Hello and welcome to the sixth session of Maths and Ideas with Raisa Negrasani. I'm going to pass the mic off to him now. Thank you. Okay, so as I was saying that because we are kind of behind, I'm going to straight delve into definitions and common notions and postulates in Euclid's elements. Then we will look at some of the examples and then we start to analyze some of them. And also regarding the, because again I know that we are behind, yes I will add a couple
of sessions to our eight sessions in order to cover some of the other stuff that I was supposed to talk about. Nevertheless, let's start. And if it is possible, we can cover questions and start discussing at the end of the session. So I mentioned a little bit about Euclid systems And in which what we are dealing with is not simply
a diagram-based reasoning, but it's a diagrammatic reasoning. I mentioned the role of the so-called axioms, which are more in tune with kind of like the intuitive idea or perceptual idea of mathematical definitions, rather than being them logical axioms. Also I talked about the relation of Euclid's elements with the philosophical discourse of the time, that ultimately, even though Euclid's elements is a mathematical book, nevertheless, in order to fully appreciate it
in order to understand the conceptual scaffolding upon which Euclid's methodology is built, you definitely need to look at Euclid's elements as a philosophical book, particularly given that Euclid attended Academy. And even though there is no evidence that he ever met Plato, but nevertheless, given the fact that Plato's philosophy was dominant during this time, so was the school of Eliatics, and Plato is simply a refinement, but also a revolutionary
revision of Eliatism, then we need to look at Euclid's elements precisely within the context of Eliottic doctrine of dialectics and Plato's doctrine of forms. And I mentioned, you know, some additional commentary on the idea of generality, particularity, and how the metaphysical role of mathematics is being illustrated within the Platonic systems to which Euclidean system fully commit.
I'm not going to go further into this Platonics idea of mathematics. Hopefully the next course, the spring course, I'm going to give a full series of lectures on Plato in relation to Christocratics. So let's start. So the first thing that we are going to look at is the role of axioms, qua the Phoenicians in Euclid systems and what they are. So Euclid realized that not every geometric fact can be proven. Because every proof must rely on some prior geometric knowledge.
Therefore, any attempt to prove everything is doomed to circularity. In this sense, Euclid knew that it was necessary to begin by accepting some facts without proof. And these facts that are being accepted without proof are called definitions in elements. So he began to choose postulating five simple geometric statements, the most basic definitions. So Euclid postulate one, it says to draw a straight line from any point to any point.
which is, we know, is the construction of a line segment. Postulate 2, to produce a finite straight line continuously in a straight line. Postulate 3, to describe a circle with any center and distance. Postulate 4, that all right angles are equal to one another. Postulate 5, that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than the two right angles, which is, this is precisely connected to parallel postulate. So, within this rudimentary definitional system, the first three postulates, namely draw a straight line, produce a finite straight line, continuous in a straight line, and describe a circle, are constructions and should be read as if they began with the words, it is possible to draw a straight line from any point to any point.
For example, as I said, it is positive late to produce a finite straight line continuously in a straight line, or it is possible to describe a circle with any center and a distance, namely a radius. So the first three postulates are postulates of construction. They are generally understood as describing in abstract, idealized term what we ought to do concretely with the two classical geometric construction tools, namely a straightedge and a compass.
The last two postulates, however, are different. Instead of asserting that certain geometric configurations can be constructed, they describe relationships that must hold whenever a given geometric configuration exists. Now, within this framework then it is important to understand why postulate 4 is needed and what was postulate 4 that all right angles are equal to one another now the
postulate 4 is important because Euclid's definition of a right angle applies only to an angle that appears in a certain configuration one of the two adjacent angles formed when a straight line meets another straight line in such a way as to make equal adjacent angles. It does not allow us to conclude that the right angle appearing in one part of the plane bears any necessary relationship with right angles appearing elsewhere. postulate for can be thought of as an assertion of a certain type of uniformity in the plane right angles have the same size wherever they appear and we will
see that basically this is possibly is used in part of the most important problems and that's when two straight lines intersect the angles on the opposite sides are always equal which is you know predominantly used in not every but you know so many of the most important complex proofs of theorems. Positively 5 on the other hand, and what was positively 5? Positively 5 was saying that
if a straight line falling on two straight lines make the interior angles on the same side less than two right angles the two straight lines it produced independently meet on that side and which are the angles less than the two right angles so possibly finds instead of zers that certain geometric configurations can be constructed they describe relationship that must hold whenever a given geometric configuration exists now again from this perspective possibly five says in a more modern
description that if one straight line crosses two other straight lines in such a way that the interior angles on one side have degree measures adding up to less than 180 namely less than two right angles then those two straight lines must meet on that same side of the first line Intuitively, it says that if two lines start out pointing toward each other, they will eventually meet, because it is used primarily to prove properties of parallel lines. For example, in Proposition 29 in Book 1, to prove that parallel lines always make equal
corresponding angles with transversal, namely a straight line that cuts to other straight lines. So Euclid's fifth postulate is often called parallel postulate, and we'll see it is one of the most important postulates in the rank of definitions, precisely because it is being exploited in order to prove Proposition 27 and its converse Proposition 29, both of which are required for Euclidean system, which is based on the invariance of distance between
two parallel lines. Any questions before I move forward? I mean, is this clear that's what PASTIV-LATE 5 says? PASTIV-LATE 5 says, so let me just turn on this. Reflector.
interior angles on one side have degree measures adding up to less than 180, namely less than two right angles, then those two straight lines must meet on the same side of the first line, namely this. Okay. Are you seeing the screen or me?
Sorry to ask. Right now we see you but we did just see the screen there. Okay, super. so in addition to definitions namely rudimentary postulates we have something else which is extremely important and ultimately is what you might call to be the inferential licenses of construction and these are called common notions in elements and I will talk about what do I mean by that they give inferential licenses in the process of construction and Euclide proofs so following the
basic five postulates Euclide also states five common notions which are also meant to be self-evident, namely definitional facts that are to be accepted without proofs. The first common notion is that things which are equal to the same thing are also equal to one another. Things which are equal to the same thing are also equal to one another. Notion 2, if equals be added to equals, the wholes are equal. Notion 3, if equals be subtracted from equals, the remainders are equals.
