Hello and welcome to the seventh session of Maths and Ideas with Reza Negra Asani. I'm going to pass the mic off to him now. Okay. Hello everyone. So, can you hear me? Because everything went silent suddenly. Okay, good. Good. So we continue with Euclid's. I can't hear myself. Someone is not muted. I think it's you. OK.
So we continue with Euclid's elements, And hopefully, we go over a couple of more demonstrations. And then I talk about some of the more, like, I think some of the cons of Euclidean system and gets a little bit into the mathematical conceptual structure of Euclid's elements. After that, if there is time, I will move to a rather back to a new topic to use in the following
sessions and that's the correspondence between analytic geometry and particularly the study of mechanical machines. I will talk a little bit about this correspondence between geometrical problems and mechanical problems which will prove useful when we get into a study of Urim's diagrams and the invention of mathematical graph and discovery of uniform acceleration and how Aurim solves these problems via going back and forth between mechanical machines of the medieval age and analytic geometry handed down to him via
Greek tradition. So, before starting, any questions, any philosophical discussion, mathematical discussion from previous session? Jake? You were looking at me. I wanted to give, so this is like, this jumps back to, I think, to the analogy of the divided line. But I was reading the epistemology of the divided line analogy and sort of getting to the issue of where the mathematicals go in the allegory of the cave. And so he gives like the, or one explanation, which is that the mathematicals fold into
the Edei and are just and rather than being something that separately mediates between the smaller line of B-E C-E and the Edei is actually just a particular way of training oneself to start to perceive them and I was wondering if we could extend that point in the allegory of the cave to say that like when you set up this when you construct this setup where you have objects producing shadows as seen by people in a particular set of positions, and you've coordinated these elements within the cave to produce a particular optical illusion. There's math in there. Someone had to have understood the process by which objects can generate appearances
via the geometry of the interaction between light, shadow, and the eyes of the prisoners and so forth is like sort of the coordination of the relationship of objects and their images. So of the second smallest and smallest portions of the line. Could we say that encodes or unfolds the mathematicals on the side of the sensible in the same way that the sort of learning radiocination on the side of the ideal involves mathematics into understanding. Did you watch the previous session when I had a brief comment on the role of mathematics
and mathematical entities in the dependent line? Yes. Yeah, I was talking... Okay. Yes, okay. So the thing is that, you know, what Jake is talking about is I mean just because the analogy of the divided line comes right before an allegory of the cave this doesn't mean that the divided line has anything to do with the allegory of the cave but in fact it does other analogy of the divided line is simply what you might call be the transcendental ladder, the diagrammatic scheme of escaping from the cave. Right. And so I'm talking about the Nicholas Rescher on the epistemology of the divided
line where he talks about this at the end and I don't know the problem that... I haven't read that. Say again? Can anybody hear me? I can hear you I think resis signals pretty now no Okay, no, yes It does. Jake?
Reza, can you hear me right now? If you can hear me, it might be better to log out. Sorry, go ahead. It might be better to log out and come back in and then try lowering your bandwidth. Oh, never mind. You're back. Okay. Hi. Hello. Could you? Reza, can you lower your bandwidth to maybe just the very low?
How can I do that? Up in the top bar where it has all the icons of like the hang up button and video button there's the triangle of ascending bars just click on that and Then it should drop down to a scale where you can lower to either audio only or High quality bandwidth Hmm I can't where is it exactly? It should be at the top of the hangout box so as you go to the top there's like Maybe Seven or sometimes there's six buttons up there You mean control room?
No, no, well It's it's on the top of the window not at the left hand side. Oh, okay adjust bandwidth usage, okay? And then go should I do go to very low? Go to the very low. Okay, is that, can you, can you talk now? Maybe we lost you now. Hello. Okay, it still seems a little spotty, but Okay, you look good now. I think that's okay.
Okay, so in response to Jake very quickly is that mathematics is essentially. which they interact, they produce multiplicities, but also unifications. you know commentaries have been written on this idea of what the principles are
and how much matter to basically these principles so when I'm talking about the one and for example the infinite I'm not talking about number one I'm talking simply a what you might call to be a metaphysical law in Plato. Number one is in the domain of mathematics whereas oneness is something that is neither ideal nor mathematical. It is simply a principle what you might call to be the principle of integration and disintegration that allow in the analogy the divided line for us to have multiplicities and identities differences
and saying this now I don't want to get into the details but precisely because mathematics is generated by these principles it has a an intermediary vision within the analogy of the divided line and it's intermediary in the sense that it is immutable it is fixed hence it has something of the ideals in the divided line named in the last segment but it's also multiple so you have multiplicity of a structures of mathematical straw and hence it is also
something of the sensor of objects so precisely because of this you know being at the same time has something mathematics both enjoys some of the attributes of the ideals riders and enjoy some of the attributes of a sensible realm it because of this it's enjoyed very special metaphysical status in Plato's analogy of the divided line in the sense that mathematics understood broadly can be seen as a model that mediates between the forms
the ideals where there is a domain of reason dialectical reasonings and the domain of the sensible. So and also one more thing is that when we are talking about and precisely when you compare analogy of the white line with the allegory of the cave you see that as soon as you start to see these connections in terms of mathematical structures, you know, the objects casting shadows and optical illusion, you have already diverged from pure sensibility and pure empirical facts
in the domain of objectivity afforded by you via mathematical models. But of course mathematical models are what you might call to be inferior within the analogy of the divided line to the ideals. In so far as they are and the inferiority of them is precisely because of their the transition between particularity and universality whereas in the realm of idols the realm of pure forms all we have is unconditional universality what you might call to be the pure transcendent wise so I don't know if this
answered you Jake but nevertheless I will hopefully when I'm teaching the next course on Plato I can make this far more clear but also one one more thing that you need to have in mind when looking at the analogy of the white line is simply not just an epistemological model within Plato's work it is a model and an analogical model that's glues metaphysical ontological and epistemological psychological and ethical problems in Plato's Republic, so it's basically the diagram, not just a diagram, the diagram of the entire Platonic
Okay, cool, thanks. That definitely helps me at least refine my question. I need to go and sort of look at the end of that paper again, and then I'll maybe post a a better version in the classroom thank you sure sure so let's start with um um our problems we went to the panel of postulets we um now i think it's time to look at um little bit more into the idea of the parallelogram just power drum something
Hi, the signal is pretty spotty. I'm going to ask if all the students can lower their bandwidth. I'm not sure if that helps, but I might as well try right now. It seems that even the iPad doesn't go through. So maybe instead of doing outright, just going to the idea of parallelogram, we should save it when we get to the idea of .
