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A C O SIMON
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I'm mainly here to learn about topoï,in order to check the thesis I'm working on.

This thesis is as follows:
1 / There is no change in the way of thinking since man lives in a group. 2 / To prove it, it is necessary to define what are the “thought patterns” of any Subject. 3 / The theory of categories can serve as a universal language model making it possible to identify these different patterns of thinking. 4 / This makes it possible to differentiate 3 levels of reflection, leading to

- The logic and elementary topos of Lawvere,
- A "topological approach", initiated by Évariste Galois and leading to Grothendieck's topos,
- A 3rd level where equivalences or "bridges" are established. 5 / This general scheme would make it possible to consider a "unitary theory" in physics. I leave on my blog the traces of my own evolution in this program. I try as I can, to discuss this subject within the Categorical Logic group led by Anatole Khelif at Paris Diderot. Not being a mathematician at all, I try to find relays to improve my approach. For a short summary of this approach : http://www.entropologie.fr/2021/01/resume-hec.html

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Morgan Rogers
(@morgan-rogers)
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Bonjour,

It's a shame that many of the symbols you use on your web page don't seem to work on my browser (I get a lot of boring boxes amongst more interesting-looking symbols). Is there any way this can be fixed?


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A C O SIMON
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@morgan-rogers

I have a copy in pdf, it could be easier for reading ?

By the way: I had translated my article in english.

 


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A C O SIMON
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@morgan-rogers

I changed the symbols used in my last article in English, I hope your browser can read it : http://www.entropologie.fr/2021/01/presentation-abstract.html.

 

 


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Morgan Rogers
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Dear A C O Simon,

It seems to me that you have read widely, and have managed to synthesize many ideas that you have encountered into some kind of symbolic formalism, which I imagine must be quite exciting for you. However, I must admit I find it mystifying to the point of invoking strong skepticism about its content. Therefore (to stay on topic with the forum) I hope you might humor me in explaining what part you think categories, and eventually toposes, have in your system.


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A C O SIMON
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@morgan-rogers

It's quite a long story.

  • First of all indeed, the place where it install the initial and final objects (starting with Set category, of course). With the "universal property" etc.
  • 2 : Very difficult for me was to express the difference between morphism and co-morphism (related to the difference identity/ idempotence), and it leads me in my development.
  • 3 : it appears, after reading Lawvere "Conceptual mathematics" that all the matter spreads between the 2 first levels.
  • More difficult, and a long process, was to understand the Galois's revolution as a regression from one process of higher level (I means the difference of approach local/ global) down to the second level (i.e.: group concept related to ex ante posture of the Subject).
  • Apart of that, my presentation reveal quite easely how to represent morphism, functor and natural transformation, with Yoneda lemme (my presentation to Khelif's CLE in 06/2019, even if I change a lt my approach since that time). I have to review this paper.
  • About toposes : it the next subjet to take into account in order to clarify my approach (why I am here) and allow me to link (bridge) it with physic (i.e.: relativity and quanta theory) and Alian Connes geometry.

Apart of the general evolution of my approach, rise quite interesting questions for the mathematician, such as :

  • At which level we need an axiom or a property and at which one it cess to be relevent,
    • meanly : existence axiom, continuity, infinite
    • concept of succesor, order
  • My presentation points out the absolute difference between "elements" and "parts" of an object, and, according to what I read, it seems that it is not clear for everyone !

Incidentally, my approach help me to detect some "implicite thought" in the papers I read, related to the fact that the posture of the author is not clear, otherwise he expresses irrelevent propositions... Smile  


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Morgan Rogers
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Addressing your points in order:

