 Okay. So it's a pleasure to present all of you to Stephanie Fonsard for those who don't know her. She is a BAG student at our institute. I'm the supervisor. And she's working in general on the foundations of mathematics. The questions of what the foundation is, are supposed to be and whether a set theory is a better foundation or a category theory or none of those or both. And for what purpose we would want foundations so all these very interesting questions in philosophy of mathematics but I think go way beyond mathematics because we might want a foundation or not want it elsewhere too in the development of science and knowledge. So I think these questions go broader and it's interesting that she's now looking at the literature on grounding and phenomenality to see whether some insights in metaphysical grounding or phenomenality can be gained also for the foundations of mathematics which is a non-professional task and a question. So it's a pleasure to introduce you and we have for one hour and then we'll have questions. Thank you. So good afternoon everyone. So as the editor introduced it, this talk is about the foundations of mathematics and we will see whether they can be expressed in terms of fundamentality or of grounding or in terms of the two of them and we will extend that to the two main potential theories for the foundations of mathematics. Set theory is not a potential theory, it's accepted but there is always the debate around category theory so I will try to apply that to the two of them. So the overview is the following. I will first introduce what I call a purpose oriented description of metaphysical fundamentality and of metaphysical grounding. It's purpose oriented because these two topics are very wide and there are different varieties, subtleties in different definitions for the different authors. So I will pick or build my own definition, conception or understanding of each of them. I suspect Peter has mentioned that that's not the main topic of my PhD so that's rather a side question. So I've been looking at the primary literature but at the secondary literature from the Stanford Cyclopedia of Philosophy with the two main topics. So I will describe my own conception of metaphysical fundamentality and of metaphysical grounding. Then I will give my personal view on foundations of mathematics and then discuss foundations of mathematics as fundamentality and or grounding. Also set theory because that's the main accepted theory for foundations of mathematics and then of course this category theory which is kind of a challenger of a set theory for the foundations of mathematics so also in terms of fundamentality and grounding and then propose some conclusions. So a metaphysical fundamentality. Why do we talk about metaphysical fundamentality? It comes from the idea that there is something which is basic or primitive in the world so that means that it's something that is ontologically independent or which is ungrounded. The conception is related to our conception of science can be related probably to our conception of science which is related to the special role of particle physics as the fundamental level of reality. We have the works as the basic building blocks. There are roughly speaking two main types of metaphysical fundamentality. The first one that is called absolute in the sense that there is an ontological independence or a non-groundiness full stop so some facts are fundamental and that's it. The other type is relative fundamentality which means that there is a hierarchical structure in the reality. So for example we would have particle physics at the fundamental level on which the elementary chemistry is built and then biochemistry. So that's the underlying idea that there are some facts that are more fundamental than others. So as a result of that conception there is a kind of priority in the ordering. So there would be a fundamental level which needs to be well founded and which gives rise to a view that reality is also well founded. Fundamentality has two main roles. The first one is to say that there is a foundation of being which consists of independent entities and that these entities constitute a complete basis on which everything else depends or from which everything else can be derived. There are different ways to account for that fundamentality either from fundamental entities or from fundamental properties. In order to express these ideas there is the conception of a complete minimal basis that's the conception that is used to explicate fundamentality as the fundamental entities acting as the basic building blocks of reality. So these fundamental entities would determine everything else and determine is taken to mean ground, realize or build. So you see that already if we try to describe more precisely metaphysical fundamentality we already introduced some golden terms. So from that conception with the complete minimal basis the idea is that by giving a complete list of fundamental entities we have a minimal complete description of reality. So fundamentality as a complete minimal basis must include all the fundamental entities but only these fundamental entities. Of course with such a conception there is the question of the uniqueness of that complete minimal basis which from what I've been reading in the literature seems to remain an open question. Another slightly different but also interesting conception of metaphysical fundamentality is primitivism. So the main idea behind primitivism roughly speaking is to say that we cannot define fundamentality but we can characterize it. So in this sense the reality should be understood as objectivity. Primitivism is closely related to the complete minimal basis conception but we can express a slight difference in the sense that the complete minimal basis means what is real in itself rather than with primitivism that's more a characterization of what may be true. So we can express in the complete minimal basis the primitivism in this case which would understand the entities in the fundamental basis as basis in the sense that they play a role which is analogous to actions in a theory. So that would characterize fundamentality rather than define it. Another important concept in metaphysical fundamentality is the concept of well-foundedness. So the one key task of fundamentality is to capture the idea that there is a foundation of being and that everything else depends on these fundamental entities and that would need to be expressed in terms of well-foundedness. So the origin of well-foundedness is from sect theory where an order on a given domain is said to be well-founded. Is every non-antisubset on that domain as the minimal element for that order relation? So it means that we do not end up with an infinite number of descending chains. There is a fundamental level from which things can be built but we cannot end up with something circular. So the consequence of the definition in terms of well-foundedness is that the fundamentality is given relative to a given order. So that's a given dependence relation that must be made explicit. And much of the literature on well-foundedness focuses on well-foundedness. So from all these characteristics and the description I've given we could characterize metaphysical foundationalism as the