How long does it take to delude yourself, excuse me, to """school""" yourself, into believing this equality to be true...

How long does it take to delude yourself, excuse me, to """school""" yourself, into believing this equality to be true? Do you have to use cult like methods?

Other urls found in this thread:

en.wikipedia.org/wiki/Thomson's_lamp
twitter.com/SFWRedditGifs

I wonder if this is a genuine question

You don't think it's crazy to believe that if you keep on summing positive numbers you'll arrive at a negative number fraction?

Are you too that delusional?

Maybe you can then answer this question. And no, this is not a troll in the slightest. I just can't even begin to understand why anyone can think this way.

once you take a complex analysis class you learn about analytic continuation

i'm assuming you're still a freshman, so just wait a while before tackling subjects beyond your comprehension

I've watched videos on that. I understand how it's explained by using Riemann zeta function.I still think it's insane to think of that as equality.

> I understand how it's explained by using Riemann zeta function.I still think it's insane to think of that as equality.
its not an equality, you obviously don't understand it

To accept is a mathematician's rite of passage--and like most rites of passages, it doesn't ask you to do something sensible because it wouldn't be much of a passage then would it? To show your loyalty, you must accept an obvious falsehood, that is, assigning obviously false values by analytic continuation to divergent series. Other well-known rite of passage is set theory, but that has become too accepted and so others have taken its place.

Even in calculus,
[math] \sum_{n=1}^\infty \dfrac{1}{n^2} = \dfrac{\pi}{6} [/math]
You sum rationals and get a non-rational number.

This is not to say that [math] \lim_{m\to\infty}\sum_{n=1}^m n [/math] has a limit in analysis.
It doesn't have a limit in analysis and literally nobody disagrees.

The sum of integers giving -1/12 is one in an entirely different theory (again, everybody agrees) and it has (as physicists know) applications in quantum electrodynamics.
So it's something that has physical applications, just like pic related is a physical application of
[math] \sum_{n=1}^\infty \dfrac{1}{2^n} = 1 [/math],
which is provable in analysis.

Fact of the matter is: You subscribed to mathematical Platonism, a centuries old philosophy that makes you uneasy with interesting and relevant math. Sort yourself out.

>mathfags
>admitting mistakes
lol

stale meme senpai.

>You sum rationals and get a non-rational number.
That's pretty weird too. But not as weird as summing natural numbers and getting a negative rational number.

Is this supposed to be an arguement?

Approximately 1 to 22 years.
Yes.

OP getting told.

>sum all positive numbers
>get negative number
>get a fucking fraction
oh, sure he got told.

Do you even know what's problematic with you platonism and why the mathematical community has moved on?

>what's problematic with you platonism
No. Please do explain what's wrong with the reality around us. I'm all ears.

Math doesn't model reality anymore? Well ok, nevermind then.

The issue here isn't "equality", it's summation. Ramanujan summation is simply an alternative definition of summation, just as a Zariski topology is an unusual definition of a topology or the Hamming distance is an unusual definition of a metric.

No it doesn't, in the same way as natural language doesn't per se. You can use it as a tool to model reality though. Point is, you need to understand the power and limitations of you tools as precisely as possible, as is done in the modern axiomatic method.

That's not what platonism is about now is it.

if you're not a Platonist, you're not a mathematician.

It never did.

Analytic continuations used to upset me too until I thought of a simpler example.

Imagine a regular polygon. Its area is equal to n/2*R^2*sin(2*pi/n) (no I don't know how to use LaTeX), where n is the number of sides and R is the radius of the circumscribed circle. Imagine now the area only as a function of n with a fixed R. Now, this function is defined only on n being a positive integer greater than two. However, you can evaluate this function pretty much at any real (or complex for that matter) number. This is how you can get the area of a regular 3.5-agon, 2-agon, 0.5-agon, some of which are negative! Negative areas of partial polygons do not make sense, do they? Moral of the story: if you continue a function analytically beyond its useful domain you lose any meaning of the result you get from it.

You're free to prove it wrong if it's so obviously wrong.

I think this idea is a failure of philosophy to understand the modern development of maths. The -isms are simply no longer relevant for a working mathematician. That doesn't mean he has no idea about his ontology or epistemology.

Indeed, it is in the practice of maths one sees that Mathematical objects are, I think, understood to exist in a very pragmatic sense: a mathematician introducing new ideas is very aware of the fact these things don't exist 'out there', hence the need for careful definitions. But neither does he think that 'everything goes', hence the care for beauty and usefulness.

>Math doesn't model reality anymore?
It often "models" reality in the sense that you get a syntactic framework that you can use for stuff you encounter in "reality". Both classical geometry as well as quantum electrodynamics is sucha use-case. Just because you personally don't use quantum electrodynamics, doesn't make it not physics.

Whenever you add two positive integers x and y, the result x+y is positive. Formally proving stuff for infinite sums is most likeleven beyond the capability of OP, it's epsilon-delta-shizzle.

