View Full Version : Zeno's Dichotomy Paradox

solomon levi
06-30-2012, 02:15 AM
Simply put, if you want to get somewhere, you must first get halfway there.
Once arriving halfway, you must pass the next halfway point, etc...
Since there are an infinite number of halfway points, one never arrives at their destination.

This is true, and yet we do arrive at our destinations.
How are both true?

One is doing and one is thinking.
Thinking can't do or explain a lot of things that acting can do.

"A man of knowledge lives by acting, not by thinking about acting." - Castaneda

06-30-2012, 12:07 PM
This is making the assumption that one cannot go further than half way at a time.

If I need to travel 2feet I can do this in one step therefore passing through the halfway


06-30-2012, 12:25 PM
If you want to get somewhere, you must first get halfway there.

How so? How long is the distance to here? And how long is half of it?

yet we do arrive at our destinations.

Do we?

Here = There = Everywhere = Nowhere = Now Here

Quoth Christian Sheppard from the 'Lost' Series Finale:

"There is no 'Now', 'here'... (https://www.youtube.com/watch?feature=player_detailpage&v=dL26K6T3IOw#t=127)"

solomon levi
06-30-2012, 02:39 PM
It's a game. If you don't want to play, what can I say?
Ask Zeno. It's his freakin' paradox.
You guys are just being difficult.

If you want to ignore or argue the halfway thing then there's no paradox.
You might as well argue the rules of poker and say your 2 fours beat a full house.

It's responses like these that make me not want to post anything.
People can't listen; can't assume other vantage points.

Am I taking this too personally? :D

You can't figure out that things have a halfway point, but you learned to write posts and type??
English makes sense, but 1/2 of two feet boggles your minds?

Stop f**king with me. I'm very fragile right now.

I'm just kidding.
I'm always fragile. :D

06-30-2012, 02:54 PM
I'm not arguing that it is possible for things to have a halfway point, assuming a linear outlook.

I'm arguing the premisses of the presented 'paradox', I'm arguing the assumption that one must go 'half way' before one goes 'all the way'.

I am not aware of any paradox that can't be undone (un-paradoxized) by a change of perspective, by calling into question the very axiomatic foundations upon which paradoxes are usually formulated...

Nobody's fucking with anybody :)

solomon levi
06-30-2012, 03:16 PM
Well, the ways of getting somewhere without going halfway first are kind of off-topic, don't you think?
You want to throw the dichotomy out of this topic on dichotomy?
What is a post on Zeno's dichotomy paradox without dichotomy?

I love the bigger picture stuff, but it doesn't seem very relevant to this topic.

06-30-2012, 05:12 PM
Doesn't it all go back to 'walk the talk'. Thinking/saying one thing and doing another gets you lost, right? So this is how I see your Castaneda quote in this thread.

Walking the talk any halfway point always arrives quicker, IME (in my experience).

So the paradox comes from the fact that when we live life in this way our destinations changes. New unknown destination appears. Only in death are we finally arriving. That old thing about the quest is the key not finding the grail.

We are all Percival.


07-01-2012, 12:27 AM
It's here (http://en.wikipedia.org/wiki/Zeno's_paradoxes) on Wikipedia Sol.

I get it now :) I like the tortoise one too.

Here is one...but it cheats a bit.

3 guys buy a TV, but a friend goes to the store for them.

The tv costs £30 so:

Guy 1 gave £10, Guy 2 gave £10 and Guy 3 gave £10 = friend has £30

Now when the friend gets to the store the TV has been reduced to £25

On returning to the guys the friend is honest and says he has £5, but as it is awkward to share between the
3 guys the friend gives each guy £1 back and keeps £2 himself as he went and got the TV.

Now as they all got £1 back:

Guy 1 gave £9, Guy 2 gave £9 and Guy 3 gave £9 = £27

The friend kept £2...27+2 = 29. what happened to the missing £1?

I know this is not a paradox; I just use it to show that information can be confused.

The next one I think I have posted here before, but I donít think this works online so try it on some friends...I have rarely seen this fail.


You have to get the person to watch as you write the numbers down one at a time

On each number they must say out loud the total...this includes the first thousand.

so they should be saying:

One Thousand
One Thousand and Forty
Two thousand and Forty
etc...to the end

Try it and see what total your friends arrive at.

Again I would like to say that the relevance of the two puzzles above is to show how things can get confused, so in the case of the paradoxes
coined by the old masters, perhaps it is just confusion that makes it appear paradoxical.


solomon levi
07-01-2012, 05:05 AM
Thanks Ghislain. I probably should have posted a link to start with.
The thing with Zeno is that he taught Oneness and these paradoxes were his way of "proving" it.
He also tried to show that motion doesn't exist.

