Cycling is a varied experience including many different reasons to ride, such as commuting, fitness, or racing. Experience is further shaped by the type of bike and environment. The book Bike Snob tries to distill what cyclists have in common and presents the definition of a cyclist as someone who chooses to ride a bike even if they don't have to. The book further argues that there's no such thing as "bike culture". However, it also summarizes (in a humorous fashion) some of the types of cyclists and how they often view each other.
Even though there's no uniform bike culture, many individual people have their own philosophy. You may be able to find some biographies or autobiographies about cyclists who describe their own attitude toward cycling and how it changed them.
Of course, there are many groups of like-minded people who might share an approach to cycling. One website that often gets cited online is Velominati, which declares a list of rules for road cycling which they claim to be philosophical. This type of "philosophy" seems to be more about creating a sense of belonging than about any inherent truths about cycling--people want to be able to feel "in the club".
Given the variety within cycling, any universal philosophy would be limited to the vague notion of self-powered locomotion.