view that reality as a foundation but in the sense that that fundamental level must be specified. And so a metaphysical foundation means that every non-fundamental effect is fully grounded by some fundamental effects on which everything else depends. So that's for not the grounding with respect to fundamental effects but in terms of entity, a non-fundamental entity are dependent on some fundamental entities or properties that give the full account of the being of non-fundamental entities. And the most common way to specify this idea of having a foundation is in terms of well-foundedness. I have one question if you don't mind. So you say in the first paragraph that it needs to be specified but there is a fundamental level in the sense that it needs to be specified but confused by that. We are the ones who specify stuff while what is metaphysically fundamental doesn't seem to be fundamental to us. So what does this need to mean is that it seems like an obligation for us to do it but that cannot be in the definition of what a foundation is of reality. It's reality that has a foundation not us. So does that mean that there is a uniqueness of foundation or a uniqueness of a fundamental level in the fact that it is specified and that it is specified well that the principle we should be able to specify we have perfect knowledge of that. So what do you mean by it needs to be specified? So if we have that contraception of a metaphysical foundation either in science or in mathematics we need to specify if you want to build something on this foundation this foundation must be clearly specified because otherwise you could say that there is a fundamental level somewhere that you don't know what it is but then you don't specify the entities on which you build the construction. So we can argue two different things for the purpose of founding a domain we need to specify it but it can be reviewed as I would say the knowledge evolves. For example, I think that two centuries ago if we had been talking about the fundamental level in physics we had been talking about the electrons, protons and the atoms why now people speak of quarks so it's up to us to specify it but it's not something that will not evolve any more in time and I would not say that it's unique because from the literature and theoretical point of view in metaphysical fundamentality it's still under discussion but also I will argue that both set theory and category theory can be considered at the fundamental level so it's not necessary that the basis is unique. So now for the grounding grounding can be defined as realist metaphysics so the grounding studies what grounds what so which are the facts that ground the ground other facts there are too many two types of grounding one which is explanation based and the other one which is determination based the explanation based grounding for example if we want to explain if we talk about the brittleness of a ball we will say that it's grounded in the ionic bonds of the constituents so that the ball is brittle because the bonds are unique so that's the bonding that accounts for the brittleness while for determination based grounding we would say that the ball's brittleness is grounded on the ionic bonds of the constituent atolls which lead the ball so that's more determined by the properties of the constituent of the ball that we have come to the brittleness of the ball it can be full or partial so for the partial grounding we will say that the Q fact is grounded by other facts if these facts contribute either to explain or determine the fact that we want to ground so they partially contribute while in the case of full grounding these two facts are the only one to ground the fact that must be grounded so nothing else needs to be added to have a fully adequate explanation or determination of the fact and the two central features of grounding are the strict ordering so grounding is a strict order so it means that it's transitive, irreflexive and anti-symmetric and the notion of well-foundedness is taken in the sense that we do not have no more non-terminating chains of grounds there is really a fundamental grounding level so as I have mentioned there are lots of links between these two concepts but these links between fundamentality and grounding can be understood as the fact that these two conceptions want to express a non-preservation between two things for example when we say that a given act is heavy because it causes harm the because is there not to express a cohesion between the two facts but it tells us what grounds the heaviness of the act that's the fact that it causes harm so that's I would say the meaning between the two of them because that's the main aim from the interpretation I am giving you and the interpretation I will base the rest on the talk on is I would take as something explanatory fundamental as grounding so what are the facts that are at the fundamental level for the explanation as grounding why fundamentality is more considered as something compositional so what are the entities or the properties at the fundamental level so as I said the focus is on foundations of mathematics so discussing foundations of mathematics in terms of fundamentality and grounding the foundations of the means for foundations of mathematics took place in a given historical context mainly to avoid paradoxes in the formulation of set theory so to avoid these paradoxes the axiomatic formulation was adopted and also at the same time there was a debate on the status of the consistency of the proof so these are for me really the two main features from inside mathematics to the foundations of mathematics so usually the foundations of mathematics are expressed with as six features ontological, logical, semantical, epistemological, methodological and metaphysical so for the ontology a founding system of a given theory consists of the entities the theory is talking about then at the logical or formal level we require from the founding system that it provides a deduction theory and basis concepts that are involved in the theory and from the formal point of view that was the axiomatization of set theory in which the set entity is central to the theory and is defined or rather characterized by the axioms of the theory then we have also semantical requirements for foundations of mathematics the semantics gives the link between the ontology and the formal system and I think that the construction of proof using mathematical logic is not purely formal so for me the mathematical proposition to which the formal system applies has a definite meaning and the proofs are meaningful arguments they are not a pure assembly of science so in some sense it means that mathematics is not purely a formal game the metaphysical aspect of foundations of mathematics that's really the unifying power of the founding system other important characteristics of foundations of mathematics are the epistemology so that's to characterize the type of knowledge is it analytical knowledge? is it based on self-evidence? is it purely deductive? and the epistemology includes the justification of a theory justification of the axioms that we choose for the theory it can also be the reduction of the theory to the founding system and then we have the last aspect which is the