Growth and Decay, Law or not? The differential equation for decay that applies to radioactive decay is: 
$$dN/dt=-kN$$
for a positive constant k and number of particles N. My question is: is this, strictly speaking, a "Law"? I have seen this differential equation refered to as a "Law" and sometimes not, so what is the deal?
Thanks in advance for your help.
 A: A scientific law is a statement that concisely states an observation about nature that is true for a wide variety of situations. The important part is that the statement is about observations and experiments; it is not an attempt to explain the phenomena. That's a theory's job.
For example, the law of conservation of mass is true for all chemical reactions. Kepler's laws of planetary motion are also examples. I think your statement, that the rate of decrease of the amount of radioactive substances is proportional to the amount of the radioactive substance, is general enough to be called a law since it applies to all radioactivity. The equation does not explain how radioactivity works, but summarizes what experiments tell us about it.
One criteria for being a law would be if the statement would constitute a strict test for a theory to explain. It took Newton's theory of gravity to explain Kepler's laws. It took the Standard Model of particle physics to explain radioactivity.
In any case, words like "law," "theory," and even "science" can have definitions that have fuzzy boundaries and edge cases. Newton's theory of gravity is summarized by the inverse-square law, itself needing Einstein's General Relatively theory for explanation.
A: Maybe your are looking for a more physical explanation. 
Imagine you have a material that consists of identical atoms. The atom cores are not stable and therefore we call the material radioactive. With quantum mechanics you can calculate the probability that a atom core that has not decayed yet will decay. If you do this you will find a constant probability! 
In order to clarify this imagine the situation where I have one atom core, which can decay at any moment. Now I wait 5 minutes and look if it decays. Imagine it does not decay. Now I take a take another atom core at but it beside it. Is the probability that the first atom core decays sooner that the second one greater than 50%. No, it is exactly equal to 50%,  because their decay probabilities are the same. 
So, historically the differential equation in your post, might have been an empirical law that some experimentalist postulated because it's solution fitted the data well. However, it has a strong theoretical foundation. 
(I would call emperical and theoretical formula's both laws of nature by the way.)
A: Physics is about quantified observations of nature, modeled mathematically. The mathematical models are rigorous and self consistent but in order to connect to measurements extra postulates are needed, which define the connection of the mathematical formulae to data. Laws are parts of these postulates. In the same way that axioms are not provable within a mathematical theory, physics laws are not provable through the mathematical model, they are part of the physical axioms which pick up the solutions that are relevant to the data.
Your example may be called a law by some, but it is not crucial to the model, it is not an axiom or postulate, it is part of the output of the mathematics, in my opinion. 
