[godelski]

  > In some sense I always considered programming to be more trustworthy than maths arguments without the certainty of a solver proof.

But programming is a subset of mathematics. They are both formal languages. I suspect the trustworthiness is more in your comfort level than the ability to verify

[zmgsabst]

That depends on who you ask.

Type theory can also be an independent synthetic foundation atop which you build mathematics.