  1. You know that the category of Sets (as classically conceived) is a Grothendieck topos, right? What are you using the initial and terminal objects of Sets for, exactly? It seems, from the blog post you linked, that you have a misconception of what a universal property is; that much we can help with here!
  2. What morphisms/comorphisms are you referring to?
  3. What "levels" are you referring to?
  4. How do groups relate to 'postures' and 'subjects'?
  5. In this blog post, you give a fairly standard presentation of functors (more specifically, of Set-valued functors) and natural transformations, and then in the conclusion talk about the initial and terminal object relating to 'observations' and conscience, even though the initial and terminal object do not feature in the preceding text... 
  6. I appreciate the optimism that you'll be able to build a framework that unites all of these things, but in light of the lack of clarity or precision in much of your writing that you have directed me to so far, I should warn you that it may be hard for anyone to take you seriously. I am sorry that I have to be so blunt, but if you insist on pushing your work under the noses of others in a public space such as this one, you must either meet the standards that the community expects or be prepared to do a lot of work to bridge the gap between your own understanding and theirs.

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A C O SIMON
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@morgan-rogers

Thanks for yours fast reply, but each of yours points need a long development ! 

1/ In the Set category, one can identify the elements within each objet (I represent that as a "ex post" or "global" posture of the Subject, able to see and define the elements as well as de overall objet. Generally speaking this is a "rational" approach when you are able to report an observation or a judgment to some critea). But, in a topology, you appraoch the objet by its parts (in an "ex ante" or "local" posture), before you make the connexion between parts and elements see "Matrice", which is the subject of the "analytic topology". Here is the confusion, I think. I have not yet claerified everythink, but you can follow the point here "Le point #8 logique-topology". About the Universal property, yes I have a lot to learn about it, but I already start to check that point here : "De la propriété universelle".

2/ For morphism and comorphism, I refer to the introduction in "Conceptual mathematics" by Lawvere. After a long reflexion, I came to the point that on its basic aspect (*) a "morphism" is a gap "up" from one level to the next, and a co-morphism is a gap "down". That suggests two differents postures to express this difference : ex post for the morphism (such as identity) and ex ante for the comorphism (leading to indempotence).

3/ By "levels" I refer to those defined in my paper, no more no less, that is meanly why I use specific glyphs [∃] [⚤] [#] [♲] [∅] to refer them, in order to avoid any "implicite thought". By the way, regarding the need to use new representation for new ideas, please refer to the paper I wrote yesterday "accepter de s'exclure".

4/ This point is really very interesting, but we would go back on it, if I can really interess you before...

5/ The discussion is over the "rational" aspect of the Imaginay of a Subject. The point is that, from my point of view, and this is not relevent here, the final point could be seen as the Imaginary part "in touch" with what Lacan call "Real", and the initial object the best representation (Imaginary) of what is the "Symbolism", over the Imagination of the Subject. The point is that the Subject is always "ex post" in the first case and "ex ante" in the second one. But this discussion is not within the category theory...

6/ When all this will be clear, I would have finish my development, and I would not need anymore to discuss about the matter, in order to clearify my ideas and improve myself...

(*) Basic aspect : I mean before we identify the "element" (*) (on [∃]) as a set element {*} (on [⚤]), with the first identification. I write it : (*)↑{*}, when the Universal property leads to {*}↓(*).

Of course, as soon as the Subject is able to imagine (*)→{*}, on the second level [⚤], the arrow is not anymore a "diachronic" concept but a "synchronic" one, and we are able to define "Graph category", another discriminant object etc...