In the real of analysis, this is easy. In that theory, for finite L,

[math] \dfrac {1} {\sum_{n=1}^m a_n} = L \ \ \ \rightarrow \ \ \ \sum_{n=1}^m a_n = \dfrac {1} {L} [/math]

and if L=0, the inverse is undefined. And

[math] \dfrac {1} {\sum_{n=1}^m n} = \dfrac{2}{m\,(m+1)} < \dfrac{2}{m^2} [/math]

goes to zero for m to infinity.

If the infinite sum would go to -1/12, the other limit (in analysis), would go to -12.

its not that weird if you consider, for example, the sum
[math]\frac{3}{10^0} + \frac{1}{10^1} + \frac{4}{10^2} + \frac{1}{10^3}+...[/math]
It is pretty clear that it will converge to [math]\pi[/math], an irrational number, yet each part is completely rational.
This is more a weirdness of infinity

Guys, what should I start reading. Math is so interesting and there is so much to learn. Where do I start? My level is around calculus.

Tell me something... as I start summing up NATURAL numbers from 1 through to infinity, the sum has to cross 0 at some point since the sum is, supposedly, -1/12.

At which point does the sum cross zero?

Can you really accept the fact that as you sum natural numbers and as the sum grows larger and larger, the sum is actually zero at some point?

>You don't think it's crazy to believe that if you keep on summing positive numbers

I'm not sure where the disconnect is coming from, so I will cover everything.

The thing that involves all positive numbers and -1/12 is not the sum of all positive numbers. The sum of all positive numbers is infinity. The thing that involves all positive numbers and -1/12 is the zeta function regularization of the equation 1 + 2 + 3 + 4 . . .. The zeta function regularization of the equation 1 + 2 + 3 + 4 . . . is -1/12.

I'm not on OPs side, but I'm also not on yours. The limit of partial sums isn't any more "the sum of all natural numbers" than the Riemann zeta function continuation is, or any other sum. The theory of analysis doesn't have prevalence of being THE sum of all natural numbers over any other summation technique. Some are just more useful to us than others.
Arithmetic, e.g. as captured by the Peano axioms, has nothing to say about sums of infinitely many numbers. Just as the fact that "+" in arithmetic always maps two positive numbers to another positive number doesn't say anything about how infinite sums behave.

What interests you?
For a shot into the dark, I'd say "how to prove it" by Velleman. It's online.
If you have any specific interest, you're a fool not to mention it in your quesiton from the beginning.

Stuff like Taylor series. A lot of things discussed in this thread. International Olympiad level stuff. Lots of properties. Stuff not learned in HS.

not an argument.

If you truly understood, you would know that it is not a question of loyalty. It is acceptance of one's limits after futile attempts at grasping the universe, followed by a real transcendence by the way of, almost Kierkegaardian, leap of faith by means of complex analysis. Seeing, accepting the world for what it is. You cannot grasp it rationallt, yet something inside you shouts that yes, indeed, sum of all naturals, defying everything, like Camus' Sisyph, revolting and finding happiness and meaning in it, equals -1/12.

This is literally false. Zeta function as an infinite sum of residuals is defined only on numbers whose real part is greater than one. The sum doesn't hold for all the other cases, such as one your picture depicts.

BUT WHY

OH YOU DONT KNOW?

THEN I HAVE BAD NEWS FOR YOU CHAMP!

As I said, it requires leap of faith to accept this and embrace the love of complex analysis, the way that GauB envisioned. There is no need fo "how".

there is but ok.

I like the idea behind structuralism. I need to read more about it some day.

Yes if it's a point at infinity. :^)

>Yes if it's a point at infinity. :^)
sure thing kid. keep on deluding yourself.

or the gamma function, which lets you "prove" that

[math] (-1/2)!=\sqrt{\pi}[/math]

>true
It isnt.
At least not in the standard way of how analysis is done.

It is obvious that this series diverges.

Wildberger is fine with projective space

It's an extension of the concept that gives it a different meaning in a broader context.

>no I don't know how to use LaTeX
Genuinely stopped reading

Memes aside this guy has a point. The way it's often written uses the equals sign to relate a function to the analytic continuation of the function with no further explanation, which is pretty sloppy. Especially since it's being seen by normies who don't know what analysis is.

I don't like this kind of stuff because it perpetuates the idea that math is a bunch of magic nonsense.

>The way it's often written uses the equals sign to relate a function to the analytic continuation of the function with no further explanation, which is pretty sloppy.
If you think that's bad then I hope you stay far away from category theory, or you'll go mad from the sheer volume of sloppiness from people who define everything in terms of set theory for the sake of "rigor", and then writing an equation that is provably false because it confuses equality with isomorphism.
(A simple example is the proposition of additivity: the disjoint union of two ordinal numbers is isomorphic but not equal to an ordinal number.)

>I don't like this kind of stuff because it perpetuates the idea that math is a bunch of magic nonsense.
Until the set theorists start holding themselves to higher standards, or mathematicians start adopting category- or type-theoretic foundations, it will continue to remain nonsense, and refutable nonsense at that.