methodology which is requiring the clear formulation of the principles, the methods and the concepts that are used to analyze but also to construct the objects of the theory so we will now discuss the relation to be a foundation of mathematics both in terms of the fundamental entities and properties but also in terms of what is ground in what so the foundations of mathematics as metaphysically fundamental the fundamental aspect in the foundations of mathematics is really the axiomatization of the theory to avoid paradoxes at the fundamental level of course if you have paradoxes at the fundamental level then you cannot do anything with the theory and in principle all the mathematical knowledge that is derived should be expressed in terms of the fundamental axioms I said it's in principle because in practice the theorems and the proofs of theorems in geometry or in whatever field are not expressed in terms of the axioms of set theory so that's the axiomatization as more of a characterization than the definition of the entities and their properties so this fits with the definition that we have given of the minimal basis in terms of primitiveness so we can consider the axioms as a basis as I said in principle because in practice that's not really what is done is it a minimal basis so there are really the minimum so there is nothing of which we could get rid of and still keep the same theory that's not clear whether it's a unique basis but we know that it's not a complete basis because it's intrinsic to set theory and to mathematical knowledge so the foundations of mathematics as grounding the idea is that every non-fundamental mathematical position can be grounded on more fundamental proposition with descending chain and a very central feature of grounding is of course the well-foundedness of the foundations that we would give which is of course what we need for mathematics so now the discussion of set theory as metaphysically fundamental or as metaphysical grounding for set theory so I will give each time an interpretation of set theory as metaphysical fundamentality and as metaphysical grounding so for the interpretation in metaphysical fundamentality I would say that the axioms of set theory are really at the fundamental level so as I said in principle from which all mathematics can be formulated and so as I said that the interpretation I have of metaphysical fundamentality is a kind of compositional fundamentality these are the notion of sets and membership are fundamental so set as fundamental entities and membership as fundamental property and the axioms of set theory as the basis of the fundamental level so as I said in principle as a minimal basis but we don't know if it's a unique basis and we know that it's not possible to show that it's complete as grounding so as already mentioned the notion of well-foundedness is crucial to grounding and to fundamentality and the set theoretic formulation of well-foundedness corresponds to the notion that is used for well-foundedness of grounding so applied to chains of grounding the well-foundedness of set theory would rule out an infinite and non-terminating grounding chains and so set theory as a metaphysical grounding can be interpreted in terms that every non-fundamental and mathematical proposition is grounded on more fundamental propositions that are given by set theory and mainly by the axiomatization of set theory so we have another set theory on which there is real debate whether it's a founding theory of mathematics that's category theory so I will also propose an interpretation of category as metaphysical fundamental and also as metaphysical grounding so the category in mathematics is defined as something with objects and arrows the objects of the category have the status of a place order while the arrows are really the basic building blocks for the definition of a category these are the arrows that allow to develop mathematical constructions but they express the properties of the mathematical objects and the mathematical entities and in particular in the axiomatic definition of a category that's the characterization of the properties of the collection of objects of the category that's not by reference to the members or by the objects but by the relationship between them so that's the composition relationship this must satisfy the axiomatic definition of the category and the links inside the connections so inside category the link between the objects are given by the arrows that's the equivalent of functions in set theory and the axiom of the category for being a category derive from the properties of these functions or the arrows in their composition and these are also the arrows what we call the funtors that allow to define the relation between collections so between categories so category theory in terms of metaphysical grounding now so one very strong point of category it was historically developed not as a theory of foundations of mathematics but as a theory to allow to express algebra and geometry so it was already developed as an abstract background of framework to unify two theories and category theory succeeded also in identifying and explicating extremely fundamental and powerful mathematical ideas such as what is called the universal property in mathematics category theory also offers a new theoretical framework in which mathematics can be expressed and there is a very strong feature of a category that is unifying the power in the sense that the category can in some sense be considered as a basis as a basis allowing to express the intrinsic structure in mathematics so the conclusions I have tried to show that foundations of mathematics can both be expressed as metaphysically fundamental in the sense of compositionally fundamental by the fact that we have the axiomatization which allows to avoid paradoxes at the fundamental level that's a requirement from my point of view of foundations of mathematics and also the characterization of the fundamental entities and their properties these are the ontological and logical or formal requirements to be a foundation of mathematics and it can also be interpreted as grounding in the sense that every mathematical proposition or every mathematical effect should be is grounded in more fundamental proposition with wealth of wealth and descending chain so for set theory fits in this requirement for metaphysical fundamentality for metaphysical fundamentality so the interpretation of axioms of set theory at the fundamental level are characterized by the clearly defined entities which are sets and a property which is the membership relation to belong to the set in terms of grounding it's that every non-fundamental mathematical proposition is grounded on more fundamental proposition with descending order to set theory and classically to ZFC and for category theory I would say that it's metaphysically fundamental from the interpretation of the roles as basic building blocks and from the axiomatic characterization of a collection by the relation between its member but also by the relationship between different categories and in terms of grounding I would interpret category theory as allowing to explicating grounding very powerful mathematical ideas and also providing theoretical framework for expressing mathematics so