Hello and welcome to the fifth session of Maths and Ideas with Raisin and Marisa. I'm going to pass the mic off to him now. Okay, thank you very much. So, before starting, does anyone have any questions? particularly regarding the Egyptian method well I'll throw in my fraction here so Uh-huh. Is this the Sylvester one?
Yeah, the Sylvester one. Yes. Uh-huh. So it seems that it was really a kind of a convoluted one. Okay, I can see that. How many steps do you require to get to the unit fractions? One, two, three, four, five, six, seven. Seven. Seven. Okay. Four lines. Seven lines I have. I'm not sure how many steps that took. Would you talk a little bit about when the Sylvester method is more appropriate than
the Egyptian multiplication division? Well, I mean, obvious that Sylvester method is, you can use it when you have odd numbers as your denominator. You can't use the regular Egyptian method of having. Obviously, you have to use the table, the table of 2 divided by n. But then also you see that if you have basically even numbers for your numerator and denominator,
it seems that not always, but most of the times the Egyptian method is much better than the Silvester. Yeah. Also another thing that Silveston method is not good is when, I'm sure that you have noticed that, when your denominator is smaller than your numerator. I can't hear you. Oh, for the Egyptian method. No, for Sylvester. Imagine, for example, 7 divided by 2.
Now you see that finding your largest units fraction that is smaller than given fraction, you find it difficult. So, Silvester method, first of all, is appropriate when you have your denominator larger than your numerator. Can we do a quick example, maybe, with the Egyptian method? Sure. Unless people think that that's a waste of their time. I'm not taking it. So you guys tell me what should I do.
Then we start halving. So what would be the halve of 34? Okay. Now you see. that you already get to an odd number so that's where you have to consult the the two divided by n table it's impossible to move forward so this is
interesting that basically this is that Egyptian method well the ones that for example if you had 26 divided by 33 you couldn't move forward by any means but there are also any kind of number as your denominator that when you are having them you reach an odd number at that point you basically using the method of having wouldn't be possible and requires consulting with the Egyptian 2 divided by n table. Now I think it would be really interesting, I haven't done that myself, to come up with
some sort of formula to argue or to show exactly what kind of numbers can't be employed in the regular Egyptian method of having. possible I'm thinking of what exactly happens so basically you need it should be some sort of so you just probably require some sort of formula for prime numbers has anyone I haven't really checked to see what would be the formula that shows
exactly what kind of numbers can be used within the regular method of having in Egyptian algorithms I haven't I haven't seen it and I think at some point I searched but I couldn't so would be great as a kind of like a home or conceptual homework if you can come up with something some general overall formula for the kind of numbers that can't be employed within regular Egyptian algorithm precisely because at some point you get odd numbers. Can it just be any odd number with a factor that is an odd prime?
Yes, yes, odd prime, yes. Yes. Mm-hmm. Which of course this can be 26 divided by 34 can be done quite easily using Silvester method. Your denominator is already larger than your numerator. So that's fine, you can easily implement the Sylvester method. So just in case some of you weren't around, we were talking about this idea that it's
What is really I think interesting in the Egyptian algorithms, so you get an overall idea of recursion in the most rudimentary sense of it, that you extract a long distance rule and this long distance rule has something to do with the really what the concept of addition how can you understand the concept of addition as multiplication or how can you understand multiplication in terms of addition and how can you understand division in terms of subtraction once which was that idea of grouping that we were talking about once you capable of conceptually laying out
division in terms of subtraction multiplication in terms of addition then you are capable of singling out a long distance rule. Once you sing out this long distance rule, you are capable of re-implementing it and applying it as an algorithmic recursive rule. But also, I think another interesting aspect of recursion is that It's really interesting that they use doubling or halving as their general overall approach to multiplication and division and this is something to do precisely with computational
optimality. They could come up with any like times three rather than times two. You can see that the idea we were talking about that other recursion other than having kind of like creating these loops. What is really interesting about recursion that it's based is simply about computational optimality or computational cost. And again it is interesting that a culture very intuitively develops this algorithm on
the basis of the computational, optimal computational cost of doubling and halving rather than any other times three or times four or any other forms of grouping. amount of time that is being spent, you know, the amount of steps, the basically effectivity of finding doubles or halves for a number. So I think the Egyptian algorithm from this perspective is, you know, even though it is very rudimentary, it really reflects not only the overall conceptual problems of recursion
but also its underlying computational cognitive undercurrents. So what was I going to say? Okay, any questions before I move to the new topic for today's session which is axioms and theorems? Okay, so I suspect we will be engaging with Euclidean geometry, particularly Euclid's
elements for not only this session but also next session given that analytical geometry is absolutely the most hegemonic form of mathematics from antiquity well into Renaissance. majority of mathematical innovations are being done within this period through analytic geometry, namely analysis, mathematical analysis laid out in terms of various constructive relationships
between Euclidean diagrammatic representations. And we will, as we move forward in the future sessions, we will see, for example, some of the most important mathematical discoveries which enabled, for example, something like the Copernican Revolution or Galilean Revolution couldn't be done at this time without heavy use of analytic geometry. Now I think, so I'm sure that you have some familiarity with Euclid's elements, but if
you are interested I highly suggest this book. You can easily download it from Internet Archive. I think it doesn't have any copyright anymore. It's called Euclid's Elements by Gabrielle Byrne. It's a kind of like a color book, very heavy on the graphical representations. And the really fantastic book about this is that rather than trying to, you know, using the regular proof line for Euclidean postulates and theorems, it's simply using graphical,
you know, colored graphical representations. And you know, one of the most interesting, another interesting thing about this book is that it was being taught at schools for quite a long time and so many of the avant-garde modernist artists that came in early 20th century were in fact influenced by this book. And you will definitely see the influence of this book on the style movement, like people like Mundrian, the late Cubist, so on and so forth. So it's just an absolutely astonishing book. But also, as I said, it is really true to the spirit of Euclid, rather than using any mathematical or rhetorical demonstration, it simply uses graphical representations in
the true diagrammatic reasoning spirit. So definitely download this book as your guide and I will, next session we will go through some of the theorems using this book. So back to Euclid's Elements. Real quick, what was the book called? Euclid's Elements by Gabriel who? Byrne, B-Y-R-N-E, I think. I thank you.
Yes. Oh, Oliver Byrne. Yes. Yes. Oliver Byrne. Yes. Sorry. Cognitive designance. Okay, so I think one of the most important things starting when we are looking at Euclid's elements, we see the introduction of so many new concepts into the system of mathematics.