Either we consider him a fool, or we have to look deeper at these proofs.
I doubt he actually believed he never arrived anywhere.
So what was he trying to show?

If we dismiss these as parlor tricks, I think we miss something.

Something from Castaneda seems relevant again to me.
Often don Juan would try to trick CC. I recall the branch that was blowing that seemed like some
odd animal; or DJ hung a cloth from a branch without CC knowing and then sat him in position
to look beyond it at a mountain range to see the color.
When CC would figure out what was happening and resore his certainty of his view of the world,
DJ was always disappointed. The idea wasn't to solve the problem... the idea was to enter an unknown.
Maybe Zeno was after the same effect.

07-01-2012, 11:35 AM
While looking up Zeno of Elea I noticed that he was a great supporter of Parmenides.

Zeno of Elea ( 490 BC Ė 430 BC) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides.

Source: (http://en.wikipedia.org/wiki/Zeno_of_Elea)

So as not to confuse this thread I have started another on Parmenides Here (http://forum.alchemyforums.com/showthread.php?2989-Parmenides-Poem&p=22751#post22751)


07-01-2012, 12:01 PM
A lot of people has solved the paradox in different ways.... the most interestng way coms from Henri Bergson:

Which is a fantastic way of understanding time in a very different way.

The best way to explain it in a rush, 'cause I gotta go, is that Time isn't digital, it's analogic.... so the paradox actually comes from conceiving time as a digital space.

Seeker of Truth
07-01-2012, 06:46 PM
There is no movement
There is now here to go
What is one, does not move
And still these words appear in my mind as I write them
I see with my own eyes that there is movement on the screen
But it is an illusion, I am doing nothing, I am nothing, there is nothing I can do
And yet I do all
Both are true when you see that they are true
True, but not seen from someone not willing/ready to see that they are true
Truth hides from no one
Love hides from no one
Always there - Always unconditional


It is a depthless ocean containing everything and more

It has nothing and yet it has all

Be = The being of what IS Unconditional
Unconditional allows all to be and that includes its discovery of itself through you/me/all.
It includes everything and excludes nothing
It is itself and so it only moves in the imagination of beings within itself.
Thought is a manifestation of the unconditional
Everything is, Unconditional is the Source, and We are the Imagined beings within this unconditional self.
It cannot move and yet it moves these words, do these words move you?

solomon levi
07-02-2012, 12:10 AM
Thank you Zoas 23! That was very helpful for me - digital and analog, being two ways to perceive the world.
I think Newtonian physics and quantum mechanics are similarly different.

Seeker of Truth.
That one brought me to tears. :)
Very beautiful.

ps - I apologise for my uptightness the other day towards Androgynus' and Ghislain's responses.
I know threads can branch into all kinds of areas, and I don't mind that.
I can't say why it bothered me the other day, except that I've been feeling like I'm not
communicating with anyone.
Androgynus, Ghislain, I am sorry. Nothing personal. I love you two. :)

solomon levi
07-02-2012, 11:09 PM
time - duration and dimension

this will be kind of speculative, but there's a lot of stories that seem to support the idea.
the idea is that a portal or dimension opens up if we stay in the same place or position for a duration.

some say one has to wait three days for the portals of the inner earth to open.
I've heard a story, from Ramtha ;) that Solomon lay in his own piss and shit, refusing to move until God gave him wisdom.
Consider Yeshua in the desert for forty days.

The longest I have been still was in Ramtha school, sitting in the lotus position for days, blindfolded. It was the only time
I truly experienced the Void, though we focus on it all the time - the difference between direct experience and reverse engineering.
In this meditation it actually dimensionalised and I was in it, as one among many "stars", like a dream becoming lucid.

There's something to this duration - dimension connection.
It's not the only way, but it is a way.

Bel Matina
07-03-2012, 02:20 PM
The solution to Zeno's paradox is that there's also infinitely many moments between the start and the finish, so there's a one to one correspondence between moments and locations, and it takes as long as it takes. In the academy, it's considered fairly pat (by most people, anyway) but that doesn't mean I disagree with everyone else.

Zeno's paradox is actually what prompted the discovery of the set of real numbers, which has some interesting properties quite germaine to our art. Its most important property is that between every two real numbers is another real number. It's also interesting in that there are as many real numbers between one and zero as there are in the entire set of real numbers, which is more than in the likewise infinite set of counting numbers. I can look for them at greater leisure if it's demanded of me, or if some kind person knows of some off the top of their head I would be most obliged.