that's the end of this talk I thank you for your attention are we doing a break or not? we have time that's for sure we wish it's a short break we have a lot of questions so let's open the room seminar for questions you know for now none online ok do you have any people? no it's ok there's questions around yes thank you for your talk my question was so so the presentation just gave was trying to apply this notion of metaphysical fundamentality and grounding to mathematics and I actually wanted to rephrase the question it got me wondering like to what extent is mathematics being used in these metaphysical discussions of fundamentality and grounding so to what extent has mathematics been used as an example and many slices have looked at mathematics as some kind of ideal so you already mentioned that the idea of the fact that there's a fundamental level has its origin in that theory so that's one question and the other question was so isn't there trying to mimic what is already present in mathematics or is it that they're just using these mathematical theories for example I know that there are some recent papers now the relative fundamentality of different leftist signs and they're applying the mathematical theory to that to somehow formalize all of these languages so it seems that mathematics is a really good playing field as one of the materials you may want as a metaphysicist it makes sense to have a mental grounding in science so I wanted to give the first question and see if you can say about that and if that could explain maybe the things you can see here as well for the first question I'll say I'm not an expert on the metaphysical fundamentality and grounding but I don't know to which point mathematics have influenced that but from that point of view it seems that it's clear that it doesn't influence I think the fundamental level I think actions to have a proper fundamental level being sure that there are no contradictions so I think that it must have an influence and also for the mathematics and science and the intertwining between the two of them that's also I think that they also evolve a little bit together because the axiomatization historically in the foundations of mathematics when foundations for mathematics were needed and that Inbert came with this 20 list of problems to provide good foundations for the future of mathematics and this idea was also to form properly what was at that time the most advanced theory in physics which was the kinetic theory of gases so it also pushed toward an axiomatization of other disciplines and there was also of course as in physics, biology the quantitative aspect which is also based on mathematics so if there are issues at the fundamental level of mathematics it means that I think that's the whole scientific building that would collapse so I think that's true that there are links between the need for foundations of mathematics and the link with science also the fact that in mathematics with as I interpreted metaphysical fundamentality with the entities and their properties and the grounding with facts so for example what are the facts also the actions that they see around the rest of mathematics and the I just said entity or the membership relation, if you look at science you also have that kind of my role with particle physics so the basic elements of the theory but also facts fundamental physics which rounds elementary chemistry which rounds so I see a lot of my roles but I'm also not a specialist of the field I think that Peter Sprex knows more on the interaction of what mathematics brought to grounding so well as an example for the first case there is clearly an example even the first example that Kip Klein always gives for grounding is that the fact that socrates exists grounds the fact that singleton socrates exists so singletons are set theory even though maybe socrates is not objective set theory a singleton so we could generalize that and say that the fact that something exists grounds fact that it singleton exists which it would be a saturated principle so even like the most basic example is kind of from set theory it has been used as an example but this is not a very mathematical way to look at set theory of course so there is something like and this could be another discussion later there is some kind of different way to look at mathematics than the way mathematicians formal mathematicians like to look at it because there didn't seem to be these levels why would the existence of objects be more primitive than the existence of their singleton I mean they are generated at the same time so to say so it's a very mathematical feasible way to look at mathematics that is a little bit in apparent conflict with mathematics as a formal enterprise so it's not very clear whether that's an example for mathematics as such and then the other point about these tools like you mentioned a category theory being used for this fundamentality discussions one could use also notions of neurology of set theory obviously like well foundedness but I don't know whether this goes beyond just the fact that these are the formal tools we have and like mathematics is a good field to study them but then that's just our way of doing like precise metaphysics by using these tools for mathematics that we have and not by remat everything but that might be like a rather risky business to influence to make that influence discussions on the foundation of mathematics itself you know because it's just a tool and we could have used any tool and so it's not clear whether the well foundedness that set theorists typically use is also the best for groundedness I've argued that it isn't well so that's so that's a bit of an answer but that you know whether you have a follow up or something No, I was going into this question what is the target into which is what is actually influenced by looking at mathematics or by not just by looking at mathematics but by using mathematics and the fact that they're working within every team which they're working in really easy ones Interesting Thank you for the talk I have two related questions both category theory and set theory can be both good for a long time for a long time basis and a running basis for mathematics but one of the questions I have is if you look at the way of the category theory to do 80 to set theory there are a lot of people who say the way to extend the project of set theory is the way to set theory in a different way for example and so there is a way to see category theory as a more general version of set theory so would you say that both as good as a set theory basis or one is better at your work is more general if you are actually busy then you can go back to what Beto was saying earlier if you are taking both to be as good basis do you see this as a complementary choice which one you think and then are you arguing against a kind of Platonism or Realism with mathematics because at some point you talk about running as a Realism with a statistical Realism or something like that so I guess that if you take both to be the more basics of both or both then I guess you can be a Platonist or you can say that but I think you are speaking very true basics as a human right or at least we are able to say because I'm listening to you for the first question