And probably one of the most important contributions that Euclid does in writing elements is a significant pair of concepts that even though are not formulated in terms of modern mathematics, nevertheless taken up by later mathematicians to conceptually develop these pair of concepts further, turn them into, you know, formalize them, come up
with philosophical problems, finding possible mathematical alternatives and And the pair of concepts that I'm suggesting is the so-called coupling between the concept of axioms and theorems. What are axioms and what are theorems? Well, the first thing that we should know is that Euclid, of course, doesn't use the word axioms. Axiom, the attribution of the concept of axioms to Euclidean definitions
is something that is done late in Middle Ages and then later mathematicians start to see what exactly Euclidean axioms are. Are they really axioms or they are not? So a question before starting, what kind of axioms Euclidean definitions represent and what are these Euclidean representations? At their most basic level, for example, the definition of a continuous line, a definition of point, the definition of surface.
We know that basically these axioms are not logical. The first thing that you should know is that the Euclidean system is not by any means a logical system. The word demonstration in Euclid's elements doesn't exist. So this is a Latin later Middle Ages commentary on Euclide elements when they are using the word proof or demonstration, which is actually so many historians of mathematics today question this application of the word proof or demonstration to Euclide elements. The accurate Latin translation of the Greek word that Euclide uses in elements is monstratum
rather than demonstratum. demonstratum is illustration, literally illustration rather than demonstration. So this is something that from a historical perspective we should have in mind that Euclid's elements is not by any means involve logical axioms or a logical system. The word proof is a misinterpretation of what Euclid tries to do in elements. There is no such a thing as proof as we understand it in a logical context, nor the word demonstration.
You know, QED is a mistranslation of original Euclidean term for basically coming up with the adequate illustration rather than coming up with the adequate demonstration. This idea of illustration is really important in Euclid's axiom. Demonstration is a logical thing, you know, when you have some logical axioms, which of course in Euclidean system we do not have logical axioms, then by way of some deductive
procedure you reach your theorems that can be said to be necessarily true, given the basically trueness of your premises. So this kind of link between theorems and axioms doesn't exist in Euclid's system. In Euclid's system what you are dealing with is a diagrammatic reasoning. A diagrammatic reasoning that even though has a deductive framework but by virtue of of its axioms not being logical cannot be said to be really any form of proof in the traditional logical sense of proof or demonstration. So with that said, what kind of axioms do
you think Euclid's elements represent? Well that text you assigned said that they are Well that text you assigned said that they are rules, they're operational rules. Uh-huh. And so you can use them to discover things. You use them to build up theorems. Yes. Yes. Any more commentary on this before I answer Hunter? It seems to me that axioms are sort of like intuitive roots that subsequent transformations are built from.
Yes. Any more? Any more? So yes, that is completely true. One axioms are what you might call to be operational units, operational rules that allow for a system of construction. But exactly what kind of construction. Obviously this construction is not purely logical and this is fundamentally
due to this fact that Euclid's definitions are what you might call to be intuitive axioms, purely in its Kantian sense, in the sense that their roots are in manifold of intuition. Coming back to the idea that the manifold of intuition, its limits are perceptual mechanisms. One of the best commentaries written on differentiation of Euclid's axioms from other forms of axioms
and we have different kinds of axioms, not only logical but also mathematical, interactive axioms and so on and so forth is Lorenzo Pantel's structure and being. I think I have mentioned it one time. The axioms are a mode of mathematical inquiry. Yeah, so definitely check if you are interested in this elaboration of exactly what it means to have intuitive axioms and what distinguishes intuitive axioms from logical axioms of different
forms. As I said, A Structure and Being by Lorenzo Pantel. It's a massive book and it's absolutely fantastic treatise on the nature of formal languages and theoretical languages. called a structure taken by Lorenzo Intel.
So, given the intuitive nature of Euclidean's definitions, the kind of construction that it affords also, is very different from logical computational, regular logical computational construction. And I'm sure that you have read the suggested reading for this Danielle Macbeth's essay on Euclid's elements and diagrammatic reasoning.
One of the first things that we need to know about axioms is that axioms are from a system building perspective are your building blocks, your operational units. What they are supposed to do is that they should afford a range of construction, a system of construction. And this system of construction should be, or should allow you to, one, introduce new axioms into the system, and two, being capable of,
diversifying the range of problems or simply coming up with new problems that from the perspective of axioms themselves you couldn't see such problems, simply excavating hitherto hidden problems. This is the task of any axiomatic system. But also as I mentioned, introduction of new axioms into the system, so these axioms can be mobilized, create further range of constructions within your system. Now, I mean, I'm sure that any of you remember the, you know, school times and working on Euclidean geometry, remember that precisely the kind of construction that involved in
geometric illustrations in elements are quite actually sometimes counterintuitive. You know sometimes for example like the basic theorems like for example two triangles with the same base and the same height can be said to be equal, equal in terms of their area area can be said to be the same. This from, you know, graphical representation actually is quite counterintuitive. I mean, you know, I remember that the first time that I saw some of these things, particularly the ones require, the ones that involve the equivalency of area of geometric representations,
is that you ask yourself, I mean, how is it possible? They absolutely, from a visual perspective, they look to have fundamentally disequal areas. So Euclid's system, even though it has this very, what you might say, intuitive axioms, but precisely because of this idea that it mobilizes, this axiom mobilizes a system of construction, a construction that introduces new axioms and those axioms diversify the range of construction for the new modes of illustration. From a broad perspective you can say that even though its nature is in intuition, its construction is quite broadens
its construction is quite broadens the scope of imagination and this is usually what's supposed to be the case with diagrammatic reasoning. Its roots is intuition, but its constructability demarcates the limits of imagination in the sense that the problems and their solutions are quite counterintuitive from perspectives
afforded by previous definitions or intuitive elements. So this is an interesting thing that for diagrammatic reasonings, essentially what you are dealing with is a function of productive imagination in Kant. And what is the function of productive imagination? a dynamic unstability between the units or the unity of intuition or the manifold of
intuition and understanding namely categories or pure concepts of understanding. in the sentence that understandings re-basically stabilizes or creates new unities of imaginations and precisely because of the constructability of imagination the way that you can form new unities of intuition you can renegotiate the limits of understanding, limits of categories. This is what But essentially Kant is defined in terms of the second function of imagination, the so-called
productive imagination. And essentially all forms of diagrammatic reasonings are like that. though they have parochial roots in manifold of intuition, their systematic constructability allow you to broaden the scope of imagination, namely coming up with new unities of intuition. So any question before I start to get into the Euclid's elements? Yeah, but one of the sidebar, is it just that in the construction process you're encountering,
it's through the construction process that you're encountering the new problems, sort of like a material constraint? Yes, yes. You actually, it's not that you, I don't think that probably the word constraint is not a good word in this context. It's what you might call to be, you excavate new relationships between intuitive components you couldn't imagine if you did not have a system of illustration, which of course from
a medieval perspective is called the system of demonstration, the system of Euclidean proof, which we said that's a mistranslation, the system of illustration, and that is simply very idea of what diagrammatic construction or diagrammatic reasoning is, illustrating new relationships between your existing intuitive components. Now of course I will elaborate exactly what kind of relationship we are talking about when we are referring to Euclid's elements.
