Not understanding theories

Have you ever witnessed or experienced the moment when someone finally understands a scientific theory? In German, one uses phrases like „the coin has fallen“ or „she has got risen a light“ to describe it. What does happen in that moment? I think it is basically the second, where the picture used in the theory gets apparent. The backbone of theories are always pictures. Mathematical pictures in most scientific theories. A mathematical entity to represent something in the real world.

Understanding scientific theories is therefore based on understanding the properties of the entities used. Like for example a harmonic oscillator, which is a ridiculously simple picture used to represent thingies „swinging“. As a result, people don’t understand scientific theories if they lack understanding of the used mathematical pictures.

But even if one owns knowledge about the properties of a specific picture already, there still can be problems in understanding a theory using them. The very same mathematical object can appear in quite a lot of different forms. You can talk about matrices as operators transforming vectors or you can see matrices as bilinearforms mapping 2 vectors onto a number. Those are completely equivalent descriptions, but they impose completely different pictures in our mind. Even if you own a certain intuition for vectors, you might lack intuition for bilinearforms.

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