And I started analyzing some of the stuff that we have been talking about, Euclid's diagrams. And maybe if the connection gets a little bit better, we switch to iPad. Sorry for this. Probably at some point I need to get a whiteboard so I can write this stuff on whiteboard. We talked about, when we were talking about Euclid's elements, particularly when we came across common notions, where basically, for example, was saying that the whole is always greater than its part,
two things are equal to one thing, are equal to one another. We mentioned something called exact and co-exact properties of the diagrams in Euclid's elements. Now I'm going to talk about a little bit about this exact and co-exact properties, which basically sheds light on exactly the inferential licenses that the diagrams in Euclid systems can afford us.
So I've mentioned that the five postulate express in elements that the first five postulates express facts about geometric configuration. For example, what is exactly a circle, what is a triangle, so on and so forth. But the common notions express facts about magnitudes. you might say they are objects that can be compared added subtracted provided they are of the same in current now this idea the entirety of information afforded to us by diagrams in
in Euclid's elements has something to do with precisely how these magnitudes are compared to one another. Now, equalities and equalities, for example greater and less, among objects distributed throughout the diagrams are of great importance in Euclid elements, an inference principle for both are available in the discursive text of the book. In order to allow these notions to function also in a diagram-based geometrical argument, the elements makes it first order of business to secure transfer around the diagram so that
direct comparison may be made, both so that inequalities already available in the discursive text may be realized as proper inclusions in the diagram, and so that inequality in the diagram may be related to explicit inclusion there, which then may be read off or made explicit. So, definitely speaking, coexact attributes are those conditions which are unaffected, and this is Manders' definition, Kenneth Manders, who first started to look at precisely the
kind of information diagrams in Euclid's afford us, exact to coexact information. Now, in Manders' board, coexact attributes are those conditions which are unaffected by some range of every continuous variation of a specified diagram. Paradigmatically, one includes another which is unaffected no matter how the boundaries are to some extent shifted and deformed, or the existence of intersections, points such as those required in Euclid proposition number one.
Now, in a more rudimentary way, exact relations are the ones applied to the comparison of magnitudes hence the idea of co-exact exact and our party of your rules involved all who exact where we basically construct
manipulate the done the given diagram the given data so in a way that it allows us to compare the information I need magnitudinal information with one another resident can you repeat that just that last bit sorry we lost you a that sure I was saying that the majority of Euclid's elements involved with be co-exact information namely the constructive comparison between magnitudes
For example, you remember how you constructed an equilateral triangle, where the radius of a circle was constructed so as it constitutes a side of a triangle, allowed us to make a comparative analysis of what it means to be a side of a triangle a magnitude as such magnitude one and to be at the same fine array and to two
well for example you remember last session when we were talking about how we have been given a line segment and point outside of it how from this point can we draw a line segment that would be that can be said to be equal with our given data namely our line segments and Philip said that you know we use the compass and open it size of a line segment and transpose the compass on to the point out so you create a new line but then we discussed about this idea
that in Euclid's element this is an illegal move precisely because there is no we should not anticipate that when we first that we have the vehicle first that we have the vehicle of such a transfer nor that throughout the process of transferring the compass the size of the compass namely the distance between two arms will be preserved these are intuitive anticipations that doesn't mean they hold into the quasi-formal system of elements so we had to invent a diagrammatic proof that allow us to construct such a new line segment that can be
said to be equal to our given line segment. So remember, we drew. Oh, well, it's going to go on. So what I was going to say is, so there are no numbers in Euclidean geometry, they're just lengths. And the lengths are represented by ratios in Greek geometry. Yes. And you can't lift the compass off the paper into another plane and put it somewhere else. So what you do is by figuring out ways to compare magnitudes by making continuous drawings built on whatever lines or points you have,
So you're not entering that plane and re-entering at another point, as you do later when you're talking about transformational geometry. Absolutely, absolutely. Brilliant. Yes, yes, that is. That is, so basically what you are getting is that you might say that coexact information are about how variations range over a continuous given set of data. For example, your idea of a plane. because for example and what is for example in in in the diagram example that I made creating a new line segment that can be said to be equal to our given
line segments what is exactly the geometrical configuration that allow us to carry out the construction and making explicit making explicit of these co exact information it's a circle if you remember we we drew two circles and these circles can be said to be like a copy machines that they do in in the diagrammatic system what we intuitively . Even segments of the line into another.
So if you remember, it's in fact whenever you are using a demonstration of a proof in Euclid, circles can be thought as a copy machine, precisely because of this property that the radius of a circle is always constant. So if you manage to construct a circle that has some relevancy with your given magnitude or your geometrical configuration, you can copy it somewhere else. And the same thing with equilateral triangles when it comes to line segments and angles and we saw that in that demonstration we dealt with both basically
the vehicle of our transfer the vehicle of ranging or variations as smoothly where two geometrical configurations circle and an equilateral triangle that allows us to extract coins that co-exact information and carry out with the demonstration. Any questions? And so this is something based on, based directly on the gesture of drawing with the compass and observing that at every point in drawing, at each point in drawing the circle,
the distance between the legs of the compass hasn't changed and so then you have like you have a brute gestural fact that the radius is constant and that through using a constant radius you have produced a circle on the page and that's sort of the gesture diagram data for natural deduction line of production? I would probably don't go so far as it called the gestures precisely because Because what we are dealing with here, yes, I mean we can talk about in terms of like diagrams being the frozen avatars of gestures and for example, a circle really freezes the gesture of drawing a circle by way of a compass. here if you remember is is a part a part of relationship
Can you hear me? The signal is really, really choppy right now. can you hear me yeah ok sorry I have no idea why the signal is I mean I can hear you pretty well at my end I don't know why you can't hear me so I was saying that the diagram in that sense is something more than gesture at least in the Euclid's element, precisely because it is about a certain stabilized particle relationships. For example, between a circle and its radius,
between a triangle, its sides and its angles. Hello? Okay, yeah, no, that makes sense. But this is a process by which those part-whole relationships become something that can be inputted as data into natural deduction or into… yes yes yes and and what are exactly the in this sense you know co-exact data co-exact act as I mentioned are
you I'm guessing we lost was a just now being contained in another thing Hey Reza, I think we missed all of that for maybe a minute or more. Sorry, I don't know. I was saying, let me, one second, let me, let me, something, one second.
