Chemistry - Is anode actually ever negative?
It is a very good questions from a student who seems to be thinking deeply. I have been thinking to write on this topic for the last 7-8 years. In short the poles of the batteries are electrostatically charged and one can actually "sense" this charge with the help of sensitive electroscopes which were known in the time of Volta.
The label anode or cathode is not defined with respect to the electrostatic sign as explained in the post by Maurice.
Think of a large parallel capacitor. Connect one terminal of the plate to one positive terminal and the other one the negative terminal. The capacitor plates would show a electrostatic charge.
By all means, using one single cell you would not be able to detect the electrostatic charge, like a charged comb. Only specialized electroscopes can sense that.