Categorical theory-building (contd.)

0:00 I'll give some general theory of what is form and what is category. I'll give a definition of form. All items falling under a given concept are isomorphic. In other words, these items are defined up to isomorphism. I should call such concepts form concepts, or simply forms. The colloquial expression up to isomorphism apparently involves a systematic ambiguity between identity and isomorphism, but it doesn't matter here. so a different paper think about any Euclidean geometrical form shape like that of a circle a circle allows for Euclidean motion and scalings I leave this side and take it I only point to the fact that all Euclidean circles are isomorphic in the sense that for any given pair of circles there always exists a reversible transformation, isomorphism, motion scaling of their composition transferring one circle into the other in that sense all circles are isomorphic through an appropriate modification of the class of admissible transformation, one may modify a given form concept, what Klein did in a single line. Thinking about a circle up to a reversible continuous transformation, one gets a more general topological notion of circle, which is not distinguished from whatever. For a further example, consider the traditional notion of natural number, conceived as a shared form of isomorphic finite sets like pairs two is common form of all pairs here it's clear that we have the colloquial notion of algebraic form provides yet another class of elementary examples of form concept take expression like i don't know x plus plus y all expressions update from this one through any admissible substitution of symbols x and y by some other variables, constant or algebraic expression, whatever, are told to share the same algebraic form. And isomorphism associated with this algebraic form are such substitution. Notice the reversibility of substitutions. Substitution is something almost reversible. And I think that really interesting thing, substitution, should really work more on that. It seems something obvious, primitive, but it's kind of basic concept, implies what and for more modern examples of form concepts think about any concept built up to as a structured sets okay that that's explained yesterday so the above examples are so various that one might think that in fact all mathematical concepts are

2:30 form concepts this view has been held by plato who however made the distinction between mathematical forms and forms strict to sensor in today's philosophy mathematics this view is known under mathematical structure and a little bit rough. Mathematical structure will exist in a number of different versions but that structure that thing determined up to isomorphism seems to be a common assumption. Understandably Hilbert is often referred to as one of the founding fathers of mathematical structure, it's paper-based. No substitution is always involved. I have, for example, A plus C, and I will substitute A by C, and I will get C plus C, but how can I get back That's a good question, but But actually, if we think about it as a form, what is form here, just two things in class between them. So we have something like places, and that's what we call form. But exactly, a lot of things in mathematics, we just don't have places. So this idea, there is some kind of analytic slangle that plays hope, plays feel, whatever. And all that is rather specific, and a lot of things are mathematics, but I don't know exactly how to treat that example, but just absolutely clearly that it's done possible, yeah? Okay. But the view according to which all mathematical concepts are form concepts is obviously wrong. think about the general concept of group not any particular group like the group of clinton motion but the concept of group as such there are certainly isomorphic group but not all groups are isomorphic so the general concept of group is not determined up to isomorphism hence it's not a form concept similar general concept of set natural number etc are not form concepts they sound like a trivial point however one may still argue that all mathematical objects are instances of form concepts however important the general concept of group might be anything like general group, all and about all particular groups. Since these particular groups are all form-forms, so the structural view remains plausible. But the situation can be seen differently. And actually, probably I should revise a little bit this passage,

5:00 but what I mean that this idea of reification, hypostatization, is like with set theory, allowed think as the natural number, in general, as one object, N, meaning the set of natural number. In the same thing, category theory allows think as a group, category of all groups, we just have this reification of general concept of group, and kind of what verification. And, of course, probably the idea to think of a category as a class, or a huge underlying class, that's probably just wrong, but that's how we define it today, there is no other interesting suggestion. But there is, I think, something wrong with it, it's kind of an intentional notion, just the idea is that we reify the concept, and that way, what is mathematically very, I would say, effective, very productive view, then we can play with it. We have this different status, we have groups as things, and concept of groups as just concepts, you know. We have this distinction, yeah, or whatever, but here we just reify again. And mathematically, it's not Platonism, at least certainly not in the sense of Plato himself, But a mathematician would always need to verify concepts, basically what mathematics probably is about. By analogy with form concepts, I shall speak about category concepts. General concept of cells, group, manifold, topological space, or the general concept of category itself. That would make a difference, by the way. You can say there is no form of forms. But you can conceive of a category or a category. At least there is no explicit contradiction. So in this sense it is more coherent. It allows for self-application. The notion of category concept is more general than that of form concept. Any form concept is a special case of a category concept where all morphism are either morphism. Some people, like Avody, argue that the notion of category is a typical example of structure, which appears to be a different name for, or at least a special case, or what I call here form.

