Traditionpartner, mathematics research projects are carry outed by a minuscule number (typicpartner one to five) of expert mathematicians, each of which are recognizable enough with all aspects of the project that they can validate each other’s contributions. It has been challenging to set up mathematical projects at huger scales, and particularly those that comprise contributions from the vague uncover, due to the need to validate all of the contributions; a individual error in one component of a mathematical argument could invalidate the entire project. Furthermore, the sophistication of a normal math project is such that it would not be down-to-earth to anticipate a member of the uncover, with say an undergraduate level of mathematics education, to give in a unbenevolentingful way to many such projects.
For roverdelighted reasons, it is also challenging to include aidance from conmomentary AI tools into a research project, as these tools can “hallucinate” plausible-seeing, but nonsensical arguments, which therefore need compriseitional verification before they could be compriseed into the project.
Proof aidant languages, such as Lean, supply a potential way to defeat these obstacles, and permit for huge-scale collaborations involving professional mathematicians, the expansiveer uncover, and/or AI tools to all give to a complicated project, supplyd that it can be broken up in a modular style into minusculeer pieces that can be attacked without necessarily empathetic all aspects of the project as a whole. Projects to createalize an existing mathematical result (such as the createalization of the recent proof of the PFR conjecture of Marton, converseed in this previous blog post) are currently the main examples of such huge-scale collaborations that are helpd via proof aidants. At conshort-term, these createalizations are mostly crowdsourced by human contributors (which include both professional mathematicians and interested members of the vague uncover), but there are also some nascent efforts to include more automated tools (either “excellent elderly-styleed” automated theorem showrs, or more conmomentary AI-based tools) to aid with the (still quite tedious) task of createalization.
However, I apshow that this sort of paradigm can also be used to scrutinize novel mathematics, as contestd to createalizing existing mathematics. The online collaborative “Polymath” projects that disconnectal people including myself orderly in the past are one example of this; but as they did not include proof aidants into the laborflow, the contributions had to be deal withd and verified by the human moderators of the project, which was quite a time-consuming responsibility, and one which restricted the ability to scale these projects up further. But I am hoping that the compriseition of proof aidants will delete this bottleneck.
I am particularly interested in the possibility of using these conmomentary tools to scrutinize a class of many mathematical problems at once, as contestd to the current approach of centering on only one or two problems at a time. This seems appreciate an inherently modularizable and repetitive task, which could particularly advantage from both crowdsourcing and automated tools, if given the right platcreate to rigorously set up all the contributions; and it is a type of mathematics that previous methods usupartner could not scale up to (except perhaps over a period of many years, as individual papers sluggishly scrutinize the class one data point at a time until a reasonable intuition about the class is accomplished). Among other skinnygs, having a huge data set of problems to labor on could be beneficial for benchtaging various automated tools and contrast the efficacy of contrastent laborflows.
One recent example of such a project was the Busy Beaver Challenge, which showed this July that the fifth Busy Beaver number was equivalent to . Some elderlyer crowdsourced computational projects, such as the Great Internet Mersenne Prime Search (GIMPS), are also somewhat aappreciate in spirit to this type of project (though using more traditional proof of labor certificates instead of proof aidants). I would be interested in hearing of any other extant examples of crowdsourced projects exploring a mathematical space, and whether there are lessons from those examples that could be relevant for the project I advise here.
More particularpartner I would appreciate to advise the chaseing (confesstedly synthetic) project as a pilot to further test out this paradigm, which was eased by a MathOverflow inquire from last year, and converseed somewhat further on my Mastodon account lowly afterwards.
The problem is in the field of universal algebra, and troubles the (medium-scale) exploration of basic equational theories for magmas. A magma is noskinnyg more than a set supplyped with a binary operation . Initipartner, no compriseitional axioms on this operation are imposed, and as such magmas by themselves are somewhat unclever objects. Of course, with compriseitional axioms, such as the identity axiom or the associative axiom, one can get more recognizable mathematical objects such as groups, semigroups, or monoids. Here we will be interested in (constant-free) equational axioms, which are axioms of equivalentity involving conveyions built from the operation and one or more indefinish variables in . Two recognizable examples of such axioms are the commutative axiom
and the associative axiom
where are indefinish variables in the magma . On the other hand the (left) identity axiom would not be pondered an equational axiom here, as it comprises a constant (the identity element), which we will not ponder here.
