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Below is a proof that straight lines cannot exist in the coordinate plane. Where is the flaw in its reasoning?

It will be shown that the equation of a straight line leads to a mathematical inconsistency. First, write the equation of the line in the form Ax+By=C, where A, B, and C are scalars. Second, choose variables u and v so that u=Ax+By and v is any arbitrary non-constant function of x and y. The equation of the line in the uv-plane can be written as u=C. Differentiate implicitly to achieve the equation 1=0, which is a mathematical inconsistency and hence the line cannot exist in the first place.

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The explanation is very similar to "suppose we have $x=1$ but if you derive it you will get $1=0$." How can that be? – Oria Gruber Apr 22 '14 at 15:47
This easily generalizes to disproving the existence of circles and hyperbolas. – Alex R. Apr 22 '14 at 16:29
The equation $u=C$ of the line is not an assertion that $u$ is equal to $C$, so it doesn't follow that some derivative of $u$ is equal to that derivativve of $C$. If $u$ was identical to $C$, then the derivative of $u$ with respect to any variable (note that then $u$ would not be variable) would be $0$, not $1$. – Steve Kass Apr 22 '14 at 16:33

1 Answer 1

$u=C$ means $u$ is constant

so after differentiation it should be

$0=0$ not $1=0$.

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