paradox question (1 Viewer)

[afta???]

ok ppl here's some algebra...

let w = a bottle full of water
a = a bottle full of air

(1/2) w = (1/2) a​

Now times both sides by 2.........

and you get: w = a



Therefore, a bottle full of water equals a bottle full of air

 

[Archman]

...i think i'll go back to sleeping.

 

[table for 1]

that only works because YOU LET half a bottle of water equal half a bottle of air. and when you say they equal each other, exactly what are you referring to? yes, if you were referring to volumes, then a bottle of water would have the same volume as a bottle of air [assuming they are the same sized bottles, same conditions such as environment, etc]. but if you are confusing matter with volume, then you're just confused

 

[withoutaface]

ok ppl here's some algebra...

let w = a bottle full of water
a = a bottle full of air

(1/2) w = (1/2) a​

Now times both sides by 2.........

and you get: w = a



Therefore, a bottle full of water equals a bottle full of air

The volume of 1/2 a bottle of water and half a bottle of air are the same, and so are the volumes of a full bottle of air and a full bottle of water. That post was dumb and the argument pointless...

 

[:: ck ::]

har har har i get jokes

 

[gordo]

a glass thats empty = a
a glass thats full = b
1/2(a) = 1/2(b)
x2
therefore a=b

therefore an empty glass is the same as a full glass

 

[SeDaTeD]

There's a slight difference between half an empty glass and a half empty glass.

 

[KeypadSDM]

A is an empty glass:

1/2 (A) = (A/2)

Therefore half of an empty glass is a glass which is half empty.

 

[Trev]

A is an empty glass:

1/2 (A) = (A/2)

Therefore half of an empty glass is a glass which is half empty.

*sweeps hand over top of head in gesture of stupidity"

 

[SeDaTeD]

Define axioms for glass algebra if anyone's really bored.

 

[table for 1]

i think we all agree this thread is a waste of time, and that you cannot say that a bottle of water equals a bottle of air. don't try to manipulate something logical with algebra. your logic is incorrect, which has made your algebra incorrect

if you let w = a bottle full of water,
a = a bottle full of air

then when you say
(1/2) w = (1/2) a

that is incorrect. why would half a bottle of water equal half a bottle of air? just becasue a bottle is half filled with water, does not mean that the other half must be air, and visa versa. when you say it like that, then the other half must be assumed to be a vacuum. and obviously, these two thngs are completely different
- a bottle half filled with water, half vacuum
- a bottle half filled with air, half vauum

ok, if we scrap the above paragraph and assumed that a bottle half filled with water is ALSO half filled with air, then
- a bottle half filled with water and other half is air = w/2 + a/2
- a bottle half filled with air and the other half is water = a/2 + w/2
and only then would they be equal. you were missing something from your equation

w/2 + a/2 = a/2 + w/2

therefore, a bottle half full of water and half full of air equals a bottle half full fo air and half full of water

x2

w + a = a + w
so, a bottle of water plus a bottle of air equals a bottle of air plus a bottle of water

in conclusion: let's close this thread and give the thread starter the evil eye

[i can't believe i bothered]

 

[PtolemysPenguin]

In other words, half empty (which is the same as half full) is not the same as half of empty, because half of empty is still empty.

 

[who_loves_maths]

Originally Posted by table for 1
i think we all agree this thread is a waste of time, and that you cannot say that a bottle of water equals a bottle of air. don't try to manipulate something logical with algebra. your logic is incorrect, which has made your algebra incorrect

if you let w = a bottle full of water,
a = a bottle full of air

then when you say
(1/2) w = (1/2) a

that is incorrect. why would half a bottle of water equal half a bottle of air? just becasue a bottle is half filled with water, does not mean that the other half must be air, and visa versa. when you say it like that, then the other half must be assumed to be a vacuum. and obviously, these two thngs are completely different
- a bottle half filled with water, half vacuum
- a bottle half filled with air, half vauum

ok, if we scrap the above paragraph and assumed that a bottle half filled with water is ALSO half filled with air, then
- a bottle half filled with water and other half is air = w/2 + a/2
- a bottle half filled with air and the other half is water = a/2 + w/2
and only then would they be equal. you were missing something from your equation...

actually, it doesn't even matter whether or not the a bottle that's half filled with water must have the other half filled with air or vice versa. in fact, it doesn't matter whatever the hell 'w' or 'a' equals to, just as long as one doesn't get confused by the difference between the abstraction of algrebra and the physical context in which it is applied here.

