Why does math matters?

My research is typically related to solving some practical underlying engineering problem, e.g. modeling a specific system or process (e.g. speech modeling), building a better recognizer, controller, etc.

I often get the question of why do I think that mathematical analysis does matter (because I think that it does matter). (Some) people say to me: "At the end, it is engineering knowledge that matters. You have to try out the algorithms in order to decide which one to use."

I do not claim that engineering knowledge does not matter. Actually, I believe that when it comes to solve a specific problem domain knowledge (if available) matters a lot and one should use as much of it as it is feasible. Still, I think that mathematical analysis has an essential role: My view is that it is only mathematical analysis using which ``we will ever be able to identify promising or solid algorithms from the bewildering array of speculative proposals and claims that can be found in the literature''. The above citation is from the NDP book of Bertsekas and Tsitsiklis -- and I cited it because I could not agree any better with it.

To give you a (simple) example, when it comes to evaluate an algorithm, until there are no known conditions for its correctness it may well turn out that it is just not doing what it supposed to do (on specific examples), no matter how many simulations "proved" its correctness. And there are examples in the literature of when it turned out that a specific algorithm did not work as expected under the conditions that it was expected to work.

When conditions for the correctness of the algorithm are known, one can start checking if those conditions are met in the practical situation of interest. One can then design tests or use some known facts to prove or disprove if the conditions hold. If one is lucky, the conditions are met. Otherwise, one should better start thinking seriously about if the algorithm is still correct in the specific situation considered. One possibility is to derive the correctness of the algorithm that covers the case. Another possibility is that one identifies the reason of why the algorithm will not work under the specific conditions considered. In this case one may start looking for a better algorithm or design one by herself!

Another criticism that is also often heard is that the conditions of mathematical results (concerning physical systems) are often unrealistic. It is true that mathematical models typically provide only a crude approximation of the underlying physical processes. Of course, the belief is that our models capture useful insights and hopefully the most important characteristics of the underlying processes. If this holds, then we are on the safe side of the road. At this point, one must rely on engineering intuition and experience.

Can mathematical analysis be useful when the underlying conditions are not realistic? One should never forget that doing mathematics is a process similar to building up castles from bricks. So if one day it turns out that some conditions are not realistic then the next day we should try to extend the theory to cover more. During this, the results derived earlier (or at least the ideas) are most likely to be useful. Given all the exciting developments in mathematics during the last centuries and especially during the last decade, my belief is that even if our present models are sometimes overly simple, it is possible to extend current mathematical analysis to the level where someday it will prove to be useful - or at least I see no a priori reason of why this would not be possible. This is my working hypothesis at the moment. In the past, some successes have been achieved in science using this hypothesis and I don't think that the existence of some hard problems should make me change my view.

Having said all this, it should now be clear that I also believe strongly in the beneficial interplay between engineering insight and mathematical analysis. At this point it is also appropriate to recall the words of Ludwig Boltzmann: ``There is nothing so practical as a good theory.''

Last update: April 26, 2004