Another interesting thing is, you know, the word apodexis in Greek, which is translated as proof, has in fact much broader meanings. An example is the first sentence of Herodotus' history, Which essentially begins, this is the apodexis of the inquiry of Herodotus of Holocarnassus. And apodexis means showing forth, manifesting, illustrating, rather than proof. Yes. And, you know, it's another of those words like logos, which has 50 meanings in Greek,
and is just philosophically narrowed down to mean one thing. Yes, absolutely. And, you know, I think that the example that Beth gives of the equilateral triangle, which is not there implicit in the straight line, as, for example, the logical relations are in a Venn diagram, but comes out, manifests itself, shows forth when you apply the appropriate construction procedures and you're kind of, you know, in an almost Spinoza-ristic fashion, showing what a body can do, which is not, we don't even know everything that can come out of these lines. Yes, yes, I mean, that is exactly what Kant would call a productive imagination, namely, bringing forth new unities of intuition.
And these new unities of intuition from a contemporary perspective, what you might call possible relationships between your local spatial invariances. Because what are really point, line, surface? These are invariances, spatial invariances. how these invariances can be illustrated differently that's the whole point of Euclidean system and from a Kantian perspective if you think that the function of the manifold of intuition is organization of local invariances so you can give a representation of something, of an item in the world through time, you
can say that the function of productive imagination is like a simulation of engine of how these local invariances can be put together differently. Namely, how can you create or generate new unities of intuition? On this note, I was kind of curious about where exactly you can draw the line between deduction in Euclid's system system where what you're deducing is necessary versus like a sort of like a
hermeneutic interpretation you know like like you know for example of like like like one of Lacan's diagrams or something like that you know where where where there's sort of not a, you know, not a, or just interpretation of a work of art. Uh-huh. So like you can create new unities of apperception that are sort of purely aesthetic, and, but these are, like there's a right answer, you know, like so, like there's, there's only, there's each thing you discover in the system is specified and is necessary and I was curious about how did all the line
I think I think okay two things one is that even though Euclidean axioms are not logical axioms, nevertheless you assign, you take them to be true in your system. And that's really the function of axiomatization. You take them to be true in your system. Now given, but of course their trueness is not logical, it's intuitive. Simply it's a perceptual posit, it's an intuitive posit. So if you take them through, namely introduce them as the axiomatic elements of your system,
then the way that it amounts, this Euclidean elements amounts to a deductive system, even though it is not logical system, it has something to do precisely that... So first, you have defined some intuitive components to be true in your system. Now given this, you have extracted some additional minimal rules for constructing them. And you might say that these are, you know, like the common definitions and rudimentary postulates in Euclid's elements. You can compare these with minimum procedure rules of classical logic, of natural deduction
or any deductive system. Again, these are not really logical in the pure deductive logical sense, but nevertheless, they are true by virtue of them being derived from intuitive components that you have taken to be true in your system, namely you have axiomatized them. So once you have these, you have the basic components, namely the axioms and minimum illustration rules, then what makes this deductive is that, at least in this sense rather than
the pure logical deduction, is that conclusions or new constructions cannot by any means violate the criteria of trueness of your axioms or be constructed outside of your procedural rules, namely rudimentary common sense definitions and postulates, the minimum rules for illustration. And from this perspective it is very much in tune to the idea of deductive logic in the sense that deductive logic is like this. If your premises are true and given the fact
that you have used procedural laws of logic, of classical logic, then your conclusions necessarily be true. So you have, I mean, you already see that the idea of deduction from the logical sense has something to do with two things. One, abiding by the procedural rules of your system, of your logical system, given the fact that your premises are true.
It creates a new connection inside your system, a deductive connection. And that's the logical necessity of the system, that your conclusions are necessarily true given that your premises are true and you have constructed a proof from your premises toward your conclusion using your procedural rules of your system. You have exactly, even though it is not logical, you have exactly the same sort of connections inside Euclid's elements. Even though the axioms are not logical and are intuitive, nevertheless you have assigned
true values to them within your system. So you have axioms. Now you have kind of an equivalent of deductive systems. Your procedural rules again are not logical, they are what you might call to be illustrative diagrammatic rules. But nevertheless precisely by virtue that you have assigned truth value to your axioms these rules also can be interpreted logically. Once you have procedural rules and the truth values of your rudimentary axioms, then by virtue of that, by virtue of abiding within this system, all of your constructions, your
proofs or your illustrations should enjoy some, what you might call, diagrammatic necessity, some, what you might call, diagrammatic necessity rather than logical necessity that they ought to be true. So it's this necessity of truth between your conclusions and premises that determines a deductive system. And the connection between the two, as I said, should be laid out in terms of the truth of the premises and restrict compliance to some procedural rules that are absolutely strictly
defined within your system. You should only use these kinds of rules if you are within this deductive system. So from this perspective, even though Euclid's elements is not a purely deductive system, precisely because it's not logical, it functions like an equivalent of deductive system by virtue of the truth values that you have assigned both to your intuitive axioms and to your rules of construction or illustration. So yes, so this is I think from this perspective we can really say that Euclid's elements functions
like a deductive system. Okay, thanks. Yeah, so the difference is, the difference between that and the cultural truth is that basically that classical logic is still in there. You're obeying it during the procedure, and it's translatable into a logical truth. I wouldn't say that classical logic is there. I would say that's... It's hard to say that elements really abide by classical logic or logic at all. I think it's really this equivalency that needs to be taken into account, that Euclid's
develops an equivalence a representational or diagrammatic equivalence of logic that it functions the same without being reducible to classical logic because precisely once we reduce Euclidean elements classical logic then it completely distorts the meaning of your constructions. But yes, I mean from the perspective of how it functions, yes, you can see that it functions equivalent to any form of natural deduction.
