Terence Tao: AI is ushering in an era of proof abundance in mathematics
Terence Tao: Math 2.0 [pdf]
Terence Tao argues that AI is disrupting mathematics by exploiting proofs' objective verifiability and digitizability, leading to an era of proof abundance. He warns that blindly optimizing problem-solving is now harmful, as open problems serve as lighthouses for exploration. Tao proposes "Math 2.0": using AI to strengthen human understanding, expand the research frontier, and decenter problem-solving.
Further blind optimization of problem-solving alone is now actively harmful to the long-term health of mathematics.
- fnordpiglet
Some of this feels like at times a public frenetic grieving process. Some of the rationalizations are fairly tortured, the seeking to place the world in some definite order is bordering on obsessive.
We don’t know where this technology advance will lead or settle, so it’s absurd to try to establish a working paradigm at this point. It’s like any system, the initial conditions can be extremely chaotic and impossible to model, but with time often a stable state emerges. But the stable state is impossible to identify from early initial states.
I know a lot of folks feel this, to torture other physics metaphors, sensation of jerk - acceleration of acceleration. It’s an unpleasant and dislocating sensation. A world that felt safe and stable suddenly isn’t, and not in a micro tragedy sense but in a global realignment sense. This happened to factory workers who had enjoyed generations of stable work, farmers more slowly and just as surely.
This is what the late stages of scarcity feels like. Labor of various types devalues rapidly. Our exchange of meal and health coupons for toil cracks, and people realize their labor wasn’t godly as great books told us, but simply needed for want of an alternative. The realization that our labors might not be valued any more, and that our sense of purpose is shaken, coupled with the fact we’ve tied bare survival to our toil in our labor, is mortally tightening. No wonder people are grieving publicly.
But maybe our purpose isn’t to toil? Maybe we’v […]
- dekhn
I work in drug discovery and AI and his slide about cancer medicine isn't very well informed. We have models and narratives about how drugs work, but they are woefully incomplete.
I ask nearly every doctor/researched in the medical field: if you had a AI-created drug that tremendously improved cancer treatment outcomes for your patient, would you hesitate to prescribe it because nobody understood how it worked? I have yet to hear "yes, I would hesitate", most people say "it would be cruel to deny a person a treatment that worked".
I think Tao is focusing too much on second order effects of AI on math and other fields; humans are terrible at reasoning about second order effects, especially ones that are happening dynamically, in real time, using the most advanced mathematical models the world has yet created.
- squidmaster
> Before injecting this cocktail into your bloodstream, would you want to know that there is at least one human cancer researcher who understands the mechanism behind this cure.
Tao is out of his lane; lots of medicines don't have understood mechanisms. We don't even have the mechanisms behind general anesthesia nailed down; do you want to forgo it when the docs cut you open to remove your tumors?
- samuelknight
Slide 12 and 13 sum up my conclusions from the Oct 8 100+ reactions to 100+ solutions. Several reported OpenAI drop included answers put a wrap on problems they had been working for years. Several said that they have to completely rewrite grant requests that they had just submitted. Others described learning of the solutions problems they had been working on like losing an old friend or a lover. The general sentiment was to bemoan the loss of a field, as if it would have been better be born in the early 20th century and conclude their career arc before reaching this point. My take away is that the field needs to get it through their heads that their old problems are no longer ambitious, and that their job in the short term is to find the new frontier.
One idea Terrance Tao conjectures, which is highly doubtful, is that spamming the AI button will solve open problems without producing insightful new methods. But the OpenAI drop would seem to disprove this. The sub O(nlogn) proof for DFT for example violated very old human assumptions. Decades of work in the field was incremental progress on sub optimal method that nobody questioned hard enough. More generally, we should always be able to go back to a super-human AI and say, "Attack this problem, but don't use a method tried before."
- theturtletalks
I just watched Primeagen’s video on this and Tao’s point is that juniors no longer have the path of solving a proof to earn Field’s medals. He also argues that the community part is being hurt by AI discovering proofs because in the past people used to get invite to talk and collaborate. Now all that is being taken away. The community must adapt because Pandora’s box cannot be closed.
Prime also mentioned that software development is different. In Software development, the product is what you’re building towards, so the means to get there can be disrupted without the industry being cannibalized.
In math research, the process is the product. You take away the researching part and not much is left. But my question is, these math proofs OpenAI released, will math shift to actually using the proofs to change the world instead of just finding new ones?
- gste
> Using AI to find and highlight new principles, methods, or
insights, rather than merely new proofs.
I would have assumed that, by producing new proofs, the AI has either validated existing principles, or discovered new ones? Isn't that worth studying?
Are mathematicians complaining that reviewing AI's proofs is not as fun as writing your own? Try being a programmer... welcome to our world!
If AI lacks imagination and is not discovering new principles, then it's doing us a favour: it's crossing out the problems that don't need new principles. So the problems/conjectures that are still left are the more interesting ones.
- alecst
Putting aside the cancer question: I still don't understand how TT seems to be fixated on what models can do today instead of tomorrow. It's realistic and even conceivable that the models will also become better at explaining and presenting proofs too. Maybe he's not emotionally ready to accept that there may not be a future where his (and to some extent, my) skills are relevant and valued. It breaks my heart. I hope I'm wrong, but it feels like we've run out of higher ground to run to.
- math_dandy
From the slide Beyond Problem Solving:
"Reaching these lighthouses [resolutions of open problems] prematurely by automated tools can disrupt the exploration of the paths not taken,
and sterilize the surrounding field."
This crucial issue is centered in mathematician psychology and the incentive structure of academic/institutional mathematics worldwide. For mathematics to flourish going forward, we will need to realign our brains to think differently about the nature of mathematical progress. And we need to reorient our institutional incentive structures towards the promotion of meaningful mathematical progress itself rather than targeting proxies that are no longer faithful.
Regardless of the precise nature or the causes of the "sterilization" Tao refers to, we (the mathematics community) can only rely on ourselves to repair it. Though, since it will involve fundamental change at the level of ossified academic institutions with many stakeholders and divergent vested interests, any such repair will be slow, frustrating, controversial, and lacking any guarantee of success.