so yes I would argue that both set theory and category are the two of them are good candidates as theories or foundations of mathematics if you just accept a category as category theory as a generalization of set theory I think then you would say that category can be just expressed in terms of set theory so that in the end set theory is your fundamental level in which you build something more general so I would not argue in that direction I would say that so there are different requirements for the foundations of mathematics I think that there is one really one part which is really the ontology the formal aspects which is the something which is internal inside mathematics while around we have other expectations so the unifying the plural the epistemology the methodology and when you start to discuss these aspects then I think that set theory and category theory have different things to bring in and so the debate what are the requirements more from a metatheoretic point of view then you will pick one theory or the other depending on what you put in your foundational program that's rather the point for me the debate is there is a debate but the people everything is not always very explicit in what is really expected outside of the theory from the foundation and I think that the main reason why the debate is there is for that reason I would say that it's more historical or programmatic on what you expect from the foundations yeah I remember when I asked you should explore grounding it's because I became more and more convinced by your study of fundamentality and mathematics so it really depends on a certain criterion so it's not in the same sense that set theory is a foundation and category theory and it's clearly not in the same sense that we use fundamentality in metaphysics or in physics so I said maybe grounding is the way to go maybe the thing that you could clarify and my impression in your talk today but I didn't follow everything is that probably it's much better to discuss about grounding in mathematics because you could say what is true or dependent in virtue of and you have the tool between this explanation notion of grounding and determination put a precise criteria it's a grounding in that sense and it's less metaphysically charged because you can't talk about grounding without talking about metaphysics you just say it's determined in virtue of it could be now that you explore grounding a little bit in mathematics when you say but this is what I would like to say but I'm not sure if you agree with me that grounding is the right concept to study this fundamental notion of mathematics but it's the right philosophical concept to analyze this historical program that you well explained in the beginning of your dissertation, this program that shows a theory or a category theory I think that your question is a little bit different from the one I dealt with here so the one here is to show that foundations can be both interpreted into the two of them I also agree when you say that grounding would be it's more flexible because that's true for describing the metaphysical fundamentality we directly use the terms of grounds, we use well-foundedness we use all these terms that relate to grounding so that's a little bit as if we would need grounding for the metaphysical fundamentality and so your question is more is grounding more the tool for exploring the questions of foundations in some sense I've been thinking in these terms but now that you ask the question it could be because as I answered the question about the two programs so as I said there are usually these are expressed with the six feature components my idea is that there are ontology, the formal and logical aspects which are really intrinsic to mathematics and the methodology, the metaphysics the epistemology, the semantics these things are more outside than in the program and when I discussed my talk with Peter at some stage he said yeah but in my new discussion you confuse two things the mathematical facts and the facts about mathematics and then I thought he really put the right words on the distinctions on ontology, formal and logic which would be the mathematical facts in the foundations of mathematics and on the other hand the facts about mathematics so the epistemology what kind of knowledge is it for the semantics is the unifying or something and so if I go for that conception which is the one with which I ended for what is the foundation of mathematics I would say that the metaphysics for the first part for the mathematical facts makes sense but if I want to talk about the facts about mathematics then I don't think I can talk about them in terms of metaphysics but more in terms of groundings but so what grounds the fact that we have that mathematical knowledge is analytical so that's probably interesting to to say that perhaps grounding is more flexible and allows to deal with the two aspects of the foundations yeah because when we talk about fundamentality in metaphysics we talk about entities it's a metaphor for what is independent in the being sometimes property is your right but and that's more ambiguous but when you say set theory is the foundations of mathematics and you describe that well or category theory is the foundation of mathematics are you saying that's the basic level or something like that or are you saying that's the thing that inverts you up everything is okay or that's the thing you have to accept and you build the rest that I was always confused but maybe it's me and set theory what is the basics of set theory is it the axiom or is it the sense and for me two completely different questions my first reflex would be saying it's the axioms so it's the ground but if someone says no no no I do not understand it's the concept of set and that's good like or not it looks like foundations in the being and it's why I'm so confused when the project of category theory as a foundation is clearly not the same thing apparently then set theory from the axiom yes but from entities vision when they say the morphism or the object and you say yeah in a certain sense yes but in another sense no and you have the discussion you just had yeah but you can put everything in language of set theory but that's not the point the point is that what do you buy as the thing in virtue of everything will be okay and I want to use the two buttons but it's my you were much more in neon these things so I'm very talking from outside the concerning the set theory or category theory the issue you raised with the axioms is it the set entity or the axioms I tried to in the presentation not to put too much weight on set entity as the funding element but more of I think that's not just my idea but if you read through literature you can also interpret foundations of mathematics in set theory and also of the membership relationship so that would not really so that would be the property which is axiomatized both in set theory and also in category theory that's the composition of the requirements around the composition or relation that is axiomatized and I think that if you interpret the foundations of mathematics in terms of membership property then it opens the subject and allows you to put the two of them on equal food I think but so that's one part of your