I was trained as a physicist, not as a mathematician, so my proofs tend toward intuitive hand-waving instead of mathematical rigor.
Leslie Lamport, How to Write a 21st Century Proof, 2011.
This appeals to me because I enjoy the dream of easy machine-assisted proofs. Ideally, I’d work with a computer to explore the conditions and results needed to obtain a proof, using an extensive database of existing machine-checked results (rather than relying on a collection of textbooks with mutually inconsistent notation, or the brains of my colleagues), and produce the final result in some rigorous machine-checkable language. But since formal proofs aren’t nearly at that level of usability, I’ll settle for writing very careful hierarchical proofs in Lamport’s suggested style.