What did Alan Turing mean when he said he didn't fully understand dy/dx? Alan Turing's notebook has recently been sold at an auction house in London. In it he says this:

Written out:

The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find
  extremely difficult to understand in spite of it having been the one I
  understood best once! It certainly implies that some relation between
  $x$ and $y$ has been laid down e.g.  \begin{equation} y = x^2 + 3x \end{equation}

I am trying to get an idea of what he meant by this. I imagine he was a dab hand at differentiation from first principles so he is obviously hinting at something more subtle but I can't access it.


*

*What is the depth of understanding he was trying to acquire about        this mathematical operation?

*What does intuitive differentiation notation require?

*Does this notation bring out the full subtlety of differentiation?

*What did he mean?


See here for more pages of the notebook.

We, who learn differentiation based on calculating the limit of quotients of differences, consider the Lebniz notation to be only a notation. We do not try to grab the intuitive meaning of $dx$ and $dy$. We don’t even care.
I guess that Turing was trying to understand how Lebniz could reach his results based on this vague notation. I guess that he refers to two things:


*

*For Leibniz $\frac{dx}{dy}$ must have had a special intuitive meaning if the relationship between $y$ and $x$ was given. So for Leibniz it was not only a notation then.

*Turing perhaps had gone through that leibnizian vague understanding before learning the standard approach. Turing must have been thinking about his own losing this once existing transcendental understanding that got destroyed in his mind by the mathematical discipline developed in the 19th century.
