We can explicitly construct such numbers using 2x2 matrices. These are called "dual numbers"[0], and the matrix representation is essentially an analoge of the

We can explicitly construct such numbers using 2x2 matrices. These are called "dual numbers"[0], and the matrix representation is essentially an analoge of the same for complex numbers. So if you accept the complex unit, i, as a reasonable algebraic thing, then the "dual unit" shouldn't be too much of a stretch. Anyway, all that is completely fine in standard foundations. The LEM contradictions only come in when we impose other axioms. For example, in Anders Kock's Sythetic Differential Geometry[1], we essentially start out by assuming that every function is differentiable. This seems sorta crazy at first blush, and it does give contradictions, but the interesting bit as that those problems hinge solely on LEM. Thus, in a way, we're left with chosing between universal differentiability or LEM. Kock explores what happens if we pick the former, and it turns out to be pretty deep, interesting and useful. [0]:https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number [1]:http://home.imf.au.dk/kock/sdg99.pdf http://home.imf.au.dk/kock/sdg99.pdf