Zeno's paradoxes, identity of whole with parts / Jean-Louis Hudry: Boundaries, parts & wholes in Pierce

0:00 The motion pass is a methodically from the representation of the complete motion as the union of that former union, and the set only containing the final wheel number of the final point of the motion of the complete motion. Thus, performing all these motion paths brings one just as far as performing the complete motion. Just reaching every point before the final point, then that clearly is something which is feasible, be it in due order. Suffices for that. In order to traverse the required distance, then there's no need to demand with C-note that there be a final step. Thus applying some rather basic mathematics would be enough to show that there are no paradoxes to be found here. Now the first point I want to make today is that this mathematical approach is more or less irrelevant as the force of these paradoxes does not derive some kind of mathematical mistake, but from a certain conception of a whole within class. that there is something non-mathematical some non-mathematical perception at work in these paradoxes appears most clearly in the case of the argument to be that it's not possible to complete a motion divided in a synonym or equal lack of a final step when we imagine ourselves performing these motion parts one by one we are aware at least on the reflection that it is possible to perform every single one of these motion parts. And we may get ever closer to the finish. This is the awareness which is codified in the mathematical point, that there is no magical difference. Yet despite this awareness, we immediately think, and I'm calling upon intuition here, some kind of intuition, we immediately think that it's impossible to complete a motion for lack of a final step. We are aware that it is possible to perform every single motion pass, but we refuse to accept the possibility to perform all motion parts, incapable as we feel we are breaking out of the infinite series of motion parts and going beyond it to the finished points. That seems to be the picture. The intuition of the system possible can be strengthened by using the paradox a little.

2:30 For example, by going to the staccato version, which is also in the handheld. So after each motion part, there's a period of rest. So that the final point where we would have arrived at our destination is always separated that at least one theory of rest from every single cell. So it's very clear that there's no, that they've confirmed to that that there's no last part of the run. I think this intuition is even stronger on the reverse version of the same paradox, where we move through the infinite series the other way around. So we start with the small ones and get to the bigger ones. then so forth, then one sixteenth, then one eighth, one fourth, one fourth, half, and then with that there. According to this version, it is impossible to start a motion for lack of a first half. Being aware that there's a completely continuous mapping, even in all those derivatives, from points of time onto points in space, describes the motion from start to finish, and Grunbaum has established, with some help of a mathematician's friends, that there is such a matrix, such a function, does not help us in any way to dispel the thought that we cannot get moving. Now, the question is, when is this strong, a very strong intuition? The question can be answered by having a look at the latest serious attempt attempt I know of, to resuscitate Zeno's paradox, and that's by Michael Burr in a paper called The Stikathler Run. It's also mentioned on the handout. Burr wants to argue that Newtonian mechanics devices to make it dynamically impossible to perform an infinite series of motion parts in a synonym way. The reconstruction here is in mind is a stikathler reverse version of Zeno's argument. So after each, after Peer progressed at the point of departure, you start with the infinitely small ones, well, you can't do the first one, of course. So, but after each motion part, the Peering progressed for an equal Peer to the country. actually in there twice as long

5:00 Burke accepts the point made by Grunbaum that there exists a mathematical function from time to distance which describes the motion from start to finish even as the counter-reverse version in such a way that it's discontinuous in all its derivatives including so there's no discontinuity in the level of velocity and the second, there's no discontinuity on the level of acceleration. Still, we argue that it's impossible to perform such a motion on the ground that there is no force which would explain the object's ceasing to be addressed by causing it to accelerate. There's no force which lasts long enough, namely more than an instant, because every acceleration must last longer than an instant. and no force short enough because for every period of time this force might last there is already a period of rest in the run and during this period of rest there is no acceleration, so no forces at line so that's the impossible one as well. Therefore, as Burke concludes, it's impossible to get started. The point now I want to make is that Burke can only reach its conclusion by slightly altering the meaning of the terms acceleration and force. While on the mathematical representation there's always a value, to be sometimes zero assigned to these factors, and moreover these values vary without any discontinuity of the time. Berg only talks about there being acceleration and force when their values are non-zero. That is, he divides the motion into discrete motion parts interspersed with theos of rest, and then demands that there be a force explaining the object's initial seating to be addressed, If there is such a discontinuity to be explained, when it is agreed, and Burke also agrees with that, there is a continuous function describing the motion from start to finish. Thus Burke surreptitiously introduces the perception of the whole motion as consisting of a series of discrete and consecutively ordered motion parts, and that's another