What's really interesting is that the limit of 1/x as x approaches zero is the cardinality of the set of reals (let's call it |R|) as it approaches from the positive side, and the -|R| as it approaches from the negative side, and conversely the limit of 1/x as x approaches the both |R| and -|R| is 0. This is confusing and probably hopelessly inaccessible if you don't understand calculus and aren't inclined to teach it to yourself (at the end of the day, it's not _that_ hard), but it implies (but does not to my knowledge prove) that 1/0=|R|=-|R|.

This has been a terrifying detour into mathematics (for me at least, I don't do this very often and between struggling to remember and flexing neglected brain muscles my head hurts) and I will bring it back to the point by observing that the hermetic significance of 1=0(|R|) ought to be fairly compelling to everyone. Also remember that the main reason division by zero is not allowed in conventional mathematics is because all numbers end up being equal to 1 which ends up being equal to zero, which from a neoplatonic perspective just makes it more appealing to me. Ow I have to stop now, I should really play with math more often, if I left any holes I will respond to comments.

07-03-2012, 04:56 PM
Hello Bel Matina and welcome to the forum.

In relation to your post above I have always been led to believe that as a divisor tends to zero that
the quotient tends to infinity.

Also in multiplication 3 x 4 for example could be written as 4+4+4 or 3+3+3+3...a sort of
mathematical shorthand for multiple addition. Therefore whenever one of the multipliers is zero
then the product must be zero. 3 x 0 = 0+0+0 or zero amounts of three, which in both cases are zero.

No matter what |R| represents, if it is multiplied by zero the product must be zero.

So how does 0(|R|)=1...am I missing something?

I am not a mathematician, but in laymans terms are you saying that "1" can have an infinite number of fractions and if the number of fractions of "1" added
together tended to infinity the sum would tend to 1?

or that 0 x infinity = 1?

Did that make sense lol. Crude, but I couldn't think of any other way to put it :)


Bel Matina
07-03-2012, 07:11 PM
Thank you, Ghislain. I've been lurking for a while, and I very much enjoy your posts.

I did get very sloppy with the mathematics. Just digging all that up from my memory was a bit taxing, and I wasn't sure how much rigor was meaningful outside the context of a formal proof. I suppose if I keep bringing math in to things, I'll get a feel for what's proper.

It sounds like you get the main idea, but I'll go through the math a little more carefully; certainly there are nuances that grow deeper the more you understand.

The short answer is that jumping from lim 1/x as x approaches 0 = |R| to 1/0=|R|. This is valid for any continuous function, though whether 1/x is continuous at 0 is difficult question to answer, and I skipped a lot of steps. So I'll start from the beginning and try to be a bit neater this time.

We started with the fact that the limit of 1/x as x approaches 0 is infinity. Actually, it's an infinite quantity which is larger apparently than an infinite number of infinite quanities - when I learned about this as a teenager there was only one. |R| has also gotten a new name since I learned about it, which I can tell you makes it so much easier refreshing myself on this stuff. They label orders of infinity using Hebrew letter and numerical subscripts, but also now apparently in some arcane alternation with variations on omega that wasn't relevant enough for me to try to place it.

Alright, enough complaining. There are almost certainly people who will be interested who don't know what limits are, so let's start over. If you plot 1/x on a Cartesian grid, it will form a smooth line that as you go down from one gets higher and higher and no matter how long a line you draw, you'll never quite reach the zero line. On the negative side, it gets further and further down in an inverted mirror image, also never reaching the zero line as far as you can draw.


As a result you get not one but two lines which are both infinite in both directions but apparently discontinuous. But the thing is maybe they're not: x=y is a continuous function, an ordinary diagonal line, and the inverse of a continuous function is usually a continuous function. There's also nothing particularly saying it can't be continuous, since after all our objection is that we can't find the point of continuity.
Assuming it is continuous, there are two possibilities: a) the point of continuity is outside the realm of space and time, i.e. not a real number, which is the consensus that was settled on in the eighteenth century, although even Leibniz and Newton were exploring other possibilities from the moment they defined limits; or b) the value of the function at 0 is all the numbers simultaneously, which short of some exceptional trick of philosophy breaks the rules for functions.

I was precisely alluding to that trick of philosophy with my 1=0(|R|) flourish. This should also remind you of the issue where if you allow 1/0 to be a valid number then every number is equal to every other number. So my answer to the dilemma is both. But let's not skip ahead.

