For example as a student of math, the fundamental way in which I view things could be more or less summarized into three phases:
- Emphasis on 'objects' (high school): E.g. what are the properties of this specific real valued function? Solve this specific geometry problem. Where does this specific function attain a maximum?
- Abstraction to spaces (undergrad): The focus changes to abstract spaces and structures (e.g. topological spaces, groups, function spaces). E.g. Instead of focusing on the properties of one function, we study properties of a function space. A theorem proven about groups in general applies to any example of a group.
- Categorical abstraction (grad school): We can abstract further from specific structures to categories. Many structure-specific constructions (e.g. direct sums of groups, disjoint unions of sets, wedge products of pointed space) can be unified into categorical constructions (e.g. the co-product for those just listed). The language of functors provides a uniform way of talking about mapping one type of structure into another.
Another unrelated major viewpoint change I've experienced is that I've resigned myself to the fact that the world is much more random and uncontrollable than we believe it is. Pretty much all major events in my life have been unpredictable and essentially due to chance. This realization has provided me with more peace of mind, than before when I was much more worried about "making the right decisions".
I am interested in what meta-viewpoints people have experienced, and any changes they've experienced. They can be related to anything: computer programming, science, life, relationships ...
