the math

[megadan]

The math behind Cesaro Summability:

let us consider

                (S_1[f](x)+S_2[f](x)+S_3[f](x)+...+S_n[f](x))
     lim       ----------------------------------------------
    n->oo                      n

(oo=infinity)
in other words, we're looking at the averages of the S_n[f] to see if it converges. If you have taken any university calculus courses, this may look slighly familear. Otherwise it may just look like gibberish.
(disclaimer: Calculus is not my strong point, i'm only taking it becuase i have to, so if i get this wrong, my appologys)
This is pretty simple if we look at it, the lim n-> oo of E f(x)/n essentially shows whether a certian funcion, f(x), is onverging or diverging, that is, whether it's sum is approaching zero, or infinty (because any number /oo must be either 0 or oo by definition).
If we take this in the context of music and Tool, it seems to be a sort of futility - either you will approach nothingness, or you will approach infinity, which is the same thing because it's ungraspable to any human (at least those of us with 44+2 chromosones?). Since this segue comes right before Ænema, perhaps it's commenting on the futility of wishful thinking - despite having a perfectly good function, in the end it will always end up as nothing or it's diametrical equivalent. Or perhaps we are again reading too deep. (Lateralus again comes to mind)


in any case,
ref:
http://mathforum.org/library/drmath/view/52118.html
http://www.idealibrary.com/retrieve/doi/10.1006/jath.2000.3488

[ragna16]

So basically any infinite sequence either diverges on a constant or goes to infinity.

Cesaro summability determines the summability of a sequence when it is not summable by standard means.

The easiest example is a function where the number continuously alternates between 1 and -1 starting with 1.

This is not summable because the summation of terms is either 1 or 0 alternating. It is Cesaro summable howeve, at 1/2 which does not really show the relationship well, or have a practical use.

The real application of Cesaro Summability however lies in divergent series that approach a number on two different sides at two different times, and attempts to attach a meaningful constant to something that is not easily or particularly useful as such.

[Comebackrunning]

Do any of the members of Tool have math degrees?
First this, and then Parabola!
Come to think of it, a hell of the lot of the tracknames require either an english degree, or a dictionary, to understand

[YGOfvn]

hah... Schism.. Lateralus... Parabol... 46 and 2.... Ænima...

THE GRUDGE!..... uh.

yeah maybe what the band meant by the cesaro summability thing is that even though humans tecknicly equal 0 in the infinity of creation, it's possible to reach infinity considering that. I think the baby just symbolizes human life and like.. how big it can really get.

[Alx_Ev]

This is all very confusing to me...Math has never been my subject....and I'm still distraught because I just found out that 1 + 1 does not necessarily equal 2. Which is just great because that was the only mathematical calculation I was almost sure of!
Oh well

[forty6n2]

I'm in Algebra int. 2

lol

I sure wish I could understand that. I bet I'm the youngest one on this opinions site. By the way I'll be 15 in a month.

[eek]

ok, the explanation of the math helped quite a bit... to my mind it sums up fairly simply as a theory i read some time ago (omega man theory), which goes something like...

once a species is born (crying baby) it has two possible fates:
1. extinction (zero)
2. immortality (infinity)

the omega man theory goes on to propose that an imortal species would continue to expand the amount of information it can store (human memory, history, culture and later shared consciousness) and that since information takes a certain space to encode a species must expand physically to accomodate that. extrapolate that to infinity and you have a species, or being, that KNOWS everything because it IS everything. at this point the species/being is God.

[paraflux]

and look at the end of the parabola video. Tricky slowly becomes one with all, and then becomes all. notice at first the eyes are all around him, flowing out and up, and then suddenly they are all flowing out of the top of his head from the crown chakra. I love the way Tool can use and switch multiple perspectives, like starting with a person and ending up with a light-being that represents us as a race joining with other light beings that represent other races' collectives, and so on and so on.

[Scobularni]

The real application of Cesaro Summability however lies in divergent series that approach a number on two different sides at two different times, and attempts to attach a meaningful constant to something that is not easily or particularly useful as such.

