IC16: An Alternative Way To Share Mathematical Knowledge

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IC16: An Alternative Way To Share Mathematical Knowledge
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We will reflect on the status quo of how mathematical (research) knowledge is usually shared through articles and books, and speculate on how the rise of the internet might allow us to develop alternatives to the current practice.
Area Mathematics Group representation Presentation of a group Goodness of fit Inference Imaginary number
Presentation of a group Mathematics Term (mathematics) Real number Projective plane Imaginary number Resultant
Mathematics Mathematics Ideal (ethics) Resultant
Mathematics Mathematics Group action Term (mathematics) Direction (geometry) Forcing (mathematics) Multiplication sign Ideal (ethics) Algebraic structure Mathematician Sinc function Resultant
Arithmetic mean Mathematics Network topology Projective plane Modulform Algebraic structure Energy level Theorem Mass Graph coloring
Mathematics Computer programming Building Existential quantification Network topology Network topology Mathematician
Mathematics Addition Mathematics Network topology Network topology Moment (mathematics) 1 (number)
Mathematics Military base Network topology Universe (mathematics) Content (media) Archaeological field survey Universe (mathematics) Line (geometry)
Ocean current Point (geometry) Mathematics Event horizon Product (business)
Ocean current Statistics Functional (mathematics) Direction (geometry) Multiplication sign Archaeological field survey 1 (number) Parameter (computer programming) Perspective (visual) Rule of inference Sign (mathematics) Positional notation Modulform Integer Theorem Predicate logic Set theory Descriptive statistics Physical system Standard deviation Model theory Projective plane String theory Cartesian coordinate system Proof theory Predicate (grammar) Network topology Order (biology) Object (grammar) Physical system
Area Point (geometry) Mathematics Natural number Multiplication sign Moment (mathematics) Theorem Physical system Power (physics)
Standard error Group action Momentum Multiplication sign Materialization (paranormal) Hierarchy Mass Polarization (waves) Theory Mathematics Many-sorted logic Different (Kate Ryan album) Cuboid Theorem Körper <Algebra> Physical system Standard deviation Gender Forcing (mathematics) Projective plane Model theory Morley's categoricity theorem Volume (thermodynamics) Line (geometry) Degree (graph theory) Proof theory Formal power series Iteration Mathematician Thermal conductivity Directed graph
Point (geometry) Ocean current Computer programming Finitismus Multiplication sign Direction (geometry) Archaeological field survey Hierarchy Theory Mathematics Causality Different (Kate Ryan album) Analogy Energy level Arrow of time Diagram Körper <Algebra> Set theory Projective plane Expression Computability Connected space Degree (graph theory) Proof theory Formal power series Category of being Exterior algebra Logic Mathematician Resultant Maß <Mathematik>
Group action Different (Kate Ryan album) Combinatory logic Model theory Knot Mathematician Imaginary number Food energy
Mathematics Imaginary number
but people who work in the area and the I and the good afternoon everyone and as the last speaker I would I like to take the opportunity to say that I'm really amazed by what I've seen the and the last few days the the then I don't know that presentations really blew my mind and and also so thought of it and control because I felt representation cannot stand up
against all these magnificent the visualisations everything so yeah the and they also by the my presentation is a bit of an but the ducklings in our sense that all presentations before fold being about projects which have already been finished or like are in a minute in the far stage of development but that's in my presentation is more like a pitching an idea rather than there and the talking about some great achievements but I will I will do my best and then another thing I want to say is that term what I find striking is that the the exhibition and and uh conferences gold imaginary but I thought this could just as well be culture realization realize because it's all about turning abstract mathematical ideas into something real and tangible and then yeah so my presentation is about something that is not central atom not really
and so excuse me for that so what what is there my presentation about uh and so I've written as some goals I want to achieve would this presentation and if I can only achieve like the very 1st goal and Mori very happy and that is to let people think about different ways to share research results and the I think I know a lot has happened also as you can see in this conference about 2 to make
mathematics more understandable to the and to to do what to wider audience but um among researchers and the way they share their results in a knowledge is X you still quite classical and and there I hope to of young but
make a lot of people think about other ways what what would be what we could do about uh and improving the way people share their mathematical research and then and and also what was the this is a a personal uh gold mines is to learn to create awareness and enthusiasm about ideal all comprehensive public database of mathematical knowledge and then it is actually
an idea which is not undermined that already exist among some groove mathematicians and most people when they hear about it they think lots semantic idea no it's it's too complicated how can you make a database containing all of no mathematics of course and but measures infinite but no mathematics is not infinite and and then so so people use if they if they heard about it think it's it's impossible and small group of yeah you could say lunatic people 0 think is possible but I think it's um I don't buy it that would be possible because know there are people who believe that in the next decade to can put you was on Mars so if we could do something like that's why notes who really achieve something which is actually easier than than than putting people on Mars a probably also the more valuable in my opinion n the so my long term goal is to learn to work on the only during the the