>pic related
Not sure why it is okay to shift 2S^2 over by one place (4th line), maybe because it is an infinite series (think of the infinite hotel room problem), but there is probably some other simpler mathematical answer. If someone can clear this up then it stands to reason that the answer is correct. I don't hold out too much hope for this happening to be honest.

>correction
*(5th line)

why is s1 equal to the average of its two possible sums

en.wikipedia.org/wiki/Thomson's_lamp

How the fuck does the statement

S - 1/4 = 4S

make any logical sense.

For summation to a given n, S will be greater to or equal to 1 (1, 1+2, 1+2+3, etc).

Following that, 4S will be greater to or equal to 4 for a given n (4, 4+8, 4+8+12, etc).

For any n value, S - 1/4 will always be much less than 4S.

1 - 1/4 < 4
1 + 2 - 1/4 < 4 + 8
1 + 2 + 3 - 1/4 < 4 + 8 + 12

Consider
S' = 4S - (S - 1/4)
S' = 3S + 1/4

As n increases, S' will increase monotonically, as will S and 4S. The difference between 4S and S - 1/4 will only increase with n, they can never be equal.

question,

if they wrote it like this

would you retards stop with this bullshit and understand that its not the normal sum?

In fact you find that kind of sloppiness everywhere in math, and it is certainly not a lack of rigor. Other foundations won't change this.

why are you implying that it changes anything?

Oh you dumbass... you can use your shitty equations to also "prove" that 1 = 2. Go ahead, give it a try.

~ is used to mean approximate. they should invent a new symbol for continuation """equivalences""".

no one has ever used ~ to mean approximately, that's [math]\approx [/math], the latex command even alludes to this with the name \approx. the latex for ~ is \sim, for similar. Its traditionally used for a binary relation that's an equivalence relation but not equality.

>why are you implying that it changes anything?
because in math different symbols have different meanings

you didn't understood the question.
why are you implying that it changes anything?

>no one has ever used ~ to mean approximately, that's ≈≈\approx , the latex command even alludes to this with the name \approx. the latex for ~ is \sim, for similar. Its traditionally used for a binary relation that's an equivalence relation but not equality.
you're a motherfucking retard. you should get off /g/ now and never come back. ~ is used all over to mean approximate. can't you fucking use a search engine?

>Even in calculus,
>...
>You sum rationals and get a non-rational number.
Wrong, it's [math]\frac{\pi^2}{6}[/math]

And the square of a irrational number can be rational (e.g. 2^0.5)

ζ(2) is still irrational. Powers of π can't be rational since it's a transcendental number.

because then retards wont think that '=' means equality on the real numbers defined by [math]∀x∀y[x=y→∀P(Px↔Py)] [/math] instead of a different binary relation between a divergent series and a real number.

ive laterally never seen this use of ~ to mean approximately in my life.

>ive laterally never seen this use of ~ to mean approximately in my life.
what 3rd world shithole are you from? can't you find a native image board for your lang/country to shitpost to?

>retards wont think that '=' means equality
but it does.

>lol i have no argument against this, but luckily he mistyped 'this' instead of 'the' so I can call him out on not knowing english and invalidate his point!

no, I'm calling you out for not knowing what ~ is used for.

not as usually defined on real numbers in OP pic, and not in most textbooks that use '=' between a divergent series and real value. It means something completely different then.

Hey, I am from a third world shithole and I've seen plenty of ~ for approximations. Also for some equivalence relations. So please do not disrespect 3rd world shitholes.

That said, I prefer ≈.

so 3rd world countries dont use non standard notation? what kind of mangled thought process did you use for that?

you still haven't said where you're from. must be fucking hell. do you even have running water? toilets? sewage system?

sorry, but no.

>THE SUM OF THE NATURAL NUMBERS IS NOT ONLY NEGATIVE BUT ALSO A FUCKING FRACTION

LOL WAY TO TAKE ``````INFINITE'''''''' SUMS SERIOUSLY

>implying the limit of a rational sequence being an irrational numbers is anywhere near the limit of a series of positive integers being a negative rational
>in an entirely different theory (again, everybody agrees)
Wrong. Numberphile has been teaching it as *the* answer for years despite the fact that all the "alternative hip'n'cool theories" that entail it are contradictory.

I don't care what anyone on numberphile says

>what is a transcendental number

sorry hun, but your wrong. dont let it get you down though, im sure youll understand it someday, just dont give up!

>understanding flawed logic
why?

You are also dealing with differentials, zeros and infinities. Things get weird at infinities. This in itself is not a problem. The problem is why 2S^2 is shifted by one space on the 5th line.

I can do far, far worse than that. I can prove that every number is equal to every other number.

But, as predicted nobody was able to suggest a reason why 2S^2 should be shifted on the 5th line.

>~ is used all over to mean approximate
Only because it is found on a standard keyboard whereas ≈ is not. But both symbols are well defined and you should use the correct one if possible.

>you should get off /g/ now and never come back
Wer're on Veeky Forums.