question that's the second part of it you said that you were confused with the fact that it's set not the axiom I would say that for me that's not the membership property that is axiomatized what can belong or not to a set so that we don't end up with paradoxes but that that's a chicken in the egg or you say or we say we build the axioms of geometry to capture these things that in a certain sense exist in a cladian geometry or we say the axioms themselves generate domains and I don't know what is the right way to think but I think from the people that are engaged in a foundation project that makes a big difference that their stuff and the axioms are just techniques so what is important is these properties that you want to map to capture or it's a formal domain and it's just language open and intelligent sometimes very surprisingly you don't control well what happened and it seems that it's a different project because when I learned about category theory and I read many things you wrote about the history of category theory these guys were not in the project to find the basic block of the thing they were looking for the universal language of mathematics which is not the same thing as set theory because I have a Frenchman in addition the name of a general of this group the Bobacchi project is like you have set theory and you rebelled everything even the notion of structure is not defined by set it's more complicated than pages and it seems a different project perhaps I will change my mind but at this stage I would say that it's more that you have stuff there and that you axiomatize it because when you look at history the notion of set was not properly given it was something that was really built and developed and fine tuned with the issues of contour and you also have paradoxes with Russell and then we will try to refine it like this and then you had set FC and also there were all the mathematical facts and they were not what was in the mathematics before and the concepts before the that formalization and the axiomatization of mathematics were not thrown away they were incorporated in a way from the axioms so I would rather go for the fact that there are stuffs and mathematics and at a different time there were different tools there were issues and they solved the issues with the tools they had and they ended up with that axiomatization the definition of a set but historically it had been different if categories had been there perhaps that would be category theory that would have been that taken as some of the foundations I don't want to take time for someone else but you're absolutely right that the object comes first in the practice as a group we had the notion of a group much before axiomatization and after that once you have the axiomatization you have new problems and new notion there was some part in Genoa about the transformation of equations it's not a full group at the beginning and after that it becomes a full notion of group there was a notion of suction transformation and it's very close but group the axiomatization came a long time after I understood for a long time because it was 7C ZFC is not that easy to find that I agree and there's the notion of tools in Mademadex we built entities to do tools group theory now is a tool in Mademadex to study geometry it's also part of Mademadex as an object you're right but why the foundation should be related to the beginning of a discipline we start from entity and we get to something else the foundation would be entity first just because it's the way we do it I give you an example in metaphysics you would say linguistically and conceptually we are much better for substance, substance is close to language and it's the way fix stuff fix change but it's always a thing that we capture a lot like that but it's completely wrong that's the real thing why should it be dependent on the way you get to it I know Mademadex is not discovering something outside so it's something we built so maybe it's more dependent I don't know maybe it's something outside so maybe Mademadex is a special case but why the path we develop to which is arbitrary like you said maybe if we develop we have developed a category of interferes but why the question of foundation should depend on the history I'm very confused about Mademadex all these questions really depend on the history I don't know but I think that there was really a turning point in mathematics when the need for foundation arose because for thousands of years before people would not I think even need for foundations and it was not that's not a top down approach then it's not like a house where you build a foundation then you build a house there you bring the foundation afterwards to save the building historically that's what happened so I think for the historical point of view that's to better understand why it was needed because it was not always needed now I agree with you that history must not guide everything in foundations because then if it's the case we would say okay that's just set theory because historically that's a theory that was designed and chosen as a foundation so we should not even discuss about the status of category theory if it was really a story that would be the criterion for choosing the theory of foundation fair enough so you have the house and you're looking for maybe the house will fall and science will collapse with the rest so it was when you look at the text people were really not in panic but it was really important for Neumann was really but if there is that issue with mathematics or the rest physics we can throw everything away and we started to do some work into that because there was really a very big issue there but I don't know if I really understood your point what history is to do with foundations why should it have something to say but yeah because there was historically an issue that we do not know but it's true that's not also something that is fixed and that prevents anything else to be a foundation okay so I guess you've already looked at it I mentioned the rest of mathematics which is the next one we are not going to get back to what is reguar to get to mathematics but my question is you are very right to say that it's a very big concept in metaphysics most of the time there is an underlying idea of explanation explanation mathematics is a very political subject so I would like to ask you what is your position on the explanation mathematics and take it as different proof and I have a question as a formal question and I would like to know what is the definition of explanation mathematics now that's it what I would take is a definition of explanation in mathematics here in the talk I'm sure that there are two interpretations of grounding, one in terms of explanation and one in terms of determination. And so I buy into more the explanation, the grounding is more explanation than determination. But what is explanation? So in this sense, the facts that would ground mathematics should have some explanatory power for explaining some features of mathematics. And perhaps one example for that is when I said that I interpret category theory as a grounding, also a possible grounding theory for mathematics. And for example, because it's a law to give an interpretation for universal property. But