terence tao is offering a reward for the most powerful brain on the internet! ai humans subvert mathematical problems? versailles netizens are gone
2024-09-29
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[introduction to new wisdom]recently, tao zhexuan launched a challenge to the majority of netizens and mathematics enthusiasts: can popular mathematics enthusiasts, proof assistants, automated assistants and ai join forces to prove mathematical problems that extend by several orders of magnitude?
want to participate in the "crowdsourcing" mathematics research project launched by terence tao?the opportunity has come!ai-assisted proof mathematical research is becoming more and more feasible
each of them is familiar enough with aspects of the project to validate each other's contributions.but if you want to organize larger-scale mathematical research projects, especially projects involving public contributions, it is much more troublesome.the reason is that it is difficult to verify everyone's contribution.at the end of 2023, terence tao announced that the lean4 project, which formalized the proof of the polynomial freiman-ruzsa conjecture, was successful after three weeks (the picture shows the latest status)be aware that a single mistake in one part of a mathematical argument can cause the entire project to fail.moreover, given the complexity of a typical mathematics project, it is unrealistic to expect members of the public with undergraduate mathematics education to make meaningful contributions.from this we can also know that incorporating ai tools into mathematical research projects is also extremely challenging.because ai can generate arguments that appear reasonable but actually make no sense, additional verification is required before the ai-generated portion can be added to the project.fortunately, proof-assisted languages such as lean offer potential ways to overcome these obstacles and enable collaboration between professional mathematicians, the public at large, and ai tools.this approach is based on the premise that projects can be broken down in a modular fashion into smaller parts that can be completed without having to understand the entire project.current examples mainly include projects that formalize existing mathematical results (such as the formalization of the pfr conjecture recently proved by marton).these formalizations are primarily done through crowdsourcing by human contributors (including professional mathematicians and interested members of the public).at the same time, there are also some emerging attempts to introduce more automated tools to complete the task, including traditional automatic theorem provers and more modern ai-based tools.it becomes possible to explore new mathematical problems
and,terence taoit is also believed that this new paradigm can be used not only to formalize existing mathematics, but also to explore completely new mathematics!in the past, he and his successor organized an online collaboration "polymath" project, which is a good example.however, this project did not incorporate a proof-auxiliary language into the workflow, and contributions had to be managed and verified by human moderators, which was very time-consuming and limited the further expansion of these projects.now, tao zhexuan hopes that adding proof auxiliary language can break this bottleneck.he is particularly interested in whether it is possible to use these modern tools to explore a class of mathematical problems simultaneously, rather than focusing on just one or two problems at a time.in essence, this approach is modular for repetitive tasks, and crowdsourcing and automation tools can be particularly useful if there is a platform in place to rigorously coordinate all contributions.this type of mathematical problem would not have been scalable using previous methods. unless you slowly explore one data point at a time with individual papers over many years until you get a reasonable intuition about this kind of problem.additionally, having a large problem data set may help perform performance evaluations of various automation tools and compare the efficiency of different workflows.in july of this year, the fifth busy beaver number was confirmed to be 47,176,870.some earlier crowdsourced computing projects, such as the great internet mersenne prime search (gimps), are similar in spirit to these projects, although they use a more traditional proof-of-work mechanism. , rather than proving an auxiliary language.terence tao said he would be interested to know if there are other existing examples of crowdsourcing projects exploring mathematical spaces, and if there are any lessons learned that can be used.tao zhexuan proposes new projects
to this end, tao himself proposed a project to further test this paradigm.this project was inspired by last year's mathoverflow question.soon after, tao zhexuan further discussed it on his mathstodon.this problem belongs to the field of universal algebra and involves a medium-scale exploration of the theory of simple equations of the original group (magma).the original group is a group equipped with binary operationsthe set g.initially, this operation o does not have any additional axioms attached, so the original group itself is a relatively simple structure.of course, by adding additional axioms, such as identity axioms or associativity axioms, we can obtain more familiar mathematical objects such as groups, semigroups, or monoids.here we are interested in the (constant-free) axioms of equality. these axioms concern the equality of expressions constructed from operations o and one or more unknown variables in g.two familiar