Notion 4, things which coincide with one another are equal to one another. five the whole is greater than the part so while the first five postulates express facts about geometric configurations you know a point a line a circle so on so forth the common notions on the other hand express facts about magnitudes and of course these magnitudes the way that are being expressed are also can be divided in different approaches which I will for
now I just call them and later on I will describe them what they are and so the approaches to magnitudes which common notions cover divide to two forms of information they talk about two different types of information the so-called exact information and the so-called co-exact information these are the two terms that have been used by philosopher of mathematics kenneth manders so very briefly exact uh information are really about the magnitudes as such whereas coexact information about magnitudes are about
geometric or topological information for example the idea of entailments sorry containment the whole is greater than the part and we will see how for example this kind of topological rudimentary topological information is used by way of common notions to for example prove some of the most elegant theorems in Euclid's elements. As I said I will get back to this difference between exact and co-exact. So most important thing for now is to know that magnitudes are up
object that can be compared, added, and subtracted, provided they are of the same kind. So we had definitions of rudimentary positive related, we had the second order of positive were common notions. Then on the third we have propositions in Euclid's elements. So what are propositions? Now Euclid refers to every mathematical statement that he proves as a proposition. So this is very different
from you know the usual practice in modern mathematical writing where a result to be proved might be called a theorem, which is basically the result of a complicated proof procedure. Now a proposition, which is the result, that requires proof but is usually not important enough to be called a theorem. A corollary is an interesting result that follows from a previous theorem with little
or no extra effort. a lemma is a preliminary result that is not particularly interesting in its own right but is needed to prove another theorem or proposition namely you can understand lemma as it is intermediate between different problems geometrical problems so even though Euclid results are all called propositions the first thing that one notices when looking at them is that like postulates there are of two distinct types some propositions describe
constructions of certain geometric configurations you remember that we talked about how you construct a equilateral triangle using circles and the line segments now traditionally these kinds of propositions are called problems but in order to you know distinguish them from the common sense idea of problem for now we can call them constructive problems because they describe constructions of certain geometric configurations for example the configuration the construction of an equilateral triangle and these are
usually stated in the infinitive to construct an equilateral triangle on a given finite straight line one must to do such and such but also like the first three rudimentary possibly namely definitions they should also be read as asserting the possibility of making the indicated construction it is possible to construct an equilateral triangle on a given point of a straight line other propositions which are called theorems assert the certain relationships always
hold in geometric configurations of a given type for example its proposition for in the book one the side angle side congruence theorem that two triangles whose two sides and the two corresponding sides and the angle between the corresponding sides are equal can be said to be congruent namely equivalent. Or for example proposition five in loop one, the base angles of an isopic triangle are
They do not assert anything about the constructability. Instead, they apply only when a configuration of the given time has already been constructed. So some of the propositions which we call constructive problems are about how to construct or the possibility of construction. Whereas the kind of propositions that we call theorems, they are not about construction as such, but they are talking about the kind of configuration, or the kind of links that emerge between geometrical configurations that have already been constructed.
So in a sense, rather than elaborating the constructability or the procedure of the construction of a certain geometrical configuration, they talk about relationships that always hold between constructed geometrical configurations so for both construction problems and theorems Euclid's propositions and proofs follow always a predictable path most proposition have six distinguishable parts Proclus which
I mentioned you know he has that famous commentary on Euclid's elements differentiate these predictable patterns in both constructive construct construction problems and theorems so the first one is enunciation the stating in general from the construction problem to be solved or the theorem to be proved for example on a given finite a straight line to construct an equilateral triangle so this is an enunciation then we move to setting out choosing a specific but
arbitrary instance of the general situation and giving names to its constituent points and lines for example let a b be the given finite a straight line. So you set out, you basically demarcate these generalities. Then we move in the process of the third step, which is the process of a specification. And that entails announcing what has to be constructed or proved in this specific sense. For example, it is required to construct
an equilateral triangle on the straight line AB. So we specify the aim of the problem, or what we have to make explicit, make manifest, or demonstrate. Then once we do these first three, which basically are really what you might call to be the framing of what we are doing in the first instance, then we move to a stage four. And a stage four is construction stage, adding points, lines, circles as needed. manipulating the given configuration and by virtue of destabilizing it
diagrammatically we are capable of arriving at new relationships so construction problems this is where main construction algorithm is described in in a Euclidean system. For theorems, this part, if present, describes any auxiliary object that needs to be added to the figure to complete the proof. For example, those of you who are familiar with the Pythagorean theorem, Pythagoras theorem, you know that the first thing that you do, from the top vertex, you draw a line
that divides the square built adjacent or abutin, basically the bottom side. So again, for theorems, this part of construction, if present, describe any auxiliary object that need to be added to the figure to complete the proof. If none are added, it might be omitted.
Namely, we won't have any of this kind of constructive manipulation of our given configurations then the fifth stage is the stage of proof namely arguing logically that the given construction does indeed solve the given problem or that the given relationships do indeed hold we talked about this that even though Euclid system is diagrammatic system nevertheless it has also deductive components and these deductive components are encoded into the diagrammatic influences and hence there is always a logical stage in
proof of Euclid's theorems and that's it's about either solving the problem or showing that a given relationship indeed holds then the sixth stage is simply the conclusion namely restating what has been improved and that's usually that's the line where in Euclid you get QED code or at demonstrator which as I said it is a
wrong Latin translation of the original Greek word in the original Greek word we have monstratum, not demonstratum. Monstratum is very different from demonstration. It is about making something manifest. It is also, again, very tangential philosophical comment on this, that this sixth stage of procedures are very much in tandem with the dialectical procedure of Plato, nonus compostratus,
Can you see the whiteboard? Yes. ok so I mean I have selected a few you know simple but also elegant theorems with some construction ones and sorry propositions some are what you might call to be constructive problems and
some are theorems so the first one that could be and of course there are some missing between the proofs of these propositions but nevertheless you know we can escape some of them and so first I will start with proposition 15 and then we move toward some of the properties of parallelogram and the reason that I am I have used just go so far to the parallelogram because we will come back parallelogram when we are talking about the scale of products in linear algebra. So proposition 15, book one.
Oops. So what does proposition 15 says? Proposition 15 says, if two straight lines cut one another, they make the vertical angle equal to one another. In other words, they say it follows that if two straight lines cut one another, the angles at the point of intersection make four right angles.