Any question? Oh, another thing that I mentioned to say, so yes, the word that is used originally in Euclid's elements for illustration is quite a very, you know, has so many different meanings. And the best Latin translation of this, as I mentioned, is a word called the monstratum. So it's not the monstratum, monstratum. Monstratum is again very different and has that plastic meaning in Latin.
Meaning representing, meaning illustration, meaning to make manifest, to make explicit. And essentially what you get in Euclid's system is to make manifest, to make explicit hidden unities or hidden relationships between your intuitive components. And you see that to make, to rendering these relationships explicit amounts to a kind of
de-stabilization of intuition. As I mentioned, like so many of the proofs or so-called illustrations in Euclid's elements from a graphical perspective you see that they are absolutely counterintuitive. How can this be true? How can, for example, two triangles with the same base and the same height under any condition can have equal areas or any, you know, basically complex mode of construction. You see that it is absolutely the case that making manifest hidden relationships between
basic intuitive components also demarcates, in a Kantian sense, new unities of intuition, the sense that these new unities can be said to be counter-intuitive from the perspective of your previous intuitions. And that's what Kant assigns to the function of productive imagination, the de-stabilizing dynamic loop between the function of understanding and the function of intuition. I just have a basic question. Oh, I'm sorry, you can go first because you had posted this
up in the side part. It's not a question. I was going to lead into another topic, which I might not want to go in at the moment. But I'm fascinated by, in Macbeth's essay, by the idea that the thinking is not about the diagram but it takes place diagramma by means of or through the diagram or in the diagram. The notion that intuition as it's sometimes presented in mathematics textbooks is not a sort of tool for the logically impaired to help them sort of get closer to to what is essentially a logical proof in the end, but is in fact part of the very essence of mathematics
that you do it with a pencil in hand, that that's what doing it involves. And so my natural tendency is to try to project ahead and figure out how, it's kind of obvious how that works in Euclid's Elements, you know, that there are all these virtual figures that are not sort of implicit in the line that can be revealed by this productive imagination. Yes. But I'm trying to figure out a way, since I find this much more aesthetically pleasing than the notion of starting out from the propositional calculus
in the first chapter of a math book, and then sort of drudging with drudgery just sort of eking out one step after another. It doesn't have what I feel are the leaps of mathematics that I enjoy when I go through it. So, and I do understand that this stuff happened only at the beginning of the 20th century, so mathematical discovery was going on for 19 centuries before that. but I'm still trying to get over the hump of saying alright you can do this for planes and lines and possibly even three dimension but when you go up into n dimensional manifolds you leave the realm of intuition and go to where only some other form of reason can take you
if you're following all this stuff through space time and yet at the same time I want to recuperate that too for some sort of diagrammatic conception of mathematics. And I don't know whether it's the fact that you still have your pencil in hand and you're doodling or you're doing things that are part of the process of thought as opposed to this strictly logical presentation which mathematics is reduced to afterwards. So I'm interested to see how we're going to extend what we find clearer in Euclid into other areas of mathematics. Well, I would say that this is exactly what essentially you get mathematics of 21st century
developed out of category theory, out of shift logics, out of topos theory, which they try to revive the geometric components or diagrammatic components of early mathematics that bring basically create a kind of a coherent unified integration or synthesis between what you might call to be its intuitive constructive components and extremely abstract formal potencies And this is exactly, really, I genuinely think that is really the fundamental gesture of category
theory. And then you see that how category theory, precisely because it has its own de-stabilized dynamics between the geometric components and formal components, you get again new forms of intuitions within this higher space. They are no longer like have any kind of obvious connection with the kind of intuitive components that you get in Euclidean system, but it's just happening in a different mathematical space, a higher forms of intuition that is probably even inaccessible from the, what you might call to be the primitive realm of intuition that you get into in the Euclidean
system that is you know it's obvious that this primitive rudimentary realm has something to do with perceptual mechanisms whereas now mathematics has been liberated from the limits of perceptual mechanisms no i i can see that that makes sense with category theory and and uh um the attempt to take something like homeomorphism and find a gestural or even graphical, a higher graphical. You know, I mean, I'm also thinking of topology where, you know, there are these books that start off intuitively to try and introduce notions of connectedness and neighborhoods
in topology, but then it moves into some sort of point set version of topology where the intuitive is left behind. But I guess there's another level similar to category theory where one can reintroduce the gestural into it. Yes, yes, yes, yes. I mean, again, great, great, you know, kind of a study of these gestural components of like tapas theory or higher categories is you know that's a classic book by Gorino Mazzola the tapas of music particularly the first four chapters of it okay should we should we have a very quick break and then come back and start to
talk about as I mentioned there are these hidden relationships that the Euclidean construction make explicit now we are going to look at exactly what kind of relationships these things are and how they can afford these kinds of monstrous literally monstrous constructions so let's have a very quick break and then come back and continue our topic 10 minutes sure super see you then see you
Thank you. So, before starting to talk about the exact kind of relationships that Euclid's elements tries to render explicit, I think it would be helpful to also make this philosophical
commentary on the origins of Euclidean thought. I mean it is obviously written in the context of philosophical thinking that is dominant in Greece. you can see that if you read one of the greatest commentaries I think written on Euclid's elements is by Proclus, the no-Platonist philosopher. And Proclus' commentary is precisely about
situating Euclid's elements within the philosophical context of that time and the kind of topics that are prevalent within this philosophical context. And these philosophical problems mostly pertain to what you might call later by Plato outlined in terms of methexis participation. How particularities participate in universalities and what kind of universalities Euclid systems
try to outline or represent. So within the Greek philosophical universe you get three forms of universality, what you might call detached universalities, universalities for and in themselves. You can think of these as transcendental universalities. Then you get mediating universalities. What are these mediating universalities? You might think of them in terms of broader particularities or possibilities that come
in between transcendental universalities and particularities. Then also you get the third level, the most inferior level of universalities. These are called coming late or coming later universalities. This is exactly the translation of the original phrase by Proclos, coming late universalities. What are these kinds of universalities? These are universalities that you can construct from your particularities. So you get essentially kind of like a very conceptual problem that you use, for example,
in advanced mathematics like Schiff's theory. is shift theory is that you have some local places, local topoi or particularities. Given these particularities, how can you synthesize them and recreate global construct, namely universalities or generic concepts, generalities that are fundamentally transcend these particularities. So first you need to come up with a mode of synthesis, usually topological or Schiff's theoretic, of how to glue these particularities together to construct them and come up with
more generic particularities. You might say these are in Proclusian term can be said to be coming late universalities. In these coming late universalities, you can think of how to broaden the scope of these universalities that you have, or generalities that you have managed to construct by gluing your concrete particularities together, coming to a different elevated level of universality or generality. These are mediating universalities. Once you have these mediating universalities, then you need to come up with, again, a different
mode of synthesis in order to get into what you might call the prior universalities, the true genericities, the true universalities, universalities that are in and for themselves. And those universalities are the ones that can never be reduced to the particularities. So it's the whole idea that the dialectics between universality, particularity, genericity and specificity is about different modes of integration, different levels of synthesis, with the understanding that the universal, the true universal, the transcendental universal,
never be exhausted by its particular instantiations. Given the fact that intuition from a Kantian perspective starts with singular experiences, what you might call particularities, and these are what really the manifold of intuition is about. Given the fact that the system starts from this, then the construction can be said to be about constructing different levels of integration or synthesis that allow you to reach to the level of transcendental universal, from particularities, parochial, rudimentary particularities of experience and intuition
toward transcendental generalities. Which again, you see that this is quite a very straightforward Platonic doctrine. This is the doctrine of forms. The doctrine of forms starts with the idea that you posit the reality of transcendental forms, transcendental universalities, the ideals. which are not given, you posit their reality, you speculate that their reality. Now given the fact that we start with rudimentary forms, what you might call, what Plato might call Iconos.
