Grant Sanderson (@3blue1brown) — Past, present, & future of mathematics

Dwarkesh Patel opens by asking whether an AI model that wins gold at the International Math Olympiad should simply count as AGI, given the creative problem solving and chain of thought such a result requires. Grant Sanderson declines the premise. He says that if you ask ten people what AGI means you will get ten slightly different answers, and that the word seems to presuppose a discrete threshold he does not think exists.

I have no idea what people mean when they use the word AGI.

His alternative picture is continuous. GPT-4 already feels general because one training algorithm applies to a very large set of tasks, which people in the sixties might not have expected of how artificial intelligence could be programmed. What people usually mean by saying it is not AGI, he suggests, is that it beats most people on some benchmarks without beating the best. On this reading, gold at the IMO would be one more domain ticked off. Sanderson asks whether AGI is supposed to mean that every domain anyone could be good at has been enumerated and the machine now beats humans at all of them.

Patel pushes the economic angle: such a system might replace a substantial fraction of human jobs, and having seen problems from Sanderson's channel he thinks it is coming for podcasters and video animators. Sanderson calls this orthogonal. He compares an IMO gold to superhuman Go or chess, where the moves carry what he calls the air of creativity, and he notes that once image generators appeared, everyone else could see something deserving the word creative as well.

He then guesses at how the result would be reached. It would probably look like AlphaGo's self-play, because math lends itself to synthetic data in ways other domains do not. A model could produce many proofs in a checking language such as Lean, train on whether each is valid, and be balanced against English-language versions. The likely output has two possible flavors, according to him. It could be a proof that is correct but unmotivated, the kind where you can follow every step but do not see why anyone would have come up with it, versus papers where you morally understand why the result should be true.

Sanderson separates that achievement from job replacement. The obstacles to replacing jobs, he says, are different: a context window longer than a small fixed length so a system can build relationships over long periods, understand where someone is coming from, and then do the problem solving. An IMO-capable model would be a more helpful tool, in the way Mathematica already helps with math problems.

Patel objects that chess programs can narrow possibilities by search, whereas the IMO problems on Sanderson's channel depend on finding the right abstraction, one not evident in the problem or in any other test. Sanderson replies that he would want to hear from Go and chess experts, because a player as good as AlphaGo is not doing tree search exclusively given the combinatorial explosion, so some higher-level structure must be involved in its understanding. He expects to be very impressed and grants that creativity is involved. His only claim is that it is another box to tick, a historic moment analogous to chess and Go, more than to the Industrial Revolution.

Patel finds this surprisingly uncompelled and asks whether Sanderson doubts that spotting an isomorphism between two ways of thinking and carrying it through fifty steps is upstream of many valuable tasks. Sanderson says he is compelled and compares it to being impressed by Stable Diffusion rendering mountains made of quartz and gemstones, which captures the essence of a landscape without being literally one. Solving a creative math problem requires lateral thinking rather than working from definitions, the same joy we get from good analogies. He places it in the same category and reiterates that he sees no sharp pre-AGI and post-AGI line, much less one marked by a math tournament.

Patel asks where applied mathematicians could do the most good in society beyond computer science and IT, naming transportation, logistics and manufacturing as candidates. Sanderson says he is perhaps the worst person to ask, and that he will instead explain why the question matters. He has considered making an atypical video aimed at graduating students.

His reasoning is that people who fall for math or a technical field are by default apprenticed into it. University physics and math programs are taught by academics whose research is in the field, so it is hard not to train as a successor. Students tell him after talks that his videos are why they studied math or are going to grad school. He is pleased but unsure he was wholly responsible, and he asks himself whether he wants that outcome. He unequivocally wants more people to identify as liking math, but those who do get shuffled toward traditional outlets.

He agrees with Patel that math academia, finance, and computer science or data science are very common destinations and that talent is almost certainly over-allocated to them, though all three are valuable. If he were playing God, he says, he does not know where he would redirect people, and he is restating the question rather than answering it. He suggests talking to people who made the switch, naming Eric Lander and Jim Simons as examples of people who moved from purely academic work to something very different.

He has floated a forcing function: pure mathematicians could be required, for example by an NSF grant condition, to spend about 10% of their time collaborating with another department, on the thought that they are strong problem solvers in one category and that spreading them could help. Mathematicians' reactions are mixed. Some doubt they would be very useful. The aesthetic of a good math problem is rooted in its purity, and assuming such skill generalizes better than other people's abilities could be a little elitist. He asks why applied biologists should not equally go work in logistics, though a part of him still feels mathematicians are special because there is something general about math.

Sanderson says he lacks answers and would like to hear from people who studied math or an adjacent field and then chose what to do by asking what impact they wanted to have rather than by following the academic itch of being good at school. Such stories could be compelling for his audience, who are probably on track for traditional math fields. He cannot say what dispersing them more effectively would look like, because as a math YouTuber he has no career in logistics or manufacturing and no feel for where abstract problem solving is needed. Patel closes the segment by suggesting that it would help to speculate on how a young math whiz might begin to see where they could have an edge.