 Okay, we're live now. Moment for the settings. Okay, we are live. Fantastic. So yes, as I mentioned, as I mentioned in introducing Mel's talk, this is the sort of formal block of the conference, which I really like. I'm very excited that we could have this set of approaches from machine learning and mathematical practice, both in the in the meetings. So it's my real pleasure to to introduce a talk that I have been really excited to see since the moment I read the abstract for the first time, actually, from from Henrik, Henrik Roksortzen on digital humanities for philosophy of mathematical practice. So with that, please, the floor is yours. Thank you so much. Thank you, Charles. Thank you, everybody for sticking around. What is late in Europe. So good afternoon to Europe and good day to the to the Americas. It's also a great pleasure for me to to be speaking at this conference. It's, it's, yeah, Charles introduced it as the formal section. It's also, to me, a unique opportunity or a new opportunity to speak to to you as an audience steeped in in in digital science and in digital studies of digital science, something that that we are not really accustomed to from where I come. So let me tell you a bit about from where I come. So this philosophy of mathematical practice is a relatively newcomer to the philosophy of science, even to the philosophy of mathematics. We think of ourselves as empirically informed philosophers of mathematics. We don't study classical ontological, it is political questions about mathematics so much as we study how mathematicians function in those philosophical frameworks. So we are interested in the practice of mathematicians. And of course, many of you are also and I know you from many of you from the from the philosophy of science and practice. And you can think of this as a slightly derived version of that, but aimed to mathematics with some drastic differences with come mainly from from from the methodologies that we'll have to to adopt. So I want to say a few things about what is this empirically informed philosophy of mathematics that I'm talking about, and then try to address how digital humanities can aid us in what we want to do. And in a sense, you can see the tension of my talk in this little diagram over here. This is this depicts the number of reviewed. Let's let's make it simple. This depicts the number of article articles published per year within the field of mathematics. It comes from a database called the mathematical reviews, which is a service that was introduced in the in the late 1930s to get mathematicians an overview of the development within the field. So so the mathematical reviews will abstract or review each of these papers make an online record of it. And and you can use that to sort of navigate the field. So that explains the drop. It's not that mathematicians have stopped working. Two years ago, it's simply that there's a delay, a lag in the reviewing of their papers. So you can see steep annual increase and you can see, therefore, aggregated increase in in the mathematical literature. So any kinds of of philosophical empirically informed philosophical questions we may ask about current day or contemporary mathematics will somehow have to take account of the fact that there are, as of yesterday, about three million eight hundred thousand papers registered within the mathematical field. And of course, the standard methodological issues that we have long long that we have used in in in asking philosophical questions such as case studies or or specific important papers, those simply dwarf against such a huge practice. So we need something else to get at this. And there my my claim and my my hope is that that we can learn from you guys from the digital humanities and digital studies of science to to get at at better tools. My slide, and it's not a promise, but it's at least a an idea to get you interested is that there might also be a feedback mechanism. It might be that studying formal sciences like mathematics with the natural language processing with other machine learning tools might offer something to the machine learning community, for instance, or simply based on the fact that mathematics is such a syntactically and semantically restricted language, where some of the text, the formulas, the figures play into a more perhaps a more strictly confined space that we can actually get more fine grained natural language processing tools to to work at that. At least that's the that's the pitch I try to sell to my my computer science colleagues. I have a background myself in mathematics and computer science and have been a professor for history and philosophy of the mathematical and computational sciences here at Copenhagen. So I this is a project that sort of spans all my my interests. Here's the graph again. You can see the the increase that was just if you couldn't see it before. But since I can't hear whether you can see it or not, I'll just skip and think that you could. So what is this empirically informed philosophy of mathematics? So usually I say that ideally, if you want to study the kinds of things that we are interested in in philosophy of mathematical practice, some of them are beyond our reach, we might want to figure out how mathematicians think and develop knowledge. But that is at least often a private process that would require us to peel their brains open and and look at at the electrode. We are not allowed to do that for legal and ethical reasons and also because it's it's it's not what we do. But we could at least and that's one of the new directions we could subject them to fmri scans while they solve mathematical problems to see whether