4 ms·

I'm not sure I unterstand. Are you saying addition was not a natural transformation, in contrast to shifting? I do have a Problem reasoning about binarry addit

• OscarCunningham 9y ago
  I mean that addition is a group operation, and that it's not easy to think of it as a transformation. The natural way of thinking about 2+2=4 is that two things and two more things make four altogether. No transfomations in sight. Of course by Cayley's theorem you can think of this group as made by transformations. The element n is the transformation that "shifts" the numberline n steps in the positive direction. Then 2+2=4 means that shifting by two and then two again results in a shift of four. But this is a very forced way of thinking about addition.
  • posterboy 9y ago
    Oh, you mean apple counting? Well, you have to shift those apples close together to recognize them as a group of things.
    • OscarCunningham 9y ago
      Right, so it's still visual, but it's not an example of the original suggestion to only define groups as groups of permutations.
      • posterboy 9y ago
        It isn't?
  • mbrock 9y ago
    That does seem like the intuition you would like to capture by talking about integers as a group, maybe? Kind of like how if you talk about points as elements of a vector space, it's useful to think of them as translations from the origin...
  • gowld 9y ago
    Why do you need group theory at all when thinking about addition? When you wonder "why does subtraction work" (guaranteed by the group structure), then thinking about shifts is a nice visual illustration, especially when you have modular arithmetic.