ideal of all comprehensive database mathematics into reality so so talk a bit about some current developments in this direction and so good the the the sharing of articles has already changed a lot since due to the internet and then and the yeah so so I think I don't need to spend a lot of time alone when when talking about that but what is interesting is that although everything has become electric and force should everything else paper journals and and it's it's I'm not against journals on paper are so Everest adjusted the freedom to do now to do the what he likes but in the and it is interesting that the format of of reaches articles hasn't changed a lot in a couple off of centuries basically handed the and they all just the new printed so idea RDI designed to be printed and now what of what I find interesting about some of the recent the projects on on the internet is a that's actually in the article I can mathematical results do not need to be structures structured in the
form of articles I got a project like stacks of and that prove wiki you could say Wikipedia's well where the Terence and and definitions and everything are structured in in a different way and how different structures in the form of trees you could say I put trees between quotation marks because you technically you could say they're are not many trees want us and open it or whatever but so so there arranging in trees and they
are granar at the level of definitions and theorems what means is that an but I think I can illustrate with this picture which actually is taken from the stacks website this shows uh lemma in in the very middle and the dependency of that land mounted on a land mass and like each color has its own meaning blue is definitions and so you really see how you you know how the mathematics is built up from from them as the rooms uh definitions and and so I I find this a very interesting way of of structuring of mathematics and and but you do not see it's for for research mathematics and
like Amendment mathematicians developed that you did not building a tree they're writing articles and that's said then did they do not really follow the same procedures like people in computer programming hand building libraries for themselves some
and them good and I find I find it interesting but as I said it's it's that these projects and although they are successful in in what they do and they would never cover all of mathematics because in the not grow along with active research and of course if something's should be all comprehensive uh of should
should contain all of mathematics should go along with their contributions research
contributions so so imagine so I will come to the to the difficulties in a moment but imagine that it would be possible for men additions to publish their mathematics as trees of course that it it's it's it's fine to publish um the articles alongside at to make things more understandable that uh and they're they're they're basically but publications are as trees and then and make these available once once they have developed as instantaneously not wait for the ones as a journal and agrees with it or or scenarios of archive but the the just this once they day-to-day road in new
dierent immediate publish it in their own line researchers and of course a donor makes sense because otherwise you will get a lot of debarkation if they could somehow relate to merchants which automatic dishes research
trees and and ultimately just like that will be the big goal as create 1 universal public research trunk and so 1 read public research tree where all the contribution all of contributions I edited and so dire why would this be a interesting but it would for example um make that much more easy it to to develop your your your mathematics because you do not have to write technical articles about all the things you're doing of course you should have provide survey articles with that but and and I think it would would uh and eliminate some overhead to which you which you would see nowadays in developing mathematics and then the researchers researchers do not have to bother to buy structuring contents and so and the it doesn't matter whether a landmark because starts in the beginning of an article or at the end which is now sometimes like gives a headache uh the 2 2 of depends a lot of course but in in when I was writing the Jesus of really give a give a headache when you were writing uh which you actually you would like to use in the 1st chapter but um but it was kind of contents it's a of it in the 2nd chapter they don't want to to refer to a which is no appears later on the writing so and it I think it and it makes things easier uh and as a senator in news also are available instantly only cells can be updated and um yeah it would it would may get that goal for all comprehensive database it may be the and there are a lot of challenges you put the yeah you could mention of decide how to how to design software to create these research trees saying in in a convenient way uh and what bases role scientific
journals and I think of all these the challenges of our are solvable but the question is like what is the best solution there's only 1 question which is really what makes things difficult and I think that's really the bottleneck why currents attempts that do not when you achieve that goal for for creating a database of all mathematics and that is that the social logical
puzzle II when got the terminology from in an article that describes why most of these events filleting point of keeping organization multiuser products without restricting the freedom of contributors the and there's a was so what do I mean when um so although Madame as an exact science the there's actually no other universally accepted notions and universally accepted annotation or terminology uh or framework so and this often does not the they a big role in the