7:30 important position, and nothing more. There is no individual motion part which explains the objects leaving the starting point for lack of a person of the series. So, according to Burke, there is no explanation at all. This is the crucial premise of this Thunonian-style argument. The whole of parts is nothing more than the sum of these parts. According to this conception, these parts have ontological priority. dependent for them, for its identity, as well, I think, and for all its qualities. Beyond the parts, there is nothing which may be called upon to explain some feature of the whole. Thus, an infinite series of motion parts, such as the one in physics by Zeno, can only be completed if there is one particular motion part which brings the mover to the finish, which is of course, which of course there isn't. This same conception of the whole with Kant could also be used to understand Sino's inference in the first paradox, that such an inference series is infinitely long. He seems, and this is more the historical Sino than I would take for myself, It seems to argue as follows. Because the series does not have a final part, it does not have a limit, and therefore the object which consists of the whole series does not have a limit, and therefore it's unlimited, infinitely large. Of course, according to mathematics, along the sum definition, this theory is of finite length. But Zeno does not argue in a mathematical way. Rather, he argues that if something is to be limited, it must have a limit for itself. But an infinite series of parts does not have a limit as a part. Moreover, on the conception that the whole is nothing more than the parts together, the whole consisting of the infinite series of parts cannot rely for any of its qualities, including its being limited, on something from outside the series.

10:00 It must take full care of itself, so to speak. It is this conception of a whole with parts which is ignored in a mathematical solution to this synonym argument. What this mathematical solution in all these various guides is hammers on is that underneath the discrete series of parts there is in the rather malicious words of one of his proponents called the Nasserat, a paper of 1962, an underlying continuum, that's what he says. It may be possible to discern an infinite series of discrete parts in the whole, and it may be possible to demarcate this whole from its environment, but in fact this realm of discrete entities is ephemeral, according to that picture. In reality, there is one continuum without any real discontinuities. Thus, there is one continuous mapping from time on to distance, describing the motion, even the scatal version. It's the same principle, from start to finish, and even before that and after that. And there's nothing more to the motion. Similarly, one will take this underlying continuum, from this underlying continuum, the limit to this infinite series of parts. in the case of Xena's arguments in 1 and 2a. The infinite series does not have to take care of itself in this respect, given this underlying continuum from which the limited world. Indeed, the whole concept of representing a magnitude, be it a motion or an extended object, remains of an interval on the real, especially this concomitant choice between an open, half-open, and it goes into all presupposes such an underlying continuum, namely the real number line. But ignoring the crucial premise of the synonian argument can hardly be called a solution. The second point I want to make today is that the synonian paradox, I don't know how much, five minutes or so, um, in fact the synonian paradoxes are only going to disappear if we are able to convince ourselves that these three suppositions can be rejected. And there are only two, namely this part-whole concept and the infinite divisibility of things with magnitude. In a realm of motion, I think it is possible, perhaps with some happens, but

