I'm sorry if this is in the wrong place but I believe the answer is geometric in nature somehow. So as a young physics student it is banged into your head that vectors/tensors are genuine geometric and algebraic "objects" and the coordinates by which you label them are metaphysical, so to speak, and not at all fundamental. The same object has many different labels depending on the basis. So this got me thinking about functions. Dirac seems to want us to imagine functions (say over just the real numbers) as an uncountable set of coordinates, analogous to vectors, one for each element of R. So just like we can think geometrically/algebraically about vectors, is there a way of thinking about functions without projecting them into any coordinate space? I suppose the thrust of my point is this: if it is non-sensical to say "(1,2,3) is a vector" then ought it be equally cavalier to say "f(x)=x^3" is a function?"