they use specific spatial areas of the brain. So sort of importing methodologies from cognitive science or from other other disciplines. I also like to this picture because it it shows the sort of old image of mathematics as a private and individual enterprise as as the mathematician as an autonomous agent creating new knowledge. And of course, in the history of mathematics as in the history of science more generally, we have had a social turn in the sense that we have become increasingly aware that there are social components to knowledge production and recently also this practical term. So what we're now interested in in this emerging field of philosophy of mathematical practice is an empirical grounding to ask questions about the the the broad process of practice of doing mathematics from the heuristics and the external representations using computers using diagrams using communication means both for for heuristic but also for for proof generations. How do mathematicians communicate in formal settings through their papers but also in in less formal settings at the at the blackboard. And these are issues where we sometimes have empirical access in the sense that we cannot well we could monitor a mathematician 24 seven and see what other science are produced from from the creative process. But it we lack the the we often lack the the group meetings that that you can that you can attend in the empirical sciences to get sort of an anthropological or sociological view of that. So in a sense what we are if I were to characterize it what we are interested in are the properties of proof that go beyond simple derivation. So so we are interested in it's a that is a statement argued against something. Typically people have considered the philosophy of mathematics to be about how proofs generate certain knowledge and and we're interested in in broader aspects of proof proofs as vehicles of communication proofs as vehicle of insight proofs of explanatory proofs all these sort of extra things that proofs can have and we we want to study that and to get out get at that I think we need to get out of the armchair and do observations interviews we have done those I have even done embeddings and collaborations working with mathematicians in producing new knowledge and and then reflecting on on how that could what that entailed in terms of division of labor in terms of of interdisciplinarity in this or trust social epistemology in the sense that that that you have to trust each other both for the epistemic claims but also of course for the for all the other immoral elements of a group collaboration we you could do focus groups or delphi studies you could many have done historical case studies but the closest we get from say an n equals to one study or at least small n studies to bigger n studies have been have been questionnaires and and even those are at a relatively small size and a relatively small small sample but luckily recently there has also been a much more interest in driving the methodological discussions about this new field of philosophy of mathematical practice and I want to to flash the entrabidine and Matthew English book or anthology that is sort of setting new new ideas into how we can approach this methodologically and you can see what I want to say as a specific addition to to to that developing new models or new new methods for for for doing getting at the philosophy getting at the practice that we want to study in PMP okay so that's that's the pitch that's what I want you to sort of hook on to there are these guys doing philosophy of mathematical practice their practice is out of their reach either because it's unethical or because it's it's it's what Matthew English often scornfully calls exemplar philosophy so if we want to get at a grounded empirically informed philosophy that does not only relate to a few exemplars then we need to develop new tools and digital humanities in this broader sense could could be such tools and when I say digital humanities it's because I don't want to be very specific about exactly what it could be it could be big data just analyzing statistically analyzing big data it could be more refined natural language processing it could be image detection and image classification and we have we have tried all of these or we are trying and developing all of these approaches in in our program so let me say a few words about the the thing at stake here so I see myself as asking and answering philosophical questions and when I say digital humanities at least in the old days in the in the literature branch of digital humanities it was often a matter of simply stating facts and counting how many when when the word boat and and and ship is used in Moby Dick and counting those or counting which characters enter which sections of a drama that's the that's the that's getting the empirical material but I wanted to be also sellable to my philosophical colleagues so I want to or the design that we are that we are envisioning is to start from an established philosophical question or concept that we then want to explore using all kinds of statistical tools and then we can go one of two ways we can either use the statistical exploration to pick out exemplars for for qualitative analysis that can be these exemplars will then hopefully be more representative or at least have certain qualities that are beyond just picking a random example or a frequent example and then based on that qualitative analysis refer it back to the philosophical discussions or we could do I have a more sort of scientific method approach with a hypothesis test if we can somehow instrumentalize the the phenomenon that we are