Geometry — a paragon of mathematical deduction?

all right well why don't we why don't we start um so welcome to the oxford lectures on the philosophy of mathematics i'm joel david hapkins and this is week four and the topic is geometry um i want to start of course by talking about classical euclidean geometry it's probably an ageless paragon of deductive mathematical reasoning of course it's aiming to elucidate the fundamental truths of geometry the mathematics of space and euclid starts with points and lines as primitive notions and he has 10 axioms there's five essentially algebraic common notions and five essentially geometric postulates and then he proceeds to build this monumental edifice proving one statement and then another and then another and in doing so he taught us all the axomatic methods so he showed us how to do mathematics and not only did he show us how uh give us a method for doing mathematics the axomatic method but also an unequaled example namely euclid's elements which of course was the most successful and influential textbook ever written and it was the principal mathematics textbook for over 2000 years okay so let's just talk a little bit about his methods so he has two classical construction tools the straight edge on the compass so the straight edge is given any two points then you can construct the line that joins them and a compass is this device like a v shape thing and you put one point at the center and one point on a and the other at the other tip of it at another point and you can make a circle uh with the given center and containing the given point on the edge so you can construct a circle with a given center and radius okay so these two tools suffice for all of the applications that are given in euclid's elements and i want to just show you um for example uh how to do that so uh let's just construct the perpendicular bisector of a line segment so we're given um we're given two points a and b and we want to construct the perpendicular bisector uh on the segment from a to b and so what you do is first of all you pick this point as the center of a circle and you make a circle that looks like uh that contains this point so this is part of a circle and then you do it the other way so using this point you use your compass to make a circle going like this and you get these two intersection points yeah and then using the straight edge you can draw a line uh connecting them if i had these large geometry tools for using on the chalkboard i would have them here and i'm sorry i don't have them but i hope we can understand it anyways okay so uh we built this line here at this point here so if this was the point a and this is the point b and this is the point p and this is the point q then we've constructed the midpoint c of the line segment a b and this line here is perpendicular to a b okay this is called the method of apollonius it's not actually how euclid proved how to construct the perpendicular bisector but let's talk about that a little bit first i want to say well to construct the perpendicular bisector it's not quite the same thing is to prove that it is the perpendicular bisector so let's let's give an argument for that so why is this point the midpoint why is this point why is this line perpendicular to a b and for that what we can do is we can use these other triangles here let me draw them in yellow so i'm going to make that triangle and i'm also going to make this triangle and each of those yellow lines if you think about it is the radius of the circle because for example this line is the radius when we have center a yeah and this line is the radius when we have center b so each of these lines is congruent to each other because each of them is congruent to the segment a b and now if we think about uh if we think about this this large triangle here so qap you know then uh and in comparison with this large triangle here yeah then those two triangles have uh have all congruent sides so these two sides are congruent to these two and they have the same base there and so therefore by side side side those two triangles are congruent and from that it follows that this angle is the same as this angle because congruent i mean the associated parts of congruent triangles are are also congruent okay and now if we think about this triangle here and this triangle here then they uh they have the same angle here and the same hypotenuse i mean the same side and this side is also common so by side angle side those two triangles are congruent and from that it follows that this side is congruent to this side so therefore c really is the midpoint of a of ap but secondly we also know that this angle here is congruent to this angle by the congruency and therefore uh but they add together to a straight angle so therefore this is a right angle so therefore we found a perpendicular bisector of a b okay so that's basically a proof using those those triangle congruence theorems of side side side and side angle side now maybe if you study geometry in high school then you might have seen the sort of two column proof format which is very common particularly in 20th century american schools then you make a proof by making a list of geometric statements in the first column and for each statement you have a reason why it's true in the second column and there's only certain reasons that are allowed and this is very common style of undertaking geometric proof but it's not at all euclid style if you read euclid's elements then it's written in a sort of open pro style it's quite readable even today i mean in translation of course and uh so it's not in this two column proof style this is a very recent uh innovation okay um now i'd like to talk about maybe how contemporary geometers think about this fact for example that uh that if you do the apollonius construction that it gives you the perpendicular bisector of the original segment because they wouldn't necessarily feel required to undertake uh a proof by straight edge encompass in that style using the euclid's theorems rather to see for example that that line segment p q is perpendicular to a b then we could consider the symmetries for example given a line segment a b then consider the the reflection through the actual perpendicular bisector the reflection of the plane through the actual perpendicular bisector whatever that bisector is that is going to that reflection is going to swap a and b and therefore it will take those circles uh to each other so therefore it will fix the points p and q so therefore the reflection on the actual perpendicular bisector doesn't move the points p and q so therefore the points p and q must be on the on the line that is the perpendicular bisector so this is a more abstract way of arguing uh instead of about the details of the construction side angle side and all of that we can just see abstractly about symmetry that peq must be the perpendicular bisector on those grounds so uh so that way of thinking really leads to felix klon's erlongen program by which you study a geometric space by means of the group of transformations and invariants that it gives rise to so geometric properties are precisely those that are preserved by those transformations of the space so in the euclidean plane for example well we have we have translation we have rigid translation you know you can move you can move figures around in the plane we also have rotation so you can rotate figures in the plane and we have reflections um and in fact any any one of those three kinds of transformations of the plane can be achieved by at most three reflections so you really only need reflections to generate all of those symmetries and so those three transformations of course preserve distance they're preserving the metric properties of the plane and therefore the congruence notion that comes from that group of symmetries of the plane is going to be distance preserving um but if you if you if you want a different kind of geometry where you're not required to preserve distance for example if you allow scaling of the point then the congruence notion that you're going to get from that group of transformations is going to include similarity it will be similarity two figures will be uh congruent with respect to that concept of geometry just in case they're similar rather than actually the same size so for example you could also study the group of transformations of the plane by not allowing reflection so this would be a kind of orientation preserving notion of congruence so two figures would be congruent in that concept of geometry just in case uh you can rigidly translate one to the other uh by by translation and rotation but not allowing reflection which is not orientation preserving okay so uh so ultimately that perspective leads you to a picture of geometry by which you specify a geometry really by specifying a group of transformations and then a concept counts as geometrical exactly when it's a concept which is preserved by every transformation in that group of transformations it's a much more abstract way of thinking about what a geometry is all right let me return to this classical case now though i want to discuss a certain controversial issue in the use of the compass so if we think about how a compass is used so maybe maybe we have a line segment a b here you know we know the compass enables us to build a circle around a but the way a compass is often used is that you you measure the segment a b you know with the compass and then you pick the compass up and you put it down over someplace else at some point c and you make a circle of that radius over at that other point c yes so what we really want to do is well we know how to build the circle uh with radius a b but what we want to do is build a circle of that same radius over here at this other point c and the question is are we allowed to or not because of course i mean cautious geometers in in euclid's day objected to this use of the compass because the the the axiom the postulate about the use