in in survey articles or in uh in books but it does uh play a role in in recent articles because you know each researcher prefers his own notation its own approach and no 1 would not like to be dictated by someone else and what what's annotation to use what amounted to use so you should be entirely free in choosing his own approach and so would that's why I think that that although the wiki model is very the powerful for for many applications but it could not uh work if if everyone would get the freedom to to to publish what it wants that have order we do on you you need some consensus among among the people who a who contribute but and then and there's the the a good model
uh which is for example a Houston in online work but as I get up where there is full freedom but it will lead to fragmentation and if there's not enough consensus denoted projects get forked and then people start to work separately and and there it it will be the lead to fragmentation and and so ideally would have assistant is about free and coherent n and so what I do right here is just my own personal idea what might be a solution in that direction and then I'm I'm I'm open to to uh lot of people serve you in that so what and we could see is a solution that is what I would call a perspective istic database and then what what what do that's so like any database it it can contain any kind of data types like integers strings file object pictures of and very important the user ideas because they wanted to use the spear portant role in this concept and the and then to objects like predicates and so if you if you are familiar with predicate logic you might sort of um already have seen this you have a data variable what I'm writing here and so that's so there are n ary uh predicates which as arguments can have any uh object including made objects that's important and then our functions like 4 basically pieces of code to people can upload stats during these uh MIT objects into other MIT objects and then but and everyone is allowed to into 2 and the contribute any kind of of of these objects to database but very important um every objects is tied to the user it added so there there is where the perspective is take a principled comes in and so at anyone can do whatever he wants but every contribution is really tied to the user which contributed and and my claims that these simple rules could lead to a free and currents collaboration model so the much how much time I have to explain how this all would work out but that and the basic idea is that OK so you have to the ontological relationships uh which are of which exist mathematically help but on long definitions Terence etc. and then the so I give just a few examples like the steering has this title uh uh this theorem has formulation and and description that era I am and then relationships like this this tumors generalization that during this prove prove that during this picture is stressed approve uh things like that but and as a standard the predicate could also have made objects as arguments so you could for example it contributes to the the things like this proof is wrong so I get that the capitalized words denote the predicates and that the the lower case words denoted good objects the or for example heights all contribution of this user or I agree with its dis this predicate which might have been a up there and contributed by another user and for example I agree with this sign convention you know when when when different dishes using no sign convention and um or I follow you know the settings of dis expert users that is the gist of a couple of examples and if you would want to get a full working system you would have to add new many more uh predicates and perhaps in in a in a better way than just a once I've written here for 4 for pedagogical for for 4 reasons to make it understandable and that's a yeah uh so if you build up the database in this way what you would get is that uh all the knowledge is is is there but and what people get to see it depends on their own preferences so so people have another perspective on the same database so to say today they browse the same database from another perspective based on their preferences and a which also of course a lot of freedom to contribute something in there the form you want yourself well and so the question the reason like you did scale up to a whole big project like for example Wikipedia and I think this is this a tree the main concern but I think it does because uh um you know and as as I'm not appointed as wikis of everyone can be an editor I can be acknowledged and recognized by others and the ones he
covers his own research area hand edits his own research in the area and other people think the desert in in a good way like a super editors so discerning good do some of them as a cult picking uh along different editions set for each research area I could create large additions and for for for larger research areas until you go all the way up to like an addition of of older that that goes all of mathematics and of course you have many parallel additions and due to the nature of the system but anyone can finds
his own way it's own the preference In incited the whole system when and then yeah so as some final remarks it's the it's the disturbance the the in in the 15 minutes is not easy to to to explain how this would all fit together and then I've been thinking about it for a for a long time but then if people are like the the topic and are interested so I'm very happy to explain in more detail on how how I think the it could it could it could work out and then at the moment myself working on a prototype and and young them I could use any suggestion or any help from others to to make it so working and that's all I wanted to say be it questions no thank you I think it's very very interesting the I've been thinking about something like that and I formed that I'm not the only 1 and not we are not the only 2 people thing on that I've been seen i've seen set out broad itself and uh organizing mathematics and they depends on the point of view of the Feature power from because those people trying to make a big database and title right these theorem and every definition in a matching friendly way sure that in some years in some markable the case maybe