which of course, and co-exact information in Euclid's elements are covered mostly by common notions. For example, a whole is always greater than sparse, or for example two things that are equal to one thing can be said to be equal to one another or for example that you can subtract equal amounts from two things that are equal and that would result in equal amounts or equal magnitudes now the majority of these co-exact information when you look at Euclid's demonstrations involved with relationships of containment for example you remember in the demonstration of if
we extend one side of a triangle the exterior angle can be is always greater than the opposites and we saw that the last stage of the proof involved with the idea of constructing a new triangle that is equal to our given triangle and the angle of one of the angles of the triangle that corresponded to alter your interior of our given triangle was inside the new external angle and we that that was the last stage but precisely because of this precisely because whole is always greater than its parts if we have managed to construct a
magnitude the given magnitude that was our interior angle we transferred it onto our what we wanted to prove which was the external angle and we show that it is in fact a part of the external angle hence by virtue of the common notion where we can say all to the whole is always great part we can say that the external exterior angle is always greater than the positive the opposites interior angles so we see here a relationship of a containment part and a
whole that we have to our process of construction even if it wasn't given to us by way of construction by way of manipulating the diagram destabilizing its particularity we managed to transfer what was given to us the magnitude the interior angle onto our problem which was the external angle that always should be greater than interior angles of a triangle and once we did this transferring then we could make explicit a set of co-exact relationships you can
see in any of the demonstrations particularly as you move forward in Euclid's propositions you can see you use common notions a lot and you can see that these co-exact information basically become more and more prevalent comparison of magnitudes the relationships of containment with and the mensory topological information. So, I really like your notion of it being a quasi-formal system, because of course it's always presented in high school geometry as if it's a purely deductive system,
you know, modeled for thinking logically or something. And it's quasi-formal in the sense that it is step-by-step. proceeds from common notions and things that have been demonstrated earlier, like equilateral triangles or parallels, to derive the next notions. But, I mean, as you pointed out, certain concepts, or as Hilbert pointed out, things like betweenness, or what exactly an intersection is, are not defined. Which is why it's quasi-formal rather than formal, right? yes yes but i would say that hilbert would say that from a hilbertian perspective euclid system is in fact non-formal but once we look into it that we see that um even though it does not have
a well formulated syntax even though that there are gaps of reasoning when we are manipulating are given datum, there are still inferential classes of inference that we have to follow within the diagrammatic reasoning. But the appeal of you comes from it being a hybrid form. Yes. And the attempt to correct its deficiencies is misguided. It's a totally unusual space that is quite congenial to exist in because you do exercise logical and inferential capabilities, but you're also doing other things that imply diagrams, constructions, and building.
Yes, perceptual cues. I mean, as I mentioned, remember that I said that Euclid's elements needs to be read as a philosophy book, particularly within the context of the ideas that are prevalent in academia, in Plato's academia. And precisely this, what you might call to be the amphibious quality of Euclid's system, to confirm the metaphysical status of mathematics as a whole, being a mediator between the perceptual mechanisms in the analogy of the divine line and the realm of forms, pure forms or ideals,
which are absolutely formal. so when we look at the very briefly again when we look at Euclid's diagrammatic constructions we discover that if one looks at the elements closely is that Euclid only allows the co-exact properties of a diagram to ground the inference in a proof. To establish an exact relation, Euclid carries out an
argument in the text, but from the perspective of a diagrammatic construction, we only deal with co-exact information, rudimentary, geometric, called topological information that are being enveloped or encapsulated within our common notions. Now how no matter how precise are techniques for the production of a diagram satisfying exact conditions there will be always be a range of variation in what's produced. So Euclid can be understood to account for this by restricting himself to diagrams co-exact properties that allow us to make
these transitions between different variations of a diagram because from a perceptual diagrammatic simple diagrammatic perspective a triangle as such any form of diagram that you make is going to be a particularity of the idea of the triangle as such so if we are going to construct and articulate what it means for something to be a triangle we need to allow us we need to come up with a method of extracting certain kind of information that allow
us to move or make transition within these particularities of particular individual triangles through which we can extract the universal information about what it means for a geometrical configuration to be a universal idea of a triangle, a triangle as such. Hence the idea of, you know, again what we were talking about, the dialectic between the universal and the particular that Procluse talks about in his commentary on Euclid's elements. Of course the soundness of Euclid's co-exact inferences from such diagrams is
not always obvious the construction is always performed on a particular diagram though the diagram is representative of a range of configurations ie all configurations with the same core exact properties it cannot avoid having particular exact properties and these exact properties can you can be seen within co-exact relationships between multiple diagrams
namely variations when the same construction is performed on two diagrams which are equivalent with respect to their co-exact features but distinct with respect to their exact features there is no reason to think that the two results will be equivalent this leads to doubts that the co-exact relations Euclid expects us to read off of an augmented or manipulated diagram hold for all possible constructions and there is nothing in the diagram itself to remove these doubts the main challenge in formalizing Euclid's diagrammatic method then is coming up with rules which license the diagrammatic inferences
as needed for Euclid proofs, and ensure the geometric generality of all such inferences. The the role of a circle as a copy machine well it I mean it's quite simple we know that circle is defined by its radius a
constant distance from the center to the circumference. Now precisely because of this constancy of a radius, if for example we are given a line segment, namely a given datum, we can we can always introduce a circle into our construction that can replicate these line segments. so all you remember in the for example was it proposition to where making an equilateral triangle we were given a aligned segment now how are you we are
going to copy it if we take this line as a radius of a circle and precisely because it's a radius of a circle it has the potentiality to be if we manage to add a circle to our diagram to our given diagram or geometry configuration we are capable of replicating the data that is given namely the line segment onto a different place on our in our on our plane and that's Exactly what circle was supposed to do in fact we created two circles because one circle couldn't produce this results We created just two circles these two circles managed to like a copy machine copy our given line segment as the idea of the
radius of a circle and transpose it somewhere else and then once these circles are being intersected being drawn the line segments intersected with one another namely two radiuses two opposite two circles became two sides of our triangle and created an equilateral triangle you have this a lot in Euclid's demonstration whenever you you see a circle in in Euclid's demonstration think about it as a copy machine as a copy machine that copies your given a given line segment and by virtue of it
part of the relationships namely the relationship between the circle and its radius it has the capacity to take this line segment translated into a radius and produce somewhere else within the circle another radius of the circle hence sharing your given datum within the diagrammatic inference license by your particular type of geometrical configuration here a circle questions