of the compass doesn't say for any line segment and for any other point somewhere else you can make a a congruent circle it just says for any for any point to be used at the center at any point uh specifying the radius then you can make the circle with that center and that radius so this sort of picking up the compass and putting it over there isn't allowed according to the axiom and so um so some people built special compasses that collapsible compasses so that they they work but only when they're under some slight pressure so you have to be pushing down on them in order for them to hold the distance and as soon as you pick them up then they the pressure is lost and they kind of fall apart i mean they they lose the distance so you can't pick it up and put it over there and the question is is a is a rigid kind of compass that that where you can do the measurement and pick it up and move it is it equivalent constructively to this collapsible compass and and in fact it is and you could prove that it was this is called the compass equivalent here the compass equivalence theorem so any construction that you can do using a uh you know what i view as an ordinary compass that holds its distance you can also achieve it with a collapsible compass so let me prove that i'm going to give one proof but then i'm going to give you a reason to not like my proof and to prefer maybe euclid's group so if i have three points a b and c and i want to construct this circle then what i'm going to do first is is use the straight edge to make a line that connects a and c and then i can use the compass to make this circle but also we saw how to make a perpendicular so i can make a perpendicular at a perpendicular to the line ac so i can make this line perpendicular and therefore i can find this intersection point and now i can make another perpendicular here all the way over here and i can make a perpendicular at c going up and i can find this intersection point and then that line segment i can use to make a circle yeah and the point is that because this is perpendicular and this is perpendicular and this is perpendicular it's there for a rectangle and therefore this length is congruent to this length which is congruent to a d and so i've done it okay so now there's two reasons to not like this argument that i just gave you so this is a proof of the compass equivalence theorem in euclidean geometry but one reason not to like it uh is that i also use the straight edge i didn't do it just with the compass and so the argument doesn't show that this kind of compass is equivalent to a collapsible compass just by itself i needed the straight edge but the second reason to not like this group is that i'm using euclidean geometry i'm using a parallel postulate in order to build this rectangle i'm really using the fifth axiom which we're going to talk about later when we talk about non-euclidean geometry but this argument just doesn't work in non-euclidean geometry because i'm using these right angles and the fact that i made a rectangle okay but meanwhile euclid gives us a proof that doesn't use a straight edge and doesn't require the parallel postulate and uh it's using only circles so he builds i don't know half a dozen circles and then you can show that you can build that other circle uh on the other side okay so that's the compass equivalence theorem i'd like to think about what what's going on with these constructions so let's talk about the universal construction procedure so so if i have if i start with two points a and b then the question is well what can we construct from a and b and what i want to do is systematically do perform all possible constructions using straight edge encompass so if i have two points a and b well then at first i can make the line that that joins them and then secondly i can make two circles one centered at a and containing b and one centimeter b and containing a so this circle is going to be something like this and then the other circle is going to be something like this i'm sorry my diagram is not perfect so i got to make that better i'm sorry let me this circle's just not good enough it's too small i guess okay so let's make it better i should have yeah okay oh my gosh is that better okay maybe so all right now when i did that i did my straight edge as much as i could and i did as many circles as i could draw using just those two initial points and what we find are more points we got this point here and this one and this one and this one so that gives us now uh four new points so we have six points all together and now we can start drawing lines and circles using those six points so i could make this line and i can make this line and i can make this line and so on and this one there's a lot of lines and i can make all possible circles using those six points and if you make a diagram uh you draw all possible lines and all possible circles using those six points then and you look at the intersection points then you get uh 203 points so it's this mess of a picture with a lot of circles and lines in it okay so the sequence so far is uh we started with two and then we had six and then 203 points are arriving next you can start to see some of those points here that are that are intersecting so uh when i uh when i drew some of those lines and some of the circles for example this is going to be a circle here so they're going to have more points coming from up there and so on even those 203 points so we could imagine carrying out this universal construction process and and systematically generating all possible um points in the construction and okay if we take these 203 points well i i was wondering uh a year ago when i was first thinking about this how many points and lines you would get in circles you would get at the next stage after that and uh and i knew it would be quite a lot for example if you have 203 points then basically you have over 40 000 pairs of points and any two points are going to determine a line some of those lines will be the same and any two points determine two different circles so that's giving us some upper bounds of 40 000 lines and 80 000 circles that we can could construct at the next stage um and so i but i i knew that to count them of course the 203 points i could count by hand but if i have forty thousand points then i'm not gonna be able to really count that by hand and also i'm not going to be able to use computer algorithm to to count them either if i'm using just floating point approximations because maybe two points are very very close together and i need to detect whether or not they're exactly identical or whether they're just very very close to within my round-off error and so i need to be performing sort of exact calculations and so i posted this question on math overflow and we got some answers so pace nielsen found that the 203 points formed exactly 17 562 distinct lines and exactly 32 719 distinct circles but he he didn't know how many intersection points all those lines and circles had and then it was someone here in oxford a graduate student uh teophil kamarasu found that when you intersect all those lines and circles and you perform exact calculations not up not approximations with roundup so there's no round off in these calculations you get exactly one billion seven hundred twenty three million eight hundred sixteen thousand eight hundred sixty one distinct points of intersection and that calculation took six continuous days of computer time running uh on six different processors on three different machines so it's really quite an intense calculation okay so the next number on the sequence is what did i say 1 723 oh my god 723 million 816 000 uh and then uh 861 number of points okay and then you do it again right this sequence never stops because all of those points any two of them determine a line and any two of them determine two different circles and we would want to construct all those lines and circles and intersect and so on this is the universal construction sequence um and and at each stage in the construction we're producing finitely many additional points and every constructable point will arise at some stage in this process right no matter what construction you're doing from those initial two points eventually if we're doing all possible constructions we will have done your construction and therefore every constructable point every point that you could construct by any process will appear at some stage in this universal construction process so descartes had a really great idea so what he proposed is that we should look at um look at the line and start with two points and let's call them zero and one we think of them as specifying a unit distance yeah and then the question is which numbers if we think of every point on this line as a number the number line then which numbers can we construct by the classical geometric construction methods of straight edge encompass of course i can immediately construct the number two you know if i use one as the center of a circle then i can construct the number two and therefore i can construct three and four and so on yeah also i can find midpoints so i can construct one half and one and a half and so on all these points in between i can also construct the square root of two because i can on this unit interval i can construct a square and the diagonal of that square has length square root of 2 and then i can use a compass to bring it down here so i can construct square root of 2 on the number line so there's quite a lot of numbers that i can construct so a number is said to be constructible by a straight edge and compass if you can find a point on the number line using strategic companies you can construct a point on the number one whose distance from zero is exactly that number okay so uh of course one might ask well maybe every number is constructable are there any non-constructible numbers at all and one easy way to see uh that there must be non-constructible numbers let's first give a soft proof which