You can work the language a little to make them analogous, but that's not the point Gowers is making. Consider this instead:
"There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study."
I imagine Gowers's point to be that sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to "intriguing new ideas", but sometimes it just indicates that your assumptions are wrong.
Consider the equation x + 1 = x.
(Edited again: this example is wrong, and thanks to Kindly for pointing out why. CronoDAS gives a much better answer.)
Curiously enough, the Peano axioms don't seem to say that S(n)!=n. Lo, a finite model of Peano:
X = {0, 1} Where: 0+0=0; 0+1=1+0=1+1=1 And the usual equality operation.
In this model, x+1=1 has a solution, namely x=1. Not a very interesting model, but it serves to illustrate my point below.
...sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to "intrigu
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