Devlin's background is in mathematical logic, so it's natural that he'd choose it as a focal point, but it's unrepresentative very few mathematicians do research in mathematical logic. You might be interested in the great mathematician Henri Poincare's essay "Intuition and Logic in Mathematics"
http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.htmlI don't know of good video courses that give an accessible introduction to more representative mathematical thinking, but some books that do are
Groups and Symmetry: A Guide to Discovering Mathematics and Knots and Surfaces: A Guide to Discovering Mathematics by David Farmer and Theodore Stanford.
The Knot Book by Colin Adams.
The Shape of Space by Jeffrey Weeks.
Yearning for the Impossible: The Surprising Truths of Mathematics by John Stillwell.
Elementary Number Theory by Jones and Jones.
Galois' Theory Of Algebraic Equations by Jean-Pierre Tignol
Integer Partitions by George E. Andrews and Kimmo Eriksson
each of which require no more than high school algebra.