is not the original proof that people had for this fact but there's a soft proof and it follows just by considering the universal construction sequence we already described how to generate all the constructable numbers just by systematically carrying out all possible constructions but the point is that at each stage in that process we're only adding finitely many additional points so therefore the constructable points these constructable numbers uh are are obtained by a countable sequence where we're adding finally many points at each stage and therefore there's only countably many so by canto's argument there must be some points on the number line that are not constructable because there's uncountably many points on the number line but only countably many constructable points okay so that's the kind of cantor style soft proof of the existence of non-constructible numbers but in fact this fact was known before canter well before kendra using some quite deep ideas in mathematics so let's discuss that um let's think about the nature of constructable points so when you whenever you're adding a new constructable point it's because you were intersecting two lines or you were intersecting a circle with a line or intersecting two circles so in each case it's because you're solving a quadratic equation because the circles have have quadratic uh equations x squared plus y squared equals r squared or you know if you translate it to a given center um so there's a degree two polynomial equation uh that that you're solving for each of those intersections and so it follows that whenever you add a new constructable point that point will fulfill a degree two polynomial over the previous points that you had and therefore it follows from that that the the algebraic degree of the um of the constructable points must always be a power of two because each time it it's a quadratic over what you had before either you already have it or else it's going to be degree two over what you had before and so every constructable number must have order 2 to the n for some n over the rational field so therefore if i have a number that doesn't have order 2 to the n i mean degree two to the end then uh then it can't be constructable so let's let's use that in connection with some of the classical non-constructability problems so one of the classical uh um construction problem was given a given a line segment a b then um we think of that line segment as determining a cube and i want to make a line segment whose cube has exactly twice the volume okay so i want to make another line segment over here cd so given a b i want to make a line segment whose cube has exactly twice the body this is called doubling the cube you know and basically if this distance is one then the this cube should have volume two because this is one by one by one has volume one so this one should have volume two so each side should be the cube root of two yeah but if x is the cube root of two then x cubed minus two is zero so x is solving this irreducible cubic polynomial it doesn't have degree 2 to the n for some n and therefore the number cube root of 2 is not constructable by the reasoning that we gave earlier so this solves a problem that really stymied mathematicians for for hundreds of years um well almost 2 000 years um and it uses this theory of algebraic extensions of the rationals and the and the degree of roots of polynomials and so on it's quite sophisticated uh mathematics let's look at another example the problem of trisecting the angle so the problem is given an angle you know so we're given a point and then two other points so what we want to do is construct a trisection so that this is exactly one third of the angle and and the question is can we can we construct a point which is going to realize that trisection now of course sometimes we can for example if the original angle was a straight angle then the trisection is 60 degrees but we can construct a 60 degree angle because that's what the apollonius construction provides those triangles that i drew in that fruit the first roof we did those were equilateral triangles they have 60 degree angles so i can construct a 60 degree angle also if you start with a right angle then the trisection of that is a 30 degree angle but you can also construct a 30 degree angle because you're just bisecting a 60 degree angle so therefore sometimes you can construct the trisection of an angle but let's consider whether you can construct say the trisection of a 60 degree angle so you want to build an angle of 20 degrees okay but it turns out that if okay if you could trisect a 20 degree i mean if you could translate a 60 degree angle and make an angle of 20 degrees then uh then you would also be able to construct the cosine of 20 degrees that's the length if you have a unit circle with and the angle is 20 degrees then if you project onto the under the horizontal you're going to have cosine 20 degrees but it turns out that if x is 20 is cosine of 20 degrees then 4x cubed minus x equals a half just follows from basic trigonometry and the triple angle formula um and therefore it has degree three not a power of two and therefore you cannot construct the cosine of 20 degrees and therefore you cannot construct an angle of 20 degrees and therefore you cannot necessarily trisect an angle so here's another example squaring the circle so given a circle uh construct a square with the same area so a circle specified by the center and a point on the edge the radius yeah so given a line segment thinking of it as determining a circle construct a square meaning construct a segment who's such that if you build a square on that segment it would have the same area yeah but of course if the original segment would had length one then the area of the circle would be pi and so what we really want to do is construct the square root of pi because the square whose side length was the square root of pi would have area square root of pi times square root of pi which is pi so to square the circle you would have to construct the number square root of pi but this the square root of pi is a transcendental number it doesn't satisfy any algebra equation at all over the rationals and therefore it's not constructable that's a deep fact though i mean to prove that pi is transcendental is not an easy thing so all of these results came very late along in the in the 19th century using quite advanced mathematical ideas so let's now move to a different topic namely alternative tool sets so of course classically we have only the straight edge on the compass but we can mix this up a bit right so what if we allowed compass only so the question is what can you construct if you only have a compass and no straight edge of course you can't draw the line anymore but if we're talking about which points are constructable the question isn't which figures can you draw it's just the question of which numbers can you identify on the number line or which points in the plane can you construct and point at if you only had a compass yeah and it turns out it's quite shocking but it's the more machiaroni theorem first in 1672 that compass only construction is fully powerful it's equivalent any constructable point that you can construct with straight edge and compass you can also construct with compass alone and how would you prove something like that i mean you're proving something about all possible construction methods so what what you need to do is something like what we argue with the compass equivalence theorem you need to argue that anything that you could have done with a straight edge you can also achieve just with a compass yeah so for example suppose we had two points and another two points you know and i'm thinking you know i'm thinking of the line i can't draw a line without the compass but i'm thinking of the two lines that they determine and i want to construct this point here yeah and so there's a construction given these two points then there's a construction involving a compass only and it has a couple of dozen circles that you draw it part one step in the construction also is to pick a point that's not on either of the lines so you pick a sort of random point you build a bunch of circles and ultimately the construction finds the intersection point here and so therefore if you wanted to have intersected two lines then even if you only had a compass you could still find that intersection point okay so what about the dual situation to that for example suppose you have only a straight edge um so it turns out that that theorem is also true a little bit so you need to have at least one circle so given given any circle so maybe over here on the left you're given a circle and with a known center and radius then if you have a straight edge you can construct anything that you could construct from a compass and distribute it so you you only need the straightedge so you only need one circle one instance and the way the proof goes is that you need to argue that anything that you could have done suppose so as you were carrying out some construction and over here you would have wanted to draw a big circle then using the straight edge what you do basically is you make similar triangles and bring that picture back over to this circle and do what you wanted to do over here and then carry using the straight edge only you you can translate it back by the similar triangles to to find the corresponding point that you wanted to have over there and what this theorem is it's an improved version of what's was known as the rusty compass urine suppose suppose that you had a straight edge and a rusty compass what's a rusty compass it's stuck you can't you can't you can't move it so a compass is you know these two pincers right but it's rusty so it's re it's fixed you can't the radius of the circles that you can draw is completely fixed and the point is that it's nevertheless fully powerful so if you wanted to have drawn a big circle then well the rusty compass says you can only draw that