What are the Iconos? It's the realm of the shadows in allegory of cave or the analogy of the divided line. The realm of Ikonos, you might say, is something like a sensory organization of items in the world. Now the mode of construction in Platonic Doctrine of Form involves moving from the realm of iconos, the realm of shadows, the realm of images and images about particularities in
Plato, moving to the realm of ideals, namely transcendental forms. And of course, the gap between the two requires to posit new realities and these realities can be interpreted as modes of synthesis, modes of integration. Every time that you reach the limits of that particular synthesis you have to posit a new reality, a new mode of integration. And positing this new mode of integration, you can progressively but also procedurally construct from these rudimentary particularities, which you can say then the building blocks
of forms to the realm of transcendental forms or true universalities. Who of you know anything about the analogy of the divided line? I think it's absolutely one of the most sophisticated accounts of integration and synthesis, if If not in the entire history of philosophy, at least definitely in the history of ancient philosophy. It's a diagram, a very basic diagram, that Plato uses this in order to define three things.
The system of forms, the doctrine of forms, how particularities participate, the word methexis in universalities and how universalities afford particularities so the doctrine of which is the doctrine of forms the doctrine of the good the form of forms and also the doctrine of intelligence so for Plato these are tantamount one another forms the good and the nature of intelligence It starts from a very, you know, almost like cryptic diagram.
It says take a straight line, divide it to two, equal parts. So imagine you have a line, create a segment at the center, a mark at the center that divides the line to two equal segments. Then he says from the center mark, again create two more equal segments. So you essentially get four segments. The first segment is the realm of iconos, the realm of shadows in allegory of cave.
And the whole thing is that in Republic, analogy of the divided line comes right before allegory of cave. It's a diagrammatic representation of the ascension of intelligence or enlightenment or what you might call to be escape from the cave. So the first is the realm of the iconos or images. What is this? Is what you might call and Plato talks about this is the realm of sensory impressions. Sensory impressions that give us these snapshots, confuse snapshots of items in the world.
Then the second segment is the realm of objects, things existing in nature. The third realm, the third segment is a realm of analytical ideologies, particularly logics and mathematics, what you might call in today's contemporary term theoretical systems or theoretical entities. The third realm is the realm of forms or the realm of idealities, dialectical thinking, justice, beautiful, so on and so forth. The whole thing is that for Plato, the good is not in the fourth segment. The good is an ideal. It's not in the fourth segment that
pertains to idealities. The good is the entire diagram. That's why good in Plato is called the form of forms, a mode of integration of all these segments, sensory impressions, intelligibilities pertaining to the nature of objects in nature, theoretical constructs like logic and mathematics, and social idealities. Now the really interesting thing is that at each segment, so basically, sorry for this
digression into, because I want to make this, to situate Euclid's elements as in fact a philosophical construct rather than just purely mathematical system. Plato's allegory of the divided line starts with the basic gesture of philosophy, that there is no such a thing as a given form, that there is no such a thing as a given reality, in fact. Philosophy starts from this assumption that there is nothing given to us in advance. So, each segment, even the most rudimentary one, level of sensory impression or iconos
or images or shadows, is not something that you can think of it as a given thing. These are all the realities posited by what you might call to be the soul, the craftsman. So as I said in analogy of the device line, sorry? What did you say after soul? Soul or the craftsman. Oh, craftsman, alright, thank you. The craftsman, yes. I mean, this is, you get in Timaos or Theatetus and some of the later dialogues, even in Republic, one of the most fundamental concepts, the soul.
The soul you can think of it as mind, as intelligence. And Plato always switches this with another word, craftsman. So it's the nature of a craft, crafting forms, crafting the mind. Mind is basically what you might call to be always preoccupied with the craft of itself and the craft of itself is the craft of the forms. So it starts, which is the nature of intelligence, how Plato understands it. So this craft starts with positing new realities, realities that are not given to us in advance.
At the first level, sensory impressions. Sensory impressions, you might say that sensory impressions actually exist, but according to Plato, no. These are posited by the mind. A first level of reality that is posited by the mind in order to craft itself and to craft the world according to itself. And what is really this reality? This is the reality of what in Kantian terms you might say the organization of sensory impressions. There is no perception without organization of sensory impressions.
in Selaresian terms you might say the spatiotemporal relations between sensory impressions, because sensory impressions are just a confused flux. These particularities are given to us in a very confused manner. The very fact that we see them as sensory impressions, as sens datum, precisely we have given them an organ given an organization to them we have ordered them at least from you know of course sellers would say that it's not the mind that orders them but the cause sufficient causal structure you can think of it as nervous system that orders them in a specific spatio-temporal way so sensory impressions can be said to have certain orderliness
about them, a minimal spatiotemporal orderliness. Then the second level, mind, so every time that it's, you can see that what Plato calls the mind or intelligence, each level of its ascension to the realm of forms, to the realm of it being the intellect, the pure form, requires a leap. And these leaps are being made possible by positing a new reality. So the second segment, which is the realm of objects or things in nature, again requires positing a new reality by the mind, a reality that doesn't exist in advance.