We know that two right angles are always equal. That amounts to 180. So we say that let AB and CD be two straight lines that cut each other at the point E.
We see, as we see in this diagram, AED is common between them, and hence we can use a common notion. And the common notion is that when two things basically share something, we can subtract it, subtract a common thing. and the common thing here is aed so once we do that we see that these two become equal so this and this become equal the same procedure can be used to show that this
and this are also equal so this is the most basic form of construction that we used common notion you'd be used a rudimentary proposition the one that talks about that went to a straight lines basically cut each other the sum of the angles is 180 that amounts to two right angles we use this plus a common notion in order to show that when two lines intersect they produce equal interior
opposite angles so AEC equals to DEB and AED equals to CEB now let's move to another example this is and this one I think is one of the most elegant and also monstrous in the sense of showing what is at the stage in order to construct is and
and what is this one so we know that we only have a straight edge and a compass So if we are given a straight line, if we are given a straight line, for example called BC, imagine that there is a point outside of it, called a D. now how are we going to make from this external point a straight line that can be said to be
equal to bc how are we going to construct a straight line segments that can be said to be equal Now, any input as what we should do in order to construct such a thing, given that we have only two things, a straight edge and a compass. Any answer? Well, by equal I mean, for example, imagine that we are creating a line called
dx how can we construct from d one endpoint d a line that can this line can be said to be equal to BC how can we construct simply how can we construct equal line segments Well, one could do it with a compass by just putting the point at B and C and then transferring the part of the compass to D and marking it off on the next line.
Yes, we can do that. But there is one problem here, and the problem is that Euclid doesn't like that kind of proof. Proofs that require transportation of the compass, or superimposition of one figure on top of another figure or moving the cutout that's you are simply moving a cutout by basically moving your compass these are considered to be weak proofs by euclid and euclid tries to as much as possible mitigate these kinds of proofs because there is no reason for us to imagine when we are
moving our compass from one point to another point or a straight edge, the distances are being preserved throughout this transportation. So how can we really make a proof or a form of construction, and this is basically a constructive problem rather than the theorem that can use minimum amount of such moves or none if possible So the proof goes like this.
BC, and we have something here called you H are given external point Now the first thing that we want to do Even we know how to construct it If we connect a to b and once we have a B we try to make an equilateral triangle we know how to make
an equilateral triangle so I don't go over this an equilateral triangle we call this the D making this is easy we have seen how you make equal equilateral triangle so connect a to be get a line segments and make an equilateral triangle so a B B D and a D are all equal so we do that we get the point
what we are going to do is using a compass we are going to make a circle from point D that we acquired by constructing an equilateral triangle and the radius of the circle is going to be BC our given line the one that we have to that we already have
we see that the circle cut the extension of db. For example, let's call this for now f, and let's call the extension of dA E. So the circle cuts DF and we call this point G.
Now this new circle, I'm sorry for the exact, I mean the drawing is kind of bad, but nevertheless that's the whole point and you believe you do not need to make good drawings all you need is to understand the diagrammatic relationships between the configurations so once we create this new circle we see that the circle cut the
line DE at L now the line that can be said to be equal to BC is AL of course in our diagram doesn't look equal but nevertheless logically they are equal by following the diagrams now the proof of it would be like this actually let me this has got a little bit... So B
And this circle we call it H, circle we call it K. What was the radius of K? Oh, K is, you see, is basically D. DG. L. DL. or dl so basically what we are doing is first making a equilateral triangle getting d and d would be the center of our second circle
b would be the center of our first circle which basically the radius of it is our given line bc bc and bg are as bc are equal because the radius of a circle whose center is b now once we got this this dg G, it's actually this needs to be, this is kind of like really bad diagram. So GD and GL are also equal. And that's how we achieved it by basically
creating a first circle, cutting the line BF at point G. Our DG would be the radius of our new circle whose center is D. Once we create this circle, then we cut the line D at point L. Once we do that, then precisely by using, subtracting basically and swapping between AD, DB, and AB, We are capable of proving that Al and BC would be equal. Now let me get to a better diagram of all of this. That shows how it's been constructed without the sloppiness.
So our problem was like this. given a point, for example, D outside of AB, how can we construct a straight line, one of its endpoints, D, and for example, one of its other endpoint would be F? How can we say that DF is equal to AB? And how can we construct a segment from D that can be said
to be equal to our given line AB? The first thing that we did is that we created an equilateral triangle, DAC. c once we constructed this we do something else we create a circle the center is a and its radius is ab we see that it cuts the extension of ca we also extended the line C A to E or to anything and then the circle we see that cuts C X at point E. So we created
the first circle. Now once we have that, once we have a circle whose center is A, its radius AB and AE, we construct another circle and the center of this new circle is C. Its radius is CE using the new point that we have acquired, point E. We do the compass from the center C, we make a new circle, the green circle. We see that it cuts the extension of CX
at point F at this point. Now, once we have this, We have a new set of configurations that we can investigate. We know that AB equals to AB because there are radius of a circle whose center is A. We also know by virtue of a, that we constructed an equilateral triangle CD, CA and DA are also equal sides of an equilateral triangle the definition of an equilateral triangle now CE and CF CE and CF are also equal why because these are the
radius of our second circle the green circle that we constructed so again something that we should always to do in any of the propositions in you think that we are trying to prove always try to decompose your given magnitudes and see the relationships between their parts so we know that C e equals to C a plus a C a plus a we know that CF is equals to CD and D F but by virtue so
we make a new basically two sides of our equation CD plus DF equals to CA plus a but looking at this is stuff that we have here we see that CD and CA are equal CA and CD are equal so we know using a common notion we can because these are equal we can subtract them from our equation so df and a e df and a e are equal we also have a B and a e are
equal remember a B and a are equal because they are the radius of the circle center is now once we have this we have again another common notion a e we subtracted so then what does this says it says that df and a b are equal namely we have created using a compass a straight line segments df that can be said to be equal to a b questions first things that we you do the constructive part is the one that
requires you know thinking how you can manipulate the system that requires you know some practices and thinking about how you can manipulate your configuration by adding new constructible elements. But after that, everything is simple. You start to write down all of the relationships that stand between your parts. So for example, we created a first red circle whose center was AB. Obviously, a circle like that cut the line CX at the point E. So AE and AB are equal.
