I've been introduced to the Banach-Tarski paradox and the fact that an infinite sums apparently adds up to -1/12.
How can this be provable in strict mathematical logic but not hold up in the real world? What is the difference between the laws that govern our existence and the isomorphism's constructed with mathematics? Is the math wrong or is math not always applicable to the real world?
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>and the fact that an infinite sums apparently adds up to -1/12
Stop
>infinite sums apparently adds up to -1/12
mathfags are so bored with their worthless useless non-practical degrees that they make up silly semantics tier bullshit and non-mathfags take them seriously.
Mathfags really needto find something to do with their lives.
Is that not the case? And if it isn't, how can a logically consistent proof be wrong?
Or is the general thought that we are assuming things that might be wrong?
Lol, chill dude. I was asking simple questions. Math has lots of applications, maybe not pure maths.
This guy is actually right. Because of stupid popsci youtubers, people get wrong idea of math. First of, many people how are not in math do not understand the concept of limes (trust me). Secondly, did you watch video about infinite sum equals -1/2? Did you see the proof? It was based on "assumption" or better to say "deal" that sum (-1)^2 = 1/2.
Please, understand me, or that guy above, as mathematician, I get so fucking annoyed when average guy comes and asks something like "hey dude, math is so wrong, look at this thing woah".
At some point I even stop responding. I just smile uncomfortably and change subject. Damn
Limes are gross.
Watch this youtu.be
>Inb4 it's over 30 minutes!
You asked, here's a real answer. Watch it fuck boi
Nice bait, I'm taking it. Ramanujan was a bored brown guy back in the day that dreamed this up and showed it true under certain circumstances of algebra. Faggot.
>but not hold up in the real world?
how can you know that, op? have you ever added something infinitely, or cut something up infinitely?
>Definition: An apple is a large mammal with a trunk
>Theorem: Apples have hooves
>Why doesn't this hold up in the real world?
>concept of limes
Limits?
>this meme again
yeah limits. sorry, i agree that they are somehow "unnatural" but, dude, it's like that 0.9999... = 1. Get over it :)
"the real world" is a nonsense idea, since everything we know about it is filtered through our senses.
As far as the Banach-Tarski paradox, there are plenty of ""paradoxes"" which aren't, for instance Gabriel's Horn could be seen as a "paradox" to somebody only making reference to physical objects, but nonetheless calculus still proves useful.
Remember that the ultimate philosophical stance is that human meaning is the realm which matters, and causality is inert except when it has meaning for us. It doesn't matter why or how math works, but the fact it does, is apparent and digging won't yield absolute reasons why.
this why I always thought information theory is more fundamental than physics
To answer your question:
Consider 2 bags of statements, TRUE and UNDECIDED:
Proving a statement in UNDECIDED puts it in the TRUE bag.
Mathematical logic lets you deduce statements from, given rules of derivation.
For example, the rule modus ponens
en.wikipedia.org
can be understood as saying that
A : TRUE
A=>B : TRUE
then conclude
B : TRUE
I.e if the statement "A" is in the TRUE bag, and if also the statement "A=>B" is in the TRUE bag. put B from the UNDECIDED into the TRUE bag as well.
A formal logic provides rules of derivation - they are choose by us humans, typically but not at all necessarily chosen to reflect some common sense principles.
A mathematical theory (written down in the context of some formal logic) is characterized by axioms: Sentences which you put in the TRUE bag from the start, without justification.
Consider the theory of Peano-arithmetic. It contains axioms that let us prove that
"for all numbers n it holds that n+1 > n"
Other rules let us prove that
"for all numbers n it holds that n+5 > n+2"
Those in turn, once in the true bag, can be used to prove e.g. "3+1>3". You can put more and more statements into the true bag.
Now let's say you create a new theory, called user-arithmetic: Let all its axioms be the ones of Peano-arithmetic, except you also add the statement "2=0" to the true bag.