interested in and we we're trying to do that but that is certainly further away from from the the philosophical way of thinking about it that we are that we are that we're coming from but in the end it's a it's a it's a tool what I want from digital humanities is a partial tool to turn philosophical questions into philosophical analysis through this statistical exploration that I that I have here and that sounds all very very broad and and perhaps even empty but let me give you just just a few examples so one thing that that this new approach if it's a new approach but new approach in our field has turned up is we have new corpora we have new sources of empirical information that we didn't know how to access previously and many of these corpora are of course as you as you know they are preferably born digital so my colleagues I think last year studied online mathematical discussion searching for explanatory phrases in an online problem solving say bulletin board or or social media that was born digital they could get the data relatively quickly but they lack or they had a hard time analyzing it because they were looking by human eyes for for these explanatory structures so what what this allows us to do is to look for other corpora that are born digital and of course the the mathematical review that I started showing you is a surprisingly underused corpus of of mathematical insight I think it's underused because it is what I call secondary it's not the mathematical knowledge produced itself because those are in the journal papers but it's reviews of those journal papers so its opinions summaries and connections drawn between uh or in the in the literature but drawn by colleagues and others who are experts within the field so we can mine that for all kinds of of secondary information which will somehow ask or allow us to to get at at the at the philosophical questions that that we have so that could be one uh one new corpus that is under analyzed another corpus that we are working with is of course the archive but that I wouldn't say is under analyzed but the mathematics part of it is perhaps still under analyzed and something that we can do and use as a test bit for for our analysis and our hypothesis and I want to give you two of the examples that we have done it's a relatively new project but we are we are working with students and colleagues and the collaborators overseas to to to get at this it's a small community and it's a sort of pushing our pushing our stamp on this this is from work that one of our students has I hope just turned in today for her master's thesis where we exactly use this mathematical reviews or math sign it data we do a query and then she ends up with a corpus she was interested in studying experiments in mathematics and and for those of you not in the know about that experiments are typically not a classical justificatory practice in in mathematics but nevertheless since the 1990s mathematicians have begun speaking about explanatory or experimental mathematics as a specific sub genre of of mathematical discovery so she was interested in in studying experiments in mathematics and we ran the analysis and she ended up with a corpus and she wanted to to do a categorization of the kinds of uses that this word experiment was taken to mean in the reviews of mathematical literature but she still ended up with I think after after the initial filtering she ended up with about 50 000 records that mention experiment which itself is a bit perhaps surprising given that it experiments were not supposed to play a major role in mathematics but but for the the qualitative analysis we used and we then trained and used a a binary classifier to sample or to filter out the part of the sample that was concerned with numerical verification of of mathematical claims so there's a specific large chunk about 30 of the uses of of the word experiment in mathematics that is about numerical verification of claims and and once that is established then she didn't want to see more of those for her qualitative analysis of course this is now absolutely not quantitative but but we have shifted focus here and she wanted to see more examples of the other other types of use of experiments in these reviews so we could use a a machine to filter out the part that was already relatively well understood and it actually could be trained on under I would say surprisingly it could be trained on the sparse data that that we had to to sufficient to be sufficiently effective to be of use for her in in getting at the sample that she wanted so this is one pathway of the graph that I showed you before from the philosophical question about experiments in mathematics then to statistical exploration and filtering into this qualitative analysis that then ends up in what will be her argument I have still to read the thesis but I imagine that there will be from not just imagine I have talked to her of course but there will be a new classification of of various types of uses of experiments in these reviews that we can then take back to the philosophical literature and and study as part of of that so that's one of the ways that we have tried to use machine learning agents in in in focusing our attention in this huge corpus that that we are that we are now approaching because it has become feasible to us the other part which is perhaps more to to to this I would I don't know it could be closer to what what you what you what what is done in the in this in the in the sciences in the digital studies of the sciences is I work with my colleague Mikkel Wilhelm Johansson who last I think three years ago he started he's interested in diagrams diagrams