/sci/ - Science & Math

[Anonymous No.12320533]

How can you add infinitely many numbers?

[Anonymous No.12320535]

>>12320533
one at a time, babey!

[Anonymous No.12320543]

Calculus

[Anonymous No.12320544]

>>12320533
Can you at least try switch up the pepe variety when you post these bottom-tier trashposts

[Anonymous No.12320584]

-1/12

[Anonymous No.12320680]

shitcum nigga balls

[Anonymous No.12320797]

>>12320533
Assuming you're talking about series, technically are not adding infinitely many numbers. When we say an infinite series "equals" a real number we are being a bit lazy. Instead, we are defining "equals" for a series to mean that the sequence of partial sums of the series converges to a real number.

By using this definition, we can intuitively extend our finite understanding to the infinite, but we do need to be careful because things don't always work the same.

[Anonymous No.12320905]

>>12320533
very carefully

[Anonymous No.12320926]

>>12320533
You cant. You stop at MOAN, which is the Mother Of All Numbers. Attempting to add anything to MOAN results in you sailing off the edge of reality and being devoured by sea monsters.

[Anonymous No.12321085]

>>12320533
by pressing one and the plus sign

[Anonymous No.12321213]

>>12320533
Limits

[Anonymous No.12321224]

>>12320926
I made you’re mom MOAN if you know what I mean.

[Anonymous No.12321231]

>>12320926
either moan+1 exists, or moan isn't a number

[Anonymous No.12321255]

>>12321231
Justified by what? Axioms? The original natural numbers are empirical. If it turns out you can't add 1 to MOAN, that just means the natural numbers of our experience don't satisfy the axioms, and we need new axioms. Of course, the infinite arithmetic of the old axioms will still be interesting mathematics, but it will no longer be relevant to real life arithmetic except as an approximation.

[Anonymous No.12321293]

>>12321255
>source: my ass

[Anonymous No.12321297]

>>12321231
enjoy being devoured by sea monsters, infidel.

[Anonymous No.12321301]

>>12321293
you should stay away from the big boy toys, go fix a lawn mower. You get a cracker afterwards.

[Anonymous No.12321308]

>>12321255
This. +1 , conditional to you not being equal to MOAN.

[Anonymous No.12321491]

>>12320926
What spicy new form of ultrafinitism is this

[Anonymous No.12321507]

The futility of MOAN is that it's arbitrary and therefore constructively useless.

[Anonymous No.12321730]

>>12321301
read a book

[Anonymous No.12322482]

Memes

[Anonymous No.12322655]

>>12320533
constructivism is for shit for brains. I don't need to actually perform the computation to work with the result. I can just grab it from the platonic realm.

[Anonymous No.12322731]

current mathematics relies on god. what's the big deal?

[Anonymous No.12322744]

>>12320533
Convergence

[Anonymous No.12322829]

You really can't because of MOAN. But you can add a lot of numbers.

[Anonymous No.12322850]

>>12322829
>>12321507

[Anonymous No.12323076]

Ok so hear me out. I know this sounds completely ridiculous but honest to God that's what modern mathematicians actually do when summing infinite series.
Given a series [math]\sum_1^\infty a_i [/math], they look at partial series [math]\sum_1^N a_i[/math]. You have two options
- Case 1: You know some number x such that [math]| x - \sum_{i=1}^N a_i |[/math] gets as small as you like for N large enough. Then they declare [math]\sum_{i=1}^\infty a_i[/math] to be equal to x. So for example [math]\sum_{i=1}^\infty 1/2^i = 1[/math]
- Case 2: You don't know any such number x. How then can you determine whether [math]\sum_{i=1}^\infty a_i [/math] actually exists? Suppose there actually exists an x as in Case 1, even though we do not know what it is.
Then one thing you can say is that for any positive number [math]\epsilon > 0 [/math] there exists a natural number N such that for all natural numbers n,m such that m>n>N, we have the condition
[math]| \sum_{i=n}^{m} a_i | < \epsilon [/math]. In more intuitive language, it states that that as you sum more and more terms, the difference you make gets smaller and smaller.
So we have a condition that avoids explicitly talking about the limit such that:
- All series that satisfy case 1 also satisfy this condition.
- All series of positive numbers which explode to infinity do not satisfy this condition.

