# Why are laws of physics always of product forms?

A first observation is that all the extant laws of physics are of product forms. This phenomenon is somewhat intriguing. The question is: why do law of physics always take, instead of a sum of two terms, sum-of-multiple-term (i.e., product) forms?

Counterexample exists in other disciplines of natural sciences. For example, biology has the White's formula (http://www.maths.ed.ac.uk/~aar/papers/eggar.pdf). Since mathematics investigates every possibility, certainly it has results involving sum of exactly two terms.

I am seeking after a reason other than technical reasons. For example, the importance of Lorentz transformation in physics does not lie in its mathematical necessity but in its physical implications, as revealed by A. Einstein. H. Poincare, E. Mach, and several others had realized the concept of relativity before Einstein, but, as the paper The structure of thoughts'' (For now I cannot recall the exact title) published in Nature points out, discerning a concept is not equivalent to discerning its meaning. It is Einstein that discerns the meaning of relativity.

• Definitely one way to look at it is in terms of units. You can't add quantities with different units. Also, you can't really compare biological laws with physical ones. I'm not sure how to phrase this, but biological laws are often more often trends than explanations. – jhobbie Jul 8 '14 at 3:51
• There are certainly plenty of important equations in physics that involve sums and differences... why did you think otherwise? – David Z Jul 8 '14 at 3:56
• I agree on the units idea. There is a really interesting form of problem solving called dimensional analysis that is rarely used these days. But, more than products, mechanics is what you might call second order or quadratic. The laws of mechanics involve second derivatives (acceleration). – C. Towne Springer Jul 8 '14 at 4:07
• A better question, famously discussed by Wigner, is why physical laws should have simple relationships of any sort. – rob Jul 8 '14 at 13:48
• @Brian:The short answer and simplest possible (for many cases you might be thinking about) is that the equations look the way they do because we want to know how much something there is per unit of something else (meters per second, kilograms per cube meter, charge per second, etc.). These equations are constructed in such a way in order to allow as to visualize and compare given phenomenons/variables. – bright magus Jul 8 '14 at 15:20