I am sure you have all heard of the double zero degrees question.
Well, I have an answer.
We are going to be using Celsius and Kelvin to figure this out.

1) Convert 0 C to K: 0 + 273.15 = 273.15
2) From 273.15 which is equal to 0 C we double it with a result of 546.3 K, this is twice as much as zero degrees.
3) Convert it back in celsisus: 546.3 - 273.15 = 273.15 C

Therefore twice as much as 0 C is 273.15 C, interesting isn't it?

OK, in Mathematical language, we name that a "paradox", where something is surprising because of the way you tell it. You tell the truth twice, but once something is not stated properly, and you cannot use a "barbara" deductive logical way for your demonstration.

Here is where the problem coming from.
It's not "equal to", it's "defined as".
This is completely different, I will explain you why.
This is a very known concept in so-called "modern" mathematics, namely in Group Theory.
In group theory we study "relationship".
"a=b" is a relation. Moreover, it's an equivalence relationship.
An equivalence relationship is reflexive, symmetric and transitive.
Reflexive : a=a.
Symmetric : if a=b, then b=a.
Transitive : if a=b, and b=c, then a=c.

Another well known relationship is "a>b", like "4>3".
This is not an equivalence relationship, it's an order relationship.
It's not reflexive : 3>3 is false.
It's not symmetric : 3>2 is true, but 2>3 is false.
However it's transitive : if a>b and b>c, then a>c.
Example : if 3>2 and 2>1, then 3>1.

So, your way making your deduction is based on "273.15 which is equal to 0 C". This is badly stated, because you cannot use the "=" sign which leads you to think that the relationship is reflexive.
If you say "273.15 is related to 0 C by a non-reflexive relationship" so that 0 C is not related to 273.15, your final result is completely different.

By the way, I told you you used a barbara reasonment (bArbArA), my next lesson will be (not any more a Mathematical but) a philosophical topics. Remember Logics is a branch of Philosophy. The correct ways of demonstration something are not so much.
They are named :
Barbara
Ferii
Dario
Celarent
These are acronyms for more easily remembering a logical way of demonstrating things.
The most known one is Barbara (three A-type sentences, A-A-A, "Barbara" helps for remembering).
A-A-A uses three times the same logical structure.
Example :
I am a Human (A)
A Human has to eat (A)
Then I have to eat (A again).

Sorry for all College young people who are currently learning that in Philosophy class, hope the other ones were interested.

I love these kind of problems which look (not at all) physics, a little bit Mahtematics, and fully Philosopy because of Logics.

Regards
Yordan

I think the real lesson here is that statements like "twice as hot" just do not have much meaning with temperature unlike similar statements like "twice as far".

Feynmann once wrote about a science textbook he had reviewed which had a problem in it roughly as follows:



A chart showing the average temperatures of different colored stars was attached. To say nothing of the "green stars" issue, the question is deeply flawed. There is just no way that "total temperature" is at all meaningful. Temperature is a meaure of relative average motion and things get funny when you use it for too much.

I don't understand what is so paradoxic here.
To me it's quite obvious that Celsius can't be used as a reference in these cases, since its absolute zero isn't the same as its 0 degree value. There are just two different references for 0. That's exactly why Kelvin scale is used in scientific formulas instead of Fahrenheit or Celsius, I guess.

Even though I'm thoroughly thrilled to read yordan's very enlightening explanation, I'm tempted to think it's irrelevant for this particular paradox. Or more precisely, it's not complete.


You see, the source of this paradox lies in a simple rule of mathematics, which is:


If a + b = c, then the only way we can keep the equality while multiplying c by 2 is to multiply BOTH a and b by 2 as well, in the form of 2a + 2b = 2c


So the apparent mistake in the paradox at hand is that when we multiplied the Kelvin temperature by two, we only multiplied the original Celsius temperature WITHOUT multiplying the number we added to it (273.15) as well. You see, the original formula is:


C + 273.15 = K


So when c = 0, k = 273.15, and the formula becomes:


0 + 273.15 = 273.15


If we're going to multiply the right side (the K) by 2, we have to multiply both (0) AND (273.15) on the left side by two as well. Which will result in:


0x2 + 273.15x2 = 273.15x2 ----> 546.3 = 546.3


So it's not really a paradox, merely a miscalculation .



