> In Dauben's biography of Abraham Robinson (page 103 of the first > printing) a quasi-field is said to be a commutative field with > distribution > > a(b + c) = ab + ac (1) > > replaced by > > a(b_1 + b_2 + ... + b_n) = ab_1 + ab_2 + ... + ab_n. (2) > > But, if in (2) we put b_3 = ... = b_n = 0, and apply the rules > > x + 0 = x and x0 = x > Why does x0 = x?
> we get (1), don't we?
You can't use x0 = 0 until you prove it for quasi-fields.
> So a quasi-field is a field.
No, unless n = 2. What you get is a0 = na0; 0 = (n-1)a0; a(b + c) = a(b + c + (n-2)0) = ab + ac + (n-2)a0 = ab + ac - a0.
a0 = ab + ac - a(b + c)
> Dauben writes that quasi-fields have the property > that the zero element for addition posses an inverse. > If for all a, a0 = 0, then it's a field. If some a with a0 /= 0, then 1 = (a0)^-1 a0 = 0^-1 a^-1 a0 = 0^-1 0.
1*0 = 0; 2*0 = (1 + 1)0 = 1*0 + 1*0 - 1*0 = 0.
> I'm intrigued!
What's an example of a quasi-field?
> I know nothing about these things beyond what Dauben > writes. Googling 'quasi-field' finds > http://en.wikipedia.org/wiki/Quasifield, but that is something else. > Dauben refers to: > > Robinson, 'On a certain variation of the distribution law for > commutative algebraic field' Proceedings of the Royal Society of > Edinburgh, 61, 1941, pp 93-101. > > to which I have no access. > > -- > When a true genius appears in the world, you may know him by > this sign, that the dunces are all in confederacy against him. > Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting >