Could someone possibly explain this definition (applied to fields) to me?
The principle of substitution: In a field F, we can, in any formula involving an element $\alpha\in F$, replace $\alpha$ by any other element $\alpha'\in F$ such that $\alpha'=\alpha$.
What is the difference between $\alpha'$ and $\alpha$? What is the motivation behind this principle?
Thanks.
I don't see how this differs from any other use of what has also gotten called "the rule of replacement". The difference between α′ and α could be a number of things. Generally speaking though, they will differ in their form, or how a formula gets evaluated to an element. Applying this principle, can make an expression shorter, longer, harder to evaluate, or easier to evaluate. For instance, in a field, say we have (a+((b*c)+(b*d))). Well, since distribution holds ((x*y)+(x*z))=(x*(y+z)), by the principle here (a+((b*c)+(b*d))) becomes (a+(b*(c+d))), which I think you would agree as a much easier expression to evaluate and work with in general.
If you look at certain logic texts, you'll find this principle outside the context of a field called "the rule of replacement", where "substitution" means something slightly different.