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Morgan Rogers
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  1. The final link confirms what I thought about you misunderstanding what "universal property" means in category theory. You refer to a particular functor to Sets as "the universal property", but a universal property is a property of an object in a category that defines it uniquely (up to canonical isomorphism). For example, the definition of product that you quote from Lawvere on the same page is a definition of a product of A and B as (the domain of) a universal span over A and B. Here 'universal' refers to the fact that any other span over A and B must factor through this one in a unique way. Universal properties can often be expressed 'representably', which is to say in terms of the corresponding representable functors which you also talk about on this page. For example, a product AxB in a small category C has the defining property that the set Hom_C(X,AxB) is isomorphic to Hom_C(X,A) x Hom_C(X,B), for all X, naturally in X, where the latter product is the usual product of sets. What I want to stress with this example is that representable functors should not be confused with the universal properties that can be expressed using them.
  2. So you're saying you have morphisms [∃] -> [⚤] -> [#] -> [♲] -> [∅] and comorphisms [∃] <- [⚤] <- [#] <- [♲] <- [∅]. If these are supposed to be the morphisms (and comorphisms...) of a category, it's a pretty simple one. This doesn't really tell me what these things are supposed to represent, however (in your "paper" you have little figures facing in one direction or the other to represent these things, but again, there is nothing resembling a definition I can use to get a hold of what these things are supposed to mean).
  3. Without clear definitions and explanations in plain language, how do you expect anyone other than yourself to understand the conceptual content of your symbols?
  4. We'll see if I have the patience for you to clarify what "subjects" and "postures" are...
  5. If the discussion is not within category theory, how does it bear any relation to category theory? If you can't give a clear answer to that, is there some other reason that you feel entitled to appropriate the language of category theory to add pseudomathematical mystique to your writing?
  6. Are you saying it doesn't matter whether your ideas mean anything to anyone else?

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A C O SIMON
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@morgan-rogers Tank for your patience.

1/ That will lead me to precise my point of view and clarify my discourse, I need time to think about it.

2/ I use specific representation, in order precisely to avoid this : [∃] -> [⚤] -> [#] -> [♲] -> [∅] because the gaps between [∃] -> [⚤] and [⚤] -> [#] or [#] -> [♲] are not of the same nature. There is 2 months ago, I used this writing :[∃]<[⚤]<[#]<[♲]<[∅] but doing so, I missed that difference.

The point is that we have not all the spectrum of our concepts at our disposal when we are in a specific posture. As an example if I am a 2 dimensions worm on a surface (local view) I cannot define it's curvature like if I were in a global posture in 3D.

Another point : in his own history, a Subject doesn't develop his representations as the theory record it a posteriori. As an example the baby is not previously separated from his mother, so the idea of an "object", separated from him is not first, but continuity must be. The concept of "object" or "element is acquired when he is 2 or 3 years old. That is why, we cannot start with the idea that "a part" of an object is a "collection of its elements", they are strictly related to differents postures : parts seen ex ante and elements seen ex post. The connexion between them is a question of education and experience (related to the so called "mirror phase").

The difference I made between (*)↑{*} and {*}↓(*) is only relevent at a very low stage between [∃] and [⚤], in fact even before the subject acquires a complete conciousness of himself (when he just records impacts from the Real). As soon as I can "represent" the arrow of a morphisum just like any other "object" or "element", I lose this dynamical aspect and we are starting with mathematics.

Then, when I repeate the gap between [⚤] -> [#], it is like R -> R2, and then R2 -> R3 etc. (will the repetition of the gap [∃] -> [⚤] is like 1 -> 2 -> 3 etc.) That suggested me to represent "morphisum" (after reaching [⚤] !) "functors" and "natural transformations" by means of a cube in my presentation of 06/19.

I must apologize if my presentation is so weak but it is not easy to give an overall view of a subject while providing specific definitions

3/ I thought I was giving some insight into the glyphs used, enought to follow the main point of the purpose which was to follow the Subject going up and down within his own Imagination.

I have in mind a presentation of Laurent Lafforgue presenting the concept of topos. He said he want to be quite evasive regarding the definition of a topology in order to go throught the subject... When you learn about a new matter, do you start by precise definitions or by an overall view ?

4/ In a discussion, the best is when both parts think that the exchange is of mutual interest... If I try, like in this forum, to exchange with mathematicians (even if I am not one of them) it is, of course, because I have so much to learn of them, but in return I offer another perspective and questioning on some mathematical concepts and axiomes, or connexions between them.

Example : "postures". I suggest that a mathematician cannot discuss about the initial object in the same manner than when he discuss about de final one. That can be seen as the result of a specific posture of the Subject refering to those objets.

  • 𓁜 Ex post : He "denotes" or points out the final object.
  • 𓁝 Ex ante : He just "connotes" or turns around the initial object.

