Intuitions of Number & Space / Logical Diagrams

0:00 So, if you just look at 1-1, where it's, uh, you're constructing an equilateral triangle. then there's no need to control for changes of the length of size. I mean, everything is just given in the two circles. So there's no room to probe anything to see if I change the length of the size, something else will happen. It just depends on the fact that I have a definition of a circle, and the postulate at one point, the third postulate says that given any point and any radius, I can construct a circle with that radius. no matter how big the thing is, that the construction of the equilateral triangle, nothing further has to be controlled that isn't given in the definitions or the positives in 1-1. There's nothing else I have to do. And then you go on to 1-1, 1-5, and one would hope that for every step, the sequence, I think that's what Marco is suggesting. Does the control for you? The definitions and the order. And perhaps you can find that there is an error. So I don't know. We should check. But it seems to me that it should be in this way. If it's not in this way, it means that there is an error in Euclid. It's not an independent practice. I have another idea that maybe we should consider the element and the ending point, the final point of the practice. So going back to control, imagine in a mathematical community where knowledge is transmitted by the observation diagram. So when I can imagine that when I physically draw this figure, Everyone starts by agreeing that it is a circle, or if I draw a triangle.

2:30 And in this sense, my physical abilities control the drawing. And your cognitive abilities control the drawing, too, because it's interpreted as a triangle and a circle. So, for example, but imagine that I want to do something about equilateral triangles, or imagine that I would like to speak about equilateral triangles. I can draw a figure, and I... and this agreement might spread about the fact that this figure is equilateral. So I need to add something for controlling the diagram to remember for, I think that the role of the tech is increasing the control. So, remember for, and so on. And this isn't for exact diagrammatic properties. I mean, we say that this diagram is subject to some properties because of a certain kind of stipulation within a community of geometries. So control is reinforced with the text, as you said. It may be a normal practice or a written practice, and the fact that we read Euclidian, we have to consider that it has come out of a long series of reading, how can I say? I don't know, I think. Maybe to go back to Karin's suggestion, the following is that it's genuine to postulate some kind of agreement between geometers and from public. We know that people like Apollonius, which is not a minor geometer of that time, disagree on the common notion. That means that we know that all of the common notions were empirically, I mean, that we can see them. or even the postulate for the pilot. I'll just explain that Apollonians consider that as soon as the anger are disminishing, they will live at some point, that's obvious. We won't need to put it as a postulate.

5:00 So it's very difficult to, very difficult, and focus on the corp argument, which is not very good, because it says, look at the synthesis, but there are no straight lines. So it's very difficult to reconstruct this kind of community because from the testimonies that we have there was disagreements of forms, basic forms, in what we can read of the diagram. There are some passages, for example, in Plato, where he takes the color of the figure as a geometric feature. He defines the figure starting with the color. So it is maybe an evidence is that even agreement on definition was not so clear. What I tried to do was to reconstruct, in my DNA, a conceptual community in which Euclidean geometry function, and that could explain how we could arrive at Euclidean. So the project is that, given Euclidean, Build the communities. There are many objects. And then look for some evidence. It's okay to succeed with the community evidence. Construct if you prefer. If you prefer. We can stipulate it. Yeah. Thank you very much. thank you so now to go to simply we go to the China restaurant and simply follow me I'm sorry, but you are, you know, I'm sorry. You have one seal. You have another, another tentative, and you have another seal.

7:30 You make it silver. You take your tiger seal, and you have this, and this is the one. And this one, and this one, and this one, and this one, and this one. And you have two fiancels, they have not one. I don't know. I don't know. Marko, is someone else going to chair after lunch? Ah, yes, I wanted, yes, yes, yes. I wanted to ask you, Mark. I don't want you to make do it. Thank you. Thank you. You want to come here, my chef? Okay, at the end of the road at the right, and that's the room, and you find the 45, and then you go inside, and I think it's on the left.

10:00 No, the right. No, it's not there. Wait, wait, wait, wait. Thank you. Thank you. yeah but you're right it's a good point you really need an axiom of choice in this proof in the proof of the paradox. But you also can make paradoxical composition with the group. That's right. Well, what I want to say just in my remark about the use of this diagram in the group, KD graphs are very much widespread nowadays in mathematics. They're getting more and more. They're kind of a fashion now. And there is a tendency to re-formulate or re-translate complicated mathematical proofs in terms of...