play a specific role in the philosophy of mathematics and the philosophy of mathematical practice because they are external representations of thought and they play the specific role and philosophical discussion going back to to person and and so on that they can somehow be a part of an argument they can you can do diagrammatic reasoning so we therefore they are often studied but the kinds of diagrams that are studied are often of of relatively small sample type so they'll either be commutative diagrams such as those that you see here or they could be simple not diagrams from from from to policy those are the prevalent and Euclidean diagrams construction diagrams from Euclidean those are the three I would say prevalent types discussed but there are many many more diagrams out there so what some poor Mikkel he made a sample of three journals every five years and he looked through all of them noted when there was a diagram occurring on a page so he had to he had to flip 53 000 pages by hand and once he had done that I made the stupid remark that I think I could do that I think I could help him by automating that process so so we we put this image detection routine together it's trained against a relatively small corpus and then put to use or to predict on Mikkel's hand hand drawn corpus and then put out in in the wild and it's it functions surprisingly well it needs to be tested again or developed further because it's trained on just one journal and applied to these three journals but but we are very confident and you can see it scores rather nicely with at least these diagrams of course I didn't pick the ones where it scores wrong it's actually fun it scores some images as diagrams that I wouldn't classify as diagrams and we wouldn't classify as diagrams because we have made a decision and a definition of when in the coding process when something is coded as a diagram but many of these are actually issues that are discussed in the literature for instance what is known as a chain fraction that's a two-dimensional object or it could be written on one line so this two dimensionality it apparently fulfills but which is a part of the standard definition of what is a mathematical diagram so that is picked up by the machine learner and we have to sort of say no no no that's that's not what we meant and then we have to develop the machine learner for that we also decided to rule out matrices because they don't have this reasoning potential as much as diagrams what we typically think of as diagrams do but we did include commutative diagrams although in mathematical practice they are actually very often almost simple like so so so the entire conceptual analysis of what is a diagram or not we actually had some very nice discussions between what humans what previous philosophers and what the computer would think of as a diagram so we trained that and got the sort of proof of concept that we could that we could do that but then becomes a question of what can we use it for what kinds of philosophical issues could we use it for and one of the things that we definitely want to correlate is discipline specific issues for instance in in the use of figures or the use of diagrams that's something we're working on and there's this very nice I was reminded of this in the in the keynote talk earlier today there's this nice paperscape visualization of the archive so so we we are putting the the the diagram detector to to work on mathematical papers in the archive trying to figure out whether there are more or less visual components of or disciplines in mathematics and trying to to figure out how to characterize that in a in a quantitative way working with people who also know a lot more about about visualization and and statistics that I do so so that's at least there are two there are two I gave away two pieces of of of information besides the content there one is yeah of course when you have a tool you look for what you can use it for isn't that what what tools are great for and the other is that it takes an interdisciplinary team to to do these kinds of of interdisciplinary work that that digital humanities and mathematical practice will pose and and in our team we are both mathematicians cognitive scientists and a few statisticians and philosophers so it takes that kind of setup but it's then very rewarding when when we actually get to talk about these things so one thing is how do visual elements of mathematical practice spread or how are they used in different parts of the field and we'll need to or the discipline different sub disciplines within mathematics and we'll need to to also work at visualization of how that is going on but I think I said that we could also hope to to give something back to the machine learning community and part of it is what I'm trying to sell there is this sort of four four step program what we call the tongue in cheek of course we call the Copenhagen program for machine learning and philosophy philosophical studies of mathematical diagrams one we see that there is a lot of philosophical study drawn from a small set of different diagrams we want to enlarge in that set to bet to get at a a better and more empirically informed philosophical analysis so the first task is to train and develop better detectors we are we are working at that second task is then to challenge the classifications that humans philosophers mathematicians would come up with of these diagrams and perhaps challenge them in a way that could be used exploratorily by our colleagues say if you if you if you specify the clustering algorithm and you specify a few keynotes in the clusters how would it group a corpus of diagrams that you