Cheers.

Another problem with this proceedure is the doubling of the temperature. What possible meaning could it have?

In Celsius or Farenheight the answer is not much at all. However the temperature in Kelvin is proportional to the heat energy in a material. So twice the temperature in Kelvin is twice the heat. So in that sense you can say that indeed 273.15 degrees Celsius is twice as hot as 0 degrees Celsius.

However this really isn't the whole picture because Temperature is not really about heat but about thermal equillibrium, that is, it is about the flow of heat from one thing to another. Poor boling water in a cup and the cup will soon be the same temperature as the water. but this does not mean that the cup has same amount of heat as the water even if the cup had the same mass or volume as the water because different materials have different abilities to store heat called heat capacity.

Any way, the point is that our perception of hot and cold is not about heat content but about the flow of heat. And the flow of heat by conduction is proportional to the difference in temperature. However since 0 degrees Celsius is considerably below body temperature it represents a negative heat flow from the body. So on the basis of heat conduction alone we might think it a good idea to make our zero degree temperature the same as body temperature. Then at 20 degrees above body temperature we would see twice as much heat flowing into the body as only 10 degrees above body temperture. Ha Ha...

This is funny because I am sure that everyone knows that 95 degrees F (35 Celsius) is not what we would call a comfortable temperature. This is because the human body produces a considerable amount of waste heat that it has to get rid of and so at 95 degrees (35 Celsius), since conduction no longer works the body uses the evaporation of sweat in order to expel this waste heat.

Ok suppose a comfortable temperature is 70 degrees Farenheit or about 20 degrees Celsius. So maybe a meaningful way to speak about the temperature being twice as hot would be the difference in heat flow from what we have at this comfortable temperature. So at 25 degrees Celsius (77 F) we would have about a third less the conduction of heat as at 20 C and at 30 degrees Celsius (86 F) we would have two thirds less conduction. So perhaps we could say that 30 C is twice as hot as 25 C, what do you think?

Like wise at 10 degrees C (50 F) we would have two thirds greater conduction and at 0 degrees C (32 F) we would have four thirds greater conduction, so we could say that 0 degrees C is twice as cold as 10 degrees C.

Unfortunately there is another complication and this is the fact that conduction is not the only means of heat loss. There is also radiation, and this radiation depends on the fourth power of the temperature in Kelvin. That is the heat flow due to radiation is proportional to Thot^4 - Tcold^4 (with the two temperatures measured in Kelvin).

This post has been edited by mitchellmckain: Sep 14 2006, 08:08 PM

Not really. It just adds more information for his knowledge. With 'problems' like this it's all about defining terms.
Stuff is defined, but not equal to.

I enjoyed the read, Yordan.

This post has been edited by Alegis: Sep 15 2006, 06:31 PM

Hehe... one party says his explanation is correct but not someone else's. And then someone defends the supposedly incomplete explanation. Recipe for all out war???

Nay, in fact, I think all the explanation is correct actually. They are just looking at the problem in different ways. Of course, tamer3kz explanation is the easiest to follow, since it involves easy mathematical equation. However, there is a need to involve the definition and also the logic/philosophy to fully understand such problems. As mentioned above by evought, what does "twice as hot" means? In "normal" mathematics, we know that multiplication is just another form of addition, that is 2*3 is actually 3+3. But when it comes to temperature, when you add volume A of water with 1C and volume B of water with 2C, do you get volume of water with 3C, that is 1+2 = 3? I don't think the equation is correct in the situation I just mentioned, even though 1+2=3 is definitely correct mathematically. This is just to illustrate the point of knowing the underlying logic/definition when applying mathematical equations.

lol ... Well, I know for sure that I never said my explanation was correct and that somone else's is wrong. I try to be VERY careful about these matters, since I know that neither my study nor my knowledge qualify me to make such judgements . But occasionally, I have a certain point of view that I like to share with others, and hope it'll add something to what the others have to say, not replace it. I actually learnt quite a lot reading this topic. As for war, lol, I don't see that happening any time soon .


And of course, everyone who mentioned the importance of definitions in these kinds of situations are very right. I'm a definition-freak actually, and so I have to agree with you guys .