (one grain of wheat plus two more grains) or 4+8, or 2+3? In your head, you reduce the expressions to 4, 12 and 5, respectively. How does your head do it? You can imagine to make a list of all results: Store "1+3 is 4" and "4+8 is 12" and so on. You can make that more efficient by setting up the computational rules: Store only specific representations of numbers, e.g. "5 is (((1+1)+1)+1)+1", and then the rule
"a+(b+1) is (a+b)+1"
That's arithmetic as in
http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic
and it let's you -compute- the value of 1+3:
1+3
is 1+((1+1)+1) by the stored information 3 is (1+1)+1
is (1+(1+1))+1 by the computational rule
is ((1+1)+1)+1 by the computational rule again used inside the brackets
is 4 by the stored information 4 is ((1+1)+1)+1

Now this naive addition will not provide you with a suggestion on how to deal with infinite sum.
So apriori, the sums
>1+2+3+4+...
>1+1/2+1/4+1/8+...
>1-1+1-1+1-1+...
are all just compactly describable expressions. No evaluation is stored or prescribed.

Here comes your "fallacy":
You have, probably through school, obtained an intiution for why the second sum should be 2.
That's pic related.
It's not like you can add 1 plus 1/2 plus 1/4 grains of wheat irl, but you have the geometric picture and mathematicians ought to make this computable.
(cont.)

A couple of hundred years ago, they succeed. We are able to speak of metrics, norms etc. and a sum might converge to the limit of its partial sums. That's a technique.

Now you say "The infinite sum 1+1/2+1/4+1/8+... is 2" and the inifite sum "1+2+3+4+..." is not defined, or infinite (which not even part of the set of natural numbers).
In doing so, you act as if the more well "weak" known summation technique is somehow "natural" or "given".
For one, you fail to realize that e.g. Ramanujan summation, a technique which will assign -1/12 to 1+2+3+4+... does not break any of the metric-definition-sum results. The more modern assignments merely extend the old one. AND they are taken to be "morally right", because they are actually used. They application isn't plane geometry, but just some other mathematical description of the physical world (quantum field theory, say)
You argue: "Firstly, I will not accept any value assigned to an expression where the naive of the metric-limit definition doesn't succeed to find a number value. Secondly, naive (Peano) arithmetic will never lead to two positive numbers being a negative number, and neither does Eulers method! Therefore I'm right in rejecting the method!!"

What you want to do is find mathematics which is nice and useful. Of course, if you stumble upon a system which you find to be inconsistent, like if it demonstrates 0=1, then you will reject it. But 1+2+3+4+...=-1/12 isn't an inconsistent one, and people rightfully say
"1+2+3+4+...=-1/12"
because time has shown it's morally right.
If you don't want to confuse people, you can change your language to "informally 1+2+3+4+...=-1/12, where the equality is to be understood as..." but I'd actually refrain from doing so. Because if you do, you artificially put emphasis on the notation that "infinite sum = x" has to be read with the (weak) limit/metric evaluations as default.

>>6693244
plus three more*

>>6693232
>scientific method
Shut up reddit

>>6693232
>because adding grains of wheat is totally unrelated to what we're talking about.
I giggled

>>6693249
Nice explanation. I would like to add that the existence of models of any infinite cardinality for Peano arithmetic and the act that natural numbers as constructed in ZFC are no subset of the reals, with the well-known 'counter-intuitive' notion of real numbers as equivalence classes, shows that the 'obvious self-evident' is already somewhat suspicious.

>>6693656
>the existence of models of any infinite cardinality for Peano arithmetic
You mean models with extra elements? Otherwise, if I don't misunderstand the terminology, one shouldn't be shocked about the countably infinite intended model :P

pic: unrelated funfact I just learned.

>>6693731
Cool pic, sauce please?

>>6693731
Yes but it are the uncountable infinite models that are weird. And indeed, you then have 'extra elements', but the extra part can't be seen from within the model.

>>6691073
>So you've never heard of Banach-Tarski? How's high school?

My engineering consultant job is going just fine. Let me tell you how many fucks anybody in the real world give about BT: None.

>>6691989
>>Wiki as a tutorial
>No, it's a reference.

Only in math.

On any other subject, I can look it up on Wikipedia and learn something new.

When I was doing my EE degree, I often looked up things on Wikipedia and learned new things about EE.

This never (almost never) happens in math. I can use WP as a reference work, sure, but I never learn anything new about math from WP because it is just straight shit at teaching math. It uses symbols and language unfamiliar to me without giving adequate explanation of the terms - this never happens elsewhere, anything else I look up is either explained on the page or one reference deep on a different page. Math expects me to know so much ahead of time that I could just derive the WP page on paper by myself.