fixed circle size and then you can do similar triangles and and translate or one can see that it's a consequence of this other theorem that i was just mentioning that you just draw one circle with the rusty compass and then you appeal to the other more general theorem okay so what about construction with a marked ruler so suppose that we have a marked ruler is a straight edge with two marks on it yeah and and you can use those marks in your construction by lining them up on lines or points uh in your figure and so for example with a marked ruler let's see suppose you're given a point here or you're given say an angle okay like this and then what you do is you draw you draw a circle of the radius with the mark you know so that this is the same as the mark and then what you do is you you take your marked ruler and your line you you you hold it so that one point is on here and you make sure that that this distance is the same as the the marks line up both on the circle and the line you can move the thing around of course if it's really too low then this distance will be too short but if it's too high the distance will be too big so there's going to be a point in between where you can line up one mark on the circle and one mark on the line and the and this one is also on the on the marked ruler and then it turns out that this angle is exactly a trisection of this one in that case you can prove so therefore this is an opposite kind of situation the marked ruler is more powerful than the straight edge encompass alone you know if you have a marked ruler you can trisect angles but if all you have a straight energy compass you cannot so it's strictly stronger so there's some other more fanciful geometric construction methods so consider origami this is the japanese art of paper folding so you fold the paper you make beautiful cranes or other exquisite works of art but we can think of it mathematically the essence of origami is that you're folding the paper and of course that's a fundamentally geometric thing if we had a figure on the paper and we fold it then we can see how that figure landed on the other side and if we think of that as a geometric construction tool so namely given any line that we're going to fold upon then given any point on one side we can we can construct the image of that point under the reflection we think of that as a construction method or for example another basic origami fold is given two points on the paper right you can you can fold them so as to bring those two points together you know and then that determines the line so basically that's allowing you to construct the perpendicular bisector of the line connecting them right because the folding line would be the line that's the perpendicular bisector of the segment joining those original two points yeah so the question is well how powerful is origami as a construction method and uh um robert gertschlager in 1995 proved that it's equally powerful uh this uh strategic compass construction is equivalent to a certain sequence a certain tool kit of origami foals seven fundamental origami folds and there's reductions in both directions anything that origami can do the strategy compass can do and vice versa but also there's certain specialized origami folds that are sometimes used in practice that are strictly stronger than that list of seven so for example given two parabolas well so how are you given a parabola well a parabola is determined from a point and a vertex line you know the set of points equidistant from uh from the line and the point so if you're given a two parabolas by means of their focus points and their focal points and the and their vertex lines then you can construct a tangent line to a common tangent to the parabolas so in practice this is something that a skilled origami artist can do and it turns out that if you allow that construction then you can construct a cube root of two so you can duplicate the cube um and in fact you can solve any cubic polynomial if you allow that origami fold okay well let's consider also say spirograph constructability maybe when i was a child i used to play with spirographs these little plastic rings gears and they fit inside each other and you put the pen in and you make a kind of a spiral pattern and and if we think of that as a geometric construction tool then the question is what can you construct and it turns out it's strictly more powerful than classical straight edge and compass constructibility for example one of the things you can do is make this spiral picture um that has seven loops in it and it's perfectly symmetric and if you look at where those loops intersect you can make a regular septagon a seven-sided regular polygon but that's something that we can prove you cannot construct with straight edge and compass alone okay so now all these different ways of methods of construction lead to the question about what is the ontology of geometry what what are we doing with these geometric constructions so um so is geometry if geometry is meant to be about the nature and features of the sort of ideal geometric figures points and lines and planes and so on then it seems like there wouldn't be any reason to object to augmenting the construction toolkit with these other tools provided that we thought that those tools were according with sound geometrical truth so if we think there's there's an objective geometric reality out there that we're trying to understand with our tools then it would seem there shouldn't be any objection to augmenting our toolkit with these more powerful tools i mean do we think that there is a 20 degree angle or not so if if the geometrical reality has a 20 degree angle in it then maybe we think well the strategic compass isn't powerful enough to tell us the nature of that geometrical reality and so we should augment those tools with more tools like the marked ruler or some other tools that can construct the 20 degree angle so um okay so uh on the other hand maybe we think that only uh that the only points that really exist are the ones that we can construct using the classical strategy compass tools this is a sort of conservative picture that well that's what geometers were using for 2000 years and those are the only points that really exist so if you think about the points that exist in the constructable plane that's going to satisfy all of euclid's axioms and in that plane there simply are no 20 degree angles and there are no regular decepticons so it's a different way of thinking what the subject is about okay so on that on that first way though maybe the impossibility proofs are pointing us to realize that there are flaws in our geometric theory we're missing the axioms that tell us that 20 degree angles exists and that and that the cube root of two exists and so on um so we would need to augment the axioms euclid's axioms with axioms that assert that these other kinds of geometric objects exist because that is not a consequence of euclid's axioms alone so we can consider maybe a little more carefully what is the role of diagrams and figures in the geometric proof right nearly every proof in geometry is accompanied by a figure is that figure part of the proof or is it just a helpful aid it's often very difficult to understand a proof without that figure and so it seems like it really is part of the proof um but meanwhile uh uh we also know in certain ways that that uh that figures can be misleading any any specific figure i mean any any figure that we might draw has specific lengths and specific angles this segment is shorter or longer than that one even though maybe the hypothesis of the theorem wasn't determining which way it went but the figure is is determining and so the problem is or the worry is that we might make errors in our arguments by looking at the figure and deducing things from the figure because it has a kind of over specificity you know maybe when you do the construction with this particular figure the certain point that you construct is inside a circle instead of outside but a different way of setting it up maybe in a different initial figure would have led it to be outside and so we're going to be led into error if we're not careful so i want to show you how that can happen and so let me just convince you i'm going to prove that every triangle is isosceles now of course this isn't true because we know that there's lots of triangles that aren't testosterone isosceles means that two sides have the same length right so um but nevertheless i'm gonna i'm gonna give you an argument so let's just draw draw a typical triangle okay here's a there's a triangle um and now pick pick any point you like and draw the uh the angle bisectors i'm going to draw the angle bisector so this angle is the same as this angle and now think about the opposite side and find the midpoint and draw the perpendicular bisector okay so this is perpendicular so this is equal to this length here okay and now take this one here and drop perpendiculars to the sides and now i've got a triangle here and a triangle here and they're both right triangles and they have the common angle here so they're they're similar but also they have the same hypotenuse so they're congruent so therefore this length must be congruent to this length and this length must be converted to this length and now i'm i'm gonna draw these lines here from this point to here okay and i'm gonna think about this triangle and this one so here so they're both right triangles because this was perpendicular by definition and also this length was equal to this length yeah and the other leg is also common so therefore by the pythagorean theorem uh this length must be the same as this length because it's just a leg leg on in a right triangle the hypotenuse must also be congruent okay and now i think about this triangle in this triangle you know they're both right triangles and they have the same hypotenuse and they have a common leg so therefore the other leg must also be congruent you know and so here i've