12:30 I think it is possible to reject this for all parts principle by arguing that emotion is not a discrete whole with discrete parts, a parts that just consists of a period during which the velocity of the object is above zero, without there being any discontinuities or real boundaries involved, not on the level of velocity nor on any of its derivatives, with the periods before and after the motion. I mean, just one continuous function. Then there's just one. One may distinguish parts within this continuum, but any demarcation is equally real or rather equally arbitrary. I mean, some are more interesting than others, perhaps. By thus rejecting this final whole principle, which Sinnoh seems to adopt, he may retain the infinite difference between space and time as involved in motion. In the realm of extended objects, however, I don't think such a solution seems feasible. It is not feasible to reject in exactly the same way the discrete conception that the whole-width part is nothing more than the sum of these parts. For the very notion of an extended object is of something discrete, with real boundaries independent from its environment and other objects. Still, it might be argued, and that would be along the line of Farsi, I suppose, that such a whole, this part-whole principle, Popsino, can be rejected by maintaining that any division, including the synonym on, would only be into parts which remain continuous and one with the rest of the whole. The division is nothing more than continuous service of the inside of the whole. Yet, even if one were to grant such a move, the synonym argument can still be constructed, I think. Rather than starting with a continuous whole and demarcating parts within it, we should start with the parts and work up to the whole. Suppose a world in which there are two colors of play, blue and yellow. Given the infinite divisibility of things with magnitude, and that of play, there is nothing conceptually impossible,

15:00 I think, with there being an infinite series of layers of play, infinitely decreasing size. In the following way, we start with a layer of blue, and then we attach a layer of yellow, and so on, and this is a half, a quarter, and then we work until, yeah, somewhere, actually. Such an infinite series of discrete parts does form a discrete whole of some kind. I mean, you get some type of material block. Being delimited from its environment, say, air and earth. At the same time, this whole is nothing more than its parts put together in this way. this principle, Ceno's principal parts, a whole part being nothing more than the parts that are preserved. The real discontinuity between consecutive parts is preserved by the fact that they're different colours, and if that's not enough for the discontinuity, we may stipulate that these layers of clay, because of their colours, do not stick together and say, but are just seamlessly touching each other. Now, if we look at the cube from the layers are thinning out from here, say, we can ask the question, what colour do we see? Well, there's no final part, so we cannot answer that question. There's also no answer to the question, how this cube is going to interact by way of contact with other objects for lack of a surface of contact. For example, if we want to paint this cube on that side, what are we going to put the paint on? Now, let me finish by making a few remarks about this synonym argument. Actually, I'll add two. The first remark is that this argument is for its ingredients, closely related to an argument by Dean Simerman for the existence of atomless gunk. And as I may remind you, Simermann argues that an extended object cannot consist of points as simples. For example, if it were true, it would be possible to have open bodies and closed bodies,

17:30 bodies lacking a final point and bodies with a final point. But that's problematic, Simermann claims, because that it's impossible to make sense of the possibility of contact between open bodies or contact between closed bodies. The former will always have to remain a point apart and the latter would need to interpenetrate in order to get into touch with each other. Thus, the assumption of composition from points is simple to produce to absurdity, according to Simmerman. Now, Simmerman treats the outer skin of point symbols, in case of closed bodies of course, as independent parts which make a real difference to the whole and the bodies are wholly made up of points. Therefore, I think his notion of composition from symbols is equivalent to Zeno's principle that the whole of parts is nothing more than its parts together. However, unlike Simmerman's argument, it is impossible to reject this principle as applying in this Anonian argument for this principle, this whole path principle, fabric of the arguments. And it derives from the discreteness of each of the extended objects making up this cube. And this discreteness you want to maintain because it's derived the discreteness of our ordinary objects. My final point is that one could have tried to escape from the paradox in another way. One could, for example, argue that it's based on a presupposition that layers of clay consist of homogeneous stuff, which is constantly impenetrable and completely opaque. That's what you need in order to get the questions asked. And these presuppositions happen not to be correct, because of physics. This is taken, for example, by Smith and Parksey in an article, when they describe what happens when two discrete objects interact. No genuine contact or coincidence of band groups is possible at all, they say. Rather, a complicated story has to be told in such cases as to what happens in the case of apparent contact of two bodies.