Now you can prove many things you couldn't prove before. E.g. from "5=3+2+0", which is trivial to prove, you can deduce "5=3+2+2" by substituting 0 with 2, as some of the logical rules for = allow.
The point of this rant is that "5=3+2+2" is not "false" in the mathematical sense. It's a statement that's true with respect to the theory - the choice of axioms. Whether you find that theory useless is independent of what formal logic says about that theory.
Even "inconsistent" only means that you can prove statements and their negations. From a formal perspective, inconsistency is merely a feature of the theory.
To wrap this up:
OP I think your mistake is in thinking there ought to be a mathematical theory which reflects reality best.
In some sense, this is what theoretical physics is about. But if the "sum" of all numbers is assigned -1/12 in some theory or whether this and that set theoretic truth holds is merely a matter of the chosen context, and nobody will be able to tell you which exact context is proper for mathematics.
The same plebs who disregard 1+2+3+...=-1/12 for being "wrong" will attack Wildberger for being super restrictive to not cross the made up math edge.
Classical analysis is a super useful super made-up framework, for example.
The truth of the Banach-Tarski theorem one literally depends on a single basic statement, i.e. if you start with that statement in the true bag or not (the axiom of choice).
The key to why BT is not so terribly impressive after all is that the "chunks" are not solid chunks as if you'd just sliced up an apple, but infinite scatterings of points, to lift wiki's phrase. Once you're out of the prosaic territory of solid pieces of (physical, say) matter, or their ideal geometric equivalents, then it becomes far easier for mathematicians to do their chicanery with point sets as opposed to what we would describe as "solid, finite bits of matter". If there /is/ a BT-related way of taking "solid bits" of a sphere and then decomposing them into two equal spheres, then I would like to know about it (IIRC I don't believe that there is such a decomposition).
As to the other meme, in point of actual fact, the naturals of course do not sum to -1/12. Everyone knows this, and if you read the very first sentence and the very first paragraph of the related wiki,
en.wikipedia.org
, then you will see that the people who edited that page are at pains to make this perfectly clear to a general reader: "okay, srs guise, they don't add up to -1/12 so don't get the wrong idea." Mathematical caveats are required which makes the statement far less impressive, much the same as with BT above.
So what's really going on? Where does the meme even come from to begin with? Simply put, there are a series of related mathematical "tricks" (that is, techniques. I want to be perfectly clear that in the sciences and also in math, a "trick" is just an experiment or a way of doing/manipulating things, and not meant to indicate deception as-such) which all point to the /notion/ of the sum being -1/12, or of /assigning/ that value to the sum, rather than a literal equality as-such which follows from high-school tier math. For this reason, the sum could and probably should be bracketed in air-quotes when it is invoked, in much the same way that a Cauchy Principal Value is denoted by "P.V.".
It's not. Axiom of choice is garbage. Infinite sets are garbage.
>Infinite sets are garbage.
is so where do natural numbers end then? please dont say 10^200
0.999... = 1 has absolutely nothing to do with limits.
[math]10^{200}+1 \quad :^\wedge\hspace{-0.05cm})[/math]
>I've been introduced to the Banach-Tarski paradox and the fact that an infinite sums apparently adds up to -1/12.
t. I discovered le sciencexd channels gais
>Talking out of your ass
Hello wild sausage
Good question but answer me this:
Is the answer to THIS question "no"?
No it isn't
Maybe.
what did he mean by this
Doesn't this just show a flaw in this "logic"?
>Banach-Tarski paradox
Axiom of Choice
>-1/12
Infinite algebraic operation
Both are bullshit, purge the filth out of mathematics
The penis axioms.
limits of a sequence ya turd
...
I wish I could go back in time and strangle the first person that made a -1/12 popmath youtube video
Well if you wait 1 day, then 2 days, then 3 days, etc. you will go 2 hours back in time.
Take your pedophile cartoons back to .
>you are this dumb
0.999... is literally a geometric series you turd