would have and what could you see so the so the inexplicable nature of the of the machine learning could perhaps even stimulate us to to to revisit some of the classifications that we would have that sounds that perhaps sounds sounds very far off but given that there are so many different types of diagrams in use in mathematical practice today Mieckli is glad that he stopped his study five years ago because if he were to cover the last five years it would just have exploded into into different different types of diagrams but given that we can do the we we have the diagrams we have the closest context that they are presented in and we can therefore perhaps develop proxies that would that would actually allow for some kind of of meaningful unsupervised classification of these diagrams to the historical side we want to everybody's talking about epidemiology these days we also want to do our epidemiology but of mathematical diagrams so how do these things spread in the mathematical literature it's fairly easy to to link them to to metadata and then again we are back in the more sort of traditional digital humanities approach that that would study the spread or the the correlation between content and metadata so so those would be sort of new issues where we would need to speak to and will are speaking to computer scientists to to get their interest in it as well but which would immediately inform the empirically informed philosophy of mathematical practice better and as a byproduct it can even help us do more sensitive selection of of interesting diagrams for contextual analysis a bit like the thing that I showed you with my student where we had filtered out all the similar things so as to focus attention on the on the more specific ones so it's a bit of a pitch it's it's a bit of a this is what we have done this is what we would like to do but I hope at least to have indicated show to you that we can put digital humanities machine learning natural language processing image detection to use in philosophy of mathematical practice so it can serve as an aid to us what might that's good for us that might be be less interesting for for the other side we also I'm particularly interested in how this drives methodological reflections both on the philosophy of mathematical practice side but also in the terms of of the digital humanities or machine learning side that they are is there a way in which we could actually use the the the lack of explain that explanatory power of a machine learner not not to see that as a problem but to see that as a as a question for for independent ways of analyzing the the phenomenon and therefore also that there's a there might even be a pushback to to the machine learning when I talk to to our expert computer scientists they are very interested in this limited or restricted semantic also syntactic but mainly semantic setup that is that is posed by mathematical texts as a combination of text formulas and figures because each of those they can analyze but they can get at a better comprehension of of the content or we would like to get from the philosophical side at a better comprehension of the content by combining these three elements and and possibly more so that was that was the the end of what I wanted to say so I hope you have questions fantastic thanks so much yes so I have a I have a semi clarificatory expansion question while we wait on a few more to come in although there's already some others um so I'll I'll take the chair's prerogative one more time tonight um what kind of features do you expect the classifier to be picking up on in the diagrams do you think it's mostly going to be kind of spatial shape structure type stuff do you think it's going to be able to see some sort of content content full a kind of analysis I'm just interested you know a mixture of both or are you are you really really have no idea and you're actually hoping I got a bit of the idea that you're hoping to be surprised by what you find when you apply it which I know that feeling right I get that but yeah what are you what are you thinking it will it will it will see certainly willing to be surprised but what it by what it finds um I so so the classification that we did or that migl did for for his paper that was that was based in different types of cognitive offloads so so resemblance or algebraic or what he called abstract and at least the abstract class which is free of resemblance to geometrical situations or spatial issues that is a huge class that is that is difficult to to to put under one unless you get at everything that looks as a as a as a community diagram so I'm hoping that that it could find some of the standard types of that at least that's what I'm feeding into it so that it should be able to detect known types of of of existing diagrams um it should ideally at least when when we get the the the part text the intertextuality aspect going it should ideally also be able to to combine the diagrams with the surrounding surrounding argument figure out what kind of of role does the diagram play there's a there's a there's a joke which is actually a true joke in mathematics that is an entire proof is see the diagram or chase the diagram and then end of proof so these diagrams do play an an epistemic role in in the in the communication but also perhaps in the thinking and it would be nice if it could pick out pick out those and then uh also more perhaps more subtle differences that that we could that we could see um one of the projects we're also involved in is is tracing diagrams in physics in a specific specific uh branch of Feynman diagrams that we we want to trace and there it will actually that