20:00 It's a story in terms of subatomic particles whose location and whose belongingness to either one or the other of the two bodies may only be statistically specifiable. I should tell you that they're talking about kissing. On the other hand, even Smith and Farsi want to hold on to the presuppositions that extended objects consist of homogenous stuff, that being the world of common sense. This synonym argument is a conceptual puzzle, I think, which arises because of the interaction between two presuppositions which belong to our ordinary way of conceptualizing extended objects, that they consist of homogeneous stuff on the one hand and are therefore infinitely divisible, and that they are discrete entities which are independent from their environment. We should try to remain as much as possible within the boundaries of common sense in this respect. So that's the puzzle. I don't have a real solution to the puzzle. I just wanted to lay down, explain how you get such a good puzzle going and what these synonym arguments ultimately are based on. If I would be pressed for a solution, I would tend to give up the infinite divisibility presupposition in some way or another. For example, by saying you can have infinite divisibility but at different levels, so some divisions you need only little energy to bring about others, more energy, ever more energy so that the material object gets an infinitely, can get an infinite structure in this way, but that you would not have the possibility to go all the way because there's no end to this structure, as the point is of scenic paradoxes as well, so that you cannot get the paradox going, because for that you need an infinite infinite series, but that would, I would only say one question. Thank you very much.

22:30 If you were, I'll go to another room through it. Maybe we have time for one short question. I have many. Only one short, and a very short answer is possible. No, okay, you know, I agree with what you say on the part wall, and I agree with what you say on the infinite visibility as being the, how do you say, the mechanisms or the presuppositions of Seno, but I do not agree that Seno's procedure is executed through time, because it is only when you put it as executed through time that you get this infinite series, the so-called discrete series, where an underlying continuum should be supposed. If you drop the supposition of time, Zeno's procedure, that is what I'm probably not clear enough and not completely enough, but what I try to show, Zeno's procedure generates both the countable and the uncountable. Yeah, but you were supposing a semantical division, and I am not, I'm just having an in the infant series of ever decreasing terms, which is just demean and demean and... Yeah, this is also this big thing about. Yeah, and I think there's nothing in Xenos actually... Well, first of all, this is a systematic that I take, but not a historical one. So even a systematic problem is interesting enough. But even, shortly speaking, there's enough evidence, I would say, for mind reconstruction because you cannot have a sematical division if you, it doesn't go with some of the verbs in Singer's testimonial, but, well. This is also the way to move. Thank you. Our next speaker will be from Edinburgh, who will speak about boundaries, parts, and how in her ontology. Thank you. Let's start with a puzzle.

25:00 So, the following puzzle, as explained by Perth. Let's suppose a black ink spot on a white paper surface with a boundary between both color surface corresponding to a physical contact. And the question is, is the boundary color? So we have two possible answers. If yes, then the boundary is part of one of the two color surfaces. The problem is that there should be two boundaries since nothing prevents each surface from having its own boundary. If we answer by saying no to the question, that is, if a boundary is not colored, then it is an abstract limit. And we face another problem, is that an abstraction cannot make sense of a physical concept in particular, therefore a boundary cannot explain how two colored surfaces are in contact with each other. So let's try to... OK, Perth is going to solve this puzzle. And to understand his answer, we need to understand his conception of continuity. OK, so his principle of continuity rests on synechism. And that comes from the Greek term synechase, meaning continuous. And this is defined as the absence of ultimate parts in that which is divisible. So, in other words, role and parts are continuous if and only if a role is infinitely divisible into parts. parts. And parts are defined as mere possibilities. That is, a whole is infinitely divisible into potential parts. So if we stop here, it seems that this definition is very close to what Aristotle says about continuity. Indeed, Aristotle says, a continuous whole has parts that are

27:30 themselves divisible. What is infinitely divisible is in potentiality and not in actuality. Hence, a continuous whole is potentially divisible into continuous parts. So, if we make a comparison between Peirce and Aristotle, we seem to be on the right path since Peirce makes a direct reference to Aristotle. Perth says, I made a new definition according to which continuity consists in quanticity and aristotelicity. The quanticity is having a point between any two points, the aristotelicity is having every point that is a limit to an infinite series of points that belong to the system. The connection, I mean, we can make the connection between Perth's text and Kant and aristotel as text, at least it's a loose connection, but, okay, so quanticity is a property for a continuum to have infinitely visible parts. And it is true that Kant, in his critique of pure reasons, there is a property of magnitude by which no part of them is the smallest possible, that means by which no part is simple, is called the continuity. So, Aristotelicity is defined as the property for a continuum to have a unique limit shared by all potential points. This is as defined by Perth. And it is... I mean, to try to find a connection with Aristotle's text, we can see that Aristotle says that infinitely divisible parts of the continuum have one and the same limit, that is, the limit of the continuum itself. OK. Yet, to compare Peirce and Aristotle is mistaken, and we need to focus more precisely on Peirce's text. Indeed, what say Peirce is the following. What is required is to state in nonmetrical terms that if the series that points up to a limit is included in a continuum,

