Find the Fallacy

Two mathematicians begin a discussion by assuming that A=B. Then they begin making changes. They add an equal term to both sides of the equation. A+A=A+B or 2A=A+B. Then they subtract an equal number from both sides. 2A -2B = A+B -2B. They look at this and agree that this all makes sense. Then they agree that this equation can be simplified by a simple operation. 2(A-B) = A+B-2B. This again simplifies to 2(A-B) = A-B. The result astounds them. Two times (A-B) equals (A-B)? Two equals one????

The answer is that the fallacy occurs only at the final step.

We get to 2(A-B) = A-B. But at this point we should recall that if you subtract something from itself you always get zero. So A-B = 0 and 2(A-B) also equals 0.

It is surely the case that 0 = 0, but thus stated the air of paradox has disappeared.

You obviously can’t get from there to the conclusion 2=1, only to the conclusion that

2(0) = 1(0).

The final step would be division by zero, which the basic rules prohibit. Precisely to avoid such paradoxes as this!

My point? Just that zero, and the rules governing its use are more exotic human conceptual inventions than one might think. The casualness with which we usually treat zero comes from familiarity, not from simplicity. This confirms the point I sought to make yesterday, that our most successful inventions are also discoveries, and vice versa.

This turns out to be, not especially Kantian, but certainly Jamesian, or well within the area of their overlap.