got two plus five here and here i've got two plus five here and so therefore the triangle was isosceles okay so i'm not going to tell you the problem with this argument of course it can't be right because not every triangle is isosceles but it's very difficult to find the flaw in this reasoning and i can assure you that in fact every single step about the triangles every single step of reasoning about the congruency of the triangles and the angles and the sides and so on all of that was completely correct there's no error in the reasoning i wasn't pulling a fast one on you with reasoning about the triangles everything i said about that is correct but nevertheless the overall conclusion there's some kind of subtle flaw in that and so if you figure it out i'm going to just leave it here as a puzzle and please think about it i'll post a video on youtube later and so if you figure it out just post an answer in the comments there okay it follows of course that as a corollary that every triangle is equilateral because if every triangle's isosceles then well we were able to pick which vertex we wanted so i could have done the same argument with a different vertex and those two sides should also be equivalent uh congruent and therefore it would actually be an isosceles triangle okay so let's see i'm everything takes longer than you think uh so let me just talk briefly about this error analysis so what if we analyze geometric constructions as a kind of uh as a as a process that you would carry out possibly with error right so if we think about say the apollonius construction so we have the line segment a b and now the first step in that construction was to draw these circles so so we pick b as the center and draw the circle with center a like this right but suppose that we don't you know when we put the compass down it's not exactly an a and uh and so we're gonna get maybe slightly different circles depending on uh on exactly how we put so i can imagine that the possible circles that we might construct um is this kind of smudged out kind of shape and similarly with b we get a kind of a lot of different possible circles that are all very close to one another but they're not perfect okay and then when we make this intersection we get this sort of region of possible intersections and possible intersections here and if we think about all possible lines that we might draw between them there's actually quite a bit of variation about how the construction would be carried out in practice and so ultimately what we're getting is a kind of uh not just one point for the midpoint but a kind of whole smeared out smeared out region of points that we construct and if we have this kind of view of the geometric construction as a possibly error prone maybe the arrows are very tiny but still we're only really constructing with the spectrum air and this would enable us this kind of perspective would enable us to compare different construction methods as to how accurate they were maybe one method is simply better than another in terms of its error because even if the error values are the same with each step the the first construction maybe is leading to a more more precise result or more accurate results and this leads to a view maybe of um you know looking at the geometric process itself platonistically yeah so if the subject is about the platonic realm of point ideal points in a plane then well if we think about the construction procedure itself as a mathematical object yeah something that could be carried out either imperfectly or perfectly so we have the imperfect version of the apollonius construction but then we have the platonic ideal the perfectly performed uh apollonius construction so in the platonic realm there's the ideal apollonius with the flowing white robes maybe and and taking out his compass and performing the perfectly ideal version of the apollonius construction to find the perpendicular bisector so this this leads one to take the constructions as mathematical objects subject to analysis and this is maybe very similar to how we think about computability now and we'll be talking about that in week six i think um computability okay so now there's more in the book so all of these lectures are based on my book um i want to move to the topic of non-euclidean geometry so so this is maybe one of the most interesting profound developments in geometry the discovery of non-euclidean geometry and what's at issue here is the parallel postulate this is the fifth postulate in the in the elements and what it says is the following so it says if you have two line segments and you have a transverse and this angle uh and these two angles are adding up to less than a straight angle then there then you can extend these lines and find an intersection point okay so given energy line segments given any given any two line segments and a transverse so that these angles are less than a straight angle then they're going to meet if you extend them far enough that's the parallel postulate there's a different version playfair's version of the axiom and it says that if you have a line and you have a point not on the line then in the weak form it says there's at most one line through this point uh which is parallel to the given line in the sense that it's not intersecting okay the strong version says that there's exactly one line through this point uh that's parallel to this line okay that's plagiarist formulation and depending on what the other axioms are it's equivalent to the parallel postulate okay so of course this postulate this axiom has a fundamentally different character to it than the other accents the other axe and say well if you have two points then they determine the line or um if you have a point and another point then you can make a circle and so the other axioms are sort of asserting fundamental seem to be a certain fundamental truths about the nature of points and lines and circles but the parallel postulate is this kind of almost technical thing and the view that mathematicians had for many many years was that it looks more like it should be a theorem and not an axiom and so they tried to prove it from the other axioms they wanted to get rid of this annoying parallel postulate we wanted just to have it as a theme so euclid in fact put off that axiom until the 29th proposition so he proved he proved 28 propositions using only the other more fundamental axioms and then he wanted to prove proposition 29 and he needed the parallel postulate and so he introduced it only at that stage that late stage okay so at first of course people tried just to prove it but then people had the idea of well maybe we should try to prove it by supposing it's false and trying to get a contradiction you know so we so we deny the parallel postulate and then reason from that assumption what what happens yeah and so this is a kind of shift in perspective and and uh janos bolyard had the carried out this process so and he went a long long way he derived all kinds of things all kinds of strange and beautiful mathematical consequences of the violation of the parallel postulate he never found the contradiction but what he found developing instead was a beautiful new theory and it seemed to him uh to have a kind of internal coherence and a geometric sense that seemed like he would never find a contradiction in that realm he wrote to his father this is 1823. out of nothing i have created a strange new universe so he's sort of discovered non-euclidean geometry simply by trying to have uh prove uh refute the parallel position by contradiction but instead ended up deriving uh the fundamental some of the fundamental facts of non-equating geometry so gauss also had been working at the same time but privately which is remarkable that he kept it secret so gauss had worked out quite a lot of the things that boy i had also but gauss never published it i mean he he had planned to publish only posthumously uh because he said he he was fearing the howl of the boeshians so um it's really quite remarkable for someone of gauss's stature to be afraid of ridicule but of course there was enormous ridicule for for working on this parallel possibly because a lot of crank mathematicians would would be working on on this topic also okay so one of the things that gauss had wanted to do is he suspected that um uh that in non-euclidean geometry there might be an absolute notion of scale you know a fixed absolute notion of size inherent to the geometry euclidean geometry is invariant under scaling you know but in non-euclidean geometry maybe you could define by some geometrical means an absolute notion of size which would be of enormous scientific uh uh importance if it were true in the physical universe so if the physical universe were non-euclidean we could maybe find that unit and then we wouldn't need the iron rod in paris or the platinum rod in paris um because we could just define the unit length the meter say in terms of this absolute unit of scale that comes from the inherently in the geometry okay so now both galas and boli did not have a consistency proof they just worked out consequences and they seemed to have this coherent theory but there wasn't any proof that in fact it was coherent and this didn't come really until beltrami in 1868 and he was the one who who first proved that non-euclidean geometry is consistent we can deny we can consistently deny the parallel postulate and not be led to contradiction if the original geometry was not contradictory so i want to show you how to think about that and i'm going to go over the hour time i'm sorry um so i want to show you how to think about that so one way so the method the basic method is to embed non-euclidean geometry inside euclidean geometry so if we assume euclidean geometry with the parallel posture then we want to reinterpret the fundamental notions of point and line and so on in such a way uh that so we're going to reinterpret them in euclidean geometry in such a way that we're going to provide a model for