To depict the project I have in mind, let me first begin eleven examples of equational axioms for magmas:
Thus, for instance, Equation7 is the commutative axiom, and Equation10 is the associative axiom. The constant axiom Equation1 is the strongest, as it forces the magma to have at most one element; at the opposite excessive, the reflexive axiom Equation11 is the feebleest, being satisfied by every individual magma.
One can then ask which axioms show which others. For instance, Equation1 implies all the other axioms in this enumerate, which in turn show Equation11. Equation8 implies Equation9 as a one-of-a-kind case, which in turn implies Equation10 as a one-of-a-kind case. The brimming poset of implications can be depicted by the chaseing Hasse diagram:
This in particular answers the MathOverflow inquire of whether there were equational axioms intersettle between the constant axiom Equation1 and the associative axiom Equation10.
Most of the implications here are quite basic to show, but there is one non-untransport inant one, geted in this answer to a MathOverflow post seally roverdelighted to the preceding one:
Proposition 1 Equation4 implies Equation7.
Proof: Suppose that trails Equation4, thus
for all . Specializing to , we finish
and hence by another application of (1) we see that is idempotent:
Now, replacing by in (1) and then using (2), we see that
so in particular commutes with :
Also, from two applications (1) one has
Thus (3) simplifies to , which is Equation7.
A createalization of the above argument in Lean can be create here.
I will retag that the vague inquire of determining whether one set of equational axioms resettles another is undecidable; see Theorem 14 of this paper of Perkins. (This is aappreciate in spirit to the more well understandn undecidability of various word problems.) So, the situation here is somewhat aappreciate to the Busy Beaver Challenge, in that past a brave point of complicatedity, we would necessarily come atraverse unsolvable problems; but hopebrimmingy there would be engaging problems and phenomena to uncover before we accomplish that threshelderly.
The above Hasse diagram does not equitable state implications between the enumerateed equational axioms; it also states non-implications between the axioms. For instance, as seen in the diagram, the commutative axiom Equation7 does not show the Equation4 axiom
To see this, one sshow has to originate an example of a magma that trails the commutative axiom Equation7, but not the Equation4 axiom; but in this case one can sshow pick (for instance) the organic numbers with the compriseition operation . More generpartner, the diagram states the chaseing non-implications, which (together with the showd implications) finishly depicts the poset of implications between the eleven axioms:
- Equation2 does not show Equation3.
- Equation3 does not show Equation5.
- Equation3 does not show Equation7.
- Equation5 does not show Equation6.
- Equation5 does not show Equation7.
- Equation6 does not show Equation7.
- Equation6 does not show Equation10.
- Equation7 does not show Equation6.
- Equation7 does not show Equation10.
- Equation9 does not show Equation8.
- Equation10 does not show Equation9.
- Equation10 does not show Equation6.
The reader is seekd to come up with counterexamples that show some of these implications. The challengingest type of counterexamples to discover are the ones that show that Equation9 does not show Equation8: a solution (in Lean) can be create here. I placed proofs in Lean of all the above implications and anti-implications can be create in this github repository file.
