Infinity

I'll keep this short so that it might get read.

Analytic continuation allows us to approximate an infinite series as a function. The more terms you add to the series, the more accurately you describe the function. So at the limit, when you have all the terms, that series can be considered equal to that function.

This allows us to assign meaningful numerical values to infinite series, even divergent ones.

For example, the infinite series "1 + 2 + 3 + 4 + 5... " can be evaluated. However, the result is disturbingly counter-intuitive. If you stop adding terms at any finite point, you'll have a larger and larger number as the result. However, if you "evaluate this after an infinite number of terms, you'll find the sum to actually be -1/12.

There are a variety of proofs for this, and this number is demonstrably a meaningful . This result is actually seen in physics. Furthermore this result is a foundation in string theory for the number of required dimensions.

Please, everybody, for the sake of your sanity, ignore the OP. It is complete and utter tosh.

http://www.goodmath.org/blog/2014/01/17/bad-math-from-the-bad-astronomer/

http://www.goodmath.org/blog/2014/01/20/oy-veh-power-series-analytic-continuations-and-riemann-zeta/

I saw this same thing on Numberphile:

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It seems weird, but they say it works and it's actually useful in some cases so I can't see what's wrong with it.

EDIT: well I read the first link from Mathsman and it seems my doubts weren't unfounded. But does that mean the physicists in the video were just pulling Brady's leg, or did they not see the error either, or what? I'm reading the Riemann Zeta link but I'm already feeling lost, so we'll see how that goes. 

I didn't completely get what he was saying about the Riemann Zeta, and by the author's own admission he was not entirely clear on how to explain it either. But in the end it sounds like the problem is saying that the sum of natural numbers equals -¹/₁₂, whereas it's more correct to say that it's a possible value that you can assign by a certain line of reasoning. To Brady's credit, there is also a follow up video directed to towards the people who had a problem with the first video:

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It sounds like another notion that has been taken on by bullshitters to "destroy" your common knowledge, much like pseudoscience does in using quantum mechanics.

It sounds like another notion that has been taken on by bullshitters to "destroy" your common knowledge, much like pseudoscience does in using quantum mechanics.

Scepti?

No wonder people's heads are so messed up. This stuff is just absolute silly nonsense for crying out loud. It's meaningless.

Quote from: DuckDodgers on June 24, 2014, 11:29:41 AM

It sounds like another notion that has been taken on by bullshitters to "destroy" your common knowledge, much like pseudoscience does in using quantum mechanics.

Scepti?

I meant taken out of context.  Similar to shmeggley's argument where this is saying it is equal to -1/12 while the actually proof says it can be approximated by -1/12 in specific circumstances.

No wonder people's heads are so messed up. This stuff is just absolute silly nonsense for crying out loud. It's meaningless.

"Said sceptimatic to the waiter at the French restaurant after looking at the menu for ten seconds."

Please, everybody, for the sake of your sanity, ignore the OP. It is complete and utter tosh.

http://www.goodmath.org/blog/2014/01/17/bad-math-from-the-bad-astronomer/

http://www.goodmath.org/blog/2014/01/20/oy-veh-power-series-analytic-continuations-and-riemann-zeta/

Cool blog bro. I was looking for something to conflict with my post, but really it just went on about how uncomfortable this branch of math makes him. He did mess up on a few parts though, like saying something 'has to converge' to be 'equal'.

This is a completely valid way of equating the entirety of the series which we could not do otherwise. This procedure gives us all the same answers for the otherwise-solvable series, and the correct numbers for problems we previously couldn't solve. ...Kinda like imaginary numbers. i was a 'non-classical' way of evaluating the √(-1), but it is completely correct to use them to evaluate something complex.

And just as we have seen i has perfect predictive power in reality, so does this.

But in the end it sounds like the problem is saying that the sum of natural numbers equals -¹/₁₂, whereas it's more correct to say that it's a possible value that you can assign by a certain line of reasoning.

