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Mathematics is postmodern; there's no single formal system which is the concrete bedrock of mathematics, and there will always be some true-but-unprovable theorems in consistent formal systems.

Cynics might take this to mean that maths is meaningless. On the contrary, I'd like to suggest that maths is the canonical emergent pattern; any formal system with sufficient complexity to represent itself will automatically exhibit some algebraic structure, and maths emerges naturally from there.

The biggest implication for computer scientists is that, combined with results of Church and Tarski, we should expect that meta-languages are more powerful than object-languages, whether this is by preprocessing and macros like in C/C++ and CL, or by avoiding using meta-features except when necessary like in Python and Ruby.




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