My answers to the survey What do you consider to be the main "Big Ideas" in mathematics?
It's all about proofs:
- truth != proof (Goedel's Incompleteness theorems and what follow)
- logic and math still support each other (Reverse Mathematics)
- proofs are not just algebraic manipulation. They give meaning (Combinatorial proofs)
- automation in proof assistants will accelerate mathematical progress by humans (automated deduction)
- the most abstract of abstract nonsense (Category theory, no this not particularly about proofs)
These are all the biggest of big ideas in mathematics. All the others that are specific to an area are what I would consider too... contingent. Topology, algebraic geometry, algebraic K-theory, as important as they are, don't have any far reaching ideas. OK, number three above, combinatorial proofs, I just like a lot and is somewhat like those three I just disparaged.
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