All of math is axiomatic. You start with certain axioms, and you can derive additional emergent properties. Once you lay a foundation, the math talks back.

Continuing on my analogy of imaginary numbers...

We could say it is impossible to evaluate an infinite series.
We could say it is impossible to factor x^2 + 1 = 0.

However, I'd argue that introducing the non-classical term "i" that lets us push our answers further and it's not "bad math", but yes it is an alternative way to explain an equation. That is why some people might have a problem calling it "equal", since we've adopted another axiom in the process of using "i" but frankly we still use an equal sign. And particles measures to these values which we can now calculate.

On that first Numberphile video, I'm confused why he shifts the S2+S2 second sequence to the right, but doesn't do something similar when he subtracts S-S2.  Everything about this screams wrong to me.  I'll have to watch that second video whenever I have a bit more time.

On that first Numberphile video, I'm confused why he shifts the S2+S2 second sequence to the right, but doesn't do something similar when he subtracts S-S2.  Everything about this screams wrong to me.  I'll have to watch that second video whenever I have a bit more time.

You're absolutely right for it to scream wrong to you. The manipulations are done cavalierly in that divergent series are treated as if they are numbers. It even starts with a piece of nonsense:
let c or x or S= 1+2+3+4+... (depending on which part of the internet you are).
This equation has no meaning. The right hand side, whilst easy to understand, is undefined mathematically so to say it is equal to something is preposterous. Any algebraic jiggery-pokery after that is just flim-flam and sophistry.
We might as well claim that 1 = 3 and from that find that 2 = 6 and 2 = 4.

In fact, I'm going to have a bash at this infinity business. I claim that

A prize will be awarded to the first person who works out how I did this.

The sad part is that there are some beautiful results involving infinity in mathematics which are relatively easily explained to anybody with sufficient curiosity.

The sad part is that there are some beautiful results involving infinity in mathematics which are relatively easily explained to anybody with sufficient curiosity.

That does not sound sad.  I would love to hear about them.

Here's a result that has probably, over the years, generated more heat than light.

Consider the natural numbers: 1,2,3,4,...; there are, not surprisingly an infinite number of them.

Now consider all the numbers between 0 and 1, there are also an infinite number of them. However this infinity is greater than the first infinity. How can this be?

This result was proved by Georg Cantor in the late eighteenth century. If anybody is interested I'll give my version of the proof.

Is it something to do with the fact that the set of Real Numbers is uncountable?

Is it something to do with the fact that the set of Real Numbers is uncountable?

Yes, I believe so. I don't know if this proves it, but I think it's interesting: you can always get a real number from an infinite sequence of real numbers, by taking the succesive digits in the decimal cases and adding 1 to each, and this number will not be in the sequence, like this:
0.4501...
0.6783...
0.9233...
0.1935...
------------------------
0.5846...
So, even if you try to get every real number between 0 and 1, there will always be a way to get a real number (between 0 and 1) that will not be in the original sequence; so, there's no one-to-one correspondence between naturals and reals. This also implies that there's infinite real numbers between any two natural numbers, so that the infinity of the real numbers is greater than the infinity of the natural numbers (I think he called it cardinalities, not infinities).

Is it something to do with the fact that the set of Real Numbers is uncountable?

Spot on.

If you google 'cantor crank' you can be entertained by people getting themselves in to such a lather about this.

Wow, if Phil Plait fell for that nonsense I guess we can forgive Singularity for being taken by it.  But... no, that's just dumb.  Like, it should be immediately obvious to anyone with any proficiency with math at all that it doesn't make sense.

S1=1-1+1-1+1...=1/2  seems just as preposterous. Averaging because you can't figure out whether infinity ends on an odd or even step seems like a stretched estimation taken as fact.

[delurk/]

I believe this is relevant: https://en.wikipedia.org/wiki/Ramanujan_summation

S1=1-1+1-1+1...=1/2  seems just as preposterous. Averaging because you can't figure out whether infinity ends on an odd or even step seems like a stretched estimation taken as fact.