So in Macbeth, this is where the issue of pop-up diagrams or pop-up features of diagrams occur. Like in the sense that, for example, given two lines that cross or cut one another, Euclid assumes that there is a point at their intersection. And he doesn't justify this or require an extra diagrammatic step in order to show it. It's simply that as you go through this process of drawing two circles and of deriving an equilateral triangle from them You have features that appear are these the kind of like Topological invariances that we're describing as co-exact or do they just depend on them? Yes, I mean, you know, co-exact is precisely what you might think about it. It's a mode. It's a kind of information
that a transfer between particularities, between variations. Obviously we talk about the idea of making explicit, making manifest what is already given so and co-exact information are precisely these kinds of transfers that can make explicit the relations between your given data by transferring them on to from one geometrical configuration from one that's on to another one other
geometrical configurations and pop up you know functions you can say that they are these moments in the Hegelian sense moments where your the relations the exact stabilized relations between your given datum become explicit by virtue of the right kind of transfer between variations Okay, that makes sense. Thanks. So one thing that we need to know is that
that even though Euclid's elements is not essentially axiomatic, but we treat it as if, in the Kantian sense of as if, as if it was axiomatic. this as if precisely in the Kantian sense of as if is not ever constitutive but about what we can realize out of what has already been given to us so it's
this axiomatic approach to a quasi formal system that even though it is not axiomatic we treat it as axiom is no longer can be said to be constituted in the sense that we usually deal with axiomatic systems of mathematics in which the role of axioms are constituted in Euclid systems we can treat it as axiomatic but no longer in a constitutive sense but in a transcendental sense in the sense that we can say as if this system was axiomatic then we could build such and such constructions and this is perfectly fine in understanding euclid system
as an axiomatic system in which axioms are not constitutive they don't have pure logical well-formulated syntax like pure axiomatic systems of mathematics but they have transcendental roles as if it was an axiomatic then it could create such and such relationships make manifest such and such correspondences or correlations within geometrical configurations so in this sense the axioms we choose are not completely arbitrary because the only axiomatic
systems are worth the studying are those that describe something useful or interesting for example in our you know in Euclid's elements they are interesting from the perspective of the kind of perceptual cues that they afford us. But from a strictly logical point of view, we may adopt any consistent system of axioms that we like, and the resulting theorem will constitute a valid mathematical theory. But the catch is that one must scrupulously ensure that the proofs of the theorems do not use anything other than what has already been assumed in the postulates.
If the axioms represent arbitrary assumptions instead of self-evident facts about the real world or the mathematical structures, then nothing except the axioms is relevant to proofs within the system. Reasoning based on intuition about the behaviors or properties that are evident from diagrams or common experience in the real world will no longer will be justifiable in that is your math existence. Now looking back at Euclid with these newfound insights, mathematicians realized that Euclid had used many properties of lines and circles that were not strictly justified by his own
positive rates. Now let's examine a few of these properties as a way of motivating the more careful axiomatic system that is hallmark of modern mathematics as in contrast to Euclid system. The first one, let me explain, I don't think I can turn on the iPad, I'm afraid of losing the connection again. So you remember the much discussed Euclide proposition in book one, constructing an equilateral
triangle. we have been given a line segment we use the compass open the arms of the compass with the same size of our given data namely the line segment now we put the compass on one end we create one circle we put the compass on the other other end double art line segments and make another circle. Now, from the perspective of Euclid's postulate what is already given to us,
the givens, the datum, there is no reason for us to anticipate that these two circles that we have created doesn't so are going to intersect hence creating and a trial and equilateral trial so this is merely a convenience and is not meant to suggest that points take up any area in the plane.
It seems obvious from the diagram of creating an equilateral triangle that there is a point where the circles intersect, but which of Euclids possibly justifies the fact that such a point always exists. Nowhere does Euclid give us any justification for asserting the existence of a point where two circles intersect. Of course I know that Danielle Macbeth thinks of this as a function of a pop-up. apex of your equilateral triangle created by the intersection of the two
circles Macbeth treats it as a pop-up function but this pop-up function we can only see it as a pop-up function see it as the concept of our function if we had already filled in the gaps of Euclid's positive lights anticipated that these two circles do in fact intersect but as I said there is no information already given to us that can be said to be a justification for asserting the existence of a point
where two circles do in fact intersect in this demonstration another one you remember in our proposition 16 and what was the proposition 16 in any triangle if one of the sides be produced the exterior angle is greater than than either of the interior and opposite angles the one that we demonstrated last session and this proposition is called nowadays exterior angle inequality we
talked about it it's probably one of the most clever and elegant demonstrations Euclid. From, you know, looking at it in the first glance, it's not easy to see where the gaps are within this demonstration. But there are at least two gaps inside this demonstration. Now you remember that so we were given a triangle, we extended one of its sides and then we would say that the external
exterior the external angle is always greater than the interior and opposite angle to demonstrate that this is really the case let me get a pen now that I can't turn on the iPod and draw this for you with a pen and a marker once again I have a basic question for the class and I don't if this derails the conversation we can just wait but if we can't imagine a point when we you know through the process of drawing to us two circles what it seems like that undermines the whole idea of like diagrammatic
construction because that that seems I don't know what other people think like what how would the process of a circle be drawn from the given data the line I mean that was kind of what I like so much about at least what I understood Macbeth to mean by a pop-up function was that there is this topological invariance in the process of sketching or of principles on a single plane which forces the emergence of an intersecting point before we would necessarily have
any prior concept of it right and to me that's sort of like that's a validation of the constructive process because the constructive product like a validation and an explanation of the way it relies on the topological constraints of its medium to produce new elements now right so this was our if you remember our diagram where we extended a line from B cutting the side AC at its midpoint and then extended be to point F where B and EF can be said to be equal so we
extended be and produce EF. Both the same equal size. Now, once we look at it carefully, after constructing the midpoint E that bisects the side AC, Euclid then extends BE past E and uses proposition 3 to choose point F on that line such that EF is the same length as BE.
Here is where the first problem arises. Although Postulate 3 guarantees that a line segment can be extended to form a longer line segment containing the original one, it does not explicitly say that we can make the extended line segment as long as we wish. we mentioned if we were working on the surface that wouldn't be our given surface our given plane what Philip mentioned when we were talking about this in terms of making given a line segment and point us we have to use a circle
rather than just moving the compass because that's we do not know it once we do that we enter another plane now imagine for example this is also the case here that's imagine that in fact we are not working on a regular plane but we are working on the surface of a sphere trying to construct and demonstrate this diagram on a surface of a sphere on a different plane now extending BE to EF wouldn't be possible on the surface of a plane.