[Anonymous No.12323079]

>>12323076
Mathematicians decided that this was the condition needed to determine whether a series converges or not. The problem was, nobody was able to prove that if this condition is satisfied, there is actually a number to which the series converges. So what did mathematicians do? They wanted this to be the condition so much that they DEFINED numbers by such series and declared that they always converge to these numbers. So instead of having an actual answer like 3, or 1.2334, they say that the series [math]\sum_1^\infty a_i [math] which satisfy the criterion converges to the formal object given by the sequence (a_1, a_2, ...) of numbers. Seems fake? What, you can't just define it to converge to some formal sequence which is itself the series? I agree, but mathematicians wanted an actual answer to the series so much that they conceded with giving fake answers. There's a problem here, two different series which should converge to the same number will converge to different numbers in our new number system of sequences of numbers. So some sequences needed to be identified:
for example
1+0+0+0.... converges to the same thing as
0+1+0+0+.... so we identify the sequences (1,0,0,0,...) and (0,1,0,0,...).
In general, we identify two sequences (a_1, a_2, ....) and (b_1, b_2, ...) if the series (a_i - b_i) converge to 0. But wait, you say! How is this a general method? Can we in general determine whether a sequence converges to 0? How do we know which sequences to identify? Your concerns are correct. In general, there's no way to determine whether a series converges to 0 or not. There are still lots of open problems in mathematics which can be formulated in terms of proving whether a given series converges to 0 or not. How is this a valid definition when determining whether a difference of two series converges to 0 in most cases is an open problem?

[Anonymous No.12323083]

>>12323079
The mathematician shrugs, for he is confident that either it does converge to zero or it does not, even though he cannot demonstrate either.
So in conclusion, to COPE with not being able to actually sum every infinite series they want to mathematicians have:
- Invented a completely new number system whose elements are formal INFINITE SEQUENCES which are DEFINED by these series and identifications which are in general completely indeterministic and in most cases open problems. This means that whatever calculations you were able to do in the naturals or rational numbers, applying algorithms to arrive at the answer, are no longer valid because for the vast majority of questions about the reals there is NO ALGORITHM, no process to get to an answer.
- Declared the answer of each infinite series which satisfy the precious condition which they think should ensure that the series converges to be ITSELF as a formal infinite sequence (leaving aside the questions of whether or not such a sequence can actually be completed).
- In the process completely fucked up all the size/cardinality issues in mathematics because nobody knows how big this fucking object they created is to the point that there are now mathematicians who think that asking how big it is is not even a sensible question (even though it's completely sensible when you're talking about actual deterministic objects like natural numbers or rational numbers).
- Fucked up the field of algebra which used to be finitistic and deterministic by using this monstrosity of an object in the proofs to the point where it's hard to tell which parts of algebra are finitistic and deterministic (not reliant on this new number system) and which are not.

[Anonymous No.12323087]

>>12323083
- Provided (allegedly) rigorous justification for most of analysis and geometry (the arithmetization of geometry and other parts of mathematics is also another issue which I will not discuss in my post, just know there are people who don't like it), which made essentially all working mathematicians agree on what proof techniques were valid and an unambiguous definition of what they were talking about.
Whether it was worth it, it is for you to judge.

[Anonymous No.12323098]

>>12323087
>>12323083
>>12323079
>>12323076
What they did is very similar to the horseshoe theory as in this video:
https://www.youtube.com/watch?v=j4hW7AwETZA
To find the answer, they just picked all the terms, put them in a sequence and declared that to be the answer of the series. If the actual answer is 1 or 2.5 they have no way of knowing and they're usually satisfied with their horseshoe answer.

[Anonymous No.12323106]

Finitists and determinists are copers mad at the fact that the universe is a gloopy nondeterministic mess.
The universe is not a bunch of little blocks stacked up together. It's a gloopy glob of fields and weird manifolds that gloop around nondeterministically. It's continuous and infinite.
Get over it. Your little fantasy of a bunch of little blocks stacked together into perfect geometry harmony isn't real.

[Anonymous No.12323160]

>>12323106
Tell me the answer to
[math]\sum_{n=1}^{\infty} \frac{sin(n)}{n^2} [/math]
(which according to you sums to a real number) without embarrassing yourself and then I will get over it.

[Anonymous No.12324230]

>>12323160
pi^2/4

[Anonymous No.12324242]

>>12324230
No it's not lol.

[Anonymous No.12324248]

>>12323083
>In the process completely fucked up all the size/cardinality issues in mathematics because nobody knows how big this fucking object they created is
bullshit
It sounds like you're comfortable with sequences of natural numbers as mathematical objects
The set of all real numbers has the same cardinality as the set of all such sequences

[Anonymous No.12324602]

Like this
0.999... + 0.999... = 2

[Anonymous No.12324620]

>>12324248
>It sounds like you're comfortable with sequences of natural numbers as mathematical objects
Particular sequences are mathematical objects that can be studied but there's no such a thing as "the set of all sequences of natural numbers", the notion of a sequence is just too vague for that unless you significantly restrict the definition of a what sequence means.