12:30 Um, I speak to you, I'm a, what take you? Yeah, let's open that one. The coordination of logical driver. Okay. Thank you Marco for the invitation, I'm here today to talk about work, logical diagrams. of course despite the what she calls the general precise diagrams we know that diagrams were always used in logic and their use is very old in the beginning the one today will not use the any I will use visual devices, but I will just distribute some leaflets with diagrams, but I prefer to work on the board because, like I said yesterday, these diagrams are constructed and I would like to say how these diagrams are constructed and why we make such or such choice. It is possible to make PowerPoint with motion now. I hope so. Yeah. Okay. I don't hope so. So you first present some diagrams used in logic in general, I suppose. Most of you are already acquainted with them. And then I will talk properly about the object of this presentation which is to talk about this wonderful idea which is to represent the collection of things by circles. This is a very simple idea which was used in the 19th century for logical inferences but which has been misused I believe at the end of the century and this is why it was not used in the beginning of the 20th century but before this idea would like to present some

15:00 other visual devices used in logic I'm not sure whether all these devices are better but I'm sure that they are ask for some visual properties okay so the believe that okay okay so the figure for instance is a presentation of the cherry for here it's what we call the property tree but generally logic would trees. For example, if I have something having the property A, the attribute A, I can perfectly distinguish between A to each of the property B, for instance, and those which don't have property B. So here you will have AB and A, not B, etc. For example, you may also have the tree with a which I divide in three classes for instance BCD this is a kind of trees you sometimes find in logic of course these are just polarization and you can't use them for inference as such and you should distinguish between trees which will present any division and a tree which represents one particular kind of division which is dichotomy means when you divide you always divide into two subclasses and the interest of this kind of trees the dichotomy of trees is that all others can be translated in this kind of trees Because, for instance, if I have this one, I can, for instance, do, I have H, which are B, and those which are not B, and among those which are not B, I can find, for instance, which are C and not C, and so. This is the first kind of diagram, which is very usual in the logical treatises and textbooks.

17:30 Second kind of diagram is the second one, picture number two, which is the famous score of oppositions. Well, I use this one because it's the one you find generally in traditional logic, and which is widely used, but today you have many other n-opposition figures. For instance, for the square of propositions, for people who are not acquainted with it, you just remember that in traditional logic you have four kinds of propositions, which are generally used. You have the A propositions, which are propositions of the form all X are Y means universal fermetive propositions you have P propositions which are universal negative propositions means propositions of the form no X is Y you have high propositions which are propositions of the form some x are y and finally propositions of the form all propositions of the form some x are not and it would believe that all propositions can be translated in one of these forms the score of propositions explain is the relationship between these propositions for you have, for instance, these propositions are compared to it, means if one of them is true, the other is false and you have this proposition, this is peritonation, it means the A proposition implies this one means if this one is true, this one should be true, and if this one is false this one should be false, and it's the same the same thing here between the E and O propositions, and finally you have a relationship of you have A and E are contraries, means they cannot be both

20:00 false and subcontractors means they can be both false but they cannot sorry they can be they can be both true but not false and they can be both false but not true well what you have here is just a relationship between different propositions and here of course you can you can do any inference except of course if you have this proposition of this guy that you wanted to know what's the control of propositions this one so and what is supernatural area but this is just a kind of educational aid if you want just to remember what is the relationship between all these propositions so this is another kind of diagram which was widely used in traditional logics number three and four what I'm not sure these are diagrams because they are tables but these are used in modern logic because they refer to propositions and you know in logic the use of propositions instead of terms of classes is quite recent since the works of Fregue, Marcauld, Piano and Peirce at the end of the 19th century. Number three is at the chair table. Well here again you have for instance two propositions, P and Q, and we know all the combinations between these two propositions. they're true for C for instance I have these true true false and here if you have another proposition which is a combination of these for instance P and you I can find each values depending on the values of p and p for instance you well it's a table are interested sometimes when you want to what do you have friends that's two propositions and you want to see the relationship of propositions for instance whether they are equivalence equivalence sorry for proposition not q implies not p i can with the same table table find the values of this propositions

22:30 this proposition sorry and i will see that it's exactly exactly the same as p and q I have Q, I will represent non-P, so we have false, false, true, true, Q apply non-P, so application is always true except when this one is true and this one is false, and here sorry uh here it's false true true okay and if i had a negation here negation of this one you have exactly the same values so I know that this proposition is exactly the same as this one so this is the truth tables number four is is almost the inverse operation I have a table with values and they want to know what is what's the proposition represented here this this well for truth values sorry truth tables were developed at the beginning of the 20th century there is about who invented them, but generally consider that Wittgenstein and Russell, during their collaboration in Cambridge, developed first these kind of devices. For the number four, this is what we call in computer science a can-no-map, it's used for simplification of propositions and used in electronic circuits. If you look at the diagrams we have here, it comes strictly like this. We have three propositions, A, B and C. I put one here, two here, plus if you have four, you put two here and one here. And the different combinations, exactly as we have done in the church tables, you will have, of course, eight compartments because you have three propositions.