7:30 Since the general concept of category is instantiated by sets, group, etc., an abstract category can be colloquially called a common conceptual form of all these things. However, this is misleading, and as far as one assumes the proposed understanding of the notion of form, Categories in question are not as a morph. They can be mapped to each other by suitable functors, but these functors are never reversible. So it is misleading to interpret the fact that set group has set to make categories in the sense that all these things share a form of category. This erroneous thing about categories as forms is, in my view, responsible for the infamous description of category theory as absolute nonsense. Yeah, then I have this thing about transformation instead of relation. but probably I skip a little bit yeah, I just explained stuff I explained yesterday the basic idea of formal axiomatic method can be explained by this slogan describe mathematical objects in terms of formal properties of their relations the slogan of categorical methods read slightly differently describe mathematical objects in terms of categorical properties of their transformation The assumption behind the categorical approach is that transformation of mathematical objects, including both isomorphism and non-reversible transformation, indeed essentially characterise these objects. A comparison between the two methods suggests an analogy between relations and morphism. One can say indeed that morphisms relate objects to each other in a way. That's why this claim is taken in the general philosophical sense. It sounds reasonable, but it is in odds with the standard technical notion of relation as a function sending cubos of individual related to truth. I already explained it yesterday, so I can skip this part. So it doesn't work this way. This observation point to a gap between the general philosophical intuition between the concept of relation and the standard logical notion of relation just mentioned. Morphism are relation-like, while within the standard logical formula, they should be treated as objects. To see how this is done precisely, consider a Lvivian 66 paper, where the author suggests a simple axiomatic theory of categories using formal axiomatic methods. This amounts to the form of objects, and Morphism are taking as primitive objects, holding three primitive relations, meaning domain-of, codomain-of, and composition-of.

10:00 Categorical objects and categorical morphism are treated as belonging to the same type since every categorical object is formally identified with its identity morphism. Lavier himself avoids speaking about object translation and is concentrating just purely syntactical viewpoint and after listing the appropriate action, say, by category. Okay, probably in the skip I come to my suggestion, but here I just show that actually in the beginning of his early paper had absolutely this classical music. Yeah, probably he wouldn't like it. Thank you. Yeah. Okay. As the above quote, now I will speak a little bit about categoricity in the sense of... As the above quote from Hilbert Coelho shows, he thinks of a formal theory as construed up to isomorphism of its interpretations. theory like that of Grundlagen, here actually I have a question, you can help me, is indeed formal in this sense, that is that all its models are in fact isomorphic, the desired property has been called by Weblen in 1994, categoricity, this term has nothing to do with category theory. When Hilbert was preparing his Grundlagen for publication, he apparently didn't yet see the problem, I don't know exactly, he discovered it about the time of the first In his lecture, Ubert and Zaltbegriff, delivered in 1999, published in the year 900, Hilbert first introduced an axiom of completeness, forstanding this kind of axiom, requiring from any model of the theory in question, this time it was arithmetic, this maximum property, given model M satisfying the rest of the axiom, one cannot obtain another model satisfying these axioms by extending M through introduction of new elements. In the second and following edition of Groot-Lagin, Hilbert used a similar axiom under the same name. Hilbert-Folstein-Dicht's axiom implies categoricity .

12:30 From the point of view of today's model theory, here, I'm not sure, so this axiom looks very dubious, if not plainly wrong. Hilbert's account doesn't provide any reason why a model with a desired property exists in any appropriate sense It apparently relies on the intuition which suggests that the intended model, the usual geometrical space, has this property. So you just can put more points there, which of course, things like forcing just shows Which is absolutely wrong. This is a very shaky ground indeed. The standard task in model theory doesn't allow for a model with a required maximum property. Is it true? Yeah? I guess it's true. because of upward Scholem-Bohden-Height-Theria, given that it is... It is indeed, it is dubious in a sense, and it shows that geometry in a sense is less expressive and less interesting in a mathematical sense in Ukrainian geometry. No, but you can do it, but you can do it, because you can do it, and people say, there's a discussion around here, and people say that this is due to the fact that points aren't really individuals, numbers in arithmetic are individuals, in a sense, they have a, they have a, they have a, who discuss them, but I try to find some, okay, okay, thank you. So this axiom turned to be incompatible with the later model theory, since the notion of categoristic has been formulated by Webel and the idea that this property can be stipulated by fire, by hand, has been largely abandoned. For the reason just explained, categoristic is commonly viewed as a desired property of formal theories. However, in the 20th century, people have learned to be tolerant to the lack of categoristic. For example, Frankl said theory, piano arithmetic, and some other theories commonly viewed as important to be non-categorical. To preclude the right of these theories to be qualified as formal on this ground would apparently mean to go too far. To save the situation, philosopher invented the notion of intended model, that means model chosen among others on an intuitive basis. Isn't this ironic, I ask, that such a blunt appeal to intuition is made in the core of form and axiomatic method?