they're artificial intelligence is here is so good in also and use URIs can be done by machines and as people trying to these but I of conduct in British like that and simply flip failed because mathematicians are not interested in that i you can the force people to write their theorems with some language with some matching language for a computer language that people are just so right there at the the articles in the with the the 2 so the tools that they are like a standard like archive or the mathematical revealed by being mass source of the blood of all these things that do not require you to 2 books at this so uh I don't sing this could be that the only way for writing mathematics so did the only things that I've seen are more like getting songs some momentum I like doing accommodation of mathematics polarization of mathematics for a higher volumes so I mean if you are a mathematician doing your research in 1 field maybe you are not aware of what what is people doing another field so maybe we can do of the uh some communication and categorization of mathematics of our field to people with the and all of this is possible the iterations for a very specific target and follows that specific alleles and that you cannot like this like a like a machine called so that's what what when you were writing this kind of tax and think that I don't feel that a mathematicians pollute the degree of say OK I'm going to write their research with that alright we'll this on these this year and has and maybe it's not the also ordered the real all of this group of people discover that during the past or ideas maybe somebody else and these organizations it so you just of some salt and American would come talk later because I know several of projects that have been going toward these you say the off of a hierarchy of that to your organization of mathematics with small ontologies yeah so and I and I haven't even talked about formalization of mathematics prospection most of the projects I refer to and also incorporates like a formalization of mathematics as that that of computer can verified prove and then in the previous talk and I gave about this subject and I also incorporated that's and and them but they say I can and uh yeah there you have another complication namely that know there's also there's no consensus about which a proof assistant use local which computer system which software to use to verified proofs and there you have different preferences and then you also have wider than the problem that would which have been size it's no different programming languages cannot communicates with very effectively with each other without creating of the risk of of of of box or errors so and the gender and most projects I and the wealthy needs are actually about to formalize mathematics and then that this creates another and complications but hopefully like in the end there's all comprehensive database of formalized mathematics that would be even more interesting than thank you we so you've got a very impassioned ambitious project their bill and they will come um my thoughts are on you know I'm a mathematics teacher and an an an educator and I really suffer for yeah over there I have you also looking for were rare in my view of mathematics education I really suffer from the fact that all the teacher materials already disparity everywhere a bit and you always have to pick and choose and rearrange yourself if you're not happy with the ready made textbooks so I can theory tell you that at least in the field of mathematics education each day I and II of the pipe giraffe sounds centralized database where can drag and drop by things and don't have to lose time to cut and paste um things and resold the exercises because there are no solutions or false solutions and so I think that that there might be a thought that you can maybe also um that might be a target group that even also be very receptive for such an idea yeah so uh and in an article and written about a certain idea but also in about formalized database of mathematics there and alter alters because it's it's it's Caltech QED manifestos and if you want you could have a look at it it's like 1 of the 1st papers published on on the ideal putting up a centralized database of all mathematics he re stressed like good educational value of having a centralized a database of all mathematics rather than know having to go from 1 source to another an our uh shows of course problem that if you do not have to subscription even cannot you know the see of the contemporary mathematics and so on uh but beyond that that's something I didn't mention here but that's people have already written also about educational value of having such a database the further question that like and yet so all very ambitious I like it 1 of the main things what would be the thing me about your talk his this remark you made about trying to not me both kind removes the need to write research articles like
the presented research of this is probably the most overambitious market phone in this whole talk and you somehow like I think I remember see use something like mathematicians would eventually not need to write these articles all he tried are so the articles once you somehow get some sort of system in place that the mean soul can you and you have a um can you elaborate on more concrete ways maybe of not writing research articles but still maintain the regal the of proof see how would you model ultrastructure proof if he didn't write an article but what I find very interesting is how programmers and collaborate on line you know working on some a piece of software and where they are actually
they're just you know building up the the data go together of course they use platforms around its to communicate into too young to to make the collaboration the go smoothly but in the end they they they can just you know a bill that what what what in this case would be that the hierarchy of mathematics it and they build their hierarchy of software that you young without you know communicating it through degree of the article said to say so what I have in mind is is is basically what what what happens nowadays in the software development and then and then of course you know it's it's it's I think it's it's good to have survey articles you know which which summarize you know what what what has been built and then Annals of course I I do not say that this and not that nobody should right classified cause anymore I II and you know I and I really like that the the freedom to do what they want to see what what works out best for everyone but but I'm just interesting about the book but an alternative I got alternative because this should be something that's that's it is possible besides the the things that are already possible nowadays mainly Arlington difference I see the list that programs are inherently