25:00 so the combinations are as fellow zero zero zero one one one one zero and I know that the values of this proper proposition are as fellow zero zero one one zero one zero one I want to know what's this proposition now we have just to put when I find two true values one is for two of course and zero for false or four or eight I put them together I will have to now to write the proposition for instance here the proposition remains true everyone a change since so I have this proposition should be equal to not B C plus and here the value of the proposition remains true even when C change so you have B A so here is the proposition which is represented here here is the negation of B and C or A and B it's almost exactly the other operation okay so and then in the in the table it represented the first and fixed notation have some problems in this kind of diagram because they look like diagrams but I'm not sure and I will explain why okay so just remember that P and P represent a proposition means linguistic entity so when I say for instance for P Perth represents P actually by RIT and P, and for Prager it's something like this, now if I want to introduce a negation, non-P, sorry, this is Perth and this is Prager, for Perth, something like that, and for Prager, we have to add this one.

27:30 Well, the problem is that I don't see what's the difference between this one, which is considered as a linguistic presentation, and this one, which we should consider as a diagrammatic representation, because, in fact, I just read it. I mean, I see no difference between this and this, for instance, or this and this one, because these notations refer to to a language not to the word not to think so I I think that it's more adequate to consider them as notations not as diagrams even if they look like diagram for instance if if you have this symbol for instance of course if it's in your opinion they type for instance if you have to set lines and so but if I are just a and B here this a is not equal to be this the same thing but now we buried as a simple limitation of a linguistic representation so the idea of whether this figure because everything is figure of course and a even B and so all these are not necessarily their grammatic representation or linguistic representation just because they have some special representation some special properties because everything has and spatial properties, even linguistic representations. The question is how do we use them? And in general I simply consider that when something I can read, or I can say is a linguistic representation, and this one is a linguistic representation, and this one too. because when I speak of course I can't say two words at the same time so I have to, it's linear I have to say a word after the other why, sorry diagrammatic representations are, I see them I mean it's not exactly the same thing when you have the same representation to see it or to say it when you say it it's a linguistic representation and this is why in this I'm not sure they are properly diagrammatic representation this is even you can see that even better in the for instance if you look at the what for P implies Q for instance Frege represents that like that and if you look at P

30:00 you for instance you have something like that that's exactly what pro before you have the P and Q exactly the same thing as none Q implies non B but you read them just wait to write these propositions and well of course regular consider an implication of the central operation while for Peirce he still used disjunction and conjunction this is why for instance P and Q is represented as P and Q while implication is represented as such because this means simply application is not P and not Q okay this is exactly the same as not P so here in this kind of figures they look like diagrams I think these are mainly similar okay well now I would like to talk properly about the the circular diagrams which were so well used in the 19th century and the I will try to tell you the story of these which, okay, as I said, these diagrams are based on a very simple idea, which is simply to put a collection of individuals, for instance, all men in a curve, a closed curve. Of course, it's not important, but it's a circle, maybe any other shape. most important thing is that it's great okay this is a very simple idea and we don't know exactly when it was invented and of course you can imagine that it was invented more than once or discovered more than once by different authors the oldest figure of this kind I ever seen is number one on page two

32:30 which has recently been published in the Britain of the British Society for the History of Mathematics by Anthony Edwards who is for the history of mathematics in Cambridge and this one is from a treatise on music of the 11th century and you can see this kind of figures or it is clearly said that elements individuals which are here belong to both classes the number number two are from Leibniz manuscripts in on of here again you can see these diagrams on on Leibniz, in Leibniz writing. Where is this from? Hanover, I would say it in English. Hanover? Hanover. Well, I took it from the Boczinski's Historic Logic, this picture. You took it from the History of Logic? Boczinski. Thank you. Okay, so it's interesting here to see, of course they were not published. published only at the beginning of the 20th century by but they were not published in so you can see here the circular diagrams I explain they use later but you can see are also another kind of diagrams which is linear diagrams so these linear diagrams were used by image here they were also used by other musicians all along the 18th and 19th century such as Lambert and Keynes I will explain briefly their their use because they were they were almost exactly the same manner as the circular diagram but for if I want to represent and that's what all musicians try to do I first use them for representation means for the presentation of propositions and then for inference and when I call this paper this presentation the golden age of logical diagrams it's because it's in the 19th century that these diagrams were used for inference before they were mainly used just for presentations and thanks to Euler Thurst well Leibniz probably used them for inference but he did

35:00 published them and it's Euler who made them popular but thanks to Euler and all along the 19th century these kind of diagrams were used to make inferences and find conclusions and this is why they were they are considered as the 19th century generally considered as a given age for the use of logical diagrams because they disappeared of course at the beginning of the 20th century partly because of this general prejudice position and also because the problems of 20th century logic are very different from the problems of 19th century logic we are not trying to make find conclusions and do eliminations and so on but the problems of 20th century are very different the logical diagrams have no place in in this kind of problems well i will briefly explain these linear diagrams if I want to just remember the four kinds of propositions the a proposition is all X's are white so if this is for instance why I want to say that all X's are white that means that the part of this of white is X so yeah so here are presented something on accessory are white right if you want to present no it's right to our wife so I can just look at them like that you can see that there are no segments between them some X's are Y can be represented something like because there is something which belongs to both the x and y and finally some x are not y so we have y here and x here and here I will emphasize on this part there is something here which is not y and this is different between this one and this one of course the problem say for instance that some x are y's that doesn't mean necessarily that there is something which is