And what is this reality? It's the reality of object. So it's a very Kantian scenario here. Objects don't exist in nature. Objects are being constructed. This is a very Kantian thesis, which for Plato is idea of particular ones or oneness, a oneness that is not defined in terms of universality, but in terms of its particularity. So what is the role of this oneness? Imagine that you have this piece of paper that you can say that it's like a,
it's a bundle of sensory impressions that have been minimally organized spatiotemporally. Now, if you posit something called a particular object, object, a oneness, a reality that pertains to particular, unique particularities, sui generis particularities, oneness, objecthood. This idea of objecthood allows you to further synthesize and structure these sensory impressions. For example, now I can talk about that if out that if these, for example, I have discontinuous experience of sense impression, this being
red or this being a square, being discontinued, I would say that this doesn't pertain to the same object, that it pertains to another object. So this even minimal positing of reality, just one particularity, that these sensory impressions should pertain to particularities, that creates the idea of object. Again, in the very Kantian sense, it's the idea that objects, once being constructed, they allow us to move from the level of sensory impressions to the level of the intuiteds, rudimentary manifold of intuitions or singular representations of objects, the so-called
image in Kant. This image shouldn't be confused with what Plato calls image. Image for Plato pertains to the first segment, namely the realm of iconos. So the singular representation allowed me to talk about sensory impressions pertaining to particularities. The experience of redness being continued in time and space allowed me to speculate or talk about a piece of paper, a particularity, a particularity of experience.
So this is the level of object construction that you get in play, which is entirely made possible by positing a new reality, the reality of particular ones or objects. Once you have this, then you require in order for you to move to the third segment, you need a new mode of synthesis, positing a new reality. This new reality is the reality of theoretical entities, or in what Plato attributes to logics and mathematics, or analytical idealities. At this level, so I have particular representations of objects in the world, but how can I make,
turn them into idea of general objects, namely mathematical or logical categories or concepts? That require me to move to theoretical constructs. You can think of this as, for example, the concept of an electron. The concept of an electron is an abstract theoretical entity. Its reality is theoretical. It does not pertain with something that is being experienced. But nevertheless, it represents a form of generality that allows me to synthesize or integrate different particularities of experience or observations pertaining to empirical particularities.
So the first level you get sensory integration, the most parochial forms of particularities. The second one, you get a more robust particularity, empirical particularities pertaining to objects of experience. Then the third level, which is the level of theoretical constructs, logics and mathematics, you have entered the realm of forms as generalities. And at this most basic level of generalities are generalities of theory that allow you to synthesize or integrate empirical particularities, modes of observation or observable data.
Again, like concepts of science that integrate empirical particularities. Then when you have these empirical particularities then you see that you might have in fact different theoretical generalities. You can think of this as competing theories, different theoretical concepts that are in competition with one another, exactly like today's science, different competing theories with their own posited generalities, concepts, logical mathematical concepts, then how can you integrate them to move to the fourth segment?
That requires, you know, within, for example, scientific community renegotiation, going back and forth, not only between theoretical generalities, theoretical entities, but also also seeing how they are connected to empirical particularities and how empirical particularities fall under these theoretical generalities, resolving the incompatibilities between these theoretical generalities in accordance with the kind of empirical particularities that fall under them. You might say this is exactly the pragmatic dimension of a scientific community, between
different mathematicians, different scientists, so on and so forth. They try to resolve the incompatibilities between competing scientific theories, moving to a social dimension, the pragmatic aspect. And this pragmatic aspect is the same. can be said once you take this interactive pragmatic aspect is that you enter the fourth segment, the realm of idealities, where you not only have forms pertaining to empirical particularities, but forms of social dimension, justice, equality, beauty, so on and so forth.
So people usually associate mathematics, I mean, there is such a, you know, bias against Plato in reading of Plato, which of course comes from later commentaries rather than original dialogues is that they usually associate mathematics in platonic system to some frozen realm of forms that is like in the sky and has nothing to do with our worldly experience. But the thing is that if you notice in this system the segment that pertains to logics mathematics comes between two other segments, the one before it and the one
after it. And precisely because the analogy of the divided line is the idea of ratios. Each segment is defined as a ratio. This is the idea of reason and the whole line is called the path of reason in Platonic system. Mathematical system, The third segment comes as a ratio that is between empirical particularities, where you might call the worldly experience of things in the world, and social idealities, which pertains to forms of social interaction where reason basically inhabits.
So it is far from this idea of mathematics being, mathematical forms being dissociated from society and from empirical experience. So as you see, each segment requires its own mode of integration, which Plato calls a reality. And through this, you move from elementary particularities to more robust particularities, what you might call singular representations, to rudimentary generalities or rudimentary forms and then to the most complex forms, true genericities.
And the whole idea, the whole line, is what Plato calls the form of forms, the transcendental ideality. This is really the function of transcendental in Kant. As you see, leaping from one segment to another requires a reality being posited that doesn't exist. It's a transcendental reality, even at its most basic segment, the segment of sensory impressions. These are what you might call to be transcendental constraints that you introduce into the system as a mode of organizing what you have and once you have this mode of organization then
you can construct the system, move toward a new segment. But once it reaches its limit then you need to posit a new mode of integration, a new reality, transcendental reality and then how this is how you leap from one segment to another. And this is really the whole idea that mind or intelligence makes these transcendental leaps. This is the gymnastics of a spirit in Hegel. And each one of these leaps becomes greater and greater as you move forward. Requires a more sophisticated form of integration. So Euclidean system is precisely like this.
It are, even though it starts from particularities, at least you can see in the rudimentary definitions, the most basic form are the rudimentary definitions, the so-called axioms of Euclid's system, point, line, and stuff. are simply what you might call to be a spatiotemporally ordered sensory datum, perceptual mechanisms. And once you have these, you get geometrical shapes. And geometrical shapes, you can think of them as objects in platonic system, singular representations
of objects, particular triangles. But of course particular triangles couldn't be constructed, couldn't you in fact posit their reality if you did not have general idea of a triangle, namely theoretical entities or mathematical forms that pertain to the form of a triangle. Once you have that, then you are capable of constructing, for example, starting to render explicit the relations that hold between generic idea of a triangle, generic idea of a circle,
generic idea of a line, and move toward further construction. But of course, this idea of constructability moving from generic forms, which in, you know, Platonic system is the realm of theoretical entities, the realm of analytical idealities, couldn't be done if you couldn't go back and forth between empirical particularities, singular and theoretical forms, the idea of generic triangles. So what you need to move from third segments to fourth segments requires going back and forth between the second segment and the third segment.