That is why I need to discriminate each posture by a glyph, but if you have a better symbol for that, I would be interested, of course.

5/ The reason with I place the final object close to the "Real" as defined by Lacan, and the "intial" one close to the so called "Symbolism", is not a mathmatical discussion, but a philosophical one.

This is related to the posture of the Subject, I presented in point 4/ here above. Lévi-Strauss, in his "Mythologies" remarked that a "symbol" is never precisly defined, and changes from a version of a myth to another... Then the difference between a myth and the topology is that :

  • the myth refer to the Symbolism stage over the Imaginary, and the Subject never 𓁝 reach in a ex post position. (i.e.: for a believer, he is at the "image" of God, but cannot "explain" God)
  • the topology is closed by a global point of view 𓁜 on the object, even if in the local view 𓁝, he cannot describe the overall object.

That makes the difference between a rational approach and a mythic one, even if both mecanisms are similars.

6/ Not at all, and I'm precisely here for that ! In my opinion, an idea doesn't exist if it cannot be discussed, and taken over by someone else than the author !


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Morgan Rogers
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1/ to be continued.

2/ If you're committed to employing category theory, it's time to get you thinking more categorically. Here is the first example: as I suspected, the morphisms you draw are supposed to carry more structure than mere arrows. But the category of these five symbols and the transitions between them is too simple to "see" the richer structure. There are two ways to get around this. One is that you could make this into a richer category. As an ordinary category, you could add more objects and/or morphisms to "witness" the extra structure carried by the morphisms you already have; or, you could look to *enriched* category theory, where the 'collection' of morphisms between a pair of objects carries structure encoded in another category. An alternative (which essentially amounts to the first option from a different point of view), is that you could equip your symbols with more structure, identifying them as mathematical objects in some category which is rich enough to carry representations of the distinctions you conceptualise between these states. Often vector spaces are a good place to start with this kind of modelling.

The local/global example you give doesn't quite support your point, since curvature is a local property (the worm can measure the curvature by drawing big triangles, without ever leaving the surface) but I get some idea of what you are trying to say. But I still don't have a good understanding what the difference between your "postures" is, or even what they really are besides perspectives which are contrasting in some sense. I also don't know what "repeating a gap" means.

The distinction between parts and collections of elements sounds a lot like the distinction between classical topology (endowing a global collection of elements with a distinguished collection of open parts), which results in the concept of topological space, and point-free or point-less topology, which takes the open parts as the primary data, resulting in locales. There's plenty of material around on these ideas, which I hope you will find useful.

In category theory, it's often said that the definitions are the most important part, so if you have to omit anything from your presentation, make sure that the definitions stay in! In any case, if you manage to come up with definitions, that's a great starting point for applying any kind of mathematical reasoning.

3/ If I don't know what "the Subject" refers to, it's hard for me to follow anything. Regarding presenting a subject, there's a vast difference between a talk in a seminar where there are very restrictive time constraints (such as the talk of Lafforgue's you referred to) and a comprehensive presentation of a subject. If you take any course or textbook in maths, while there might be some motivating examples, the first real content consists of precise definitions of the objects or problems which are the subject of the course. If you needed or wanted to, you could fill in all of the detail that Lafforgue omitted, using the relevant academic papers and resources. Moreover, he can include technical terms without definition as long as he knows that the definitions are commonly known enough that his intended audience of mathematicians will be able to recall the standard definitions of these terms. I cannot do the same for your writing, partly because you use language from a discipline I am not well-versed in, but mostly because you give no references and offer no more intuitive explanation of your own.

4/ I gather that Lawvere places rather too much emphasis on the role of the category of Sets in category theory. Initial and terminal objects are not concepts unique to the category of Sets; they are defined in terms of their universal properties, and there are many other categories having initial and/or terminal objects. Crucially, their universal properties are dual to one another. That means that if I take a category  \mathcal{C} with an initial object, and I consider its opposite category  \mathcal{C}^{\mathrm{op}} (the category with the same objects but where all morphisms are reversed), then that object is now a terminal object. In the plain language of category theory, we in fact can talk about initial and terminal objects in exactly the same way! It's only the structure of the category of sets itself, which is the structure of a topos (although for this we only need that it's a pretopos) that makes the initial and terminal object behave in ways which aren't as symmetric. Could you explain more precisely what these postures are/what they do/what the difference between them is?