37:30 not right so here i represented some x's are y but no nothing to say that there is something here for instance if i had this one even here are some x are y but this is uh the first proposition so some solutions have they suggested for instance doing something like that and at the line means there may be something here but we know that there is surely something here and so on this ambiguity is exactly the same that we will see in the circular diagram this is why I want to just to make small presentation of this kind of diagram which were used in the 19th century but we generally talk about the circular ones because they were much more used and they were first popularized thanks to the work of Euler and then to the British tradition then okay well now I will talk so I will now speak properly about these circular diagrams so on second and second page I have the education from Euler's book well this is a translation because Euler's book letters to a German princess was written in French where he introduces his logic diagrams okay these four species of propositions may likewise be represented by figures so as exhibit their nature to the eye this must be a great assistance to work comprend in more distinctly wearing the accuracy of a chain of reasoning consists what you can see that he introduced the the differential parts as in a general notion contains an infinite number of individuals objects we may consider it as a space in which they are all contained that's for the nation of man we form a circle in which we can see all men to be comprehend okay what I wanted to explain here and this is why I included this quotation well this may look obvious today okay is this point when he says for instance that the infinite

40:00 number of individuals and they are all contained in this curve this is something was very unusual in the time the logical processes of the time because this view of logical classes is an extension of view it means I'm saying that I have here individuals which means if this is the class of man here all the points here represent man this is very different of course from what was used at the time which is the logic of terms if I say man the class of man I have to define what man mean but here who are using classes classes and not terms means for instance if you want to define a set say what I will use a mathematical example X between 1 and 9 for instance this is of course an intentional definition an extension definition will be simply to list aren't these individuals okay this was how logical propositions logical classes were defined until the 19th century this is how they are used here and this is how they were used by Google and all the for instance when I say it's not the same thing when I say that every x is y and when I say that all x's are y's and when Bull writes for instance classes x y and so on that doesn't mean that the thing which is both x and y this is the this is the intersection of the two classes it means the collection of things which are X and which are Y this is clearer I mean it's different when you are working with this one and with this one

42:30 here we are working on individuals for instance if I want you if I this is for is the universe of all things you have all these points I think okay I want to select for instance things which are things which are nice so what I divide extension of things I have all the things and I divide the extension I can't divide the term for instance if I want to divide nice things into new things and old things for instance you have old things your new things I do not divide the term nice into old and new because has nothing to do with that I'm dividing the extension of nice things into the old things and the new things so this view is very difficult and different because instead of working here for instance is is what here for instance R is just a relation of inclusion and exclusion because you have classes extensions but here it's very different because is maybe identity and a vision and so do you have think and attribute and there are many metaphysical questions which, philosophical questions here for instance how can think be an attribute and so on but when I'm working with extensions this is much better from mathematical viewpoints because I'm working just on topological relations of inclusion and exclusion so this is why I see that this idea is very important one very powerful one because using this our propositions thanks to just relations of inclusion and exclusion and this is what I learned did I I think in the next table I presented before before propositions okay but there's a table where I represented the four propositions using Euler and Euler's notation and then

45:00 using Venn's notation Euler introduced them in the 18th century 1768 while Venn introduced them much later in 1880 I will first explain how Euler used these diagrams to represent propositions and instead of historical the sketch I will I will compare directly the two methods our errors principle is very simple the relation between the two classes is the relation of inclusion exclusion for instance if I want to say that all X are wise I have simply to represent the class X the expansion of X inside the expansion of Y here you have for instance all X are right if I want to represent for instance that no X is white I have to represent the two extensions super equally okay so I represent the actual relation be the script between a between X and Y in the van method we are using a very very different method we are still but I will explain the principle of representation which is very different in the very method I do not represent the actual relation between X and Y I first represent all possible relations between them for instance if this is the if here you have all things in this space I select I would like to select I have sorry two classes X and Y these two classes divide the inverse into four possible combinations you have things which are X and Y things which are X but are not Y things which are Y but are not X and things which are neither X nor Y so I draw first the four possible combinations so you have here X and Y you have things which are X not Y, X and Y, Y not X, and X which are not X not Y but here I didn't represent any information now I have to add some visual devices to represent a big proposition so we are just representing the classes now to

47:30 present the propositions for instance to say that all X's are Y's that means that's it