Because great circles have a built-in maximum length. Let me show you what would happen if we were constructing this on the surface of a plane. how F behaves. ...
the surface of a sphere. Can you raise it up just a little bit? A little lower? Ray just a hair more. That's good. That's good. We can see it. You see where F ends up to be. Now the second problem arises toward the end of the proof when you plead claims that the angle ECD ECD, the angle ECD is greater than the angle ECF.
ECD is greater than the angle ECF. Now, this is supposed to be justified by common notion 5, which is whole is always greater than a part. However, in order to claim that angle ECF is in fact a part of angle ECD, we need to know that F in fact lies in the interior angle of, sorry, in the interior angle of In the interior of angle ECD now this seems evident from a
The diodes there is nothing in the axioms or previous propositions that justifies this claim to see how this could fail consider again that we were constructing that this diagram on the surface of a sphere which I did with a at the North Pole and B a at the a at the North Pole and B and C both on the equator if B and C are far from or far enough apart it is entirely possible for the point F to end up south of the equator in which in which case it is no longer in the interior of angle
so if I unwrap the diagram constructed on the surface of the sphere we see that's According to common notion find, the hole is always greater than the part. We are supposed to show that ECD is within ECF, these two angles.
showing that the external angle ECD contains the angle ECF and ECF corresponds to either the interior or opposite expert in angle of the triangle the exact class of information that we managed to extract the manipulating of a dagger but in order to claim that the angle is easier is part of angle ECD hence demonstrating
the proposition we need to know that f lies in the interior of angle ECD now this seems as I mentioned evident from the diagram from you know the perceptual encounter with the diagram but from the class of inferences afforded to us by common notions by the propositions and the postulates there is nothing in the axioms or proof or previous propositions that justifies this claim precisely because we would have we can construct this diagram in the in it so
that F is no longer in the interior of angle ECD namely ECF will never be within ECD the part what we taken to be part of a whole will never be a pot Now there are so many of these examples in in Euclid's elements and These kinds of gaps and these gaps are precisely because We are dealing with a quasi formal system
And interestingly enough, people like Lobachowski, like Riemann, the people who basically moved to non-Euclidean geometries precisely looked at these gaps to see what kind of information might be introduced to these gaps and if we could find what these gaps are really about could we move out of the euclidean system or not and that was you know one of the main motivations behind the conceptual development of of non-euclidean geometries exactly determining the nature of
these gaps and the kind of information that we can come up with that could cool here and play cover up or filling these gaps within the within Euclid elements now can you explain a little bit what exactly you mean by gaps though it sounds like what you're saying is like, you know, we have these principles of construction and as we execute them, they have these hidden premises about the way that they function. So the implicit assumption was that we were constructing on a flat plane
rather than some other type of topological space. Yes, yes. That's exactly, by JAPA, mean like that that there is an assumption that is always already talking about an unwrapped flat Euclidean plane even though we haven't seen this assumption being made explicit or being contained within any of the previous axiom or propositions that we have dealt with so far in constructing or demonstrating this proposition. Now once we, as I mentioned, you know, the idea of the givens or the datum is something
what you might call to be truth candidates. So truth candidate means that we have to create a theory, a system that can, that only uses these two candidates as building blocks of construction it should not smuggle in any hidden assumptions that wasn't explicitly contained or expressed by way of these basic truth candidates or given that if we do that we basically move toward what you might call to be non-formal systems now it seems that's
the development of non-euclidean geometry can be seen as an inquiry into the non-formal aspects of Euclidean geometry in comparison with formal aspects of proper axiomatic systems of mathematics that allows us to determine what are exactly the sort of information that are missing within the Euclidean system that make the system non-formal and these are what you would call to be
the gaps once we identify these missing assumptions or information then we are capable of finding a solution how they can be accommodated within our systems but sometimes accommodating this new information require us to move into a different system altogether rather than assuming that every system that we are every plane that we are dealing is going to be a flat which of course this is not X mean even the idea of you know a plane unwrapped surface is not explicitly mentioned or given datum not given propositions in and you please elements but imagine that we
identify that this is really the implicit assumption now in order to see how the construction can be done now explicit assumption that this is like a flat plane we see that we can't really adequately investigate this unless we move we transit to a new system in a system in which the what you might call to be unwrapped flat surface is a special case of a curved wrapped surface a greater sphere and this greater sphere basically sheds light both on the implicit assumptions that we had in our
previous systems but also give us new data new givens new information so so this is maybe I don't even know how to formulate this question exactly but how would you even begin to move to a different topological space without you being the constructive elements that you've been using well as I said the The first thing is what you might call to be a procedure. The first procedure is that identify the assumptions that haven't been explicitly stated within
the previous propositions or possibly. nevertheless we are mobilizing them blindly in so once we identify these and see tips to ease a song hidden assumption the implicit assumption was that we are working on a flat unwrapped surface now imagine but we already know in that geometrical progress of that investigation people already know that
there are different forms of planes in the sense that they are different from flat wrap surface this is already implicit given to us by Archimedes you know hyperbola and a study of the great circles so once we identify this is implicit assumptions find possible alternatives to this implicit assumptions you have a flat unwrapped surface but also we know that there are non-flat wrapped surfaces the surface of a given a sphere now trying to make the
same construction or demonstration within these wrapped, non-flat planes, the surface of a great sphere. Once you do that, then you see that the demonstration doesn't hold up. Once the demonstration doesn't hold up, then a mathematician can go on and find a solution a more general solution that can in fact be said to hold but within the within the new basically plane a plane that can be said to be more general than
our flats unwrapped plane the Euclidean plane so it goes in a kind of a what might call it to be a kind of a critical procedure that starts with investigation of the hidden assumption then transposing this hidden assumption onto a different alternative given to us what you might call a new set of data that covered different types of geometrical configurations then a study whether this demonstration hold or not within this new class of entities if it doesn't hold up then find a solution that can be more general and then from that you know construct more and go back and forth between the your previous system and the
kind of system that you are developing and this is this is exactly how and Euclidean geometry was made by way of this what you might call to be dialectical back and forth that started from making explicit of hidden assumptions awesome those helpful so How much time do we have? Class is supposed to go till 3.30. But if we'd like to, we can run a little bit later if you're able or I'm flexible today.
