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Mathematics LibreTexts

2.1: Models in Science and Engineering

  • Page ID
    7772
  • [ "article:topic", "authorname:hsayama" ]

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    Science is an endeavor to try to understand the world around us by discovering fundamental laws that describe how it works. Such laws include Newton’s law of motion, the ideal gas law, Ohm’s law in electrical circuits, the conservation law of energy, and so on, some of which you may have learned already. A typical cycle of scientific effort by which scientists discover these fundamental laws may look something like this:


    1. Observe nature.
    2. Develop a hypothesis that could explain your observations.
    3. From your hypothesis, make some predictions that are testable through an experiment.
    4. Carry out the experiment to see if your predictions are actually true.

    • Yes→Your hypothesis is proven, congratulations. Uncork a champagne bottle and publish a paper.

    • No→Your hypothesis was wrong, unfortunately. Go back to the labor the field, get more data, and develop another hypothesis.

    Many people think this is how science works. But there is at least one thing that is not quite right in the list above. What is it? Can you figure it out?

    As some of you may know already, the problem exists in the last part, i.e., when the experiment produced a result that matched your predictions. Let’s do some logic to better understand what the problem really is. Assume that you observed a phenomenon P in nature and came up with a hypothesis H that can explain P. This means that a logical statement H → P is always true (because you chose H that way). To prove H, you also derived a prediction Q from , i.e., another logical statement H → Q is always true, too. Then you conduct experiments to see if Q can be actually observed. What if Q is actually observed? Or, what if “not Q” is observed instead?

    If “not Q” is observed, things are easy. Logically speaking, (H → Q) is equivalent to (not Q → not H) because they are contrapositions of each other, i.e., logically identical statements that can be converted from one to another by negating both the condition and the consequence and then flipping their order. This means that, if not Q is true, then it logically proves that not H is also true, i.e., your hypothesis is wrong. This argument is clear, and there is no problem with it (aside from the fact that you will probably have to redo your hypothesis building and testing).

    The real problem occurs when your experiment gives you the desired result, Q. Logically speaking, “(H → Q) and Qdoesn’t tell you anything about whether H is true or not! There are many ways your hypothesis could be wrong or insufficient even if the predicted outcome was obtained in the experiment. For example, maybe another alternative hypothesis R could be the right one (R → P, R → Q), or maybe H would need an additional condition K to predict P and Q (H and K → P, H and K → Q) but you were not aware of the existence of K.

    Let me give you a concrete example. One morning, you looked outside and found that your lawn was wet (observation P). You hypothesized that it must have rained while you were asleep (hypothesis H), which perfectly explains your observation (H → P). Then you predicted that, if it rained overnight, the driveway next door must also be wet (prediction Q that satisfies H → Q). You went out to look and, indeed, it was also wet (if not, H would be clearly wrong). Now, think about whether this new observation really proves your hypothesis that it rained overnight. If you think critically, you should be able to come up with other scenarios in which both your lawn and the driveway next door could be wet without having a rainy night. Maybe the humidity in the air was unusually high, so the condensation in the early morning made the ground wet everywhere. Or maybe a fire hydrant by the street got hit by a car early that morning and it burst open, wetting the nearby area. There could be many other potential explanations for your observation. 

    In sum, obtaining supportive evidence from experiments doesn’t prove your hypothesis in a logical sense. It only means that you have failed to disprove your hypothesis. However, many people still believe that science can prove things in an absolute way. It can’t. There is no logical way for us to reach the ground truth of nature1.

    This means that all the “laws of nature,” including those listed previously, are no more than well-tested hypotheses at best. Scientists have repeatedly failed to disprove them, so we give them more credibility than we do to other hypotheses. But there is absolutely no guarantee of their universal, permanent correctness. There is always room for other alternative theories to better explain nature.

    In this sense, all science can do is just build models of nature. All of the laws of nature mentioned earlier are also models, not scientific facts, strictly speaking. This is something every single person working on scientific research should always keep in mind. I have used the word “model” many times already in this book without giving it a definition. So here is an informal definition:

    A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical).

       As a cognitive entity interacting with the external world, you are always creating a model of something in your mind. For example, at this very moment as you are reading this textbook, you are probably creating a model of what is written in this book. Modeling is a fundamental part of our daily cognition and decision making; it is not limited only to science.

    With this understanding of models in mind, we can say that science is an endless effort to create models of nature, because, after all, modeling is the one and only rational approach to the unreachable reality. And similarly, engineering is an endless effort to control or influence nature to make something desirable happen, by creating and controlling its models. Therefore, modeling occupies the most essential part in any endeavor in science and engineering.

    Exercise \(\PageIndex{1}\)

    In the “wet lawn” scenario discussed above, come up with a few more alternative hypotheses that could explain both the wet lawn and the wet driveway without assuming that it rained. Then think of ways to find out which hypothesis is most likely to be the real cause.

    Exercise \(\PageIndex{2}\)

    Name a couple of scientific models that are extensively used in today’s scientific/engineering fields. Then investigate the following:

    • How were they developed?

    • What made them more useful than earlier models?

    • How could they possibly be wrong?

         

    1This fact is deeply  related to the impossibility of general system identification, including the identification of computational processes.