TITLE: What did Alan Turing mean when he said he didn't fully understand dy/dx?
QUESTION [116 upvotes]: 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.

REPLY [14 votes]: I'm no expert on Alan Turing at all, and the following will not directly answer your question, but it might give some context. Following the link provided, I also found the following page, which might shed some more light on things:

It says the following (my apologies for any errors in transcribing, I'm grateful for any corrections):

A formal expression $$  f(x) = \sum_{i=1}^n \alpha_i x^i $$ involving
  the `indeterminate' (or variable) $x$, whose coefficients $\alpha_i$
  are numbers in a field $K$, is called a ($K$-)polynomial of formal
  degree $n$.
The idea of an `indeterminate' is distinctly subtle, I would almost
  say too subtle. It is not (at any rate as van der Waerden [link added by me] sees it) the
  same as variable. Polynomials in an indeterminate $x$, $f_1(x)$ and
  $f_2(x)$, would not be considered identical if $f_1(x)=f_2(x)$ [for]
  all $x$ in $K$, but the coefficients differed. They are in effect the 
  array of coefficients, with rules for multiplication and addition suggested
  by their form
I am inclined to the view that this is too subtle and makes an inconvenient
  definition. I prefer the indeterminate $x$ [?] be just the variable.

I think one thing to keep in mind here is that at this time anything relating to `computability' was not as clear as today. After all, Turing (an Church & Co.) were just discovering the essential notions.
In particular questions of intensionality vs. extensionality could have been an issue. It might be that Turing was pondering on the difference between functions (and also operations on functions) from a purely mathematical point of view (i.e., functions as extensional objects) vs. a computational point of view (i.e., functions as some form of formal description of a calculation process, which a priori can not be looked at in an extensional way).
All of this can be still seen in the context of the foundational crisis of mathematics (or at least strong echoes thereof). Related to this, are of course, questions of rigour, formalism, and denotation. This, in turn, is where your quote comes in. As others have outlined, Turing might have asked the question, what $\frac{\mathrm{d}y}{\mathrm{d}x}$ is (from a formal point of view), but also what is denoted by it, and (to his frustration) found that the answer to his question was not as clear as he wanted it to be.