15:00 I agree with a guy called Davy, who recently argued in some form, foundation mathematically, that no one has ever been able to explain exactly what they mean by intended model. Other people question the categoristic requirement. Lim Dauer, someone says, why are you allowed non-standard models? What's wrong with having other models? Why should we make our model all smaller and not larger? I believe, in my ethics, that the lack of categoristic of theories like CF and PA is indeed a certain flaw. Because lack of categoristic undermines the very ideal form of theory according to my reconstruction. At the same time, I agree with this guy Lindauer and other people who think that the pursuit of categoricity is completely misleading. These two claims might seem to contradict each other, but they don't. Instead of forcing categoricity or looking for philosophical excuse of the lack of categoricity of formal theories, I suggest to change the method of theory-building and corresponding option of theory. But all these second-dollar theories are categoric. second order arithmetic is category, second order is category, so you can have it if you want to, but you have to quantify a lot yeah, in a sense Hubert actually is also second order, but then of course the status of this, yeah, ok, it's a big discussion as far as non-reversible morphisms are treated on the same footing with isomorphism the pursuit of categoristic has no sense any longer trying to describe a model of game theory up to arbitrary morphism rather than up to isomorphism one may get a category of models which has good categorical properties making it well manageable in the following paragraph i provide some details of how this can be achieved. As we shall see, the categorical approach undermines the usual distinction between a formal theory and its models. In the new context, a theory can be naturally seen as one of the models having this specific property that it generates all the others. This is hardly surprising, given that the notion of formal theory, as distinguished from its models, requires categoricity in a better sense. A great advantage of formal axiomatic method is that it provides a clear idea about the role and place of logic in theories built by this method albeit details can all be subject of philosophical discussion

17:30 thinking about uh talking about categorical method as an alternative to formal method we cannot avoid this important issue either so i would speak a little bit about logic I must, of course, confess, I have a big project which is not achieved, so I should somehow sound nice and just make hand-waving. The next paragraph is actually about what I call formalizing logic. Also, probably, you might correct me. Traditional logic is conceived as a general theory of reasoning independent of any particular subject matter one might reason about. I try to be rather direct here, except logic itself. On Aristotle's account logic is closely related to ontology. In particular, Aristotle's trithological law on contradiction as fundamental ontological principles. Perfect syllogism reflects the structure of his conceptual entity. On these account, logical truths are always grounded upon ontological truths, even if the former do not coincide with the later. Skip a little. Hilbertian notion of formal theory is that of framework or scheme of concepts, as we have just repeated, taking in abstraction from its possible basic elements. Formal logic, in the usual today's understanding of the expression, is formal in the same sense. Let's give the sharp distinction between logic syntax and logical semantic, which is not found in the traditional logic. Purely formal means here, ultimately, purely syntactic. As Karna put this in his work, it's called Formalization of Logic, And I don't know what he would be the first, how do I say, put in front of this idea of formalizing logic. I don't know, maybe even because it would go in his direction. He says like the tax of formalization of any theory belongs to syntax, not to semantics. However, at least on Hilbert account, logic essentially differs from other formal theories. So this is absurd that no whole tones used in the axioms of Hewlett-Pudnacken have variable meanings, meaning of tone, and, or, and so on, are not, well, so, okay, I keep saying just be supposed that logical vocabulary always has the same interpretation, right?

20:00 and in that sense probably just doesn't need to be interpreted, it's become kind of, rather how to say, illusive, this idea that we also need interpreted logic. However, this shows that the idea of interpretation of logic is distinguished from interpretation of theory, And here the distinction between logical syntax and logical semantics is in fact redundant. However, with this distinction, we also lose the modern law of emotion. Actually, this latter distinction is absent of Hilbert. The idea behind his rule is to base geometry and mathematics in general on logic rather than intuition, whatever. A formal theory in Hilbert senses is a logical skeleton or logical form shared by a class of traditional so-called naive mathematical theories. itself, on this account, is not formal in anything like the same sense, for applying the notion of formal theory just given to logic, one would need to speak of logical skeleton of logic, which is at least unclear at most senses. Carnap and other promoters of the idea of formal or formalized logic, largely disregarded this philosophical difficulty and applied formal or syntactic method to logic itself, introducing the nowadays standard distinction between logical syntax and logical semantics from a technical point of view this was a very productive move since it allowed to develop and study a system of logic on equal footing with other mathematical constructions so we got more logic and model theory but what kind of new philosophical logic is needed to replace hilbert's or fragist logicism in order to cope with all this development remains an open question most philosophers working today in logic share hilbert's weak logicism according to which logic has to do mathematics and of other sciences. At the same time, a few of them, if any, hold this old-fashioned logical monism according to this, there is only one true system of logic. For given the variety of different systems of logic presently around, this later view sounds nearly as absurd as the ones proper of view according to this, there is only one true system of geometry. But if logic like geometry allows for different incompatible systems, then the whole idea of logical grounding of geometry on other sciences, you know, logic fails. The two parts of popular view just described are incompatible with each other.

