Some thoughts and questions about the role of intuition in mathematics:
- Is intuition needed to really understand a topic?
I would say yes, since in the end we reason through ideas, of which we have an intuitive representation. Without intuitions, it is difficult to relate topics with each other as we lack in hooks, and we often lack a deep understanding as well.
- Do you feel like you have understood something even if you do not have an intuitive representation of it?
- How does formalism complete intuition?
It shows whether and how an intuition is right. Sometimes intuition can be deceitful and/or tricky, especially in high dimensions or very abstract topics.
- Can/Should intuitions be taught? Or are they only effective when discovered on one’s own?
I side more with the latter. This is bordering with Maths Education, but I deem the process more important than the result – it is the tough digestion of some math material that ultimately leads to developing an intuition what really makes the intuition strong in one’s mind. If somebody else (like a teacher) does the work for us, then the result does not really stick, albeit nice it may be.
- Can we say somebody with only intuitions (well understood and well reasoned) is a mathematician?
I would say yes. I often find the intuitive side more important than the formal one.
- Is it possible to develop intuitions for very abstract topics? If yes, what shape would they have, since there is rarely anything visual we can hook up to?