Write down the terms. We also know because we created an equilateral triangle, DA, CA, and CD are equal because there are signs of an equilateral triangle. Write these down. And then start to see what other kinds of terms you can add. As I said, make sure. This is platonic dialectics. Everything needs to be manifested. And the way that you manifest these things is by way of making explicit what is already implicit in your sides of equation. So if you have, for example, line CF and line CE, write these two lines or decompose these
lines into their basic parts. And what are the basic parts? It's CD and DF, part of an equilateral, side of an equilateral triangle, and plus the segment DF. And CE is, again, a part of an equilateral triangle plus the radius of your red circle, namely AE. Write these things down. Decompose everything. Separate the parts of a whole that you have acquired. Once you separate these parts according to their types, then I start to integrate them back into new unities. So once we acquire C to be equal to CF, because they are the radius of our green circle, we decompose them.
we separate them to their parts according to their kinds so we would say that ca plus ae equals to cd plus df now once we have separated things according to their kinds we can create new unities by saying that okay ca and cd are equal because they are sides of an equilateral triangle so we factor this out and what remains from this process of decomposition and integration is that df and ae are equal df and ae are equals if df and a are equals by virtue that also ae and ab
were equals the radius, the rating of our first red circle then by virtue of this we would say that Df and Ab are equal Ab is our given line and Df is a new line that we produced from the external point D so this postulate has both constructive elements and elements of it being just a theorem is that
right yes but basically the possibility is not interested for you to make the proof is simply ask for you to construct a line but nevertheless you need to prove why is that this line that you have produced is can be said to be equal to your given line namely ap and that's when basically it can be treated the proof of that construction can be treated as a theorem So this is actually considered to be one of the most complex forms of construction. And it is, you see, it is quite an elegant and a phenomenal form of construction.
Which number is it in the PDF? It is proposition, I forgot and I think it's, let me see. All right, I can look for it too and I can post it later. I think it's proposition two. OK. I mean, as Philip was saying, you know, you can do this by, for example, if you had a
ruler measure a b and then distribute the new measurement that you have acquired transfer this to point d and start to draw a line but the problem with this kinds of weather compass or ruler or straight edge is that you transfer information from one from your from the given to what you ought to construct. There is nothing, there is no reason for us to expect from these very rudimentary steps that the information that we are transferring is actually going to be invariant. Yes, they are invariant,
and this is the property of the Euclid system, but Euclid cannot reasonably say that. he needs to demonstrate it. He needs to come up with a procedure that does not require this kind of, basically, transport or transfer of information. It's simply, you see that there is a, like a, really a kind of a dialectical, in a very platonic sense of dialectics, a procedure is involved here. You are given certain data. These data are your given.
Now, to do philosophy, and this procedure is philosophy, you should not speculate something that is outside of your given data. You simply need to use what you have been given. Manipulate them in a certain sense by disintegrating and integrating them so you can come or arrive at new unities, new manifestations out of your givens. Questions?
proposition 16 says in any triangle if one of the sides be produced the exterior angle is greater than earlier that either interior and opposite angles so what does it exactly say for example we have a triangle ABC we extend this line to D It says that this should be greater than either interior and opposite angles.
This and this. this so how are we going to go to prove this any indication this is again one of those really elegant proofs that requires a simple manipulation but the manipulation is is quite significant of course I will show that there is a gap here in this proof that diagrammatic reasoning of Euclid can never uncover this gap but I will hold this elaboration of what this gap is until
later. Sorry, did you say this is for the parallel postulate? No, no, this is not for parallel postulate, but it basically, it creates a, you know, a significant new proposition that is extensively being used in proving other propositions. What I was saying that there is a gap in this proof the diagrammatic reasoning cannot really understand or make explicit and we will see that essentially this is one of those very unique propositions and non-euclidean geometry we will show that this doesn't really work the way that
Euclid tries to prove this but nevertheless let's go now how how are we going to prove that ACD is greater than any interior and then either interior and opposite angles of the triangle ABC Any indication of what we have to do first? Well, maybe you could start by drawing the circle that BC is the radius of.
If we make that circle, it would be something like this. How are we going to use this? I mean, this is great. I mean, that's exactly what you have to do. You will start to impose new things that you have constructed onto your given configuration. see how you can manipulate it, a meaningful manipulation that can lead you to what you require to demonstrate. Now, a circle is kind of like, doesn't really, it might serve for some other proposition, but here it doesn't really help us.
Something else helps us. And that thing is that if we create a line that intersects A and C at its middle point. We call this E. Then we extend the B, E that have cut AC to two equal parts equally. So, Be and F, Ef and Be, we draw them so they be equal.
Once we acquire the point F, we simply connect it to point C. We do not know what FC is. We are going to investigate what FC is. So essentially, we have constructed something that has the potential to make explicit the relation between two triangles.
