Société de Calcul Mathématique SA

Robust mathematical modeling -3-


We continue here the description of our joint Research Program with several Companies, Institutions and Universities.

The basic rule of real-life mathematics

There is one single rule, simple but essential: no counter-examples known !

Never try to bring a precise answer to a precise problem

Before we explain the reasons of this rule and its pertinence, let us observe that it goes against the education received by all mathematicians, and against the education they deliver in their courses. Indeed, any mathematician has been taught: “try to make things precise”, and then “try to come with a precise answer”. So there is a considerable cultural gap between any academic education and this basic rule. This gap explains why mathematicians find it so difficult to help in real-life problems, and also why they are so rarely consulted for such problems.

Now, let's turn to the rule. It says that if you try to bring a precise answer, not only you fail, but moreover you make yourself a fool. You lose all credit. You will be regarded as someone who understands nothing, at best a dreamer, at worst an arrogant confined in his equations, trying to manage the world in his own way. Don't ever do that !

Indeed :

  1. There is no such thing, in real life, as a precise problem. As we already saw, the objectives are usually uncertain, the laws are vague, the data are missing. If you take a general problem and make it precise, you always make it precise in the wrong way. Or, if your description is correct now, it won't be tomorrow, because some things will have changed.

  2. If you bring a precise answer, it seems to indicate that the problem was exactly this one, which is not the case. The precision of the answer is a wrong indication of the precision of the question. There is now a dishonest dissimulation of the true nature of the problem.

  3. If you bring a precise answer, one can build a software from this answer, and then build some automates which will use the software. In other words, your solution may become totally computerized, totally automatic. But nobody wants to lose control, and give it to some machine ! There are extremely good reasons for that: first, people might lose their jobs, if they can be replaced by machines. And second, people think in general that machines work fine when everything is fine, but what if something unpredicted happens?

  4. The Agency, or the Company, which asks the question has worked on it for some years, sometimes 10 or 50 years. What will they think if a bunch of mathematicians say “we will solve your question” ? The engineers know their jobs, and the mathematicians are new to this job. To pretend you will bring a precise answer is stupid and deeply arrogant. If you bring a precise answer, it will certainly be to a wrong question.

  5. If you bring a precise answer, it will be used against you : it is very easy to show that this answer requires precise assumptions, which are not satisfied in practice. Then, the Agency or Company which gives the contract will conclude that you did not answer the question and will terminate the contract. This answer, in general, will have a very negative effect, no matter how bright it is.

  6. If you bring a precise answer, it seems to indicate that you consider this answer as best, perhaps as unique. So, it seems to indicate that you want to take responsibility, to take the decision. But your task is not at all to take a decision. Those who are in charge will be deeply offended.
Let's now turn the rule into a positive statement :

The mathematicians' task is to help take a decision.

Considering a variety of situations (not a single one), mathematicians will try to associate quantitative information to each situation, sometimes in a probabilistic manner. For instance, they might say : in such situation, if you take this decision, the cost will be X, in this situation, it will be Y. In other cases, they will draw charts, maps, on which a large number of possible solutions are indicated, with their advantages and disadvantages. Some solutions, for instance, might appear as curves in red, others in orange, best ones in green, and so on, so as to preserve the freedom of choice of those who are in charge of the decision.

Let's take a concrete example.

There is no such thing as a BEST itinerary between Kent State University and Case Western Reserve University, Ohio. First, it depends whether you have a car, a truck, a bus, or a bicycle. Second, it depends on traffic conditions, roadwork, weather, and so on. Third, it depends on the time of the day, on the day in the week, on the week in the month, on the month in the year. Fourth, it depends on what you want : do you want to go fast ? to limit gas consumption ? to see nice houses ?

So, if you come out with a precise itinerary, it will not be “a best”, since you do not know in what sense. But anyway, you should come out with some itinerary, as all these GPS-based computers do. This itinerary is obtained by some coarse optimization, taking for instance the simplest path (as many highways as possible) ; it won't adjust to specific cases. But if you get lost, if you are blocked by some road works, you can use the device again : it works fast.

And, on this example, we see the basics of a good solution : instead of a device which takes as input infinitely many parameters and computes some optimum within hours, and will finally be useless and costly, a useful device will consider only few parameters and will bring very quickly coarse solutions.

In the next article, we investigate this question: how are decisions taken in everyday life ? Is it by means of mathematical modeling ? Please click here

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