constructive and not prove our that's the main thing like that the difference but I well you mean you mean no constructive and software does something you write something in it out something in finite amount of time finite computations risks of proof is often what we the point like getting dressed in blue infinitely many things you can't really and also like logical constructions all like save somewhat diagram some so much on a diagram of the logic of proof that's I mean mathematicians to it all the time I do it all the time but there's a big difference between join a whole bunch of arrows and like a logical connection between them actually writing the whole thing out to the level of quite for of from yeah so so and analogy I use turn it building software is not to say that mathematics and software engineering is exactly the same as of course our services although it's interesting that like formalize mathematics is very similar as even much more similar to to in there to rot writing software and then there are writing code some and and in a way that is actually constructed there are and how many are not going to in technical details but there are many things written about its that yeah its why the it's constructive in the sense that that you you describe how from 1 the set of uh expressions you go to another 1 and so yeah this is something we we could discuss further but I think it's not really a problem for the for the project maybe which uh 1 of the questions that thank you very interesting and I definitely agree with radically improving on the ancient methods of disseminating knowledge I just like to say I think optimistically that it's not so far off as as you might imagine and that the end lab I think goes further than you indicated to was doing this and for people who don't know the and others a wiki there exists for research in my field which of high-dimensional category theory and the the advantage is that the field is quite small and and so it's quite easy to create a huge database of all of it because it's not that big yet and it is in fact current and it grows all times that definitely has a very up-to-date research and sometimes I go there and discover other people's explanations of my research when I'm trying to remember what I did that that's that's how current it is I think what you said about the problem of socialization is definitely key there because instead of being a democracy it's a kind of time okresie and so 1st of all it depends on a few extraordinarily energetic people writing everything up on everybody else's behalf over time including research seminars that they've been t which is white so current but then as a result it kind of gets skewed towards the point of view all of the people who have the most time to spend on it and because of the people who spend the most time have but they have a say aquatic at the point of unit Opel in home home of type 3 point it's gone in that direction and people can add their own comments and they can offer
elaboration and appears in an extra extra combinations appear but it's still as you say the problem is that because there are certain people who have some somehow huge quantities of energy for writing this stuff up it's difficult to know how to keep it but you
have to keep it more broadly and the apart from by trying to persuade other people and you can't persuade other people to put in the amount of energy then it's not clear what to do so I think the interesting difference the whole that the interesting questions about how to regulate turn the old method we have peer review where it's kind of experts who were picked by previous experts who were prepared by previous experts in a kind of perpetuates like that and in this model we have you whoever's got the most energy and and maybe we need to think some more like you say about how to organize learning that might be the most difficult question be they are the questions I don't think it's that far off and others then lab is really extraordinary in is only been there for and I remember when the discussion started about the idea of this thing and it was maybe a few all I and who it seems like about a week ago but made it was 5 years ago or so 8 years ago or something and now it's got everything on so I yes I don't feel daunted by thinking that these things are totally far off yes a fully agree that I was so so there and that is very successful but I think it as you said because it is not the people actively right and it is a is a is a group of people which is not so big that they can and that that that you know fragmentation is is is it is it is a threat so they and they are small enough to do have some consensus along the thing they're writing up and but and I think that if a node is taken they also say that if you want to contribute that they do some kind of the the editing like some kind of monitoring has so knots just everyone can can write up a anything or you they they they they could remove something that is for example they think it's not correct or or or or be wrong so like like like Wikipedia and then and and but that does that does need to even go up to the frontiers of research and then and the only difficulties to find something that really you know this not does not only go to the frontiers of research but also I can involve like ideas speaking all the researchers know older research mathematicians around the world profit at the end of another solution
is is needed well this is a very interesting discussion I think we should quit now let's thank the authors speaks of the session that