it will be much more restricted what is looking at but we want to to quantify the prevalence of a specific form of of Feynman diagrams so it it it's not necessarily that I want a different classification come up with it what I what we meant by this part two is actually ideally to make available to our colleagues who try out different classifications a corpus and a clustering mechanism that would allow them to to sort of see what try to describe philosophically what would be the the classification and then see it run on on a on a corpus of of examples because at least some of us in the philosophy of mathematical practices are are are committed to the to the to the examples even if they show that that our classifications are are a bit of very cool did that answer your question yeah no I think yeah absolutely absolutely thanks so much um let me go to a next question here from uh from Stefan Hesburgen who asks uh is it a viable strategy to in the in the mathematics community to parse latex source for for papers or preprints and try to come at come at diagrams or or reasoning from from that side is there is there data like that available there is data like that available say and that's what we're doing so so the so so um that's at least part of what we're doing for the triangulation we're doing for the triangulation so the the image detector runs on images of course and picks out diagrams in in images uh partly because it's it I mean meekle meekle tried meekle trained it for a century of mathematical publications so I'm also a historian so I want to be able to do that uh but but what we are doing now is we are trying to correlate uh the image detection with the with the structural information that we can get from from latex and we and we we we can do quite a lot and and we can we can get structural information we can get what I call contexts so there there might be an outer context of an of a paper and and there might be a theorem statement which is an inner context and there might be a proof of a theorem which is also an inner context and so on so we can we can try to figure out where structurally different types of of text or different types of figures occur and then link that to to the to the to the images that we get and of course it's also a great training set for for for the for the for the image detector because we would know exactly where we can trace exactly where the why the images occur so yes and there's also the sigmath link uh special interest group on mathematical linguistics who uh who sort of curate a lactic dataset of of the archive but the entire archive is available through kettle and and and ds utils so you can actually yeah we are we are we we have quite a lot of data um nice nice position to be in yeah um yeah so uh one more oh several questions cool this is great uh so the next next question coming in from uh from alexander summoner who asks do you think that that these questions in philosophy of mathematical practice could offer us any insight into machine learning when it comes to uh in particular to things like explanation or algorithm interpretability because that's been a a bit of a theme through through some of your remarks i hope so uh because i also i mean what i try to present today is is if we have this tool called machine learning or or digital humanities more broadly i can apply it but i also want to give something back and the especially the the discussion about explainability in in ai is something that is that i'm very interested in and i think many of those it might not be this particular approach but many of those discussions would would could benefit tremendously if some of of the philosophers of science and and philosophically inclined computer scientists would get a i mean a bigger voice in in that discussion and i think the my my suggestion that it could be that that my classifier would come up with something that is uh that is surprising to me and therefore requiring explanation but i can't get i can't ask the classifier to give me that explanation because it's black it's opaque but it will still prompt me to to to think about and provide human explanations for humans so i think that that is that is at least that motivating force i think the biggest uh philosophical computational uptake of of what we do is we're also interested in um we have a similar paper on on what is uh so there are things called interactive theorem provers so computer programs that help mathematicians prove theorems and those are sort of a sub genre of of mathematics today they are studied by formal mathematical people interested in logic and a few who are sort of mainstream mathematicians use them for for their teaching relatively few and we have then combined the philosophical of philosophy of mathematical practice studies of what is what are the drivers of of human mathematicians well they wanted to be recognizable they wanted to be at the right if they choose a problem at the right level of difficulty they wanted to be uh within their within their toolbox uh and those criteria how do we they want to use external representations like diagrams they want to publish something that can be read by other mathematicians and many of these are exactly the i wouldn't say stumbling blocks but the great obstacles that that interactive and automated theorem provers are are coming up against these days they want they are extremely good at they are shown extreme progress in finding proofs but getting getting the rest of the mathematical communities to give up their sort of human proofs is difficult and it's it's even not completely acceptable to publish something by by computer proof right so so so we can