What is the second segment? You might call particular triangles and the third segment, generic triangle, the idea of triangle. It's a part-whole relationship. Without this part-whole relationship, which can be elaborated in terms of particular triangles, and the generic idea of a triangle, you cannot move from the idea of a generic triangle to new forms of construction, namely elaborating the relation between, for example, a triangle idea of a triangle and a generic idea of a circle.
Or for example, a line and a radius of a circle. Any question? I sort of had a question, but it might be a detour longer than we want to take, which is just what's motivating this movement from the Iconos towards form, which is the same question of what motivates the dialectic in Hegel. What you might call to be
Plato's idea that the good, with the good being this whole line rather than just one segment of it, is the idea that Plato calls the self-interest of thought. What is really the self-interest of thought? You get in Hegel it's really the doctrine of the transcendental. That thought precisely starts with a consequential even though rudimentary gesture. It realizes itself outside of nature.
It posits a reality for itself that is not the reality that is given to it, but it itself transcendentally has posited. you might call to be the self-realization of thinking, that thought starts with this self-realization, which of course in Plato you get a teleological account of the self-realization, that there is some telus in thinking that allows this to be possible, whereas in Hegel starts with the social, recognitive aspects of thinking. The very fact that we have experience is precisely because we have concepts. We couldn't have concepts unless we had social interaction to language that allows us to
to come up with judgments, judgments that elaborate the differences between individual experiences. Once we have these what you might call ur-judgments or proto-judgments, we are capable of demarcating of our individual experiences. In fact, to attribute these experiences to ourselves, in Kant you get that there is no consciousness without self-consciousness, there is no self-consciousness without judgments, or there is no experience without categories of understanding.
How does, I mean, you see, so in Kant you start with sensibility, namely receptivity to the impingement of objects on senses. Then you get imagination, namely construction of singular representations of items in the environment. Then you get understandings, which are categories of true concepts, and you get eschematizations. And then at the top level you get reason. Now the whole thing is that for Kant, in Critique of Pure Reason, quite like Plato and quite like Hegel, is that the very idea that you have experienced is not because you have sensory
impressions, but because you have higher domains, like categories, that you have applied to your sensory impressions that are unified or organized into a manifold of intuitions. And those manifold of intuitions have been again integrated into singular representations of objects, namely images. It always starts from the top, from what you might call, you know, the constitution of thinking, the self-realization of thinking, which of course according to these three different philosophers the self-realization can be interpreted quite differently. In Plato it's quite teleological.
It's the transcendental reality of intelligence. That intelligence starts with in fact a transcendental self-recognition, then Kant starts with understanding the self-realization. In Hegel it starts with reason, the social dimension of reason, which is the experience. That there is no cognition without recognition. The recognitive aspects between agents of a community, which Brandom, you know, elaborates in terms of the semantic-pragmatic interface of sociolinguistic practices.
That how pragmatics generate semantics and how the semantics, if you didn't have this, you wouldn't have any sort of experience of items in the world. Nor could you attribute these experiences to yourself, which is the transcendental unity of our perception. Thank you just to give a time check we're approaching the three hour point we can go further if we'd like yeah sure let's go for 15 minutes just this idea of relationship and I talk about it more next session and we are also start to look into a specific theorems and positive leads and we
try to do some actual math. So I mentioned that Euclid's system needs to be understood within this idea of how particularities participates in universalities and how transcendental universalities or generalities posit their own particularities and the dialectic between them can be said to be the realm of construction and the realm of construction adequately understood as the realm of integrating different levels
of particularities together to create more generalities, I mean basic generalities that you to expand, you know, what you might call the scope of synthesizing your particularities, your axioms. So as I mentioned, the third segment in Plato's idea of the divided line pertains to basic forms or basic generalities. You can think of it as the idea of generic triangle. In order for you to be capable of doing something to further construct this idea of generic
triangle and for example elaborate its relationship, for example, with a generic circle, you need to go back and forth between what it means for something to be a generic triangle and a particular triangle between its particular instances and generic forms. The idea of circle as such and a particular circle. The idea of a generic triangle and a particular triangle. Now then that's where you get what you might call manifestation, manifestation of how these
particular images or particular representations stand with regard to different generic forms. For example, how can a line ever be in relation to a circle, a generic circle and a generic triangle? How can, which of course once you make this manifest, how a line can ever stand in relation to a generic circle and a generic triangle once you make this relation manifest, then you see that there is a relationship between the side of a triangle and the radius of a
We have the idea... Oops. Well, an idea of circle. When we are talking about idea, we mean it in a pure platonic sense, idos form. Namely, we are talking about generic triangle and generic circle. This is the construction, the mode of construction of an equilateral triangle in Euclid's elements.
And how can you make an equilateral triangle with a circle? So you get this, you get the line. You use this, you interpret this as the radius of a circle. You make first a circle. Oops. Then you put your compass here and you make another one. Then connect this and you connect this.
You see a new relationship has been manifested, rendered explicit. A line within this scope of construction can be both the radius of a circle and the side of a triangle. You have created, by way of this construction, a new relationship between a triangle and a circle. A relationship that we couldn't even see if we were just having access to simply the
idea of a triangle or the idea of a circle dissociated from one another. But precisely the dialectic between particularity and generality, part and whole, part of a unity and unity as a whole, allow us to render explicit a new relationship where side of a triangle can be seen as a radius of a circle and a radius of a circle can be seen as a side of an equilateral triangle. This is what Danielle Macbeth talks about the idea that these geometrical shapes within
the Euclidean system are not like photographs or pictures. They are what you might call to be platonic forms and platonic forms always come in part-whole relationships. You can think of these part-hold relationships in terms of the dialectic between particularities and generalities, between unities and their components, their constructive elements.
