Mathematical thought can't be fully formalised
Gödel’s theorem shows that human creativity can't be fully formalised. The ingenuity of mathematicians in devising new methods can't be reduced to a precise logical form.
For example, it has been shown that humans using "informal" mathematical reasoning, can prove theorems that can't be proven by any formal means.

Ernest Nagel and James R. Newman (1958).
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Artificial Intelligence »Artificial Intelligence
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Are thinking computers mathematically possible? [7] »Are thinking computers mathematically possible? [7]
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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Theorems show limitations of machine thought »Theorems show limitations of machine thought
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Mathematical thought can't be fully formalised
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Gödelian arguments don't affect open proof systems »Gödelian arguments don't affect open proof systems
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Proof of human superiority relies on proof of consistency »Proof of human superiority relies on proof of consistency
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