As one can see, it is already somewhat tedious to compute the Hasse diagram of equitable eleven equations. The project I advise is to try to broaden this Hasse diagram by a couple orders of magnitude, covering a meaningfully huger set of equations. The set I advise is the set of equations that use the magma operation at most four times, up to retaging and the reflexive and symmetric axioms of equivalentity; this includes the eleven equations above, but also many more. How many more? Recall that the Catalan number is the number of ways one can create an conveyion out of applications of a binary operation (applied to placehelderlyer variables); and, given a string of placehelderlyer variables, the Bell number is the number of ways (up to retaging) to set upate names to each of these variables, where some of the placehelderlyers are permited to be set upateed the same name. As a consequence, ignoring symmetry, the number of equations that comprise at most four operations is
The number of equations in which the left-hand side and right-hand side are identical is
these are all equivalent to reflexive axiom (Equation11). The remaining equations come in pairs by the symmetry of equivalentity, so the total size of is
I have not yet originated the brimming enumerate of such identities, but presumably this will be straightforward to do in a standard computer language such as Python (I have not tried this, but I envision some back-and-forth with a conmomentary AI would let one originate most of the needd code). [UPDATE, Sep 26: Amir Livne Bar-on has kindly enumerated all the equations, of which there are actually 4694.]
It is not clear to me at all what the geometry of will see appreciate. Will most equations be incomparable with each other? Will it stratify into layers of “strong” and “feeble” axioms? Will there be a lot of equivalent axioms? It might be engaging to enroll now any speculations as what the set up of this poset, and contrast these predictions with the outcome of the project afterwards.
A brute force computation of the poset would then need comparisons, which sees rather daunting; but of course due to the axioms of a fragmentary order, one could presumably resettle the poset by a much minusculeer number of comparisons. I am skinnyking that it should be possible to crowdsource the exploration of this poset in the create of submissions to a central repository (such as the github repository I equitable originated) of proofs in Lean of implications or non-implications between various equations, which could be validated in Lean, and also checked agetst some file enrolling the current status (real, inedit, or uncover) of all the comparisons, to elude redundant effort. Most submissions could be deal withd automaticpartner, with relatively little human moderation needd; and the status of the poset could be refreshd after each such submission.
I would envision that there is some “low-hanging fruit” that could create a huge number of implications (or anti-implications) quite easily. For instance, laws such as Equation2 or Equation3 more or less finishly depict the binary operation , and it should be quite basic to check which of the laws are implied by either of these two laws. The poset has a mirrorion symmetry associated to replacing the binary operator by its mirrorion , which in principle cuts down the total labor by a factor of about two. Specific examples of magmas, such as the organic numbers with the compriseition operation, trail some set of equations in but not others, and so could be used to originate a huge number of anti-implications. Some existing automated proving tools for equational logic, such as Prover9 and Mace4 (for geting implications and anti-implications esteemively), could then be used to deal with most of the remaining “basic” cases (though some labor may be needed to alter the outputs of such tools into Lean). The remaining “challenging” cases could then be aimed by some combination of human contributors and more persistd AI tools.
Perhaps, in analogy with createalization projects, we could have a semi-createal “blueprint” evolving in parallel with the createal Lean component of the project. This way, the project could adselect human-written proofs by contributors who do not necessarily have any proficiency in Lean, as well as contributions from automated tools (such as the aforealludeed Prover9 and Mace4), whose output is in some other createat than Lean. The task of altering these semi-createal proofs into Lean could then be done by other humans or automated tools; in particular I envision conmomentary AI tools could be particularly precious for this portion of the laborflow. I am not quite brave though if existing blueprint software can scale to deal with the huge number of individual proofs that would be originated by this project; and as this portion would not be createpartner verified, a meaningful amount of human moderation might also be needed here, and this also might not scale properly. Perhaps the semi-createal portion of the project could instead be set upd on a forum such as this blog, in a aappreciate spirit to past Polymath projects.
It would be kind to be able to fuse such a project with some sort of graph visualization software that can apshow an infinish determination of the poset as input (in which each potential comparison in is taged as either real, inedit, or uncover), finishs the graph as much as possible using the axioms of fragmentary order, and then conshort-terms the partipartner understandn poset in a visupartner requesting way. If anyone understands of such a software package, I would be satisfied to hear of it in the comments.
Anyway, I would be satisfied to get any feedback on this project; in compriseition to the previous seeks, I would be interested in any adviseions for improving the project, as well as gauging whether there is adequate interest in participating to actupartner begin it. (I am imagining running it unclpunctual alengthy the lines of a Polymath project, though perhaps not createpartner taged as such.)