Sure, sure. again back to you know one of the things that we we have been talking about that demonstrations in Euclid's elements involve all of them involve a process the kenneth manners calls augmented diagrams augmenting diagrams in the sense that these augmentation requires some sort of manipulation using given data once this
manipulation is being is carried out we can extract co-exact information between what is given to us in a proposition and what we were supposed we were supposed to demonstrate these co-exact informations are the ones that not only allow us to carry out the diagrammatic reasoning but also in the overarching perspective demonstrate the truth of the proposition namely showing making explicit the exact relation between the final stage of
construction and our initial data exactly like the proof for demonstration of the exterior inequality of the exterior angle that we were just talking about so within this land escape the landscape of augmented diagrams ultimately what we are
doing is that we interpret everything in terms of dependence dependencies induced by construction of co-exact information within our geometrical configurations for example a construction a step the joining of two points in a segment produces an object y namely a segment from a tuple of object x prime example two which are two points now if y and the object x prime are so related say that
y directly depends on the object x prime the relation is asymmetrical and so naturally understood as an ordering relation we can extend it to all objects of the augmented diagram by taking its transitive closure and so obtain a partial ordering that records the dependencies of the diagram objects to describe how this partial ordering enters into the interpretation of the diagram it is useful to define one more term define the relation is linked as the symmetrical version of the directly depends of the directly depends for
example X and Y are linked if X directly depends on Y or Y directly depends on X so the problem of isolating the general the universal in the augmented diagram comes down to determining which of the unlinked object in the diagram display general relations. The relations exhibited by linked objects are automatically general. For example a circle and an equilateral triangle. The relations exhibited by linked objects are automatically general. It is built
into the concept of a construction step that the constructed object, for example y has a certain positional relation to the object x prime it was constructed from if a line segment is constructed from for example point p1 and p2 then it is immediate that this line passes to p1 and p2 also if if a circle c is constructed from center P and radius R it is immediate that C contains P and R. Generality or universality in Euclid systems cannot be said to be built into the relations
exhibited by unlinked objects in a similar way, however. And so it is indeterminate whether these relations depend on the party or not. The rules of the Euclidean system that are of central interest are those that remove this indeterminacy for a certain relation and classify the relation as general so this is the overarching diagrammatic methodology in Euclid's elements in terms of how constructions link objects or geometrical configurations
to one another and to this act of construction that link geometrical objects we can make universal we can basically make manifest the universals or generalities or invariances and this is very much in tune with Proclus commentary on basically on Euclid elements that I wasn't talking about you remember we talked about particularities and different levels of universality and how we move back and forth between universality as such and
particularity by way of these intermediating universalities and these you intermediating universalities of generalities are the process of construction in Euclid's elements which are carried out by linking via co-exact information geometrical configurations particular diagrams questions just to troll I guess like that isn't it a bit presumptuous to think that the
result of our construction or something that we come to through our construction would be even remotely universal especially as like acknowledging that the gaps yes yeah no it is it is I mean from the modern perspective it is promise it is it doesn't hold up from the modern let's with that okay but we shouldn't false Euclid about this. As I mentioned, Euclid's elements, before even being a mathematical book, is a philosophical book. It has a philosophical ambition. And this philosophical ambition is particularly about the metaphysical status of
mathematics rather than the exact what you might call to be exact epistemological and a structural characteristics of mathematics so of course even if from a modern perspective you might see this to be a faulty position, but we see that in fact it works if you completely forget those gaps. The gaps are justifiable within the Euclid systems precisely because the Euclid systems
It's not a formal system. It never, it in fact never claims to enjoy such a status. It always, you know, all of the, basically, every of the primary definitions and common notions are intuitive, are about not exact information, information about magnitude as such, about their comparisons if we you know get rid of the idea of gaps that can only be seen from the perspective of form of mathematics of contemporary perspective then we do see that in fact it produces universals in fact this is
really fun so what I'm trying to say is that's in order for us to look at Euclid's elements properly, within the context of Euclid's time, we should see Euclid's elements as a metaphysical treatise about the standards of mathematics. Even though Euclid has gaps at the level of methodology, but at the level of the metaphysical ambition it gets it right yes this is it should be able to move within particularities and pure generalities
now whether euclid's elements can do this or not it is entirely dependent on the kind of methodology that we are using within the euclid system As I mentioned, the diagrammatic reasoning does not allow us to overcome these gaps. Nevertheless, the formalization of the Euclid system, a project started by Tarski, allows us to do this and to show that in fact, at the level of methodology, once we rectify this, the Euclid system produces universalities. I mean basically it lives up to its metaphysical ambition.
Interesting, cool. So is that generally, maybe this is just like a trivial question, but is that generally the case that if we formalize a system and invent a fully formal notation for something that previously wasn't, that it necessarily generalizes over gaps, like, for example, the one involving a curved plane versus a flat one or an unwrapped one that were present in the non-formal system? Is that sort of like what distinguishes the gesture of formalization or a formal versus a non-formal system is generalization versus gaps? no no you see what basically the formalization of you please system allows us to do is that it does
not you remember that these gaps were more like hidden assumptions okay hidden assumptions that weren't as stated as explicitly within our previous definitions axioms and postulates what formalization does is that it does make these assumptions explicit precisely by way of well-formed syntax by way of introducing logical laws so on and so forth and within that system we still miss the whole you know the richness of none Euclidean alternative is that if for example you know moving into a different
plane but nevertheless the Euclid's proof holds up precisely because for example the hidden assumption that we are working on a unwrapped flat surface has already been made explicit by way of formalization we don't we don't simply move or overcome the inadequacy of the system but we simply make the system robust from a formal perspective. Formalization of Euclidean system doesn't necessitate transition to non-Euclidean system. But all it does is to show that the Euclidean system works, why it works, precisely because there are formal connections.