non-euclidean geometry so one way of doing that is with what's called spherical geometry so in spherical geometry so imagine us here okay so maybe here's a equator okay a giant polished silver sphere we can walk around on it you know and uh of course on the surface of the sphere we could do things like uh carry out euclidean construction so if i have a line segment there i could i could form the circles and intersect and make a little apollonius thing and i could find the perpendicular bisector on the surface of the sphere i wouldn't notice that it wasn't it wasn't non-euclidean maybe if the sphere was very large so in this in spherical geometry what what we mean by a point is just a point on the surface of the sphere and now what we mean by a line is uh is a geodesic so on the surface of this here this is a path that a plane flies uh on the when it's flying on this on the earth so it's a a great circle you know it's a so it's a it's a circle that arises uh who which is co-planar with the center of the sphere you know and that is a geodesic because it's locally the path of shortest distance so any two points on such a great circle uh it's the shortest distance that connects them locally that's why planes fly on that path because it's the shortest distance so we've redefined a notion of point and line so a line now in spherical geometry means one of these great circles on the surface so we think of this sphere as as a version of planar geometry okay so when you're on the sphere you can't look up at the cells and so on you have only the plane and when you construct a line with a straight edge you end up constructing what we can see in the embedded copy is this great circle so now that the point of this interpretation is that well locally we can see the space looks euclidean we might not even notice we can we can do straight is encompassed constructions on the surface of the sphere so if they're very tiny it looks basically euclidean but at large scale we can see that the geometry is totally different from euclidean geometry for example if i if i start at the north pole and i go down to the equator and then i turn and go one quarter of the way around and then i turn again and go back up to the north pole then well this is a right angle because it's going a longitude line and then on the equator and this is also a right angle because it was the equator and then a longitude line but this is also a right angle because we went one quarter of the way around you know and so i've got now a triangle with three right angles in it and this definitely doesn't happen in euclidean geometry so it's a way of seeing that the geometrical truths of spherical geometry are somehow different than they are in euclidean geometry okay also we can see that the parallel postulate is false in spherical geometry because all lines i'm sorry the the the euclid version of the pillow partially is true because all lines intersect but the playfair axiom is not true because if i have a if i the play for axiom says if you have a line and a point on the line then there should be a line that uh that doesn't intersect the given line but in this version all lines intersect okay so it's revealing differences in the in the geometry so there's a version of this okay this the way i described cervical geometry so far isn't isn't quite fully satisfying all the euclidean axis because some points some pairs of points have many lines for example the north and south pole don't determine a line there's many lines that contain the north and south pole so one can fix this issue by but with another interpretation so we think of the a point now as a pair of opposite points of antipodal points so given any uh given any pair of opposite points we think of that as one point in what's called elliptical geometry so it's a re another reinterpretation and in that reinterpretation you do get all of euclid's other axioms so there's another notion of interpreting so this is hyperbolic space so if we use the concrete disk model we think of the the unit circle the interior of the unit circle and points are just points in the in the interior of the unit circle but lines are now circles that are perpendicular to the to the boundary okay that that's what a line is in hyperbolic space so if i have such a line and i have a point on the line you can see that i can maybe find a lot of points i mean a lot of lines through this point that don't intersect that one yeah so now if i if i have here um i can make triangles and you can see that in hyperbolic space triangles have angles that add up generally to less than 180 degrees instead of exactly 180 so that's one difference there already so this is violating playfair's axiom because we have many lines through the given point that don't intersect the given line so the parallel possibility is false in hyperbolic space so let's talk a little bit about curvature so if you think about spherical geometry and you have a point and you make a circle of radius r so on the surface of the sphere then because in in spherical geometry so i have a sphere i have a point and i make a little a little radius and i make a circle of that radius then the the circumference of that circle is a little bit less than you would expect in euclidean geometry of course in euclidean geometry the circumference is 2 pi r 2 pi times the radius but because the space is curved the the circumference is a little bit smaller than you would expect okay so this is what this is what it means for a space to be curved it means that there are slightly fewer points at a given distance from you okay and it's the same in higher dimension so for example in in space if you're standing in in three dimensional space you could imagine all the points you could reach at arm's length you know so that's determining a kind of sphere one arm's length away from me you know if i move in any direction and in euclidean space that the num the area of that sphere would be what is it four pi r squared so that's the area of the points that i can reach at arm's length from me okay but in a positively curved space it would be less it would be less than four pi r squared and maybe a lot less if the radius is very big okay now hyperbolic space on the other hand is negatively curved which means that if you look at a circle say in the two-dimensional case if you look at a circle of a given radius um then the circumference can be bigger than you expect and much bigger if the if the space is extremely negatively curved and similarly uh in the three-dimensional version if you look at all the points that are given distance from you in hyperbolic space then there could be many many more points than you expect so this is how i write about it let's see if you are a resident of a city in hyperbolic space then the number of shops and restaurants within a few blocks of your apartment would be enormous because in hyperbolic space there are so many more locations that are this close to you if the space is extremely negatively curved then there would be there could be whole undiscovered civilizations within walking distance simply because a strong negative curvature means that there are such a vast number of locations quite nearby criminals escape easily in hyperbolic space simply by walking away a short distance it's too difficult to follow them far since at any moment one must choose from amongst so many further directions to continue for the same reason you or your loved ones may easily become lost in hyperbolic space for it is so difficult to find your way exactly back home again so please be careful and hold them close okay so i want to now turn to uh the topic of errors in euclid so so euclid is held up as this paragon of deductive reasoning and the question is well how does it fit the bill and are there mistakes in it and so of course there's some kind of uh maybe so a pendant could object to some minor omissions for example he doesn't have an axiom that says there are any points at all and he doesn't have an axiom saying that every line has at least two points on it or that say not all the points are collinear and so on so okay so we can give him these incidence axioms right this sort of non-triviality axioms but there's other maybe more serious omissions so if we think about proposition one he's he's constructing an equilateral triangle so we have a b it's just like the it's just like abalone is construction so he makes a circle centered at b and he also makes the circle centered at a and he says therefore they intersect yeah but let me put an open circle here yeah why do they intersect yeah it doesn't have any axiom that says they intersect this is a missing assumption in his argument now so why is there such a point of intersection we needed that point to make the perpendicular bisector right but maybe maybe the circle somehow passed through each other or something right or maybe you say well look come on obviously they intersect right there's a point they're just you know it's where the one circle crosses the other but it's not the the subject isn't about what's obvious geometrically to us the subject is about proving things from the axioms and so if we don't have an axiom that says that there is a point there then maybe there isn't one it's a kind of continuity assumption that's really missing and let me just convince you that there really is something to that let's consider the rational plane so we only have points whose coordinates are both rational numbers and then given any two rational points then you can consider the line of all rational points you know that the the rational ones that lie on the line connecting them and also the the rational points on the circle and so on um and and this gives a concept of geometry instead of geometrical universe the rational plane universe and in the rational plane universe there is no point of intersection there because because this is gonna have height square root of three over two uh if this is distance one and the square root of three is not a