The Wikipedia page for group theory seems fairly impenetrable. Do you have a link you’d recommend as a good place to get one’s feet wet in the topic? Same with symmetry.
“Group” is a generalization of “symmetry” in the common sense.
I can explain group theory pretty simply, but I’m going to suggest something else. Start with category theory. It is doable, and it will give you the magical ability of understanding many math pages on Wikipedia, or at least the hope of being able to understand them. I cannot overstate how large an advantage this gives you when trying to understand mathematical concepts. Also, I don’t believe starting with group theory will give you any advantage when trying to understand category theory, and you’re going to want to understand category theory if you’re interested in reasoning.
When I was getting started with category theory, I went back and forth between several pages (Category Theory, Functor, Universal Property, Universal Object, Limits, Adjoint Functors, Monomorphism, Epimorphism). Here are some of the insights that made things click for me:
An “object” in category theory corresponds to a set in set theory. If you’re a programmer, it’s easier to think of a single categorical object as a collection (class) of OOP objects. It’s also valid and occasionally useful to think of a single categorical object as a single OOP object (e.g., a collection of fields).
A “morphism” in category theory corresponds to a function in set theory. If you think of a categorical object as a collection of OOP objects, then a morphism takes as input a single OOP object at a time.
It’s perfectly valid for a diagram to contain the same categorical object twice. Diagrams only show relations, and it’s perfectly valid for an OOP object to be related to another OOP object of the same class. When looking at commutative diagrams that seem to contain the same categorical object twice, think of them as distinct categorical objects.
Diagrams don’t only show relationships between OOP objects. They can also show relationships between categorical objects. For example, a diagram might state that there is a bijection between two categorical objects.
You’re not always going to have a natural transformation between two functors of the same category.
When trying to understand universal properties, the following mapping is useful (look at the diagrams on Wikipedia): A is the Platonic Form of Y, U is a fire that projects only some subset of the aspects of being like A.
The duality between categorical objects and OOP objects is critical to understanding the difference between any diagram and its dual (reversed-morphisms). Recognizing this makes it much easier to understand limits and colimits.
Once you understand these things, you’ll have the basic language down to understand group theory without much difficulty.
The Wikipedia page for group theory seems fairly impenetrable. Do you have a link you’d recommend as a good place to get one’s feet wet in the topic? Same with symmetry.
Thanks!
“Group” is a generalization of “symmetry” in the common sense.
I can explain group theory pretty simply, but I’m going to suggest something else. Start with category theory. It is doable, and it will give you the magical ability of understanding many math pages on Wikipedia, or at least the hope of being able to understand them. I cannot overstate how large an advantage this gives you when trying to understand mathematical concepts. Also, I don’t believe starting with group theory will give you any advantage when trying to understand category theory, and you’re going to want to understand category theory if you’re interested in reasoning.
When I was getting started with category theory, I went back and forth between several pages (Category Theory, Functor, Universal Property, Universal Object, Limits, Adjoint Functors, Monomorphism, Epimorphism). Here are some of the insights that made things click for me:
An “object” in category theory corresponds to a set in set theory. If you’re a programmer, it’s easier to think of a single categorical object as a collection (class) of OOP objects. It’s also valid and occasionally useful to think of a single categorical object as a single OOP object (e.g., a collection of fields).
A “morphism” in category theory corresponds to a function in set theory. If you think of a categorical object as a collection of OOP objects, then a morphism takes as input a single OOP object at a time.
It’s perfectly valid for a diagram to contain the same categorical object twice. Diagrams only show relations, and it’s perfectly valid for an OOP object to be related to another OOP object of the same class. When looking at commutative diagrams that seem to contain the same categorical object twice, think of them as distinct categorical objects.
Diagrams don’t only show relationships between OOP objects. They can also show relationships between categorical objects. For example, a diagram might state that there is a bijection between two categorical objects.
You’re not always going to have a natural transformation between two functors of the same category.
When trying to understand universal properties, the following mapping is useful (look at the diagrams on Wikipedia): A is the Platonic Form of Y, U is a fire that projects only some subset of the aspects of being like A.
The duality between categorical objects and OOP objects is critical to understanding the difference between any diagram and its dual (reversed-morphisms). Recognizing this makes it much easier to understand limits and colimits.
Once you understand these things, you’ll have the basic language down to understand group theory without much difficulty.