that's really the role of category theory, which would allow to show that some mathematical construction are equivalent up to isomorphism. That equivalence to isomorphism is really something that is given in category theory. So from that point of view, it contributed to understand something that we can buy some objects, which they have properties, but we can build them in different ways. They will be equivalent. That's explained in some sense by category theory. So from what I remember about the debates, it seems that most of the virtues you have are not subjective, or it's hard to have a real definition of what is an effective value in mathematics. So for communication, I can see the definition known to subjective definition, but apart from that it seems very hard to have a notion of explanation in mathematics that is not a social interaction in the community. I would expect that from that explanation, one would not expect to be in the definition, to be very different from what's the other type of mathematics in the definition. So if you want to have a fundamental foundation of mathematics, that's the base of the boundary of relations. Then I guess you'd have to say something about the communities of mathematics, no? You mean in the sense that the explanation concept is dependent? If it's in reality, beyond the universal definition of what is an effective value from an explanation point of view? Yeah, that's also true. That's a concept which is very... That's quite complicated and not clear what explanation is in mathematics. We could also... Explanation is really complicated, but you could just... The formulation in mathematics that you choose to formulate things in set theory or in category theory really depend from one mathematician to an order that's always a sociological question. Yes, that's an issue. But then, yeah, with this consideration, it's just a sociological aspect of mathematical knowledge. That's something that's also, I think, much... Complicated, perhaps at the boundary of the questions and foundations. I would say that I would keep this really at the boundary because just with what we have, it's already complex. And there is also that meta-theoretic aspect, and so if you inflate it up to one different notions of explanation and so you would have different expectations on the grounding depending if you have a grounding which is explanatory in the sense of Masha, then that's true that you widen so much that... But then it goes back to the thing that you have contingency in the foundation in mathematics as if you're forced to follow a tradition that is directly driven... Yeah, up to a given point, yes, you have a contingency, now it's not... I would not say that it's subject-related and there is a gradation but at some stage, yes. Sorry to give myself a chair at the board, but I'd like to bounce back on this discussion. And I was thinking about the thing that's always said about the process of physics. Maybe we should see explanation and foundation in mathematics indeed as quite questions that go beyond the history, the sociology, the practice, not the practice of a mathematician with a coherent point of view. Their internal practice, so to say, is important and how they deal with new incoming data, how they revise their beliefs, that that is at our issues that are very important to practice, but how the fact, for example, that people see different things as explanatory shouldn't necessarily be a problem. I mean, if there is something like an explanation in mathematics, it could very well have different forms in different mathematicians because there is no general philosophy of mathematics. There's mathematics, there's formulas, and they all do the same things, but they do it in a different way. They have different ideas behind it. So maybe we should look at the formal properties of what an explanation is in mathematics rather than the fact that every mathematician should agree on what explains what or whether specific proof is explanatory. Rather, if you consider this explanatory, then you should also consider that explanatory and making explanatory relations into a topic like logic is one of which we can characterize. We cannot define, like you made that point about phenomenality in mathematics. Maybe we could have a point of view like that in mathematics. I wonder what you think about it. And this also made me think about the notion of grounding. You have the intuition that this is maybe better suited than notions of composition of phenomenality or something like that because there is an ontological feel to it. The problem is that grounding in the literature as it stands now is viewed as effective notion in your realist metaphysics, as you said, because it's fine, it's so much more intelligent than everybody else, and he has this metaphysics. So people just copy this. And of course, that's perfectly fine to be at that point of view. But maybe for mathematics, we would be interested, it would be more useful to have a notion of non-effective grounding. So where propositions ground other propositions in the sense that there is some relation between these propositions that makes the relation between them one of order, one of this, if this were true, another effect, another thing would be true in virtue of that. Whether it's true or not is maybe not even the issue. Which goes well, that's another point in the discussion with reverse mathematics where you cannot say like, well, all these things are true or something like that, but we see that this seems needed for a lot of other things. So it seems more fundamental and more explanatory whether it's true or not. There are different more advanced theories of set theory that are stronger than set theory. You can add axioms. Do we add these axioms because they are true? No, probably not, because continuum hypothesis is true or not. People don't know, but they say, well, we can do that. And then we can get to other things that are that seem less fundamental. And the relation that we have there is not one of logical consequence because that will completely not work. That doesn't have that fundamentality level. Everything is flattened in mere logic. But if we take explanatory, the grounding relation as something that is non-factive and which we cannot, we don't need to define. We just need to have to see what these formal properties are. Maybe we can get to a notion that would satisfy a lot of these doubts that we seem to have or desires or something like that. And so I'm wondering, we don't have to read that set theory is then grounds geometry or something that may be specific to, but some results seem to be much more like complicated and based on a lot of more simple mathematics taken together from different theories. And such things seem to be, I mean, there seems to be something that we would like to capture, even if we may disagree on which part of mathematics is more fundamental. Just the fact that fundamentality and explanation and explanatory influence is something that gets to some know, some cementuitions in the practice, maybe enough, maybe shouldn't like try to have this definition of a explanatory proof because the literature shows that this is like, we haven't done anywhere yet after a lot of time studying this kind of notion. So these are