Why are multiplication and division not allowed when using the interval scale? This is what I already know:
There is no true zero value on interval scales like temperature.
 The zero value doesn't state that temperature becomes unavailable at that point.
Zero is just a human defined number which represents a certain level of temperature.
This is all because temperature is a relative scale.
What I'm confused about:


*

*Why can't I say 40$^\circ\!$C is twice as heat as  20$^\circ\!$C?  

*What about weight? It's relative, but I can say 40 kg weighs twice as much as 20 kg.
 A: First, weight is not an interval scale, it is a ratio scale. It has a true 0 and you can say 40 kg is twice 20 kg.
Second, the reason you cannot say 40$^\circ\!$C is twice as hot as 20$^\circ\!$C is precisely because it has no true 0. If we convert to degrees F, then 40 = 104 and 20 = 68 and 104 is not twice 68 - but the temperatures are the same.  There are temperature scales with a real 0 - Kelvin is one - where 0 is absolute 0, where all atomic motion ceases. 
Third "relative scale" is not a term I've seen used. The correct term is "Interval" and it is different from ratio.
Finally, for more on these issues (at a very nontechnical level), see my blog post: Stevens' typology and some problems with it
A: Just to add some further intuition via an example to Peters correct answer, consider the following case:
Suppose for temperature we defined what is currently -5$^\circ\!$C as 0$^\circ\!$C (i.e. shifted the scale by 5). This is allowed as the scale is arbitrary as you stated. Well now we have 
20$^\circ\!$C $\rightarrow$ 15$^\circ\!$C 
40$^\circ\!$C $\rightarrow$ 35$^\circ\!$C
And clearly 35 is not double 15 proving that our choice of 0 dictates our interpretation of double.
A: I would write this as a comment to seraffej's answer, but it turned out to be longer than allowed for comments, hence the answer.
I found seraffej's explanation very helpful, thanks. With interval scales I know (?) that the distance between values are linear. That is, the distance between 10 and 20 is same as the distance between 30 and 40 (please someone correct me if I'm wrong). Similarly, the distance between 10 and 30 is twice as much as the distance between 10 and 20. This sounds as if we could have division and multiplication with interval scales, too. Nevertheless, as the seraffej's example clarifies, divisible distance does not mean divisible values.
In the example, the distance between 40 and the arbitrary zero is always twice as much as the distance between 20 and the same arbitrary zero, no matter where we shift the zero point. On the other hand, the value 40 cannot be said twice as many as the value 20, since their exact interpretation relies on the zero point (40 becomes 35 and 20 becomes 15 when zero is shifted). Very interesting indeed.