And what are these two triangles? A, E, B. And F, E, C. now let's look at these two triangles we had a proposition that when two lines intersect one another they create equal opposite angles AC and FB have intersected one another right so these two angles are equal angle and the way that you write
angles improves start with the side then angle e the point of the angle and then the other one the point of the angle of the vertex should always be at the middle equals to F e C but by virtue of the kind of construction that became carried out we know that be equals to ef because we extended be to the point f so that ef would be
equal to b that was part of our own construction plus the first stage of our construction was that we create a line BE so that BE cut AC the side of our triangle at its middle point E. So if that's the case it has intersected or cut AC at its middle point then we can say that AE also equal to EC. So we haven't talked about the conditions of congruence, namely
the equivalence of triangles. This is one of those conditions. They can be side, side, side to triangle that has three sides are equal to triangles that their sides and their interior angles are equal to angles to corresponding angles and the side that they basically envelope are equal also can be said to be congruent so what we have here this is a side this is a side and this is an angle this is called the condition of this is s a s side angle side by virtue of this
this type of congruence namely equivalency between two types of configurations here two triangles we can say that because of this a e b and f e c are equal these two triangles this and this
Now, if we have created two equal triangles, then we can also start to make manifest what precisely because these two triangles are not equal, what other parts of these two triangles can be equal? Obviously, all corresponding angles and corresponding sides are equal because these two triangles has been shown to be congruent, namely equivalent. And
this is B A E equals to E C F Remember, this is a mirroring condition, a congruence. AE and EC are equal. so and also a e b and f e c are equal then the angles other angles are equal
is be a and see e sorry this is I made a mistake here this is this is yes yes that was correct ECF so this and this are equal now here we have been resort to another common notion and that was the whole is always greater than its part remember what we are going to demonstrate is that ACD
the exterior angle is always greater than either interior and opposite angles. So we want to show that ACD is always greater than for example BAC. Now we start to make manifest a relationship between two angles of our two constructed triangles and that was the equivalency, the equality between angle BAE that I have marked
an angle ECF, again that I have marked. But we know that ECF is really part of a greater angle, diagrammatically speaking, and that greater angle is ACD, the ones that we wanted to demonstrate. So if ACD greater than ECF, and by virtue of the equal correspondence between ECF and
BAE we can say that ACD is greater than BAE. Part-whole relationship. Whole is always greater than its parts. The whole is the angle ACD, the one that we ought demonstrate and the part is angle ECF that we created simply by constructing and manipulating our original triangle creating a new triangle and that triangle was the triangle FEC this triangle
You see, Maria, we can't, there has been, of course, there has been an attempt in the 20th century, particularly by Tarski, to try to encode these semantically. But the thing is that it is far more complex than what you might think. You can't really put them, encode them to syntax really. In fact, again, this has a number of reasons. One of the reasons was the one that Danielle Macbeth was talking about, the pop-up function
of the diagrams. because diagrams once you treat them as constructive elements that they basically they acquire new objects or elements for example the objects that we acquired here by using by creating the diagram was point e a middle point e which is the middle point between the side AC of our triangle but and also we created a point F a new point now the thing is that's precisely because as I said Euclid
system is not ultimately logical but diagrammatic you can only arrive at this proof diagrammatically, not syntactically. Precisely because once you get rid of the diagrammatic inferences, there is no reason for you to anticipate that actually F can ever be connected to C. I will talk about this later. This is in fact one of the most unique gaps in the diagrammatic reasoning of Euclid. So there are so many reasons that diagrammatic cannot be sufficiently or consistently be
encoded into regular syntax. Cartier has done that, but the results of his proofs and his Euclidean systems are extremely more complex than Euclide system. fact it makes working with Euclid systems extremely difficult so so there is it as I said this encoding to syntax is difficult and that's due to two things the advantage of the diagrams but also there is this advantage advantage of the diagram you can think of this advantage of using the diagram at construction precisely in the way that Daniel McVeck was talking about it is that idea that they unfold configurations and make explicit implicit
connections according to the inferences that they license once these inferences are being followed you can see that new elements that you couldn't even imagine are popping out in the course of your construction. The middle point E and also the other point F. So this is the advantage of using diagrammatic reasoning that makes working with Euclid's system in this diagrammatic framework extremely easy. But also there's a disadvantage. The reason that these pop-up elements crop up is that because they are abiding or they are conforming to precisely a very
intuitive in the Kantian sense intuitive manipulations of your visual components their grammatic components but if you had moved from the imaginative a scaffolding that afford your diagrams to a new unity of imagination, let's say this new unity of imagination and your diagrams would be something like Lobachowskian geometry or non-Euclidean geometry, then there are no reason for us to expect that these pop-up elements will appear in the course of our inferences. there is no reason in fact for if we extend the point B into a segment BE this BE will
ever intersect the side of our triangle AC and that's basically the point that non-Euclidean geometry tries to show. And to show that there are in fact gaps in the diagrammatic reasoning of Euclid's elements. But these gaps cannot be seen from the axiomatic perspective of Euclid's system as such. You need to step outside of these definitions in order to see these gaps. Questions?
Anything? Philip, Hunter, Theo, Maria. Okay. Okay. Maria just posted another question. She said, is there a connection to Leibniz's situational analysis later? Yes, there are, but I mean these relations are not essentially interesting.
can be applied to any philosophy of the space, whether it would be Leibniz or Grasman, Whitehead. Yes, precisely because these diagrammatic inferences index some rudimentary topological information and essentially analysis situs also by me and analysis situs also presupposes some of this topological information but the way that they approach and they handle is geometric topological information are quite different and this is due to the fact that you know Leibniz analysis is a
more rudimentary form of topology whereas Euclid's diagrammatic inference is essentially geometrical it's not really topological in that modern sense of topology I will talk actually talk about when we get into the idea of extensity intensity in mathematics I will talk a little bit about like needs situational analysis whiteheads theory of extension and grassman's theory of linear extension they are more in convergence and you bleed and for example likeness also a really really great I mentioned I forgot I think it Dona mentioned a really great book on like this analysis it's called a
you're doing is laying out the information from what you can build or deduce from, but but there's no way to challenge the premise. Right, correct? Yes, absolutely. Well, this is the whole idea of working inside axioms. Right. You can never challenge your own axioms. Right. You can create models out of your axioms. And of course, Euclidean geometry can only make one model of three, basically three-dimensional coordinates model. but nevertheless this whole idea that you know this is very again very very idea of metaphysical idea of mathematics in Plato as you see mathematics all in the analogy of divided line between this between the
basically objects in nature, namely empirical object and the realm of ideas. Now the thing is that it's an intermediary realm between basically the immutable and also unitary always being one coming in one realm of ideas and changing and plural domain of objects so basically mathematics is has some qualities of immutability definitions
are being fixed, but also has some properties of multiplicity, like plurality of objects. So this idea that from this perspective you can see that mathematics essentially has the most appropriate structure to intermediate between, for example, ideas of forms and a structure of reality, objects in nature. And precisely because of this, so you have this, that mathematics is understood as being a model, its metaphysical status. And the way that you can make these models is by accepting
some facts. But these facts are not being given to you by simply sensibility nor by ideas as such, but by intermediary facts. And Euclid's system shows that precisely your axioms are both perceptual but also being logically handled, as if you still have some components of forms, immutability, and being of one, and also some elements of multiplicity and being of belonging to realm of the sensible, namely intuitions and stuff. So once you have this, and these are your given datum, and the whole idea that this
is essentially a philosophical thing, you should never sidestep from your given datum. You can speculate how to go beyond your given datum, but you can't carry out this form of speculation by contradicting the facts that you yourself have posited, namely your definitions or your axiom. You need to start to construct a model that show you the limitations or the alter-riches of your axioms. Once the model reaches basically its limits, then that's when you need to construct
new axioms, new facts, and show why is that these limits arose in the first place, and whether these axioms of your old system can be held within your systems or not. And then the creation of those new axioms, that's sort of the frontier of the development. Yes, it's a development, but not in development in the sense of what you might call to be the progress. It's a development in the sense that you encounter in a very, again, platonic sense, a flash of new truths.
