What you learn to do is take a bunch of nouns—1, 2, 3, etc.—and a bunch of verbs—addition, subtraction—and make sentences. “1 + 2 = 3.”
I still have no idea how to express this in a picture of objects and arrows. I suppose that 1, 2, and 3 are objects. Is the addition an arrow? But an arrow has only one start and one end...
More meta: You have already provided the readers “motivation” in the two introductory articles. It is not necessary to add more hype in each article. Yes, I already heard that you can do everything in category theory, and I am willing to suspend disbelief. Now I am curious how specifically it can be done.
It’s possible to construct a category where numbers are objects and where the arrows are “plus zero” (identity), “plus one”, “plus two” and so on. (“Numbers” here might look like it stands in for “natural numbers”. But actually, as described, it would work just as well with “real numbers”, “complex numbers”, “integers greater than three”, “numbers whose fractional part is the same as the fractional part of e to five decimal places”… formally, any set which is “closed under addition of natural numbers”. Unless you pick a different way to operationalize “and so on”.)
Then the objects in “1 + 2 = 3” are in and three, and the arrow is “plus two”.
(If you picked “numbers” above to be “natural numbers”, then there’s a one-to-one correspondence between objects and “arrows from this object”, for any object. But I’m not sure if that’s important.)
More normally, “the set of numbers” would be an object all by itself, and the arrows would be the same as above, but all pointing from this one object to itself.
Neither of these sounds like what OP was trying to describe, but I don’t have an answer that does.
But then there would be no obvious connection between the number “two” and the arrow “plus two”. Also, no obvious connection between the “plus two” arrow doing from 1 to 3, and the “plus two” arrow going from 6 to 8. That feels like we can make a diagram that somehow represents the addition of integers, but we can’t derive new insights about addition from looking at the diagram, because most information is lost in the translation.
I guess what I meant was: I have no idea how to express 1+2=3 in a useful picture of objects and arrows.
Knowing that haskell I think the pattern to turn multiparty relations to two place relations is R(a,b,c,d,e,f,g) → R(S(b,c,d,e,f,g)) → R(S(T(d,e,f,g)) … R(S(T(U(V(X(Z(g)))))))
The connection between “+2“ and 2 would then be a function of +(2)=”+2”. You migth also need =(3)=”=3“ and then you can have =3(+2(2)) = “2+2=3” and maybe a T?(“2+2=3”)=False. In another style you would set it up that only true equations could be derived. Then one of the findings would be that any instance of +2(2) could be replaced with 4 and the mappings would still hold (atleast on the T? level). Mind you “2+2” could be a different object from “4″
I still have no idea how to express this in a picture of objects and arrows. I suppose that 1, 2, and 3 are objects. Is the addition an arrow? But an arrow has only one start and one end...
More meta: You have already provided the readers “motivation” in the two introductory articles. It is not necessary to add more hype in each article. Yes, I already heard that you can do everything in category theory, and I am willing to suspend disbelief. Now I am curious how specifically it can be done.
It’s possible to construct a category where numbers are objects and where the arrows are “plus zero” (identity), “plus one”, “plus two” and so on. (“Numbers” here might look like it stands in for “natural numbers”. But actually, as described, it would work just as well with “real numbers”, “complex numbers”, “integers greater than three”, “numbers whose fractional part is the same as the fractional part of e to five decimal places”… formally, any set which is “closed under addition of natural numbers”. Unless you pick a different way to operationalize “and so on”.)
Then the objects in “1 + 2 = 3” are in and three, and the arrow is “plus two”.
(If you picked “numbers” above to be “natural numbers”, then there’s a one-to-one correspondence between objects and “arrows from this object”, for any object. But I’m not sure if that’s important.)
More normally, “the set of numbers” would be an object all by itself, and the arrows would be the same as above, but all pointing from this one object to itself.
Neither of these sounds like what OP was trying to describe, but I don’t have an answer that does.
But then there would be no obvious connection between the number “two” and the arrow “plus two”. Also, no obvious connection between the “plus two” arrow doing from 1 to 3, and the “plus two” arrow going from 6 to 8. That feels like we can make a diagram that somehow represents the addition of integers, but we can’t derive new insights about addition from looking at the diagram, because most information is lost in the translation.
I guess what I meant was: I have no idea how to express 1+2=3 in a useful picture of objects and arrows.
Knowing that haskell I think the pattern to turn multiparty relations to two place relations is R(a,b,c,d,e,f,g) → R(S(b,c,d,e,f,g)) → R(S(T(d,e,f,g)) … R(S(T(U(V(X(Z(g)))))))
The connection between “+2“ and 2 would then be a function of +(2)=”+2”. You migth also need =(3)=”=3“ and then you can have =3(+2(2)) = “2+2=3” and maybe a T?(“2+2=3”)=False. In another style you would set it up that only true equations could be derived. Then one of the findings would be that any instance of +2(2) could be replaced with 4 and the mappings would still hold (atleast on the T? level). Mind you “2+2” could be a different object from “4″