22:30 However, obvious might be the genetic link between Hilbert's notion of formal theory and Carnap's notion of formal logic. The two require, apparently, very different philosophical grounds. Categorical logic suggests a solution of this problem through a revision of formal axiomatic method and more broadly of formal approach. To show this, I shall first develop a speculative notion of categorical logic as generalization of formal logic, in the field, supporting the speculative notion, and finally make epistemological confusion. Both traditional Aristotelian logic and modern formal logic hinge on the notion of logical form. What kind of forms are logical? Forms is a difficult question, which I shall now try to answer. Let's see instead what happens to logic when the notion of form is upgraded to that category. Remind that categories are like forms, generally speaking, don't allow Given a class of balls, one may think about them after isomorphism and stipulate the ball as their shared abstract form, but nothing similar will work when objects or even class make a category. So a categorical system of logic, unlike focal logic, cannot be anything like a self-standing structure, occasionally blighting this or that particular context. Instead, it must be intonal or intrinsic with respect to a given category playing the role of such context. This raises the question of universality of categorical logic. to give the title of logic to something which implies to a particular category rather than to everything like traditional logic. Let me make three remarks on this question. First, nothing prevents one to conceive of everything as a category. This idea is behind the six paper categories, categories, of course. Personally, I am not sympathetic to this idea. Actually, I consider the local character of categorical logic as its advantage rather than The second remark is that the idea of regional logic, that is the notion of a system of logic designed for some specific purposes, has been already around during at least a few decades, and it better fits today's technical development of logic than the traditional idea of the universal logic. The third remark is that in the categorical setting, the notion of regional or local logic can better cope with the following important objection. Making logic regional breaks the rational thoughts into a number of incompatible domains, which contradicts the whole idea of rationality. The usual response to this problem is to find a weak system in view of universal logic, such that regional logic could be seen as a specification of this universal logic in particular context.

25:00 Alternatively, one may challenge the assumption about the unity of rationality on philosophical grounds. Categorical logic allows for a different solution, namely to provide means for translation between different local logic. Such transition doesn't intervene here as a new external principle, since what I call here local or regional logic is construed in the categorical setting in terms of morphism, which can be naturally viewed here as translation. To make this work, one still needs, of course, some universal principle, namely the general principle of categorical theory. from philosophical viewpoint is this. In the categorical setting, universal principle of rationality turned to be principle of translation rather than form of principle imposing universal form of reasoning indifferent to its content. One may argue that general principle of translation I'm talking about are themselves formal, but this is my views and views of the language. As far as one tries to be precise about the meaning of the term formal, it becomes clear that the argument is wrong. Observe that categorical logic assumes the possibility of multiple local logics to begin with, so that no counterpart of the aforementioned problem of formalization of logic doesn't arise in the categorical setting. Yeah, okay, now I just try and explain what this functorial semantic, so I already tried that yesterday. So probably I just need a conclusion, okay? Lavira's functorial semantics has been developed for a special case of algebraic theorems and so it cannot be immediately used as a method of theory building applicable in law areas of mathematics or elsewhere. Since then, a lot of technical work has been done in related fields of categorical logic and categorical model theory. For historical introduction, I refer to this Bell paper to the development of logic. The categorical method of theory building is a work in progress, so that they don't exist in any standard form. The purpose of this non-technical paper is to provide this method with an appropriate epistemological background, blah, blah. the categorical method outline above suggests an epistemic strategy which differs from that suggested by formal method in the latter case the general epistemic strategy is subsuing different objects under a common form this amounts to construct and reversible transformation between

