What if all the math was wrong? No, I don’t mean what if someone forgot to carry the two because they were thinking about that super hilarious meme they saw on Reddit earlier. I mean what if all the math was wrong? OK sure, two plus two will always equal four, you can check that with a table and a few spare Q-tips you have lying around (throw those out, by the way, that’s gross). There’s a lot of math you can check for yourself, but what about all the other math? The further mathematical proofs get from two-plus-two, the harder it is for individuals to understand all the finer details, and we’re a long way past two-plus-two. Increasingly mathematicians are relying on the assumptions that earlier published proofs are correct because there’s not enough time in a human lifespan to re-prove all the foundations that a new proof is based on. In the words of number theorist Kevin Buzzard, “there is a non-zero chance that some of our great castles are built on sand.” Buzzard says the chance is small but it’s still there, and that worries him.
First, what is a mathematical proof? It’s a series of logical statements demonstrating the truth of a mathematical concept. Once something has proof, it then becomes an accepted mathematical concept and filters out into other fields beyond mathematics. Once something is accepted as a true mathematical concept it is then used as an argument for more proofs. Kevin Buzzard believes that many mathematicians are basing their published proofs on prior proofs that they do not even understand. He says:
“I’m suddenly concerned that all of published math is wrong because mathematicians are not checking the details, and I’ve seen them wrong before.”
In an article for Motherboard, Mordecai Rorvig writes:
New mathematics is supposed to be proven from the ground up. Every step must be checked, or at least the reasoning followed. On the other hand, there are senior experts and elders of the mathematical community who provide a reliable testimonial guide to what is true or not true. If an elder cites a paper and uses it in their work, then the paper probably doesn’t need to be checked, the thinking goes.
Kevin Buzzard spoke to Motherboard at the 10th Interactive Theorem Proving conference in Portland, Oregon, which sounds thrilling. He says that because proofs have become so complex, the work of math elders is increasingly accepted verbatim without being checked, which goes against all the principles of mathematics. Citing a legendarily difficult math problem, Fermat’s Last Theorem, Buzzard says:
“I believe that no human, alive or dead, knows all the details of the proof of Fermat’s Last Theorem. But the community accept the proof nonetheless. [Because] the elders have decreed that the proof is OK.”
So what’s the solution to this mathematical sandcastle? Buzzard thinks it lies in AI-assisted proofs. He says that once he was introduced to a proof verification software called Lean by mathematician Thomas Hales, he “fell in love”:
“I realized the computers would only accept inputs in a very precise form, which is my favorite way of thinking about math. I fell in love, because I felt like I found a soulmate. I found something that thought about math just the way I thought about it.”
Some people are really into numbers, it’s cool. But as romantic as this software is to numberphiles like Kevin Buzzard, mathematicians think we still have a ways to go before it can be used at a large scale. But it’s something that people are working on, and the automation of pure mathematics might be a game-changer. Michael Harris, professor of mathematics at Columbia University and a colleague of Kevin Buzzard says:
“One thing I can predict is if really smart people like Thomas Hales and Buzzard continue to think along these lines, then something interesting is going to come out of it; it may not be AI but it may be whole new branches of mathematics or whole new ways of thinking.”
How many seemingly incongruent things in the universe might only appear to be incongruent because of faulty math? I have no idea and that’s probably a dumb question, but whatever. When I hear that the so-called “language of the universe” might be a wobbly house of cards, my mind starts going to weird places. But hey, it’s getting weirder all the time, what else is new?