Dr Faustus of Modern Physics - Department of Speech, Music and ...

Chapter 8
Wellposed vs Illposed
I do 10 times as many errors as my students. But I correct myself 20
times as fast. (Hadamard)
8.1 Small Cause - Small Effect: Wellposed
A mathematical model where small perturbations of input data gives a small
effect of the output, is said to be wellposed or correctly-set. In a non-wellposed
or illposed model a small perturbation of input can give a big effect on the
output, which means that the output cannot be trusted. To work with
illposed models is like walking on a string all the time risking to fall off
at any small puff of the wind.
There are illposed problems, like walking on a string, but if you want to
form a coherent scientific theory of some phenomenon, you need a wellposed
mathematical model. This was pointed out by the French Mathematician
Jacques Hadamard in the 1902 article On partial differential equations and
their physical relevance, one of the most important mathematics articles all
times.
To give up wellposedness is like giving up rational thinking for magical
thinking with invisible ghosts controling the world beyond human understanding.
To say that a small cause can have a large effect in the model you are
dealing with, is to say that the model is illposed, and thus is not scientifically
credible.
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