You are not simply progressing or overextending your own previous truths, but you encounter a new type of truth afforded by your new set of axioms. This is Plato's idea in Republic that truths are always, you encounter truths always in a flash of light. What you might call this an event. Is there a definite connection between the creation of those new axioms and the gesture that we've been talking about throughout? Yes, but I think, yes, in the sense of destabilization or constructing a system to its ultimate limits
to see basically how much expanse of reality can cover. In that sense, yes, just sharing that idea of destabilization or taking a system to its ultimate, but not in a kind of perceptual way that we are talking about, or the more kind of creative but more of a speculative gesture. Remember, mathematics is always between forms and objects. So you can always see the limitations
of your mathematical system by comparing them with the universality of forms. I was going to say that Macbeth's idea of pop-ups is very, very interesting as part of Euclid's geometry. The notion that on that straight line, it's not clear even looking at it, that the disposition towards an equilateral triangle is contained. Yes. Yes. That's precisely the sort of act of diagrammatic construction that kind of releases virtual
possibilities or whatever. And so it strikes me that both intuitionists and formalists can look back at Euclid and take a part of it out to support their view of what mathematics is. Well, I mean, this is exactly what happened in the debate between, you know, proponents of Brower, like Hayting and Kolmogorov, and Hilbert. In fact, there is this book, I will, I have it in physical copy, I don't have, unfortunately. Hilbert is quite, actually, very Kantian. And he, in fact, defends all of these intuitionistic accomplishments, the idea of imagination.
But it comes up with a different interpretation. And it is absolutely like in the debates about fundaments of mathematics in the 20th century. This is exactly what happens. They really do look into Euclidean geometry, into the idea of Kantian intuition, and how both camps in fact defends these intuitive components, but the interpretation is different. And I think the more I have read Hilbert, Hilbert is in fact far more accomplished than Brewer because Hilbert is essentially a transcendental thinker. He doesn't believe that intuition can ultimately,
basically his point is like this, that even though intuition can give us a massive arsenal of construction, but mathematics ultimately its alliance is with the realm of forms in a platonic sense, namely transcendentals. And those transcendentals are the one that we can ultimately renegotiate the limits of imagination or the manifold of intuitions, not the other way around. So what's the name of the book that you have? Let me get it. I have it here. One second.
So this is the book, From Brewer to Hilbert, The Debates on the Foundations of Mathematics in the 1920s. Okay, great. And for Hilbert, it's The Foundations of Geometry that you should read on this? Yes, yes, but also he's opened his conversations with Arant Hayting and Kolmogorov. Okay. Okay. I mean, people usually unjustifiably attack Hilbert as this rigid, you know, person who promotes formalism.
But I think the way that Hilbert ultimately, philosophically, think about formalism is quite sophisticated. Yes, his idea of completeness of arithmetics is the one that is being crushed by Godot. But nevertheless, his idea of formalism and why formalism is ultimately important, I think is quite a very, very sophisticated position, far more than any of this, you know, kind of French turn in mathematics, try to basically dissimulate Hilbert's position. And the French turn would be who?
I think all of the, you know, you can think of this, the whole legacy of Poincare. And particularly the new generation of philosophers of mathematics like Giuseppe Longo, like Bernard Taizier, like Alan Berthaud, like Stanisla Dohan, Jean Petitou, the so-called GEOCO group, Geo-Co. It's a shorthand for geometry and cognition. And where do Cavallier and Lotman come in?
I think Lotman is a very different story. Lotman is a Platonist. He's a hardcore Platonist. And in fact, in the Manifestable of Geometry and Cognition, written by Tazir Longbore, and John Pettitou, they actually endorse Lottmann, but they attack blatant Platonism of Lottmann. But I genuinely think that people's interpretation of Plato and his stand on metaphysical status of mathematics are quite sloppy. I mean, it's just like driven from commentaries of commentaries of commentaries of Plato. has nothing to do whatsoever with Plato's sense toward mathematics.
Kavai and Lottman are, you know, I think they are quite open-minded. They see philosophy in terms of plurality of philosophical positions. they don't take sight really that's that's I think really is particularly Lothman is quite phenomenal so I lost track of time a little bit but I think we're ending in just 10 minutes okay sure so I will so as I mentioned that don't worry we will extend like two or three sessions on top of our eight sessions to make sure that we cover you know most of this stuff so I'm going to
so I'm going to finish with the one of the most important but also simple propositions in Euclid and then the next session we will look into the conclusions of this proposition and also we end Euclid's elements with analysis of some of the propositions that we have been talking in terms of a more robust philosophy of mathematics like it from today's perspective why is that there are gaps in Euclid's elements and what are the consequences of these gaps also we talk about why is that the diagrams in Euclid elements work and particularly I will concentrate on Kenneth Manders exposition of Euclid's elements the one that Danielle
cheating going on at the middle of elements. So when he reaches Proposition 27, and also with Converse, which is Proposition 29, which are talking about the properties of parallel lines, he drops his rigorous method of diagrammatic reasoning. He resorts to method of reductive ad absurdum, proof by contradiction. Now proof by contradiction from a constructivist perspective in mathematics is a weak or invalid proof. For example, in Breuer intuitionism, we see
that basically ultimately proof by contradiction uses double negation, namely elimination by double negation. But in so far as elimination by double negation is weakened or suspended altogether in intuitionistic school constructed as a school of brah, proof by contradiction is not really a good or a valid proof. Because ultimately proof is not about proofing or basically showing that something exists
shorts basically circuited matter but it's about a procedure of search or finding for that which exists and this is really the modern definition of proof a search a search that can be also understood as a construction a search of of a for a generality for in the platonistic sense for a form to show that it exists rather than in a procedural manner rather than by simply showing exists we need to find this this is a proper philosophical way of proof
rather simply for going with the procedure of search or finding a particular generality and simply showing that it exists so this is this is the proposition 27 proposition 29 which relates pertain to parallelism in Euclid elements in so far as they use reductive adapts or the improved by contradiction, they show a very, very weak point in the overall scope of Euclid's elements.
