AI mathematical proof: How tools can reverse shape the way mathematicians think

When algorithms become mathematicians 'co-author

Mathematicians suddenly discovered that they were no longer the only subjects asking questions. Recently, OpenAI used Monte Carlo algorithm and neural networks to overturn the 30-year-old conjecture in combinatorial geometry. On the surface,"AI is fast and good," but in essence, it lays the foundation for mathematicians to think about problems. ** For the first time, the proof tool circumvents the boundaries of human knowledge and directly extracts counterexamples ** from a large number of samples.

Guess how the traditional mathematics community reacted? "What kind of proof is this?" Old-school scholars beat the table and cursed, believing that using random sampling to verify discrete problems was blasphemy on mathematical purity. What's more interesting is that even the author's signature of the proof paper triggered a war. DeepMind insisted on writing the algorithm as a "collaborator" in its early years, but this time OpenAI only dared to put it in an appendix for fear of being hanged as a disgrace. Tools began to participate in the core path of scientific research, but people refused to recognize their subjectivity.

Don't stare at "calculating quickly", watch it "thinking wrongly"

Those who say that AI accelerates problem solving underestimate change. AlphaGeometry is still a "regular army" in solving IMO problems, but this time it goes astray: in the face of Keller's Conjecture, humans cannot find counterexamples above the seventh dimension. ** The algorithm directly blindly touches the data: using neural networks to predict high-dimensional spatial structures, and then using Monte Carlo to violently sample candidate points **. Finally, an 8-dimensional counterexample is output and the throat is directly sealed.

The most devastating thing is this: **AI doesn't understand what conjecture is **. It is like a pangolin, making a hole in a theoretical wall drawn by a mathematician, and humans have to follow behind and explain: "Uh... this hole does make sense." DeepMind's previous matrix multiplication optimization did the same-the algorithm found a calculation solution 20% faster than humans, but no one knew why.

Don't install it, the nature of the certificate is deteriorating

Old pedants argue that AI proof is not mathematics, essentially because they fear "the transfer of interpretation power." The core of traditional proof is the traceability chain of cause and effect: starting from axioms, deduce step by step. The AI proved to be a black-box violent group, and the output results were compiled and explained backwards-this trick also appeared in Tao Zhexuan's cooperation project last year.

• "It's like hiring a talented agent and handing in an altered report after the mission is completed"* A doctor involved in the project complained privately. Mathematicians are degenerating into "Party A acceptance specialist": check whether the results are self-consistent, and then write a manual for the algorithm. ** The tool has been upgraded from an "auxiliary checker" to a "proposer"**, and humans have become annotation tools instead.

I read the OpenAI technical report-these people are more realistic than DeepMind, openly saying that "understanding is not necessary". They feed a large amount of high-dimensional data to the neural network, as long as the coordinates of the output points meet the constraints, no matter what its internal logic is.

rebutters Don't pretend to be asleep (Steelman session)

Someone must be jumping: What is the difference between this and computer-aided certification (CAS)? Thirty years ago, the four-color theorem was also calculated with machines!

Far from it. The essence of CAS is an upgraded version of the abacus: humans define algorithm steps (such as exhaustive dyeing), and the machine is only responsible for execution. This time it was the machine holding a hammer and saying,"There may be gold mines in the wall. Let me help you break it open and have a look?" The difference between > is that the former is a human commander, and the latter leads people with their hands.

The more hardcore opposition comes from mathematical philosophers: "What truth is it without proof of human understanding?" This is beautiful but hypocritical-the history of mathematics is a history of cognitive upgrades with tools: when calculus was first born, it was called "ghost theory", and the plural number was called "false numbers". Now they are not all included in textbooks?

Cross-border Demon Mirror: Physics has long been reshaped by instruments

Looking back at the history of physics, it is normal for tools to subvert cognition. In 1609, Galileo used a telescope to discover the moons of Jupiter, and the earth's center collapsed-at that time, the church denounced "phantoms in tubes as not observations." In 1932, Anderson used a cloud chamber to capture positrons, and Dirac's equation turned from joke to truth. All observation tools are thinking crutches, and if used too much, they become legs. **

Mathematics is even worse-it has always claimed to be a "temple of pure thinking", but now it is being ripped off by AI: ** The so-called intuition is just an under-trained neural network **. Combinatorial mathematics is particularly dangerous: a large number of problems are essentially ultra-high dimensional space searches, and the human brain is still dreaming about it in a three-dimensional coordinate system.

Business Accounts: Who is feeding this monster?

Don't think this is academic self-congratulatory. In 2023, NSF's funding for the "AI+ Mathematics" project will increase by 400%, and DeepMind supports a 30-person pure mathematics team. Why is capital suddenly keen on giving out dog food to mathematicians? ** Because abstract reasoning is the last bastion of AGI **-a model that can handle mathematical proof, it is not far from universal reasoning.

What is more realistic is industrial applications: wiring optimization in chip design and molecular combination screening in pharmaceuticals are all discrete mathematical problems. Nvidia used AI to reconstruct sparse matrix computing last year, saving US$300 million in computing power costs. Academic purity is nothing when tools can directly produce patent and financial report numbers.

Human Nature Experiment: Why do smart people play dumb together?

The most ironic thing is the current situation in the mathematics circle: ** While dismissing AI proving that it is unelegant, while frantically brushing the rankings **. The International Mathematical Union quietly added a category of "machine-aided proof", and reviewers of the top issue learned to use GPT-4 to check for logic loopholes. Young scholars are even more divisive-they are afraid of being replaced by machines, and they are also jealous of their opponents 'use of algorithms to publish top issues.

After all, tools never eliminate industry, only posture. Telescopes didn't kill astronomy, but they made naked-eye stargazing a performance art;AI proof won't end mathematics, it only eliminates one mode of thinking: ** Mathematicians must adapt to the transformation from "provers" to "problem curators"**-Designing better constraints and feeding smarter algorithmic beasts.

So stop asking "Can AI do mathematics", the question is:

** When we rely on tools to discover truth, is the definition of truth still in the hands of humans? **