30:00 the limit is included. It may be remarkable that this is a property of a continuum to which Aristotle's attention seems to have been directed when he defines a continuum as something whose parts have a common limit. So, in other words, he has read Aristotle's text. That's true. As they have shown, we can find Aristotle's arguments in his physics. Yet, we are dealing with two different problems and two distinct definitions of continuity. Indeed, post-continuum is defined as a series of points, meaning that its potential parts are indivisible points. In contrast, a nice-totelian continuum is composed of infinitely divisible parts. Thus, a continuous line is composed of lines that are themselves divisible into lines. So, we cannot speak of indivisible points as parts of an Aystotelian continuum. Likewise, Aystotel defines physical continuance through infinitely divisible paths. The points of time is composed of divisible temporal intervals and not of indivisible instants, and matter composed of divisible matter and not of indivisible atoms. So, in other words, we need to make a distinction between two notions of infinite divisibility. Either we say that the continuum has an infinitely divisible path, meaning that paths are themselves divisible, or we say that the continuum is infinitely divisible into paths, meaning that a whole has an infinite number of indivisible paths. This distinction is verified by the two definitions of a continuum. So, first definition is Verstappen's principle of continuity, which defines a geometric or physical continuum, whose parts are infinitely divisible. And this is, I would say, the intuitive definition of continuity. You just need intuition to understand why a line is continuous, because its parts are lines.

32:30 So, in other words, there is no... it cannot be... we cannot think of a discontinuous element as a part of a continuum. In other words, parts are themselves continuum, meaning that a continuum is intimately divided into a continuum. OK? Then Perth's principle of continuity is based on, and I say is based on, because we are going to say that it's not an algebraic continuum, But Peirce uses the notion of algebraic continuity to make sense of this definition of continuity. So, and what is an algebraic continuum? As defined by Deleking, and so he just uses the definition given by Deleking, it's a set of real numbers, and real numbers are in one-on-one correspondence with indivisible points. That's why Peirce speaks of points. But we need to understand that these points are set-theoretic entities, or arithmetical entities. They are numbers. That's why we call the set R a real line, but the real line is not a geometric object. It's a purely arithmetic object. It's a line of real numbers. So, in other words, we can define Perth's continuum, continuous whole and part, through algebraic interpretation, such that quanticity is equivalent to set theoretic dampness, such as there is always a rational point, rational number, between any two rational points. So, an algebraic continuum is dense, but denseness is not sufficient for continuity. Indeed, a dense interval of questionnaire numbers is not continuous. So, we need to define set-theoretic continuity. And again, I should say that when I speak of set-theoretic continuity, it has nothing to do with the intuitive idea of continuity. What we just expressed is the concept of arithmetical completeness. In other words, when we speak of real numbers, we increase all possible numbers, not only rational numbers, but also irrational.