rational point and so in fact in the rational plane they the circles do pass through each other precisely because this point is necessarily irrational so so therefore they're really we really do need an axiom that says that there should be a point there and this is a kind of um continuity axiom that's present in the actualizations of hilbert and tarsky and so on they use a version of dedican's idea to state a kind of completeness of the geometric figures and it addresses this flaw in euclid another flaw in euclid is what could be characterized as the missing concept of between so euclid argues with the idea of between quite a lot but he doesn't have any axioms about it so for example maybe he has a line segment and he has a b and c and he says because b is between a and c it follows that a b is less than ac and so on this kind of reasoning about between but what is the nature of this between this relation um there's no axioms in euclid about it but also let me just convince you oh here's another kind of case another important kind of case if you have an angle say like this you know and you make the ang the angle bisector so this angle is the same as this one that's possible to construct then the the question is does it intersect the line you know is there a point here and you want to say well yeah it does intersect and also that point is going to be between these two and so he's using that kind of between this idea but let's think about spherical geometry so we have these great circles and think about say on the earth if we have say new york sydney and prague they lie approximately on a great circle so i've got these three points they aren't aligned which one is between the other is prague between new york and sydney or is sydney between new york and prague or is new york between prague and sydney how does it go and if you think about it you realize well it's the concept of between is highly problematic in spherical geometry that it's not sensible to say that a is that b is between a and c if you can go all the way around any of them can be viewed as between the others if you go the other way around right or if you have two points then in euclidean geometry we say well consider all the points between them that's the segment but in spherical geometry which segment are you talking about if you have two points on the sphere you could go this way or you could go the other way and which one is the segment a b uh which which points are the ones that are between a and b and maybe it's not so clear okay so the the other actualizations of geometry sort of fix this issue so hilbert uh introduces a a version of geometry that that's explicitly talking about the betweenness relation and the s axioms that say um for example if if b is between a and c then it's also between c and a and and other axioms of that form if you have two points uh b and d then you can find points a c and e so that b is between a and c and c is between b and d and d is between c and e and okay so so reasoning in this kind of way about betweenness tarsky also introduced a version called the elementary theory of geometry so for him the fundamental objects were just all points but he had a a trinary between this relation as a fundamental relation on the points and also he had an equidistance relation on pairs of points so this this line segment is congruent to that one and develop geometry using that scheme okay so there is of course the connection between geometry and physical space the question is for 2000 years people geometers looked at geometry the subject of geometries the study of the mathematical properties of physical space maybe but now how does this this fit with this sort of euclidean non-euclidean suppose that we somehow discover that space is not actually euclidean it's non-euclidean would that invalidate euclid's theorems if we discovered that space isn't actually euclidean because say it doesn't have parallel phosphate so gauss had wanted to measure to test empirically whether the euclidean axiom is true or not you can imagine drawing an enormous triangle gauss did it between mountain peaks and got a measurement that was euclidean to within the accuracy of his measurements and lebashewski called for such experiments to be undertaken at an astronomical scale to determine if the physical universe is euclidean or not and so uh greenberg ident uh emphasizes this point that if you make such a measurement and you have margin of error then you can show you this sort of several things that could come out of such an experiment you could conceivably show that space is not euclidean because you you get an angle of say 179.5 degrees and that's far enough away from 180 degrees within your margin of error that you could deduce that space is definitely not euclidean but it seems that you couldn't ever establish that space was euclidean because if the 180 degrees was within your measurement of air then there are also some numbers that are very close to 180 uh that are also within your within your era of measurement um and so it seems like you can't definitively establish that space is euclidean but only that it is not on experimental evidence of that kind okay but none of that really affects the idea of uh whether a physical observation like that about the physical universe can refute euclid's theorems it seems like the hypothetical itself is is a refutation of the idea that what geometry is about is the the nature of physical space because we don't think that that euclid's theorems are refuted by any experimental observation at all we just think that that would show physical universe isn't modeled by euclid's axioms but it doesn't mean that in euclidean space the conclusions that euclid was making aren't still true so there's this sort of separation between the sort of platonic ideal of geometrical space and the nature of physical space and maybe it's easy at first to confuse these two ideas and to think that that geometry really is about physical space but but maybe when you think about what the consequences of an experiment would be and so on you're led to the conclusion that actually the mathematics is about its own ideal platonic realm the sort of ideal uh geometry that it's about both in the euclidean and non-euclidean cases and then the nature of physical space is just something else okay so i want to conclude by talking about a really remarkable theorem of tauroski and what he proved i mentioned his axiomization the elementary theory of geometry and uh he proved that that theory is both complete and decidable so that means that in the language of geometry that he set up his axonization proves or refutes every single statement any candidate statement is either a theorem or the negation of a theorem and furthermore there's a computable procedure to find out which one and so this is really amazing because what it shows is it can be viewed as a kind of completion of the subject of geometry after 2000 years we've reduced geometrical reasoning to a road procedure we can follow toroski's procedure to determine the truth of any given geometric statement i find that really incredible let me tell you a little bit about how how he did it it relied on his proof in the in the real close field so if you think about the structure of the real numbers the real field then he proves that every statement in the real field as an ordered field [Music] is equivalent to a quantifier-free statement a statement without any quantifiers in it and for example just to illustrate suppose i make a statement say there is an x a x squared plus b x plus c equals zero okay this is a statement has a quantifier and in the real number so a b and c are real numbers then of course this is a quadratic equation and this is this statement is saying that it that it has a solution but we know exactly um when quadratic equations have solutions it's exactly when the discriminant is positive it's not negative uh b squared minus 4ac should be bigger than or equal to 0. so this is if an online so this statement which has a quantifier is equivalent to this statement which doesn't have any quantifier in it so we've eliminated quantum parts and what tyroski proved by a quite deep argument in the 1950s is that every statement is equivalent to a quantifier-free statement and then because our geometric concepts are expressible in the real numbers because we can we can refer to lines and circles and ellipsoids and all of the classic geometric objects by their equations in any dimension any finite dimension tauroski's elimination of quantifiers argument shows that we can decide the truth of any geometric statement in any dimension about those kinds of objects so okay i find that really incredible a kind of completion of the subject of geometry okay but of course there's many counter arguments so one counter argument to that grandiose view is that the pres the procedure takes too long it's really long uh tauroski's original algorithm uh was the time complexity was not bounded by any elementary recursive function this has now been improved and it's double exponential time which is a dramatic improvement over what he had originally but it's still infeasible we can't actually use this algorithm to decide any geometric statement because it would just take too long to answer any interesting question that you write down would take the age of the universe to use this algorithm to figure out the answer and so the algorithm although it's a theoretical kind of algorithm that works in principle um it's not actually useful for any purpose because we can't uh wait that long for the answer okay that's all i had to say today so i'm happy to stick around and if people want to talk and discuss and ask questions that's completely fine so go ahead and ask questions in the chat or um uh or just unmute yourself and ask that's also fine may i ask a question sure there's one though that came in just on the chat how does the process of quantifier elimination work in general so let me just answer that first so so