The circle is not image of an actual circle in the world. The circle is only circle by virtue of its part-whole relationship. It has implicit relationships between parts, between its particular components and its unified components between parts and whole. And by virtue of it, it becomes a circle, exactly like Platonic forms. The forms are not that sloppy interpretation of idus or ideas being like these frozen things
in the sky, but they only become forms by virtue of particularities that comes before them and generalities that come after them in the analogy of the divided line. The same thing about triangle. Triangle is only a triangle not because of its shapes but precisely because of a specific relations between its part, between its unified construct, its whole and its parts. And of course in Euclidean elements for a triangle, these specific relationships can be elaborated in different ways, between for example how angles stand in relation to the
sides, how angles stand in relation to one another, you know, the idea of modes of congruence, of equivalency. These modes of congruence are exactly these specific part-hole relationships without which there is no such a thing as a triangle in Euclid's system. And once you You see these geometrical shapes no longer as iconic representations but as dialectics between unities and parts, between wholes and their specific components.
You are capable of, by way of construction, elaborating new relationships between parts of one form and parts of another form. As manifested in this basic construction, the part of a triangle, a side of it, can now be said to be a manifest or illustrated relationship with a part of a circle, namely its radius. Question.
I have one brief question just relating to the concept of gesture. To me it seems like gesture serves for the most part, but not exclusively, a pedagogical component. Yes. Can you elaborate about this a little bit? like gesture leading to the dialectical destabilization of the intuitive manifold creating the capacity to construct new cognitive affordances yes yes absolutely yes which you can
see this as modes of individuation. I mean the gestures for example in Euclid system you can see for example the idea of a triangle can be individuated via different gestures individuated via different gestures by way of, for example, the relations between angles and sides, sides and sides, and angles and angles. So for each idus, for each generic geometrical form, system, there are different modes of individuation.
And depending on what kind of gesture you choose to employ, you can illustrate or make explicit new forms of relationships, new ways of producing or individuating a form. In the Platonic system, you might say that if we completely got rid of the idea of a gesture and illustration, in Plato's analogy of the divine line, these are pathways of reason or hedon. So to elaborate on that, I know that Theo asked a really good question.
that he asked how are the intuitive axioms, how are they introduced? And it seems to me that from a pedagogical standpoint, you would have to deliberately conform to the modes of intuition that have already been established. in order to draw new components, form new axioms, and then those axioms then become subsequent building blocks, modes of intuition. Yes, I mean, this is one of the things that still no one knows how these definitions or axioms in Euclid's elements have been posited.
It seems that they are not coming with a specific gesture, but you see that, for example, in pre-Socratic philosophy and the definitions in Euclid systems, different definitions correspond to what you might call representations or equivalences of these culturally dominant or philosophical culturally dominant concepts, gestural concepts which are of course the products of for example
you know, sociocultural, you know, not, sociocultural, not gestures, but are embodied concepts. Obviously, Euclid's idea of a line, Euclidean line, is not something that Euclide invents and puts it as a definition. The concept of the line is there. And simply, Euclid tries to minimize it to its basic characteristics.
But this concept of a line, again, you see that in pre-Socratic philosophy, is something that doesn't pertain to mathematics, but it's really what you might call to be the product of the entire culture. How we were talking about that some of these rudimentary mathematical concepts are the products of natural and cultural evolution. So I don't think that we can really point to a specific phase, at least in Euclidean axioms and say that well it corresponds to this specific gesture and Euclid had this specific gesture in mind. Well we can talk about overall the gestural landscapes of these axioms within the evolution
of Greek culture, which of course again we need to talk about if we are, and there are a couple of books on this, for example the concept of a line, the concept of a point, that they represent for example Greek rituals, there are concepts that evolve from Greek rituals from, which of course and these rituals are you know lived experiences in a Longoian sense, phenomenological sense, but also with these purely embodied gestures they have also philosophical residues. For example, the concept of a point has a philosophical connotation in Greek culture.
So in Euclid elements it's not just they represent one, for example, form of gesture that has been posited at a certain point. You see that they are consolidation of different gestures, natural, cultural, and also philosophical concepts that Euclides tries to condense them or encapsulates them in a specific definition, that encapsulates all of these things. And as I said, it would be really a shame to interpret Euclides' elements as a stand-alone mathematical book, you need to always think of Euclid's elements precisely in its cultural context, particularly its philosophical context. It is really a
philosophical book before even being a book of mathematics. So Phil asks for the book recommendations on the side. I will get, let me just write it for you. I forgot the titles. I will look into it next week. Fortunately, I will be back in Connecticut so I can recover those suggested titles from my library and I will definitely find these books. But there are, one of them is a collection, I remember it's an Oxford collection within
like 1970s, commentaries on the socio-cultural and philosophical context of Euclid's elements. I can't remember the name of the editors or the title but I will find it differently. Iconos is with K, I think. So I thought that the line was sort of ascending but the way you drew the diagram was different.
Yes, well it is ascending you see. Oops. So yes, you get these. So you have this and this. You have this. Then again this and this. And as you see, these leaps can be interpreted as ratios of segments to other segments. And in fact Plato really illustrates, talks about the ratio of these segments.
I forgot the exact ratios that he puts. The idea is that this idea of a ratio in the divided line is precisely because that you can see that forms both in their basic manifestation, namely analytical idealities, and in their most sophisticated manifestation, namely ideals, can be seen as constructions that can only be done by way of how you integrate the previous segments.
And this integration of the previous segments can be seen as ratios of how the components of that segment stands with relation to other ratios. And also you see that the advent of, or the ascension of intelligence, the ratios become greater and greater, the leaps become larger and larger. Quick question, did you finally find the Google Drive password?
No I haven't. I mean that unfortunately I have it in home next week. I will be in Connecticut and I will put all of the books. in the Google Drive also I would say that that would be so tragic if you like lost the password to your oh no no no I'm sure that I have it I completely know where my password is I have it I have written it down also I one of the reasons that I didn't didn't manage to give you the uh Ono Harari's essay was because it's it's part of that Google drive and I can't find it online because I remember that actually a friend of mine
send it to me so I'm just going to give a time check we're at 340 right now we've got quite a bit of your extra time haha okay if no one has a question we can I think conclude today and you can in the meantime think about the that homework that I gave you you know discussing under which conditions the the Egyptian algorithm stops to work and also in the meantime you can read Proclus commentary on Euclid's elements great and is that
in the you should be able to find it online okay Christian posted it already on the sidebar oh super great thanks Christian oh yeah I always try to get them in the sidebar if I can thank you so much um everybody take care yeah have a good day thank you everyone thanks for a great day bye bye okay