It's not really "just averaging". You can't take the average of an inifinite series obviously. The actual methods to get that result are more subtle.

That's nonsense.  The end result of that series is plainly 0.

(1-1)+(1-1)+(1-1)+(1-1)...
   0   +   0  +    0  +   0...  =  0

See?

That's nonsense.  The end result of that series is plainly 0.

(1-1)+(1-1)+(1-1)+(1-1)...
   0   +   0  +    0  +   0...  =  0

See?

No it isnt:

1+(-1+1)+(-1+1)+(-1+1)...
1+     0   +     0   +     0....=1

That's nonsense.  The end result of that series is plainly 0.

(1-1)+(1-1)+(1-1)+(1-1)...
   0   +   0  +    0  +   0...  =  0

See?

There is no end result, that's the point of an infinite series. Doing it this way implies infinity is even, which would mean it'a a number, which it isn't.

Infinity is imaginary.  Therefore infinity=i

Wow, if Phil Plait fell for that nonsense I guess we can forgive Singularity for being taken by it.  But... no, that's just dumb.  Like, it should be immediately obvious to anyone with any proficiency with math at all that it doesn't make sense.

Meh. I haven't responded to his posts since he's yet to actually challenge any point I've made. Of course, if he wants to disprove the standard equation written in textbooks across the world and all the string theorists who depends on that axiomatic identity, he can't just give it away for free on the internet.

Also, that informal proof on numberphile is non-rigorous, but does help illustrate the problem in widely accessible terms.

By the way, I did like the thought experiment: If "+1 -1" equated to turning the lights on and off, and if each new number in the series happened over shorter and shorter time approaching an asymptote in time, than once the time is passed, you've "switched the lights an infinite number of times". Had we stopped at any point along the way, we could have determined if we were on an even or odd count. But after the minute is up, would the lights be on or off?  Or in some waveform/indeterminate state expressing both binary states?

Honestly I thought the source of discussion would be defining what it means to be considered "equal", which is highly semantic. Considering its superior power modeling our universe, I've argued it can and should be considered "equal" under a branch of math that uses analytic continuation. It surprised me to see him straight up argue that result is trash to the same extent that 1 = 3 would be trash. :\

I haven't responded to his posts since he's yet to actually challenge any point I've made. Of course, if he wants to disprove literally every string theorist on the planet, he should get a Nobel prize or something.

How could he possibly do that with an unfalsifiable theory?  It's kind of sad that that's what passes for science these days.

"1+2+3+4+5..." can't possibly equal any number.  It's counter-intuitive because it's wrong.  I don't care what kind of crazy algebraic or analytical gymnastics you go through to get there.  It is immediately obvious that it has no meaningful value.

Quote from: ﮎingulaЯiτy on June 28, 2014, 01:32:23 AM

I haven't responded to his posts since he's yet to actually challenge any point I've made. Of course, if he wants to disprove literally every string theorist on the planet, he should get a Nobel prize or something.

How could he possibly do that with an unfalsifiable theory?  It's kind of sad that that's what passes for science these days.

"1+2+3+4+5..." can't possibly equal any number.  It's counter-intuitive because it's wrong.  I don't care what kind of crazy algebraic or analytical gymnastics you go through to get there.  It is immediately obvious that it has no meaningful value.

However it's of interest to mathematicians and theoretical physicists. Also, it's been covered that series of natural numbers can't be said to be equal to some value, but it can be assigned a value.  Sounds weird to me too, but it's a meaningful distinction in mathematics. Lots of things can be counter-intuitive in mathematics. Zero was unpopular when it was invented. Negative numbers were also scoffed at. Never mind imaginary numbers, people thought those were crazy. I can barely wrap my head around i myself. But all these concepts turned out to be quite useful.

This video is also good concerning OP. And you can win a million USD.

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