35:00 And that's the core of a set-traumatic continuity. And also, we can use the same definition when we speak of continuous functions. That means we speak of functions defined over a set of real numbers. It has nothing to do with the possible continuity of motion. I mean, we are unable to know what it means to speak of a continuous motion except through perception, but we cannot have a scientific definition. But we can have a scientific definition of a continuous function. So, okay, so, in other words, i-stotelicity is equivalent to set-theoretic continuity, defined as an algebraic convergence. Namely, every real point, real number, is a limit of an infinite series of real points. In other words, an algebraic continuum is a convergent infinite sequence of real numbers. OK, now, this is not exactly true. In other words, it is true to say that Peirce's concept of continuity is based on algebraic continuity. But Peirce is not happy with this concept. Why? Because we are losing the intuitive idea of continuity. And Perth is a pragmatist, meaning that, at the end, our scientific concept must make sense of our intuition. And so, he's going to transform this definition of continuity so that we can reach a mathematical of continuity which makes sense of our intuition. And to do this, he's going to introduce the notion of topology. So, I should say that this conception of topology has nothing to do with modern topology. So, and he's going to define, so topology as a pure geometry, which includes algebraic properties, but which can go beyond. So we are going to see what does it mean. So just discrete actual paths corresponding to algebraic values. And for Perth, these algebraic values are discrete, intuitively discrete.

37:30 to speak of real numbers and such. So, he's going to replace these discrete actual paths with what he calls smooth potential paths. And he's going to call these topological neighborhoods. So, in other words, the Perth true continuum, and he always uses this expression, true continuum, as opposed to the set theoretic continuum, or classical. So Peirce's true continuum is a smooth collection of topological neighbourhoods understood as indistinguishable potential points that are actualizable through distinct algebraic numbers. So in other words, Peirce's pragmatic view makes topological continuity not only intuitively continuous, but also applicable to physical objects. such that the potential parts of a physical object are both non-metrical and indeterminate. So, let's try to understand what it means. So, consequently, what we can see is that a whole cannot be identical to the total sum of its parts, for purpose. Why? Because the potential, parts are always potential. If a part is actualized, it's no longer a part, it's a whole. So in other words, the potential cannot be identical to the actual, parts are always potential, so we cannot make sense of a whole even with the total sum of its parts. So, in other words, an actual role is in contact with actual roles, while potential paths are never in contact with each other since they are potential, they are possibilities. So and we are going to explain the contact through a boundary, that is, the boundary makes the role both actual and determinate, and if we imagine the absence of a boundary sense of an actual role. And when I say an actual role, I imply that the actual role is always distant from another actual role. In other words, we cannot speak of a unique actual role. When we define, when we speak of an actual role, we oppose this actual role

40:00 with another one, at least one another one. So, let's try to... So if there is no boundary, if there is no contact, we just deal with a part, which is potential. Potential, that is a possibility. So let's try to sum it. So first, we have the role, which is continuous and lateral. continuous because it is infinitely divisible into potential, instinct, smooth points. That is, topological points. These points have algebraic properties when they are actualized. So they are not geometric entities, but they are topological, which means, for us, that the underlying algebraic properties, even so they do not correspond to actual discrete entities, but are indistinct entities. He is going to speak of smooth infinitesimal. So a hole is actual because of a boundary which makes it distinct from and in contact with another whole. So let's compare this with a part. A part is continuous and potential, continuous because it is indistinguishable, it is an indistinguishable topological neighborhood. So, and again, meaning that it is an indistinguishable topological point. And you understand that why, I mean, a point, in order to be continuous, must not be defined as a point, I mean, an indivisible point, but must be defined as something indeterminate. It's the only way to make sense of the intuitive idea of continuity with respect to a point. Otherwise, by definition, a point is discontinuous. So the only way to make it continuous is to make it indistinguishable.

42:30 A part is potential because no boundary can make it distant from and in contact with another part. and a boundary. So a boundary is discontinuous and actual. It is discontinuous because it is an actual limit divided into broken discrete parts. But it is actual because it makes at least two holes distinct from and in contact with each other. So, in other words, I mean, we need to understand, if we say, oh, it's a hole, that is not true for, that does not make sense for us, it cannot be a hole, because in order to define a hole as actual, we must define a whole as in contact with at least another whole. That is, we must suppose this. And then we can speak of a whole as actual because it is in contact with another Or we could imagine several holes. So, in other words, the boundary does not belong to a hole in itself alone in an imaginary But the boundary is implied by the contact created by other holes. So in other words, a hole by itself does not make sense. We need to define a hole with respect to other holes. And that's why we can make sense of an actual hole in so far as it is in contact with other holes. And so the boundary does not belong to the whole, but does belong to the fact, or is created by the fact that the whole is in the middle of other holes, is in contact with other holes. And by this contact, we can make sense of the boundary. So in other words, we can now answer the puzzle. Is the boundary colored? No. The discontinuity of a boundary makes it incapable of sharing