>>6694792

I have learned a decent amount of math from wiki, in the sense that I use it as another source to supplement my reading, much like what you probably did with EE topics. Once you know how to read it, you can pick up the generalities of the topics pretty easily.

>It uses symbols and language unfamiliar to me without giving adequate explanation of the terms

Then fucking learn them. Anything that isn't standard, it usually has a link or details the notation.

>mfw the today's SMBC
Zach Weiner confirmed for browsing /sci/.

>>6694789
No one cares about it in math either.

>>6694673
Saw it in
>Computational Commutative Algebra 1 - Kreuzer, Robbiano

>-1 x -1 = 1
>start with no spheres
>multiply no sphere by no spheres
>you get a sphere

>>6694901
>Then fucking learn them. Anything that isn't standard, it usually has a link or details the notation.

Fucking learn to read, maybe?

When I need to learn new math, I get my ass on a course, I take a class. Shit, if I need it soon (This has never happened) I'll probably look up a tutorial on the net.

What I'm saying is not "Weh Weh math is hard and WP isn't fellating me hard enough," what I'm saying is that WP is singularly bad at specifically math, which stands out because it's good at everything else.

>>6694975
this

>>6694979
I have not taken part in this conversation, but as another opinion: I disagree. I've never felt this way.

>>6694979

Its not bad because you don't know standard semantics and syntax. Its like saying wiki is bad because you are spanish but you are using the english version.

>>6694998

Not only that, but wikipedia math pages generally have a good overview on the topic in english.

>>6695002
Not that I noticed.
Most I read are filled with math jargon.

>>6691923
>he can't read
Get off /sci/ please

>>6695007

Can you link to an example so I can get a feel for what you consider math jargon?

>>6694979
Have you considered simple.wikipedia.net?

>>6695010
>elitist dissing

everyone can see you are trying to compensate for something

>>6695024
I am compensating for someone's lack of will to improve. Everyone should be able to read.

>>6691937
It doesn't talk at all about functional notation.
<span class="math"> f: A \to B [/spoiler]
<span class="math"> x \mapsto f(x) [/spoiler]

>>6695011
https://en.wikipedia.org/wiki/Lattice_(order)

>>6695041

It had all that stuff in the sub-section of the first link. I agree that the page dedicated to functional notation is ridiculous, but that seems more like a coordination issue. The information is on the first page.

>>6695045

What terms specifically are fucking you up or is it just everything?

>>6691571
This. I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

>>6695046
yeah I was complaining about the second link.

>>6695049
In the first part
When I started studying math:
partially ordered set
supremum
infinum
algebraic structures

Now:
partially ordered

There's a lot like them as I noticed since studying math by myself.
https://en.wikipedia.org/wiki/Construction_of_reals
This one was of no help at all:
https://en.wikipedia.org/wiki/Dedekind_cut

etc...

>>6695058
So then look up what a partially ordered set is.
If you get confused about some term while reading that, look that term up.
Repeat until you understand what a partially ordered set is, to the extent you wish to understand it.
Then repeat with supremum, infinum, and algebraic structures.

It's like you want wikipedia to explain EVERYTHING, and explain EVERY part of those explanations, until it's all trivial, and all in the same article. Do you not realize how impossibly dense that article would be?

>>6695058

But the link to posets was not helpful? I mean, thats like me saying I go into an EE page and it uses EE jargon like current, and power, or LSB/MSB (if micro controllers).

Going into topics that require previous knowledge will mean you must have that previous knowledge.

>>6695056
I didn't get this was a pasta until now. I just thought everybody trolled in a similar way.

>>6695045
>>6695058
I just read the introduction of that page.
When I started I didn't know what a lattice was.
While reading I encountered (like you) "partially ordered set"; not knowing what a partial order was, I clicked on the link before continuing reading the lattice page.
End of the story, I now have a basic understanding of both partial orderings and lattices.

I actually think wikipedia's pages about math are (for the most part) well done and I learned a lot by reading them.
The only annoying thing that I noticed is that sometimes wikipedia's explanations and definitions are much harder than those I find in my math-related courses in university (I'm studying engineering). However, in these cases the fault is of my courses, not wikipedia's; in fact, often the math taught in engineering is oversimplified and makes a lot of assumptions. This is of course a problem when trying to study for an exam, but I'm glad I can read wikipedia when I want to learn real math.