Just opened this up in my sound editor - noticed that the two channels, left and right, are completely different. Each channel has half the song, and at least make more sense on their own, than when coupled together.

Just an observation. Probably not an explanation but something to think about

[Cronos]

So then maybe if you listen to the two halves separately the words in it will be more audible?

[paraflux]

just pan your stereo all the way right or all the way left and you can answer your own question.

[halo99]

that equation is cool and everything, but is it feasible that any song Tool creates is simply a great-sounding compilation of drums, bass, vocals, guitars, and various other neat-sounding effects?

[Cronos]

Blasphemy...

[zweitracht]

I realize that this post is several months old, and I apologize for reviving it, but I have an important correction to make.

magadan wrote:
"...because any number /oo must be either 0 or oo by definition."

This is not exactly right. By definition (and aside from a few special cases), any number divided by infinity is zero. It cannot equal infinity. Look at it this way:

1/1 = 1
1/10 = 0.1
1/1000 = 0.0001
1/1000000 = 0.000001
and so on...

So, as the denominator approaches infinity (gets larger and larger), the quotient approaches zero. I hope that clears things up a bit. Sorry if it screws up some of your theories. ;)

[spiralion]

I read in a deftones thing that Cheno went to their practice space or whatever and said that they had some calculus chart or something up and that they used that to write songs which intrigued me alot. I would like to know about that much more.

[svet-am]

So then maybe if you listen to the two halves separately the words in it will be more audible?


Or, as someone else suggested, the left and right channels represent man evolving to nothing or infinity--separately, but still intrinsically linked.

When you die, you are physically 'nothing' (as in dead, no more...) and everything (infinity) to those people that love you and don't want you to pass on beyond this realm.

[AllforUnity]

But then all the people who remembered you end up dying, and then you are totally nothing.

[spiralion]

ok, um, i have some ideas. i'll put them as points

1. You know how maynard said about how if you meet your negative self you will dissappear, and I read it in a book once, because it would be the total opposite, as in antimatter of yourself, hence the line in pushit.

2. Talking about the alternating between 1 and -1, it could be a representive of the meeting of the two selves, and who is the 1 and who is the -1 is all relative because you both are the same thing only the other self is the polar opposite of the first self and thinks they are right. Anyways you get what i am saying.

3. In understanding why this is here one must look at such one might call "filler tracks" such as 'crimes' a dn the such on other albums. There has to be a pattern there somewhere.

4. Of course, following the concept, and i'm just writing this as i go along, the baby at the beginning is the infant in pushit that is himself that he is killing, choking. Aenema is the the hatred he has from being separated from himself towards the world because he hates himself and hates everything. Third eye is him coming to terms with himself and loving himself again.

I dunno. Work from here. I'll think more on it.

[AllforUnity]

But 1 and -1 are one. Because it's math, about absolution. That's why 1 and 1 are 1.

[g-bay-be]

OK you guys are talking about something based on false pretences. no numbers are real. You are all followers of the religon I call math. 1 is not a posible experience. think about it. the numbers between 0 and 1 are infinite. I mean you start at 0 and then you go to .00000000000001 and then .000000000000002. It get even smaller man. decimals can go forever. so you never enven get to the number one. so what we just skip over everything else and go from 0 to 1 to 2. you have to have faith that one will be there when you get there. its like a religon man. and think of this!!!!!! if you cut inifinity in half... you have infinity. ok so this distance between 0 and 1 is infinite. so whats the distance between 0 and.5 its infinite. so its all a concept. now think about this. they say the universe is getting bigger. how can something infinitly big get bigger. well here it is the distance from 0 to 1 is infinite... but the distance from 0 to 2 is infinitly larger than the original distance.... Insane is what it is... insane

[AllforUnity]

They mean the Unviverse is getting more explored, l believe. Do you study Sacred Geometry?

[spiralion]

could this have to do with the liner notes where it talks about the futants and such? Reaching into space. fuck, my head don't feal right...