Right. So it sets up a loop where it makes it clear that in order For the conclusion of Euclid's proof to be true It needs to be the case that we have this assumption of an unwrapped plane and that if we have this is the That Euclid's proof correctly uses the assumption of an unwrapped plane to reach the conclusion that it does Yes, yes. Yeah, and so but and So that doesn't really that there's no necessary connection between at least like the idea of alternatives Or like the availability of potential alternatives like you know having a larger space of topology to operate in in which to say we can define what it means for there to be a flat plane which is not a curved plane it's really hard to say probably not and that's because you see I mean isn't it the whole point
of a mathematical system the point of the whole axiomatic system so we know that you please element is not axiomatic, but let's say that Tarsky does this, turns into a fully axiomatic system, in which the proofs are fundamentally different from the diagrammatic proofs of Euclid's elements. This creates a system, a system that within itself has, is always basically what you might call to be its constructive potentialities are always restricted by the kind of building blocks that it has incorporated, namely its axioms. What you might call to be its restriction with regard to possible other alternatives,
non-Euclidean alternatives, is always determined by the nature of the axioms. the axioms in if you want to move out of the system then you need to revise your access right basically coming up with a different system for constructing different axioms and that system also will be limited again by its own axioms And then you can see this quite quite In tandem with girls in completeness Okay, I think I followed that and then sort of the equivalent of having to devise new systems of axioms if you somehow wanted to do this
Like prior to or exterior to axiomatic systems if you were doing it from the perspective of a quasi formal diagrammatic proof sort of the equivalent of moving that into a new kind of underlying space or a new set of hidden assumptions is there sort of something as I don't know as clearly statable as well you need to devise new axioms what is the well you need to be able to do this in order to like within Euclid's own type of method move from a flat unwrapped plane to a curved one for example is there sort of like a similarly well expressed move that you have to make like even since you don't have these premises that you that you can just say well I need to rewrite some of these I
need to find new ones in order to move my system into a new basis well I don't think that there is a like a general I don't think that we can come up with general thesis about whether the the moves that trans the transition moves are already contained within the old systems or actually part partially require us to in fact whole thorough goingly move off of the previous system there is no I don't think that there is a general statement about this okay we can't we can't really come up with a I mean I really don't know to it from the perspective of for example
Lovachovsky and geometry and Euclidean geometry you can But I genuinely don't know for example. What would be the case? within other geometrical systems Or are there for example are the Transition moves Can you identify them within your existing system or whether you have to completely abandon I Mean abandon your system to in fact arrive at these transition So yeah, definitely makes sense. There's not I don't have like an argument here or anything. I'm just kind of exploring this relationship between like gaps alternatives and reformulations like how we are able to make these jumps from one system
The next but yeah, that's really interesting. Thank you. Well of course you see there is I mean usually you see alternatives in terms of making moving to a new system not simply for example finding an alternative geometrical object moving it to a different geometrical system they usually these identifications happen at the limits at the higher limits where basically the constraints of the constraint of the system within the
Euclidean system are really about how we postulate where basically you can see what is exactly the constraint of the system okay as the most compact form You can see exactly in the simplest way you can see either these lines are parallel or they are not and on that basis You can see the rest of the Sort of the underlying geometric assumptions going on in Euclid okay? Yes, I think Maria has a good question here too that kind of brings it back to the text What was pro proposition 27 again that was infinity or when they when the parallel lines oh, yeah, yes, yes, yes
he if you remember so we talked about this when we were talking about the the external inequality of the exterior angle we said that the point F doesn't essentially come inside that angle ACD. And hence we can say that for example ACF, angle ACF is within the angle ACD, namely part is inside its hole. Now we have the same similar situation
with proposition 27. You remember that we were saying that these, we put them let's say that they do in fact intersect then what would be the result then the result would be that the results contradicts the premise of of the of the proposition the situation with for proposition 27 is not quite similar to this but I mean it is not the same but it's similar it's like something like this
something like this this is this is a perspective and I this is a point C the intersection and a and B so this is a very rudimentary form of a non-euclidean geometry so what is this exactly in the Euclidean geometry a straight line moves and should only create one point here so if you were seeing this object you can think about
as a straight line just moving to this point C. But point C within non-Nucleus geometry can create two points rather than just one on the plane of perspective. And this creates a triangle whose sum of interior angles are slightly more than 180. This is something It is really interesting that the problems with Proposition 27, and hence moving toward the non-Euclidean geometry, weren't fully identified until the advent of Newtonian mechanics.
some of the observation of physics in fact led to postulating what you might call to be alternative geometrical planes in which the Euclidean parallel postulate no longer holds so it wasn't really for this case for the case of proposition 27 and how you prove it wasn't really mathematics that fully was responsible in identify alternative system but was how mathematical physics put forward you know in terms of observational anomalies and laws of
physics the existence oscillated the existence of alternative geometrical planes in which proposition 27 no longer holds so can I just read out sort of this question that I had I think it connects to like both what Maria you are talking about it's what what are the laws that are governing the creation of of new axioms. Is it our goal just to maintain as much of the existing system as possible? How do we move from one axiomatic system to the next? What's the system under which that transition operates? I don't think that there is a, you see,
this question cannot be answered within, with regard to what you might call to be, a quasi-formal system of Euclid's elements this is a question that Hilbert tries to answer within his formal system that's the the transition the what you might call to be the moving from one axiomatic system to another axiomatic systems are done with from the perspective of finding
Axioms, or what you might call to be axiomatic relations that can be said to be more general than what is already within your given system. So the moves, what I'm trying to say is that the moves from one system, axiomatic system, and other axiomatic systems is not discovered by way just by way of for example identifying the upper limits or upper constraints and limitations of your existing axiomatic systems but needs to be understood within the overarching program of generalization of mathematical structures with this
underlying assumptions of whatever axiomatic systems that you are currently working on, you can still find a more general mathematical structure. So this is again really, I think, I don't want to get into too much. So it's ultimately this move from the perspective of philosophy of mathematics has something to do with an assumption concerning the metaphysical status of mathematics, namely the transition between particularities and generalities.
our moves from one axiomatic system to another are warranted by our overarching goal to find more general mathematical structures. And precisely, we see that, for example, 1950s, 1960s mathematics onwards, This becomes the dominant trend of development of theoretical mathematics. That's the transitions are always being postulated by way of
hypothesizing or constructing a more general mathematical structure. That whatever, again, and this has nothing to do with what you might call to be just about, for example, the limits of our current axioms. But it is, we need to think about it also in terms of the, as I said, the affirmation of an assumption regarding the metaphysical status of mathematics. The metaphysical status of mathematics needs to be moved toward generalities. And hence, from this perspective, philosophically you can say that whatever general mathematical
structure that you are working with, whether or not Euclidean, Euclidean, so on and so forth, there is always a more general mathematical structure than this. And hence, how can we really generalize any given mathematical structure moving, for example, from generalities existing within our current axiomatic systems to a more general framework. This is a process that is done by way of all of the procedures in contraporial mathematics that deal with processes of generalization.
