Toby Ord Says Mathematics Exceeds Automated Proof
Philosopher Toby Ord argues AI proof automation leaves higher mathematical tasks untouched.
Toby Ord posted a thread on his new piece titled Mathematics is About Much More than Proof. He notes calculation was automated in the twentieth century and symbolic manipulation followed. AI systems now advance on proof. Ord states that asking the right questions and developing new fields of enquiry remain. He points out the lack of training data or verifiable reward for these skills in current systems. Ord adds there is not yet much evidence on whether AI can identify important questions among all possibilities.
New piece: Mathematics is About Much More than Proof AI systems are quickly becoming very good at automating mathematical proof. But proof isn't everything. Mathematicians used to spend about half their working lives on calculation, but his was automated in the 20th century… 🧵
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Toby Ord Says Mathematics Exceeds Automated Proof
Philosopher Toby Ord argues AI proof automation leaves higher mathematical tasks untouched.
Toby Ord posted a thread on his new piece titled Mathematics is About Much More than Proof. He notes calculation was automated in the twentieth century and symbolic manipulation followed. AI systems now advance on proof. Ord states that asking the right questions and developing new fields of enquiry remain. He points out the lack of training data or verifiable reward for these skills in current systems. Ord adds there is not yet much evidence on whether AI can identify important questions among all possibilities.
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7 postsAnd symbolic manipulation (such as integrating, simplifying, or solving for x) was automated over the last four decades. We may now be witnessing the automation of proof (or large parts of it). But there is still much mathematical work at higher levels to be done.
A key area is asking the right questions: Even a machine that could prove any true mathematical statement in a microsecond would never get up to considering the unit distance problem before all the stars have burnt out.
Somehow human mathematicians have the ability to find important questions amidst the exponential of all possible questions we could consider. There isn't yet much evidence AI can do this (or that it can't do this).
A key issue is that we don’t have good training data (or a verifiable reward) for asking good questions. Unless it has come for free when pretraining on maths texts or while learning to prove statements, it is hard to see why they would possess the skill.
And beyond asking the right questions is developing entirely new fields of mathematical enquiry. Lots of the major work in mathematics was creating the concepts and vocabulary of new fields, allowing us to ask entirely new kinds of questions.
It is possible that these parts of mathematics will also be automated soon, but I haven't seen much effort being put into tracking whether that is happening.
