Re:Visual Group Theory: What is a group? (2016) [video]
by mbrock on Saturday February 10, 2018 @06:48AM
(#16346921)
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...
Re:Visual Group Theory: What is a group? (2016) [video]
by gowld on Saturday February 10, 2018 @11:22AM
(#16347927)
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.
Re:Visual Group Theory: What is a group? (2016) [video]
by posterboy on Saturday February 10, 2018 @06:19AM
(#16346852)
Oh, you mean apple counting? Well, you have to shift those apples close together to recognize them as a group of things.
Re:Visual Group Theory: What is a group? (2016) [video]
by OscarCunningham on Saturday February 10, 2018 @06:29AM
(#16346872)
Right, so it's still visual, but it's not an example of the original suggestion to only define groups as groups of permutations.
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.