The process of generalizations, you can think about it as, for example, if we have a given mathematical structure introduce some algebra and geometrical operations that can generalize this further now once you generalize this further is your current mathematical structures that hold up within your axiomatic system then see whether your mathematical system can support this new generality. If it does not then that's when you can pose or posit to be more accurate posit this new
generality as a new axiom. A new axiom that combined with new generalities that you have constructed can be taken as a new set of axioms for a different more general axiomatic system so this is again comes back to this idea that this transition you know from one axiomatic system to another axiomatic systems can be seen from different perspectives from a Hilbert and perspectives are simply you know, finding the constraints of your systems completely within formal relations between your current axioms, or within a more overarching landscape, generalization of mathematical
structures, which is, as I said, which is something, which is essentially at its core, is a thesis about the metaphysical status of mathematics as such, that mathematics should strive for more generalities, and hence the operation of mathematics should always investigate, by way of operation of mathematics we should always investigate the level of generality of mathematical structures that are given within a specific or
certain axiomatic system if we are capable of further generalizing these mathematical structures and then investigating and confirming that the current axiomatic system does not support these generalities as it should be then we can extract these new general structures and posit them as axioms of a new mathematical systems and in fact we're looking into the history of mathematics from the time of Euclides to contemporary mathematics we see this
is exactly what is done what how the transition from one axiomatic system to another axiomatic system is done by way of a striving for broader generalities and testing the limits of axiomatic or current axiomatic systems against the emergence of what you might call to be these new generalities new general mathematical structures so is an example an example of some of an axiomatic not supporting a new generality including like paradoxes or contradictions arising from it so if we think of like having to introduce the axiom of selection to
prevent Russell's paradox from arising is that a case where we have like if we had to add an axiom that restricts your ability to designate sets further than it was before in order to prevent a paradox from arising is that still like sort of adding an axiom in order to like non contradictorily support a new level generality which is that designations yes yes I mean usually paradoxes contradictions what you might call antinomies emerge precisely because the the relationships between axiomatic constructions are not adequate to cover
the new generality that we have thus constructed but also there I think there are there are different reasons and there are different ways of for example identifying that for example a our current axiomatic system is not adequate to deal with this range of generalities and that some of these don't have essentially anything to do with the rise of contradictions or antinomies within our current axiomatic system they are just basically right like all the like
exactly like for example are you know the the difference between the Euclidean and non-Euclidean. It's not really contradiction, it's that you see that simply axioms and the kind of construction that is licensed by them is no longer adequate for accommodating these generalities within their scope of constructive inputs. Right. And then my other question, if everyone will bear with me for just one second, is
like, so in terms of different species of generality, you definitely got, like, I understand they're, like, in a single line having higher and higher levels of generality, but like in the 20th century, if we look at, like, set theory, category theory, and Gerdel numbering, all they seem to be three like qualitatively different types of generality or accounts of generality is that like is that a different a different dimension of difference in among generalities than the one you're talking about or a similar one I think they are different because the kind of for proclaim generalities I was talking about you might see them as again as
metaphysically characterized levels of generality what while whereas when we are talking about you know set theory category theory this is not the metaphysical status of general it's simply generality of mathematical structure category for example categorical objects can be seen as more generalized version of set theoretic objects precisely because they encompass more exact information but so and you don't call it actually we don't that that kind of general of higher generality maps to a higher metaphysical generality
you can but of course mathematicians are not interested to do that not all of them of course good mathematicians sometimes do i mean this is like growth index for example or albert notemann was a complete platonist and you see that this is exactly what he tries to understand to correspondence between the metaphysical status of mathematics and the metaphysical levels of different metaphysical generalities on the one hand and the generalization procedures of mathematics as such in within a completely formal characterization but
this is of course I mean you know this is something that is more in line with the philosophy of mathematics rather than how mathematicians look at their own discipline All right, that makes sense. Thanks. Would you be ordered sets, groups, categories? In terms of levels of abstractions? being more abstract than sets and then categories being more abstract than groups?
In terms of generality rather than abstractness. Okay, generality, that's what I mean. Yes, yeah, sure, yes. But do me a favor and elaborate on that distinction between generality and abstraction. you see when we're from a philosophical perspective we might say that sure generalities at least from a electronic standpoint that generalities are abstractions as such not abstract but abstractions as such act of abstraction as such rather than abstract objects and that what makes them generalities when we are talking
about for example the groups and category and the concept of abstract in mathematics concept of abstract in mathematics is not essentially related to the concept of generality abstract in mathematics I think for a good part is about pure what you might call formal relationships formal relationships that are what you
might call to be context independent to be completely devoid of any semantic dimension and that's exactly so what I'm trying to say is that yes there is a correspondence between abstract and generality well we should be careful when we are talking about abstract mathematics this abstract in mathematics can be understood as it is usually being understood in a completely different characterization given by the school of formalism by
by Hilbert, that abstract is no longer interpreted in terms of generalities, but simply in terms of pure, complete, formal relationships. Okay, I sort of get the distinction. But still, I mean, a lot of people, for example, will refer to group theory as abstract algebra, to speak less loosely then you would say that we should say general yes yes yeah abstract abstract in that sense I think it's a more of a relaxed definition of abstract that's yes from that perspective yes but it's more of a common sense idea of the abstract rather than what mathematics on the sense
abstract so I just want to give you a heads up we're 10 minutes to 3 30 If you're able to stay longer and would like to and other people want to stay a little bit longer, that's fine. If not, we can end directly. I think we are good. Next session, sorry for this connection. It seemed that the connection got better, but I couldn't turn on the iPad. Nevertheless, I will try to cover some of those things. but nevertheless you can on your own look at a couple of propositions in Euclid particularly look at the demonstration of parallelogram that the
two sides and opposite angles of a parallelograms are equal and also at a demonstration of let me see which proposition it is its proposition is proposition 35 in book one it says parallelograms which are on the same base and in the same parallels are equal to one another look at this demonstration on your own and as a kind of a homework try to specify exactly how many times
you use co-exact information which steps of your demonstration involve extraction or use of co-exact information in the definition of courts activity gave so that would be kind of homework and also you you will go and investigate what a parallelogram is and learn you know some of the interesting properties of parallelograms have and make explicit so that would be homework and then I can get back to the idea of parallelogram when I'm talking about linear algebra but next session I will talk about a little bit about this relation between mechanics
and geometrical problems between engineering and medieval times since and not specifically just in Middle Ages but from antiquity to late Middle Ages this significant correspondence between mechanics and g engineering and machine means geometrical analysis and talk about a couple of examples from our committee's that then should help us to move forward with Nicole or in this diagrams. And do you have some reading that you might recommend
for this week? Yes, I have a couple of books on this relation between machines and geometrical principles. I will put them in the Google Drive. OK, everybody. I'm going to go down and join the march. So see you in this tweet. All right, see you, Philip. Thank you very much. Have fun. Thanks, Reza. Bye bye. Bye bye. Thank you. Bye, everybody. Bye bye.