45:00 the property of a continuous wall. So in other words, a boundary is not a part. So that's why a boundary cannot be colored. Otherwise, if a boundary were colored, a boundary would be like a part. Is a boundary a pure abstraction? No. I mean, a boundary is the limit to the but it has a physical status. So in other words, the physical status of a boundary, even so it is not a physical object, but it represents a physical bridge of continuity. That is, in other words, when we speak of a continuous whole, we forget to say that the continuous whole is discontinuous with respect to any other whole. So in other words, there is a notion of discontinuity, And we need to make sense of it. And we need to make sense of it through a contact, as defined by a boundary, which does not belong to the role, but which results from the fact that the role is in the middle, in the middle, I mean, in contact with other roles. Thank you. Thank you very much. Who has a question? Well, one of my questions was about the last part, according to Peirce, a whole not only has a boundary when in contact. Is it actual contact or potential contact? Because what about the universe as a whole, and does the fact that the universe as a whole, again in the old sessions, is not in contact with anything outside itself, does it mean that it is not limited, therefore it is unlimited? I mean, it's a good question. To be honest, I mean, it's an ancient argument. I mean, if we say that the universe is infinite, so we avoid the question because we are dealing

47:30 with finite problems. So, but if it says that the universe is finite, so in this case, yes, there is a problem. I don't know, to be honest, what Peirce says about the universe itself. It must have been a way. I mean, it being an ancient argument, I think it's... but you cannot, there's no boundary to touch that for the whole universe. Yes, if the universe is infinite, yes, there is no problem, since we are at the end of course, financial, and infinite. This would make you a priori argument for the intensity of the universe. Because, well, it cannot be getting into, if human has a boundary when in contact, the universe is all, mind you. Is it? Could you deny that? There must be no role. Yeah. By definition, the universe has no role. yes so in this case i mean if the universe is infinite you cannot but it does not have a boundary then it is infinite now there is no problem it is infinite but that would give you an aphile argument for the infinity of the of the universe is it going to hold the electron uh yes i mean uh with respect to I mean, boundaries are actual. Boundaries and holes are actual. But a hole cannot be defined independently of a boundary, even so, a boundary does not belong to the hole itself, but is a consequence of the fact that a hole is in contact with another hole. Yes, that is, what is actual is the boundary and the hole. But the hole, the boundary implies at least two holes. What is this? So that I can speak of a boundary. If I just speak of the unique hole, I cannot make sense of a boundary. But is it not a bit too simple to say that holes must be finite, the set of natural numbers is infinite and is a hole, but it's a special kind of hole, an inconsistent hole, as Kant calls it. But we deal with a physical case, not a mathematical and abstract case.

50:00 But in that case, let's stick with this universe, because if the universe is infinite, it is no whole, by definition, right? I mean, otherwise, it must have boundaries to become as a whole, and if it has no boundaries, it is infinite, and therefore no whole. Which sounds weird to me, to say that the universe is no whole. Why? I mean, it's infinite. I mean, can you define infinite? I mean, physically speaking, I don't speak about abstraction. It's very important to understand that Perth tries to make sense of the physical case. He's not dealing with a pure mathematical structure. between planetary bodies and the whole of the rest of the universe is, I mean, the universe limits other holes, and therefore it's weird to say the universe in itself is no hole. I mean, by this you imply that the universe is finite, I mean, you know what I mean? So, you speak of the whole of the universe. So you already know that the universe is a whole. Do you have the answer? I don't know. I just say that the universe offers a boundary for planetary bodies, for example. So the boundary between the planetary body and the other whole is the universe, and therefore the universe is a whole. But on the other hand, the universe in itself has no boundaries and they have the definition no whole. Yes, I mean, you base this as just an hypothesis, you don't know whether or not the universe is a whole. So what you are going to say, you are going to say the solar system. And then there is a boundary, you know, to the, I mean, there is another whole. We can say, you know, I don't know, galaxies. So, galaxies are, I mean, correspond to a whole in contact with the solar system. And you can go like that, like that, like that. And maybe at the end you are going to, you know, but I don't know the answer. Nobody knows the answer.