27:30 the object and then stipulate this form as a self-standing abstract object i shall call this procedure formalization here object may stand for various mathematical construction including whole mathematical theories of the sense in which the object is abstract is relational. This leads to a traditional hierarchical structure of organization of mathematical knowledge where theories and concepts are subsumed under other theories and concepts which are more abstract, more general and more formal. Apparently the same pattern implies the interface between mathematics and material world and is responsible for the usual qualification of mathematics as formal science. I disagree with you. An alternative strategy of categorification is different. It is more general and in a sense more straightforward. Categorification amounts to taking into consideration all transformations which can be considered as categorical morphine but not all reversible ones. The requirement that object and transformation in question make a category is much weaker than the requirement that these transformations are reversible and so make a group. So formalization is a special case of categorification. Unlike formalization, categorification in the general case doesn't assume objects in question under a conceptual umbrella, but simply links them by morphine into a whole name or into a category. When objects are theories and morphies are mutual interpretation of theories, one gets some network of theories. Also this network might have no single sentence, might be still coherent and well manageable if it has good categorical properties. If it has a purpose, for example, you have a kind of universal language right there. We see that categorification-like formalization solves for integration of knowledge, but categorical integration, unlike formal integration, doesn't bring about a hierarchical structure, given that the very idea of foundations seems to imply a hierarchical structure of knowledge, which starts with foundations and branches and various specific sub-domains, one may wonder if categorization is compatible with the value of foundation. I think that at least one version of the notion of foundations remains viable in a categorical concept, namely the pedagogical one. I mean the notion of foundation as an entry into a theoretic network. Such entry should exist for any theoretical network, but it obviously needs not to be unique. I suggest that hierarchical structures can attain the longer surface universal models of organization of knowledge, just like they can attain the longer surface models of organization of societies.

30:00 But the task of integration of knowledge into a manageable whole remains pertinent as ever. So I believe that's a categorical method. Thank you. I don't know, it's a little bit, of course, I have a big pros, because after all, I don't have some real preposition. So, yeah, probably just a little bit. I understand that all these categories are very abstract and very high level framework and maybe it's really interesting to try to use it in philosophy simply as a kind of, of, I don't know, model, but not in the model theoretic sense, model for organizing knowledge, so... But why are they abstract? I mean, I think it depends, I think they appear to be abstract when you combine them with this more traditional framework, right? right then it's something like because the notion category itself is very weak you know it's kind of weak but you are simply describing what's going on by using these transformations and this is a little bit unusual In a sense, actually all the discussion between formalism, this classical discussion between intuitionism, right, and whatever, formalism, is based on that idea of substance or something, right, that you have some primary intuition, and the idea that you don't need that, that you can all somehow reconstruct, say, through relations. And in that sense, category theory runs on the sides of formalism, you know, again, we don't need. But, but, it's still very different, I mean, because in a sense, we don't have this, how to say, abstraction inside this procedure.

32:30 We don't have anything like counterpart of notion of formal theory. But whatever objects were not, I don't know, we can't abstract our category from what is category all, you know, it doesn't make sense. What we can do is we can always, having two categories, we can think about factors between them, and then consider a category of factors between them. You know, that would be, in this sense, a good instrument for comparing things, you know, and all this stuff about substance and form. I don't know, probably just not any longer. But you're holding material content, in a sense. Because you're not interested anymore in objects. Yes, yes, of course. some kind of physical application, you know, for example if you think that notion of set right, is something in some sense applicable to real world, and you can reasonably speak about a set consisting of that table, and you know, some star in different galaxies and it's rather risky in which sense, because it's just meaning that you have your, say, scale, and then you extrapolate it enormously. And I think at a certain point, like with geometry, you know, you make geometry on paper, then you extrapolate for some kilometers, and it works. And there's something great about that. You can make a plan of building, and then make a building. In that sense, Euclidean geometry is absolutely a terrific empirical theory. But, if you take a little bit longer, it doesn't work any longer, because all this idealization, all abstraction from means, from whatever, you know, you cannot even measure your, how to say, distance to the moon, but anything like you measure distance on the table, you have different instruments. And if then you apply instruments which are relevant, something like radar, right, then you get special relativity, I'm sorry, you know, so it's just risky to think, extrapolate your concept too far. And then, now, if you're trying to think, say, categorically, you need some kind of morphine, so you might think about star and the new year in terms of, I don't know, signal, light signal, whatever, in very, very concrete physical terms.

35:00 At least my intuition and my hope about that. Yeah. But the problem is, of course, that, okay, that's a kind of idea, but I confess that I don't have really, really something really workable if you can, like, maybe make formal series. It's what Bill Lavia tried to make. He tried to make it with what he called algebraic series. And that, officially speaking, that's all logic, but I think it's kind of independent. and you can just cut it off and try to consider this kind of self-contained proposal. Okay, um, what's the time now? Okay. Thank you.