examples of such axioms are the commutative law xoy = yox and the associative law (xoy) oz = xo (yoz).where x, y, z are unknown variables in the original group g.on the other hand, the (left) identity axiom eox = x is not considered an equational axiom here because it involves a constant e ∈ g. such axioms involving constants are not discussed in this study.next, in order to illustrate the research project he initiated, terence tao introduced eleven examples of equality axioms about primitive groups.these equality axioms are equations involving only primitive group operations and unknown variables—so, for example, equation 7 represents the commutative axiom, while equation 10 represents the associative axiom.the constant axiom equation 1 is the strongest, because it restricts the original group g to have at most one element; on the contrary, the reflexive axiom equation 11 is the weakest, and all original groups satisfy this axiom.next, we can explore the derivation relationship between these axioms: which axioms can deduce which axioms?for example, equation 1 leads to all the other axioms in this list, which in turn leads to equation 11.equation 8 can be used as a special case to derive equation 9, which in turn can be used as a special case to derive equation 10.the complete derivation relationship between these axioms can be described by the following hasse diagram:this result specifically answers a question on the mathematics q&a website mathoverflow: whether there are equational axioms between the constant axiom (equation 1) and the associativity axiom (equation 10).it is worth noting that most of the implication relations here are easy to prove. however, there is a non-trivial implication relationship.this relationship was found in an answer to a mathoverflow post closely related to the previous question:proposition 1: equation 4 implies equation 7proof: suppose g satisfies equation 4, thereforeit holds for all x,y ∈ g.in particular, when y = xox, it follows that (xox) o (xox) = (xox) ox.applying (1) again, we can conclude that xox is idempotent:now, replacing x with xox in (1) and using (2), we get (xox) oy = yo (xox).in particular, xox and yoy are interchangeable:furthermore, by applying (1) twice, we get (xox) o (yoy) = (yoy) ox = xoy.therefore, (3) can be simplified to xoy = yox, which is equation 7.the formalization of the above argumentation process can be found in lean.it is worth noting, however, that the general question of determining whether one set of equality axioms determines another set of equality axioms is undecidable.therefore, the situation here is somewhat similar to the "busy beaver" challenge, that is, after a certain point of complexity, we are bound to encounter undecidable problems; but before reaching this threshold, we still hope to discover interesting problems and phenomena.the hass diagram above not only asserts the entailment relations between the listed equality axioms, but also asserts the non-implication relations between the axioms.for example, as shown in the figure, the commutative axiom equation 7 does not imply the axiom of equation 4 (x + x) + y = y + x.to prove this, just find an example of a primitive group that satisfies the commutative axiom equation 7 but does not satisfy the axiom equation 4.for example, in this case, we can choose the natural number set n, whose operation is xoy := x+y.more generally, the diagram asserts the following non-implication relations, which (together with the implication relations already noted) completely describe the partially ordered set of implication relations among these eleven axioms:here, tao zhexuan invites readers to propose counterexamples to complete some of the proofs.the most difficult counterexample to find is that equation 9 cannot deduce equation 8.a solution can be given using lean.in addition, tao zhexuan also provides a github repository that contains lean proofs of all the above inclusion and anti-inclusion relationships.it can be seen that just calculating the haas diagram of 11 equations is already a bit cumbersome.the project proposed by terence tao is an attempt to expand this hass diagram by several orders of magnitude to cover a wider range of equation sets.the set he proposed was ε, the set of equations using the original group operation o at most four times, until the axioms of reflexivity and symmetry of the sum equations were relabeled.this includes the eleven equations mentioned above, but there are many more.recall that the catalan number c_n is the number of ways to form an expression using the binary operation o (applied to n+1 placeholder variables); and given a string of m placeholder variables, the bell number b_m is number of ways these variables are assigned names (which can be relabeled), which allows certain placeholders to be assigned the same name.therefore, ignoring symmetry, the number of equations involving at most four operations isthe number of equations whose left and right sides are the same isthese are equivalent to the reflexive axioms (equation 11).the remaining 9118 equations appear in pairs due to the symmetry of the equations, so the total size of ε is
tao zhexuan said that he has not yet generated a complete list of such identities, but he suspects that it can be easily done using python.using ai tools, you should be able to generate most of the required code.he said he had no idea what the geometry of ε would look like.will most equations be incomparable to each other? will it be divided into "strong" axioms and "weak" axioms?now, terence tao's message area has dozens of comments.interested readers, tao zhexuan has also extended an invitation to you.