>>6690359
This is because mathematicians made a stupid choice: they accepted the axiom of choice over the continuum, when they should only have accepted the axiom of dependent choice and the statement that all subsets of the reals are measurable. This is called Solovay's model because Solovay proved it was consistent. Paul Cohen's "Set Theory and the Continuum Hypothesis" is by far the best book to learn set theory and formal logic from because Cohen was aware of the politics that led to AC over the continuum getting support (this was mostly because Godel supported it vocally for a while).

This is the only big mistake ever made in mathematics.

>>6695064
>>6695064
>>6695076
I linked lattice because I was trying to figure out what already looking up what it what it meant.

I find it hard to juggle around 5 new words while to trying to mash it in with a new concept.

If I go to the electric current or power (physics) I can usually pick up the idea from the first sentence.

It's definitely gotten easier to study mathematics, but rarely because of wikipedia.
Textbooks are so much more informative.

>>6695144
> I linked lattice because I was trying to figure out what already looking up what it what it meant.
I was trying to look up what it meant because it was used in another sentence I was trying to parse

>>6695144

Thats because you usually dont try and learn 5 new concepts all at once. If you want to learn about lattices and you realize there's previous knowledge then you STOP thinking about lattices and you learn what you need first, then you go back.

> 2014
> still doing maths in ZFC

Get with the times, anons. HoTT is the future.

>>6695103
I could swear you're Ron Maimon.
E.g. as seen ranting here

http://physics.stackexchange.com/questions/1894/path-integral-vs-measure-on-infinite-dimensional-space/13315#13315


>>6695045
I'm >>6694985 and I have to say I find lattices hard to grasp too.
(And I also find Zorns lemma much more lifeless than the axioms of choice.)
The Wikipedia page really tries it's best, though. Maybe the examples should come earlier for you.
Generally, I think it's no shame to store concepts in your mind, just be thinking of special cases and reading of their properties.
David Hilbert said:
>"The art of mathematics is to find the special cases that contain the germ of generality"

Oh btw., at this point I have to point out that there is a radio recording of Hilbert, and he has the cutest voice, you youdn't expect ^^
http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3

I thought about that two-post argument for 1+2+3+...=-1/12, namely >>6693244 and >>6693249 and in response to
>Do you three really believe that if you sum up ALL POSITIVE NUMBERS that you end up with a negative rational?!?!?!??!?!
I should have pointed to "1+1/4+1/9+1/16+...=pi^2/6" where the "type of number" changes too, maybe in a less radical fashion.

>>6695056

ignore the pasta

bump!

>>6695066
that's todays SMBC!!!

final bump for balls

Because math is full of bullshit. Look, how to make a formal science.

>smoke weed
>create some axioms based on nothing
>make implications about axioms with no meaning and arbitrary existence
>do this for like 500 years
>????
>profit

>>6697978
>do this for like 500 years
>????
>profit

If you could provide us with examples here, please, which go back to before 1800, decades before Hilbert. Thx.

>>6691571
>after -1/12 fiasco
there are more definitons of infinite sums

let that sink in for a moment

>also, did you guys know that most math problems do not have a unique right answer?

theorems have equivalent but different poroofs
your point?

>>6697982
Are ye mad because your numbers suck

>>6690359
A point is infinitely small.
When you remove a point, the hole is size zero.
Remove every other point.
Make new sphere out of the other points.
Because the holes are all size zero, you have a new sphere instead of swiss cheese.

>>6695030
coming from someone who advocates wikipedia

>>6698400
that's why we have the restriction of sigma-additivity for measures
protip: that's not the reason why banach tarski works

>>6698400
>remove every other point
7/10 made me lol

>>6695213
I'm not Ron, but I'm a fan of his work. He's not just ranting, he's completely right about this.

>>6698544
k. He's a little nutty with some of his opinions (I don't mean physics/math here), but he's pretty knowledgeable and in fact I have email contact with him.

You should be able to solve this.

>>6690359
Instead of asking a load of people who have no clue about the details of the proof why don't you learn some measure theory and group theory and learn the relatively simple proof yourself.

>>6695103
>Set Theory and the Continuum Hypothesis
Thanks for the tip anon.

>>6698552
>in fact I have email contact with him.

Why?

0 = 1 - 1

Start with nothing and then you get a positive one and a negative one.

>>6699159
He's covering his tracks m8. That was ron. Mathematicians are pretty smart but they suck at lying.

