by Hari Balasubramanian
Thoughts on the differences in math applied to the physical and social sciences.
The quote in the title is attributed to the statistician George Box. The term ‘model' could refer to a single equation, a set of equations, or an algorithm that takes an input and carries out some calculations. Box's point is that you can never capture a physical or biological or social system entirely within a mathematical or algorithmic framework; you always have to leave something out. Put another way, reality yields itself to varying degrees but never completely; something always remains unknown that is not easily describable.
And in any case, for the practical matter of achieving a certain outcome that extra effort may not be necessary. If the goal is to put a satellite into orbit, the equations that define Newton's laws of motion and gravity, though not 100% correct, are more than sufficient; you don't need Einstein's theories of relativity though they would provide a more accurate description. But if the goal is to determine a GPS device's location on earth you do need relativity. This is because for an observer on earth a clock on an orbiting satellite ticks at a different speed than a clock on earth and if the necessary adjustments are not made, your phone's location estimate will be inaccurate.
So there is this art in modeling, this choosing of some aspects and ignoring others, trying to create the the right approximations. As Box notes: "there is no need to ask the question 'Is the model true?'. If 'truth' is to be the 'whole truth' the answer must be 'No'. The only question of interest is 'Is the model illuminating and useful?'"
Models vary widely in the amount of truth they capture. In the engineering disciplines that exploit physical laws – mechanical, chemical, civil, electronics and communications engineering – the test of a model is whether the mathematical answers match empirical observations to the degree of precision needed and whether the results can be reproduced again and again.