perhaps point to from the philosophy of mathematical practice we can point to items or perspectives on human mathematical practice that would uh that would ease say the the use of of ITPs into into ordinary mainstream mathematical practice if it could fulfill more of these concerns that that human mathematicians have so that's that's that's a different way of of being of service to to the to the mathematical community but but no this closer to us I think that's very cool it's very cool thanks um question from uh from multi misrahi us do you think that a similar approach uh can be applied to study symbolism or notation in mathematical practice in addition to diagrams sure I think I think we can do most things that we are that we are that we're interested in that's also what my computer science collaborators say yeah the mathematics is so nice it's it's a limited it's a finite alphabet you can just yeah we can do but but the thing is what would be I mean yeah notation what would be what would be interesting philosophical questions about notation that we that we could could ask of course we could use the same we could more or less take the the four steps here and say please please figure out how to to count various notations various concepts even please classify them please see how they spread across disciplines yes of course we could we could try to do that they're not they're not visual in the same sense that that the diagrams are so we would have to uh if we cannot get at the that the structural source text there would have to be a different detection mechanism but yes we could but but I think the the interesting thing is that that both for the experiments in mathematics and for the diagrams in mathematics there are there are standing discussions that we can sort of of tap into and and contribute and there's also a huge standing discussion on on notation but I'm not so sure whether whether that would I mean I don't know let's see I would love to talk about that again you you have my twitter handle sent me a sent me a message we yes we could I'm sorry that that didn't really answer the question but but it I'm quite optimistic that we can do things I'm less optimistic that that my philosophy colleagues will recognize it as as contributing to their discipline well may or may not be a perennial me perennial issue um question a bit more historical question from from lucca rivelli who asks uh do you have any hypothesis on the possible changing role of diagrams in mathematics so do you expect them to have to have changed roles say from being proofs in a euclidean context to to having a more explanatory role today definitely and that's also why megal chose to so megal chose to study diagrams in the last century uh well that's not no longer the last century but but from the end of the 19th century to to to the beginning of the of the 21st century and what he noticed was a a shift so he noticed two shifts he noticed a shift in the prevalence of diagrams and he noticed a shift in his typology which focus on these these different cognitive offloads so of course euclidean diagrams were very prevalent before of course I say but I'll tell you euclidean diagrams were very prevalent before the war second world war then came a period with few diagrams and then a period with very abstract uh diagrams in as as algebraic topology gained gained importance in mathematics and then came the explosion in the in the late late 20th century early first early 21st century with with a diverse of diagrams so yes of course it's it's it there is historical development in that um megal has done the the hand calculations or the hand counting for three journals and five year intervals the next easy thing I will do is put put the detector to to test on a on a bigger corpus then we will know more about shifts and changes do I have a hypothesis yes megal also I think offers that there's of course also again a a that he starts in the late 19th century because there's a there's a philosophical opposition to the use of diagrams right to Hilbert and posh and these people saying you cannot draw inference from from diagrams you have to formalize it in some way and he expected that that would mean that there would be few diagrams in the early period that's of course not the case because there's also a geometry side that uses a lot of of euclidean diagram then came this sort of valley that that that was when bubakist formalist mathematics really set in and then came sort of the uptake again so so in a sense for the historian in me it's not so surprising that he sees that but it's interesting what he sees in those in those figures that he sees a shift in the types of diagrams that he sees new diagrams emerging old diagrams of getting borderline and that he sees this huge explosion where it might be that might be a hypothesis the diagrams today are more vehicles of explaining something or getting the user to understand what you mean rather than this this epistemic tool for for for argument that would be at least one hypothesis that you could that I would try to to to investigate with the with the last 25 years of diagram fantastic well on that note my apologies Mo to have to abandon your last question but we are we are out of time for the evening thanks very much Henrik thanks very much everyone that is of course the end of day one of the conference I will hopefully see many if not all of you back here tomorrow same same time same website same everything et cetera 2 p.m. Central European 9 a.m. east coast USA so thanks everyone very much it's been a fantastic first day and looking forward to looking forward to more for the next for the next three days coming