The Mpemba effect is a good example of both the limits and the usefulness of paradigms.

The Mpemba effect is the unusual phenomenon of hot water freezing more quickly than cold water. Generally, if you put a cup of hot water and a cup of cold water in the freezer, the hot water will often freeze first. There are a lot of explanations that people have come up with for the Mpemba effect, but generally they are all within existing paradigms.

Here are some of the unknowns / variables a person might look at.

1) How long has the water been at its current temperature? There may be some unknown property, a "temperature memory" that makes water that has been at a certain temperature for 5 minutes different than water that has been at that temperature for 2 hours.

2) Water may have memory for some other influence. Such as air pressure or additives that have been dissolved in it (but are no longer present).

3) More mundane factors may be ignored in some cases. Such as volume of the containers, depth, shape, etc.

Here is a paragraph about the melting point of sugar, from http://www.huffingtonpost.com/2011/07/26/sugar-melting_n_909712.html

"The sugar melting study showed that the reason scientists and cooks haven't been able to isolate a definitive melting point for sugar is that sugar doesn't melt—it decomposes. This means that, rather than melting at one definitive temperature, sugar can become a liquid at different temperatures depending on heating rate. If you heat sugar quickly, using extremely high heat, it will melt at a higher temperature than it would if you heat it slowly, using low heat."

 Any particular quality of something is on a continuum, but qualities themselves are also.

If you say someone is "short" or"tall", their "shortness" or "tallness" is a useful description because it puts that person on a specific place on a continuum, from short to tall, or tall to short.

When a description does not seem to be on a continuum then it is a fair bet that there is a greater problem with the paradigm that includes the description. For example if there was a  human quality called "being exactly 5 foot 6 inches" but that quality was not associated with any other quality, there is a sign of a problem. Someone might say "Is that person 5 foot 6 inches?", and if the answer is "no", then, in the faulty paradigm, that person would "lack" that quality, it would be a yes/no, a quality that is not yet understood.

Someone trying to explain the Mpemba effect has a certain number of variables to work with.

 Basically, they are limited to the already described laws of physics. You could say there are a hundred laws, or variables that can be used to solve the riddle. Those are the boundaries of the paradigm that exists.

In the case of the Mpemba effect, obviously those laws have to be expanded a bit. For example there is the quality of "heat" which is a central variable in the riddle, "hot" water vs "cold" water. Generally, heat is in the direction (on a continuum) of instability, and cold leans toward stability.

Things that are "hot" change more rapidly, they undergo chemical processes more quickly, etc, etc. Cold things are much more stable. Something frozen sufficiently will remain virtually unchanged for a long time. So there is already a relationship between heat, stability and time. A person could rephrase the riddle of the Mpemba effect as "Why does unstable (hot) water change its phase (freeze) more easily than stable (cold) water.

In other words "Why is something that is easy to change easier to change than something that is hard to change". It is not a riddle anymore. But is it accurate as a paradigm, and useful? People have the habit of assuming that their paradigms pre exist the underlying reality of nature. This is a result of the academic mind.

If a person has never seen a dog and you teach them about dogs they will have a surprise when they finally meet one. They will realize that they were not learning about dogs, rather they were hearing someone else reminisce about their experiences with dogs, under the guise of learning.

A person can generalize the following about paradigms. "The more concrete a specific element of a paradigm, the less reliable it will be in the long term". So, for example "1+1=2" is a very concrete element in the paradigms most people use for dealing with numbers. Because it is so concrete (i.e., lacks abstraction) it is not a reliable element. You would not be wise to build too much on top of it.

So why does hot water freeze more easily than cold?

Who knows. Maybe you can figure it out.

 

 

 Diagram from http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/phase.html

See also http://en.wikipedia.org/wiki/Enthalpy_of_vaporization and http://en.wikipedia.org/wiki/Enthalpy_of_fusion.