5/ Okay, these associations sound like analogies between philosophical ideas and the behaviour of objects/morphisms in the category of sets. I am not in a position to appreciate these analogies, for lack of understanding of the philosophy literature you reference. It is not clear for me, for example, why "the Subject [can] never reach the Symbolism stage in an ex post position", or what this has to do with the initial object (in the category of Sets). Maybe you can at least clarify what property of the initial object you are appealing to?

What "topology" are you referring to at the end here?


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A C O SIMON
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I thank you very much for your reply.

2/ What a good idea to see my collection of glyphs as a category ! Thanks for that.

But I go step by step, and before going in that way, I'ld like to check my approach on concrete and physical topics, in order to verify if my approach is consistant.

Just one example regarding relativity. In this article, I came to that point : there is a difference, in the way we refer to time and to space, roughly :

  • For the space, the auteur 𓂀 can discriminate between two positions of the Subject 𓁝𓁜
  • For time, the Subject must be in 𓁜's posture, so the relativity is between 𓁜 and 𓂀.

My feeling is that difference in the postures leads to the difference between our representation of time and space (i.e. : between orthononality (realmaginary) parts of an hamiltonian, and orthogonality between the 3 imaginary dimensions)... It is not yet finished, I have to progress on that point.

Another point : I find a general way of thinking that we can see in various fields in mathematics (see "ordre et chaos".

In this general overview, I know that I have to learn about jauge theory and obviously norm theory. 

After that, and after making sure the approach is consistance and of some interest, I'ld like to follow your way : to see this set of glyphs as a category seems a great idea !

3/ That is a philosophical subject. I can just express here my own point of view : the Subject doesn't "exist" as the final object, but refer to the vacuum just as the initial one. That is more or less an Bouddhist approach... And in some way that of lacan and a lot of philosophers, for whom the Subject is definied as a being of speach.

4/ Yes I know that there is others categories, but I have to start by the begining. I define the "imaginary level" [∃] as a degenerative one, out of any other concept except the idea of "something" disturbes me. A cat that removes its paw by pricking on a heir, hurts itself to the so called "Real", and the entry in its imaginary passes throught that gate [∃].

In the same way the path [∃][⚤] is just when the cat reconizes the disturbance, in reference to its passed experience.

After that, in [⚤], the human can imagine all the discrete concepts he can, as repetition, frequence, sucessor, time, order etc. It can also represent the arrow itself ↑, and go to Graph category and have other discimiante objects than {0;1}...

Using de Saussure's linguistic vocabulary :

  • the first arrow ↑ is a "diachronic" concept,
  • the reference ex post  to that one, at level [⚤] is a "synchronic" concept, just like any "object".

Regarding the paper itself : as its title recalls, it is an "abstract" of a presentation made on Zoom... Not at all an academic paper. And, as I have not finished to check this approach, it is not time to present an academic paper, just time to check, discuss, rewrite and correct it...

5/ The similarity lies in the posture of the Subject 𓁝 which is the same in both cases.

Initial object is vacuum.


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A C O SIMON
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@morgan-rogers

Vos remarques sur mon incompréhension de la "propriété universelle" sont toujours restées dans un coin de ma tête durant ces derniers mois, et finalement, j'ai pu y revenir après une lente maturation de mes propres idées.

Vous aviez bien entendu parfaitement raison, mais ma représentation de l'Imaginaire était insuffisante pour m'en rendre compte.

Tout est chamboulé, et je sais qu'un gros travail m'attend, mais je voulais vous remercier de ce coup de pouce salutaire !

Cordialement

Alain Simon

"foncteurs et transformations naturelles"


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