'(1/2)w' or '(1/2)a' can simply be intepreted as (literally) half a bottle of water or air [ie. a full bottle of water of air cut in half by a knife].
in this intepretation, the half bottle has half the initial volume but is still filled up with water or air. so it only makes sense that 2 such half bottles full of air will equal to 2 such half bottles full of water IF it is initially true, for whatever physical reason (which is not important here), to say that half a bottle of water = half a bottle of air.

the equality sign means nothing, it's not necessary to confuse it with any physical manifestations of equalities between a bottle of water and a bottle of air. they can in fact be made to be worth the same - eg. since it's not specified, then the size of the bottle containing the air can be significantly larger than the bottle of water, such that perhaps the cost of the larger bottle + the air is = the cost of the smaller bottle + the water. (only if price is at stake here).

if it's the water that counts, then the bottle containing the air can be large enough such that the number of water molecules in the water vapour of the air is equal to the number of water molecules in the smaller bottle with the liquid water...

(1/2)w = (1/2)a -----> w/a =1 ; ie. if a physical intepretation is needed, then it simply implies that the ratio of a certain property of the bottle with water and a property of the bottle with air is 1:1, ie. in equivalent proportions.
hence, w =a is then perfectly correct if the variables 'w' and 'a' represent the same quantities for before and after.

so if the initial equality of (1/2)a = (1/2)w is true, then there is no paradox here at all. if it's not, then the question is non-existent via self-contradiction.


the whole "paradox" plays on the fact that we are supposed to accept (1/2)w = (1/2)a as a true inequality and at the same time give physical reality to the pronumerals 'w' and 'a':

let w = true ; and, a = false...
thus, (1/2)w = (1/2)a -----> x2, w = a -----> true = false
hence, it follows that if the argument for a bottle of water being equal to a bottle of air is valid, then what is true is also false, meaning that a bottle of water is then NOT equal to a bottle of air, self-contradicting the initial "paradox" - then the incompleteness theorem dictates that the "paradox" be 'indeterminate'.


it all falls upon interpretation

 

[ogmzergrush]

actually, it doesn't even matter whether or not the a bottle that's half filled with water must have the other half filled with air or vice versa. in fact, it doesn't matter whatever the hell 'w' or 'a' equals to, just as long as one doesn't get confused by the difference between the abstraction of algrebra and the physical context in which it is applied here.

'(1/2)w' or '(1/2)a' can simply be intepreted as (literally) half a bottle of water or air [ie. a full bottle of water of air cut in half by a knife].
in this intepretation, the half bottle has half the initial volume but is still filled up with water or air. so it only makes sense that 2 such half bottles full of air will equal to 2 such half bottles full of water IF it is initially true, for whatever physical reason (which is not important here), to say that half a bottle of water = half a bottle of air.

the equality sign means nothing, it's not necessary to confuse it with any physical manifestations of equalities between a bottle of water and a bottle of air. they can in fact be made to be worth the same - eg. since it's not specified, then the size of the bottle containing the air can be significantly larger than the bottle of water, such that perhaps the cost of the larger bottle + the air is = the cost of the smaller bottle + the water. (only if price is at stake here).

if it's the water that counts, then the bottle containing the air can be large enough such that the number of water molecules in the water vapour of the air is equal to the number of water molecules in the smaller bottle with the liquid water...

(1/2)w = (1/2)a -----> w/a =1 ; ie. if a physical intepretation is needed, then it simply implies that the ratio of a certain property of the bottle with water and a property of the bottle with air is 1:1, ie. in equivalent proportions.
hence, w =a is then perfectly correct if the variables 'w' and 'a' represent the same quantities for before and after.

so if the initial equality of (1/2)a = (1/2)w is true, then there is no paradox here at all. if it's not, then the question is non-existent via self-contradiction.


the whole "paradox" plays on the fact that we are supposed to accept (1/2)w = (1/2)a as a true inequality and at the same time give physical reality to the pronumerals 'w' and 'a':

let w = true ; and, a = false...
thus, (1/2)w = (1/2)a -----> x2, w = a -----> true = false
hence, it follows that if the argument for a bottle of water being equal to a bottle of air is valid, then what is true is also false, meaning that a bottle of water is then NOT equal to a bottle of air, self-contradicting the initial "paradox" - then the incompleteness theorem dictates that the "paradox" be 'indeterminate'.


it all falls upon interpretation

Nice one mate!

 

[SeDaTeD]

aaarrrghhhh.
*runs away*