Given the fact that parallelism, everything before these propositions, has been sketched out has been constructed in order for us to arrive at the parallelism properties. Now right when the climax of the story happens, Proposition 27, properties of parallelism, it seems that Eugliffe cups out. He doesn't abide by his diagrammatic procedure of finding and construction. He resorts to proof by contradiction. We have a still element of construction, but this element of construction is subordinated
to the element of proof by contradiction. logical, basically, priority of reductio ad absorptum. I will talk about, next session, about this weak hinge in Euclid's elements and how non-Euclidean geometry basically assaults this weak hinge to show that Euclid's, the gaps in Euclid's elements can in fact be made explicit by how Euclid, at the most important stage of his
book, resorts to a method of weak proof, proof by contradiction, or in modern terms, intellect proof. So, proposition 27 tells us that given two infinite straight lines which are cut by a transversal, if the alternate interior angles are equal, then the lines are parallel. In other words, it says that if a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.
and CD to a straight line in fact two infinite straight lines and let's EF to be a transversal a straight line that falls on them cuts them now the proposition 27 says that if the alternate interior angles are equal then
the lines are parallel, namely A, B and C, D, never going to cut it one in infinity. the sign for absurdum or contradiction now of course you believe can't do this
unless using a proof by contradiction now what is exactly a proof by contradiction in this case. We assume that the conclusion we are trying to prove is false and then show that this leads to contradiction of the hypothesis, the premise of our problem. In other words, to show that P implies Q, one assumes P and not Q and then proceeds to show that not P follows simply you have arrived at a premise which is the contra position of your given premise so
what does this all mean simply it means that the course of the proof goes like this you move toward you construct your proof as usual but toward this basically showing that what you are trying to prove namely that two lines of the transversal card and this transfer also has produced opposite alternative interior angles are equals produces in fact lines that
at infinity can meet one another. Namely, these two lines can be said to be not parallel. So let's assume this. So we reconstruct this. We arrive at a result and then this result shows that it contradicts what we have assumed initially, namely our given data. So basically you, instead of deriving your hypothesis, your assumption, your premise, derive or acquire or obtain the contraposition of the assumption or hypothesis or premise.
If P is our hypothesis or premise, throughout the course of proof by contradiction, we show its contraposition, namely not P. means that we prove the contrapositive of the given proposition instead of the proposition itself so this is the soul of proof of contradiction in this case now let's see how you complete does that.
So let AB, let me change the color of the pen. Let AB and CD be two infinite straight lines and let EF be a transversal that cuts them. let at least one pair of alternate interior angles be equal without loss of generality for example let's a H j to be equal to H j
A, A, J, and H, J, D. Now assume that the lines are not parallel, namely at infinity they are going to meet one another at some imaginary point called G. This. Without loss of generality again, let G be on the same side as B and D.
because of course you know we need to always say how we are going to construct it this proof by contradiction the diagrammatic way of it cannot be accomplished unless we explicitly say that we are going to construct G namely the imaginary point where the line a B and C D meet on the same side of the B and D. Simple. Simply because as I said, diagrams can only tell you so much. You need sometimes make it explicit
how you are making this construction in order for you to be capable of making diagrammatic inferences. if G is not going to be on the same side of B and D where you have extended your line AB and CD, then there is, then you can't arrive at this kind of method of contradiction. So since A8G, A8J, is an exterior angle of the triangle
from the proposition that we investigated that said external angle of triangle greater than internal opposite, the one that we just were talking about it follows that angle A H J is greater, this angle is greater than the angle H, J, G.
so this this one is greater than this one but this was the premise of our proposition that it was saying that in In fact, these opposite alternative angles are equal. It was telling us that AHJ is equal to HJG or HJD.
Hence we arrived at a position that we have created a situation that contradicts our given data, our premise. And hence we showed that proof that if two parallel lines, if two lines cut by a transversal that created opposite-equal alternative angles, if these two lines were met at infinity, namely if they were not parallel, this leads to a contradiction, a reductio ad absurdum.
And what it contradicts is the premise or hypothesis of our own proposition, namely the equality between two angles AHJ and HJD, opposite alternative angles, made by a transversal falling on two equal straight lines. So this is one of the strongest proofs by contradiction in Euclid's elements.
Similarly, you can do this as a homework. You can see, you can look at Proposition 29 and see that Proposition 29, which is the converse of this proposition, also used as proof by contradiction. So two moments where basically everything needs to be abiding by the same strictly inferentially licensed diagrammatic reasoning, we resort to proof by contradiction. Precisely because Proposition 27 and Proposition 29 are extremely important for everything
that comes after in Euclid's elements, we see that there is a kind of a, at least if not mathematical, a philosophical weakness that starts to emerge. Of course, everything resumes back after this, after these two propositions, more or less to, again, the tight diagrammatic reasoning that we have been dealing with. But really, this is the moment that we are getting into properties of parallel lines. And because everything after these propositions are really investigating parallel lines, for example, like parallelograms, and so far as these two that make explicit the properties
of parallel lines are being proved by proof by contradiction, we see that the whole justification for studying properties of more complex parallel properties then are somehow, even though they are from a, you know, the systems or from a logical standpoint are valid, but they don't have the same robustness we look at them very suspiciously and I will talk about the kind of gaps that arise from now on in our basically in our propositions precisely because this
This is the moment that Euclidean geometry or Euclidean systems become a system about certain types of parallelistic properties. But non-Euclidean geometry shows that there are other kinds of properties with regard to parallelism. And they can only be extracted once we relinquish or abandon the kind of diagrammatic reasonings that we are using in Euclidean geometry.
So what I'm trying to say is that Proposition 27 and Proposition 29 are in a sense the limits diagrammatic reasoning in Euclid's system. They are weak because they are the limits. It has to resort to proof-of-white contradiction because this is really the limit of diagrammatic with. Questions? It seems like you can probably see that because there's no pop up.