>>6699159
I'm developing some ideas regarding efficient mathematical notation* and found Ron had some tangent ideas in the past. He wanted to develop what I'd call some kind of "Lispy Mathematica". Quote
>I was sick of Wolfram holding chunks of the physics community hostage using proprietary software.
But all of this got him side-tracked into biology

http://arxiv.org/pdf/q-bio/0503028.pdf

*Here I'm thankful for any help and discussion - there are many threads: My project involved some vague idea about tweaking mathematical representation so that it's easier to find models (in the formal sense) for theories with lots of proven theorems in other far removed theories. Then one and a half years ago my mind was blown when I learned of a kind of example for this has recently been discovered - homotopy theory in intentional type theory. I saw people are developing gauge field theory from that point of view and wanted to understand it. Shortly before I had been reading into Haskells type system when I wanted to learn using the 'parsec' package for my notation stuff, and so naturally I got side-tracked too.

>you start with a single sphere
>you chop it up into pieces
So from the thread I understand the chopping is just a choosing.
The rotation leads to a duplication, but it's not like in OPs picture where things are moved through space, right?

>>6698544
following up a bit of his online activity, he sounds like a raging nut with a phd.

>>6701498
Yeah, me too.

Things getting complicated because of it isn't really a good reason to disregard a mathematical universe with non-measurable sets. He also acts a bit paranoid when talking about the axiom of choice, and I think treating mathematical axioms like political ideologies is not a healthy thing to do.

>>6701466
actually YES. The pieces are rigidly rotated and translated. Isometries. That's the freaky part.

>>6701498
>>6701508
What makes you think he got a PhD? (He doesn't.)
He knows most than many PhD holders anyway.

>Working on ZFC, not ZF
ISHYGDDT

>>6698552
I'm the guy you're responding to, and I'm also in contact with Ron.

>>6701508
Look, it's ultimately an arbitrary choice. You can do it the regular way or in Solovay's model. Both work, but Solovay's model agrees with our physical intuition and makes it super easy to formalize things like "random choice" that should intuitively be easy to use in math but aren't because we decided to accept AC over the continuum.

You can keep doing it that way, but it's stupid to do so when there's a better and equally logical alternative.

That's all Ron means. As for sounding paranoid, he likes to say that he tries to get as many points on people's crank scales as possible. He deliberately compares himself to Isaac Newton just so that people will have to see the merit in his work instead of just blindly accepting what he says.

>>6701735
>set theory
>not type theory

>>6701811
There is a variant of ZF as type theory in Paul Taylors book "Practical Foundations for mathematics". A version is online, but not in TeX
http://www.cs.man.ac.uk/~pt/Practical_Foundations/html/s22.html

>>6701791
so what's your name on SE?

>>6690359
https://www.youtube.com/watch?v=rkgLzo6qhQg

>>6702068
hmmmmmm

>>6701791
AC also agrees with my intuition and I think it's very reasonable to assume that there are sets so complicated that they don't have a measure. You could say the rational numbers being a null set is counter-intuitive, and if you talk about cardinality that's also kinda weird.

>but it's stupid to do so when there's a better and equally logical alternative.
Why is it stupid? I think it's interesting to work with different models, but like constructionists, you people get crazy and offensive and appear to disregard a whole bunch of great mathematicians because of personal opinions. I think crazy shit like Banach-Tarski paradox are cool and they're one of the reasons I'm majoring in mathematics, I don't like people that want to take that from us.

And even within the Soboley model weird things apparently can happen:
http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice/22935#22935
Mathematics is filled with this kinda bullshit, there's no way around that. And even if you disagree that that's counter-intuitive that just shows how personal our intuition is.

>He deliberately compares himself to Isaac Newton just so that people will have to see the merit in his work instead of just blindly accepting what he says.
He could try acting more like a civilized and sane person, people would probably take him more seriously.

>>6702335
(not the one you're responding to)
I have not enough intuition for the uncountable and think "too complicated to measurable - why not" is not the best arguments, and why deal with the ugly non-measurable on purpose? But I also have no super strong feeling about it.
The guy you argue here comes from a "practical position" though, that's why he rejects that which has been shown not to work.

I'm curious about the 2^\omega partition claim. Principally, why would the construction work if you have one axiom less, and not work once you add it. AC does provide something, it's not a restriction in any way I can read it.

>>6703070
>"too complicated to measurable - why not" is not the best arguments
But that's not an argument to not consider the other possibility, I'm just saying that this decision is kinda personal.

Why deal with them? I don't know, I'm just an undergrad. But I would guess it's the same reason why we deal with discontinuous functions and non-differentiable functions, because apparently there's a sort of smooth analysis based on intuitionistic logic where every function is continuous. It's a different deal though, but still...

Also, the partition seems to work because every subset of the reals is measurable, that's an axiom in Solovay's model, you can't prove it with ZF.

>>6691816

>Fact is, you summation of DIVERGENT series makes NO SENSE! It's like dividing by zero!

except you cannot divide by zero. through Riemann's theorem and Ramanujan's method, you can, however, sum divergent series.