52:30 So, but you could, you know, define holes and holes and holes. If you know them, you can define them. When you introduce one boundary, do you picture it has two, or three, or five? Where? Here there is just one boundary. Yes, but the other body also will have a boundary. OK, so I need to define it. And now I can speak of two boundaries. Yes, I have one and two. OK, I need another. In other words, I cannot make sense of a boundary if I don't have the idea of a contact. And if I just define one or two, sorry? But then you always need two, right? I mean, two or more. Then you need two. A boundary requires one, I believe. At least. A boundary requires one. Yeah, at least. I need at least one. Do you have another question? I mean, suppose here I have different contact. So I have different boundaries. I have different boundaries for each, you know, one boundary with this hole, another boundary with this hole, another... A boundary is not a property of the hole itself. If I don't have another contact, I cannot make sense of the boundary. That's not your point of contact. It's in the inner circle. So contact is defined as if I think there is a contact between one hole and another hole and I can make sense of a boundary because there is a contact. Here there is no contact because I don't know what we know. But if I define another hole, I have another hole and I have a contact so I can make sense of a boundary. But how can you make sense of the last thing being a hole when it has only a boundary at one side and not on the other? I mean, the point is to say that a boundary cannot be defined as a property of a whole. And if I just say, oh, this is the boundary of a whole, if I say this, I'm unable to answer the puzzle. Because is the boundary a color? If I say yes, oh, so the boundary is a part of the hole.

55:00 And if I say, and so that means that if I have the two, you know, two holes in contact, I have two boundaries. And at first, believe that it's... I think that we have time for one more. Yeah, I have a different question, but I think the solution to this one is that the word whole is just used ambiguously, just it's the totality and a whole object, so anyway. Yeah, my question is, what does Pürer's puzzle solve that Aristotle couldn't solve? I think Aristotle would have said, is the boundary colored? Of course not. It's a category mistake. You can say colored only from something that has an extension. So the boundary itself, not. He said something like that in Topica. And it is a boundary of pure abstraction. No, it's a cynic, it's a cut. So it's real, but in the sense of being a cut. So what does first puzzle solve more than Aristotle could have solved? Very good question. I mean, what Peirce solves with respect to Aristotle is that he integrates the notion of arithmetic continuity in his definition of continuity. What Aristotle, of course, couldn't do this since he was not aware of arithmetic continuity and setteretic continuity. So, but, you know, Peirce lives in the 19th century. He knows what, I mean, the principle defined by dating. That is, this new mathematical definition of continuity, which is no longer geometrical, but theoretical. And what he wants to do is to integrate this new definition of continuity in philosophy. So he's not against Aristotle, because he's not, that's why he uses, you know, Aristotelicity. he believes that he is just trying to modernize Aristotle, but not against him, but just trying to say, well, I mean, and I don't know if Buss was aware of this, but, I mean, I'm not sure he was aware of this, but Aristotle defines an intuitive idea of continuity just based on pure geometry, on geometry, and in the, sorry, yes? He just holds the same problem with a different concept of continuity. Exactly, exactly. So it's not against Aristotle, he's just trying to improve his principle. Ok, I'm afraid we have to start. Thank you for your presentation, thank you for your presentation.

57:30 I believe there is some tea or coffee. Do you know that this is one of the physical calculs that you know? Yes, yes, yes. But you know that you lose the logic. You lose the . Yeah, yeah, yeah. So it's an important . It's not just change. Sure, sure, no, I agree. It changes the logic and the value, so... Yeah, no, no, no, no. That somebody hasn't placed? Yes, CFS.