We have pop up, in fact. I mean, if you look at it, G is a pop-up. But you see, G is not pop-up in our traditional sense that was basically we were using this as a constructive element. We suddenly twist it, and we use G as a logical element rather than a constructible diagrammatic pop-up element. We use G and bring it back into the logical framework. We use it as basically an element that can be exploited for indirect proof, namely proof by contradiction. So, in a sense we have a pop-up element, but this pop-up element is not being exploited
or used in the same way that we were using our pop-up elements in the previous propositions, namely within the procedural diagrammatic inferences. switch suddenly we twist into logical framework proof by contradiction. And because it's hypothetical and it's also not really there. Yes, yes, but also yes absolutely and the whole thing that this idea of hypothesis building that we have created it is only can be justified within this system but if we were really using the methodology of diagrammatic reasoning,
we couldn't arrive at this hypothetical point. So basically what Euclid does, he shouldn't never outstep the implications of its axioms. But in fact, as you say, the hypothetical point G is an instance where Euclid in fact transgresses or violates the implications of his axioms. For us there is no reason, a priori, to assume that we in fact can construct a point G through which we can arrive at a proof by contradiction.
And this is basically where non-Euclidean geometry emerges by showing that there is no reason for us to anticipate that we can in fact construct this hypothetical G. um yes yeah it is it Maria is right it is not properly diagrammatic reasoning but
reasoning that is based on diagrams is a diagram basically in all of the stuff that we have been working on we have had diagrammatic reasoning but here we have a diagram based reasoning the diagram based reasoning in so far as the method of construction doesn't fully conform to the methodology that Euclid so far has been complying to. So why isn't the introduction of something like point G more like book one like the definitions where he's introducing these
constructible parts that have limitations and function in certain ways? Well, because you see G is essentially a hypothetical point. It is a point, but it's also a hypothetical point. Hypothetical point in the sense that it explains some other geometric configuration. And what does it explain? What does it try to explain? parallelism that two lines meets two parallel lines can never meet at infinity
so it is not it can never be a definition it is a point yes like any other point is a definition but nevertheless it's a hypothetical point in so far as like all hypotheses it tries to explain something a geometric configuration or some or this or sets of behaviors and what are the sets of behaviors two lines cut by transversal this transversal has produced equal opposite alternative angles and if this is the case these two lines will never meet even at infinity but we hypothesize that they do in fact namely point G and
if they they meet then what would be the properties of our system then we see that's angle a H J is going to be greater than the angle H G D or H G H and that leads to a contradiction of the premises While point G is a point, positing is a hypothesis, is positing the hypothesis, the hypothetical dimension of this point is problematic.
Otherwise, it's just a point. But nevertheless, the way that we posited in relation to the behavior of two lines caught by transversal producing alternative angles, that's where things get toward it, at least philosophically. position of this hypothesis is where is when we have a problematic component because positing this is not necessary or cannot at least be shown using the regular diagrammatic reasoning of Euclides
Okay, so if there is no question, we can conclude this session and then the next session I start to look at other propositions regarding properties of parallelism, particularly parallelograms, And then I will get back to some of these gaps that I have been referring to, but also talking about some other properties of Euclid systems and the kind of inference and diagrammatic reasoning licenses. And that will be the relation between exact and co-exact properties in Euclid systems
that I briefly mentioned today. So I just wanted to return to this last diagram that we're using, Syl. So the, it just strikes me that G as a, you know, it's a proposition put forth that has definitional characteristics, right? We imagine that these two lines converge. I guess I'm seeing a similarity between putting forth that proposition and the putting forth of the basic definitions of what a point is described as,
what a line is described as. not essentially because you see the point the definition of the point and that's the most what you might call the axiomatic definition is that where two lines intersect it produces a point the point is intersection of two breathless lines it has no dimension whatsoever so that's the point But here, as I said, it's not simply a definition. It is a definition that is posited with regard to the behavior of another geometrical configuration. And the geometrical configuration is the body of our propositions. Two infinite lines cut by transversal.
The transverse two equal opposite alternative angles. Now, can we posit this definition with regard to this geometrical configuration or not? The position is a hypothetical. Euclide does that, but from a non-Euclidean perspective, In fact, there is no reason for us to say that this is the only basically point that we can add to our system. Namely, this is not simply the only hypothesis that we can posit.
We can posit different scenarios. lines intersect at different points producing different g's or never meet in fact but diverging rather than basically preserving the distance between them so they can converge at multiple points or one point but also they can fundamentally diverge so it can so basically as soon as you move beyond your Euclidean system, you can posit different hypotheses. You can either hypothesize different Gs
or you hypothesize that there never be a G, but nevertheless, the distance or even these alternative angles are not equal. Again, what is really important, the definition, you have a definition, but this definition is posited with regard, with regard, hypothetically, posited hypothetically with regard to your given geometrical data. And what is this given geometrical datum? Your geometrical configuration of two lines cut by transversal, producing two opposite
alternative angles. So it's not just a point. It's a point that tries to make a hypothesis serve as an explanatory component in your system. Yeah, I think I understand that. I guess maybe I'm saying that the definitions themselves when you say like a point is that which has no parts, that's a building constraint from the get-go. Yes, but it is also a fundamental, you see, it is a non-controversial definition, precisely
because it says, it doesn't say that two lines will always intersect or never intersect. It says that suppose that two breathless lines intersect, it creates a point. So it's non-controversial. It is very, very elementary in its being a definition. it does not have a hypothetical dimension it says that it doesn't say that two lines basically always create a point it says that two lines when they intersect they create a point this is the very definition of point and this definition is not hypothetical in every system you get this
Also, one important thing is that your axioms or definitions should never contradict one another. So you have two lines, two lines intersecting one another, two lines intersecting one another creating a point. Your point should not contradict the criterion of two lines intersecting.
Any other questions, any observations, thoughts? Yes, all of the definitions are basically repertoire for constructions. One of the most important things is that your definitions shouldn't contradict one another. And they should have a necessity by their own. They should give basically what you might call to be definitions are your basic novelties.
And these novelties, even though they might be produced from one another, but nevertheless You're expressing different types of elements, novel elements. A point is not just a line. A point is something different. It doesn't have a dimension like a line. It has different qualities. Okay, should we conclude the session then? Sounds good. So we'll meet next week Saturday at the same time.
Sure. And if you have any readings that you'd like. Yes, I will try to upload. I have it in physical copy, but I will try to find the digital copy of it. Kenneth Mander's essay on Euclid's elements. Thanks, everyone. Thank you. Yeah, thank you. Have a good day. Thank you. Bye bye. Take care. Thanks. Bye bye. Thank you. Bye bye.