some ideas that I hope for this question, maybe, yeah. That's true that, because yes, I would say yeah, that's true that grounding is flexible for analyzing the internal part of the mathematical effects, but also the effects about mathematics for the foundation. And also, I said just before the beginning of the talk to Alexandria, and there is also like to apply this to examine the status of the multivirals in mathematics. And as you say, if I take the grounding, so which it's a grounding relation, which is purely factual for the the status of the multivirals, that's, it seems to be factual and factive. So factive is implying truth. Yeah, actual is like, about the facts. Okay. So, but if we have the multivirals in which that we consider as a tool for representing some assertions that may be possibly true, which we don't know. That's true that with that concept, the conception of grounding, as you said, more different from the one of fact that would make it an extended tool, a better tool for analyzing this. So that's, that's, that's a nice interpretation of grounding that could be very useful also for these for the statuses. Also, is the multivirals something that is fundamental in mathematics or not. So that's, yeah, that's very interesting to consider that notion of it's an extended notion, but a different notion of grounding. Yeah, so affirmative notion. So just there is a relation between two sentences, propositions, I mean, the relation is never really between sentences. We are not interested in the linguistic stuff, what they represent. So they represent propositions. And then just like, A explains B, this is or it's called grounding to make it less extremological, because that's not all we want to get at. It's not what explains it for us, but it is what, what is more basic than another thing. And all that grounding and then you find actions or rules for, for this relation and see if you accept certain things as more fundamental than others, like what, like it's probably transitive, that's something that everybody in grounding sort of almost everybody agrees on, but it can also relate to other pieces of language like modality. If you have a grounding relation, A grounds B, you probably want that it's necessary that if A then B, so it implies modality. So you have relations with many things of language, maybe with causal sentences and so on, that's maybe less interesting in mathematics, but and it's rather maybe, it's going to be good idea to look at the formal properties and the way we use grounding rather than to understand what it would be or whether it says something about facts and whether things really exist or something like that, to take it away from realist metaphysics, to try to do some more structuralist metaphysics, if I would, it's not a good term, but I guess you get the idea. I'm going to do some more in some part of mathematics, where I completely don't follow logically from the kind of system that has some part of mathematics, where you have proof that you can logically construct easily or now when you have a new use of infinitive numerical calculus and stuff like that to prove the conjecture that this guy is approving, which is clear, you got check that the proof works. I haven't thought about these cases. Because you have this kind of niche cases where mathematics can be used to logic methods even, right? Yeah, well we have like the endosteorem or something like the middle sentence would not be provable, while true. I'm always very skeptical about such claims, I think they can also be used to see the problem. Because it's contractive, so we don't care. Yeah, but then the computer analyzer has just been a huge rotation between the proof and the conjecture. But there is no proof. So I'm looking at something that is even stronger than a logical inference, right? That the kind of proofs that are explanatory. So it's a subclass. Of course, if there's things that explain in mathematics beyond proof, I'm not sure that that is actually all of them just sort of heuristic and not really very reliable. It might be a useful image to have, but not something that is deeply mathematical. But that's not what I wanted. It's even more specific than deduction. And they do that everywhere, but now let's see what is more basic, yeah? And why would we really do a proof that is showing the less evidence from the it's not about the evidence, but you get the nature, it's about levels of basicness. Of reality. Or you want to go with that? Uh, mathematical or reality, mathematical constructions or whatever. Yeah, yeah, because I don't exactly follow the trouble with ground and it's giving such overused consequences. It's difficult to follow what's at stake, but I know that in metaphysics, you are moving away from ground for various reasons. And for example, you see different that's going, building relationships to make sense of fundamental level, fundamentality levels, and they're using relationships such as ontological priority or other things. Constitution. Constitution, whatever. Something like, you know, to get some kind of a relationship story and what did God have to build in order to make this world? Basically, that's the idea that you're trying to make sense of that. In, in, in, in, in religious terms, they're very, very similar to grounding. I mean, sometimes you're, you're even a medical person, but don't, don't go into the process of grounding. So ontological priority, I think it could be, could be used in different ways. Yeah, but the, the project of Peter is to go outside that kind of question. Metaphysical grounding, metaphysical grounding. It's to, it's a relation between proposition. There could be stuff behind if you want, but you can't stop there. You know, this proposition, in virtue of this proposition. I like that because that's the way I understand the word grounding, but I'm the only one because get fine, start the thing and it's really effective. Well, actually, Brentano started a long time ago in the 19th century, and there it was another effective notion. There's more conceptual notion. Concepts are not effective. I mean, they are not. So, so there is not obviously effective. But nowadays you see explanation as effective, grounding as effective, and there's a focus on truth, but it's real. I think at, at some point, grounded in, in, in this paper, some grounding, get fine makes a distinction between a realist metaphysics and naive metaphysics or something like that. And grounding is for realism, of course. And then it gives the solution. So, it won't be the word, of course. To be naive, to not care what really exists is a something like that. What was the other distinction made in this paper? Was, was it really real or really, really real? Was somebody else supposed to, I didn't, we still have 12 minutes, if there are still some points? No, everybody's satisfied and wanted to add something. No, thank, thank you for the questions, for the interaction. I think that it helps me to also to clarify, to clarify something, to, to see exactly if the concept plan I have is something that is, that can be defended. So, so it was, it was very, very great to have your feedback on this, on this proposition for the interpretation. So thank you to, to all of you for, for listening, for contributing.