Leibniz and Newton developed limits to find the slope of a curve at any given point. It works because a curve is by definition continuous, and a function S(f(x)), which finds the slope of f(x) at x, will be continuous at all points where f(x) is continuous, with the conspicuous exception of the point where f(x)=0 (where it will do that weird hyperbola thing, just like 1/x).

If this is reminding you of other points where "scientists" would turn red and call you irrational because they can't find it but you think it might exist, it should.

So, in the very least, while we have to stop talking about it around the panting guy in the lab coat, we can tell him that well 0 is 0 no matter what you multiply it by so maybe it's just weird, but once he leaves we can admit that if 1/x is continuous at 0 it's got to be whatever the limit as x approaches 0 is, because there's no reason why that lim f(x) as x approaches n = f(n) iff f(n) is continuous should change just because the number isn't real. That we can think of right now, but if there's a rat, we'll smell it. Well, lim 1/x as x approaches 0 turns out to be infinite. Specifically, the aforementioned |R|.

Taking the leap of faith that a) 1/x is continuous and b) |R| and -|R| can be the same number, because the limit from the negative side is -|R| and if those are different the function isn't continuous, and ignoring the possibility for now that it's continuous by being simultaneously every number from |R| to -|R| because we don't want to break too much, we come to the following proof:

x->0 lim 1/x = |R| = -|R| because of the definition of a continuous function

1/0 = |R| because of that whole limit on a continuous function thing

From here we can go all sorts of funky places. We might be tempted to break down the walls between numbers with something like this.


1(|R|) = -1(|R|)

and remove |R| from both sides leaving 1=-1, but that's for nothing when we see that


which simplifying gives us


which is obvious enough, but it means that when we divided both sides by |R|, really we just multiplied by 0, and 1(0) = -1(0) is much less impressive. In plainer terms, it means that neither 0 nor |R| are on one side of the spectrum or another, which is not that surprising. We'll also be tempted to do this:

1/0 = |R|
like we said, so if we multiply both terms by zero, we get
or moving some parentheses around
but then we have to decide what 0/0 is, and that's another headache.


I'm really glad you asked that question.

I'm pretty sure this works out if you math it enough, at least I have a very strong recollection of it, but my head is throbbing. I'm sorry, but I'll have to come back to this another day, unless someone comes by in a blaze of white light and explains what I missed.

I'm pretty sure I've been satisfied regarding the proof of these equivalences for a long time, and if there's a flaw in the logic, I'm just discovering it now.

Thank you, Ghislain.

07-03-2012, 09:04 PM
Just in the back of my mind isn't it that what you say holds true only as long as x <> 0 and only while x -> 0 ?

It's a bit above me :(


Bel Matina
07-03-2012, 09:41 PM
Actually, those questions sort of prove you are following it :).

If 1/x is continuous (that is, a smooth, unbroken line) at 0, which was our hypothetical condition, then it all works out. It's a shame the picture didn't load, because the graph of 1/x makes it a lot easier to understand what's going on.

What's confusing about the negative number is that the limit of 1/x as x approaches zero from the negative side is apparently negative infinity, that is it goes down in the opposite direction from the curve on the other side of zero. This isn't a problem if the "infinity point" is the same point in the positive as the negative direction - in other words, if |R|=-|R|, effectively making the Cartesian grid a peculiar sphere or a weird.... hooped... topography. Frankly I can't quite wrap my head around it, or figure out if it would be the same infinity point on both axes, but even if we're just dealing with a Cartesian line you get a circle with two poles, 0 and |R|, both of which like to break the rules.

Basically the math works out as long as what is going on with the 1/x graph is it's going so far up as x approaches 0 from the right that it comes out the bottom once it reaches the negative numbers. You could also get it to be continuous without |R| being equal to -|R|, but I didn't want to go there and now it seems I can't remember how. Even if you did, 1 would have to equal 2 and so |R|=-|R| would hold anyway.

I see no reason to suspect I lost you at any point. That burning sensation in your frontal lobe is supposed to happen. Or at least it does. Or at least it's considered par for the course in professional mathematics. Then again there are proteins in your synapses that your nerves need to fire, and if you push it when they're out it's probably like running your car with no oil. Anecdotally speaking, mathematicians tend to burn out some time in their thirties and not be able to do professional math anymore. And there's that whole movie Pi which is really about something. You know your field is arcane when a movie about it has to use gematria as a more accessible shoehorn. So maybe haste is the enemy of the work.

I'm glad I went into linguistics instead.