x = x + x
y = y * y
z + z = z * z
The bean analogy is simply the basis of Peano's axioms, in intuitive (but accurate) terms. "zero" is an empty pot, etc.
The correspondence between 2 of anything is quite natural. If one cave man asked another how many arrows he had, if he put two beans in a pot I think the other would understand. That is 2.
Cute! Took me a few minutes to figure out what meant but I finally got it:
"x= 0" is the number satisfying x= x+ x.
"y= 1" is the number satisfying y= y*y.
"z= 2" is the number satisfying z+ z= z*z.
However, I have to point out that 0= 0*0 and 0+ 0= 0*0 also.
true enough, but one could define + and * as the ring operations of the initial object in the category of all (small) rings. presumably, "=" needs little explanation.
then we have:
(y+y) + (y+y) = y + y + y + y = y*y + y*y + y*y + y*y = y*(y+y) + y*(y+y) = (y+y)*(y+y)
indicating that z = y+y is "a" solution to z+z = z*z.
an interesting question: can one give examples of a ring R where 1+1 ≠ 0, but there is a solution to z+z = z*z besides 0 and 1+1?
Thanks for your reply. Your question* is interesting but no more relative to original question than 2 is the solution of x^2=4.
*"an interesting question: can one give examples of a ring R where 1+1 ≠ 0, but there is a solution to z+z = z*z besides 0 and 1+1?" One might give matrices or complex numbers a try.
EDIT: Personally, I thought posts 14 & 16 were the best replies to the original question.