quantifier elimination is a is a standard part of the toolkit in model theory which is a part of mathematical logic and maybe tauroski's theorem is one of the first most prominent instances of it i guess so one can show that certain theories admit quantifier elimination for example the theory of dense linear orders cantor's argument basically tells you uh how to uh um uh using those ideas of cantor's uh categoricity result for endless stance on your orders you can establish quantifier elimination for for that theory and the theory of algebraically close fields of characteristic zero also admits elimination of quantifiers generally those arguments are quite sophisticated and difficult though um and so when i didn't i don't want to with this example i don't want to give the impression that somehow that's what he did his paper is 50 pages long or something like that and so they're often quite involved reasoning to establish the elimination of quantifiers but once you get the result basically when you can establish elimination of quantifiers result for a theory it means that you basically demolish the theory you completely understand everything about it because whenever you do the elimination of quantifiers result then you're going to get a decision procedure and these other kinds of consequences so so it always comes with this extremely powerful kind of context for understanding what the theory is like when you can establish elimination of quantifiers okay so someone else was asking was it was it ahmed or yes go ahead yes thank you um i suppose that uh euclid knew that the earth is not a flight i am not sure uh i'm sorry i didn't quite catch that he knew that what uh that the earth is not a flat yes i think so yeah yeah so did he have some um uh uh question about the link between his geometry and the real world that's my first question and so uh the link between theory and the emperor's particular question that's my first question the second is that the corpus allow us to construct some numbers we have also the perfect compass which which was made to to construct conics and based on the movement yeah is there any theory of uh construction of uh of numbers based on the movement you know uh and the last so your first question really i'm not sure i mean this would take a historian i'm not really a historian and i don't know what euclid did or didn't know about whether the earth was flat or not i know the ancients there were some experiments in ancient days that that measured the curvature of the earth quite accurately um using the length of a shadow at day in cities at different latitudes and they use that to calculate how big the earth must be and it's quite amazing how accurate that calculation was i mean to within a few percent i think um and uh but i think even the classical geometers would separate the idea of the earth being flat from what they were doing in plain geometry they would say well if the earth isn't flat then our planes aren't wrapping around the earth but they're you know going off uh into space or whatever and so we're talking about the figures on an actually flat plane and so i think that i think that they would separate that although i would defer completely to the historians because i just don't know what they actually knew or wrote about that matter uh your second question i'm not sure i understand it but uh i guess you're talking about conic sections and using them as a method of construction um and uh i guess okay this is of course the if we actually have cones and so on then we're out of the realm of of straight edge and compass which is in the plane only right this this is the concept of construction in planar geometry but when you're doing conic sections then you have cones you're already in three dimensions um and so you're in a bit of a different realm there already um and uh and i just don't know if those tools are are more powerful or not i tend to think that they're not because the conics are all quadratics they're all described by quadratics and therefore they're not going to bring us out of the degree to uh realm but meanwhile uh any number that's degree that's degree of power of two over the rationals is constructable and so therefore my expectation would be that those methods wouldn't bring you outside of what's uh what's constructed with straight edge and compass alone okay are there other questions thank you sure we can come back if if other people get a chance any other questions or comments okay so if not then go ahead i mean if you had a third question no my question was about the perfect compass which was made by el cui in 10th century and it's permitted to construct hyperbola and parabola and my question is a technical one uh i don't know enough about it i'm sorry yes i i don't know if there is so in england there is some museum where i can found it i see okay i know that this this other construction tools which i didn't mention and which aren't in my book but uh there's this uh this hammer one which is um allows you to trisect an angle so it's really quite remarkable uh this this sort of uh hatchet-shaped thing and you lay it on the on the on the figure and it ends up exactly constructing the trisection um it's really quite amazing and so there are these kind of tools that you can design that go strictly beyond straight edge and compass constructability i mean we already saw several you know origami and spirograph and marked ruler and so on but there's a whole assortment of these tools that are strictly stronger than straight edge and compass to me the interesting philosophical question about that is um to what extent are those tools somehow changing our conception of the ontology of geometry right if we think that geometry is about this sort of ideal collection of points then if we're willing to admit these other um i mean if we have that picture of how the ontology of geometry works then we should be willing to admit these other construction tools and so why is it that mathematicians were so unwilling to do that uh for such a long time um was it because they had the idea that somehow only the points constructed by strategic compass were the ones that really existed or was it because of some other sort of formal idea in terms of uh what what were they a person with that point of view i would want to know what is it that they think the subject of geometry is about or is it about these sort of ideal points and lines in in a plane that may may have features that we couldn't achieve or maybe it was related to the idea of sort of hope that ultimately strange encompass would be as powerful as any of these other tools turned out to be okay we have another question here let's see i can't quite see it me know this um let's see i can't quite read this right okay so um okay so the talk we have these three different concepts of geometry so first of all this sort of group action on the power set of some points or inductive generation points in a space in terms of construction or aximizations i think that's really quite a good way of describing what uh what the some different views that we've talked about are so they sort of the the geometers the contemporary geometries don't work from construction methods or from maximizations really they specify what the group of transformations is and in your london program that group of transformations determines what are the geometric concepts namely any concept that's preserved by the by that group is a geometric concept if and only it um and that's quite an abstract picture of what geometry is whereas um if you think about in terms of construction then uh it's fitting much closer with uh with maybe the ancient picture of of what's going on in geometry um sort of augmented by these aximizations okay so the question was have you introduced any of these aspects really as a part of some integrated concept of geometry and i haven't tried to do it although i guess informally they're all integrated because they're all geometrical and so these are different facets of geometry and we're trying to tease them apart so i view the project really differently not to unify them but to point out that there are these different ways of looking at what geometry is that's really what the point of of making those distinctions is okay uh oh okay there's a question here about decidability so what would you say decidability of a theory is what makes mathematicians move on uh if euclidean geometry was not decidable would mathematicians still be interested in okay that's really interesting so so i remember okay so this tarsky's theorem i just find it incredible and when i was a postdoc in berkeley i taught the logic class there and i really went overboard with extolling the importance and uh of taski's theorem about the disability of geometry and the phrase uh geometry is finished may may have been uttered uh in my lecture and okay so then i went back to my office and it turned out that the very next class taught in in that room and some of the same students was by charlie pugh who is a famous um dynamicist and geometer and a little while later after he taught his classic there was a knock on my door and it was charlie pugh and he was saying what is this that you're telling your students that geometry is finished what is that all about and so we had a really nice discussion about taraski's theorem and um and and it's exactly this point you're making so if we have a decision procedure for a subject then does that mean the subject is finished if you can reduce all questions to a road procedure in principle then that's the sense in which we've sort of polished it off right okay and this was the kind of basis for my enthusiastic remarks and so one thing i learned from this is that well it's sort of surprising to me that often times you can find geometers who don't know about tauroski's theorem and they don't know that the subject is decidable in principle um and maybe they're surprised to learn it and uh and i think that's a pity i think every everyone studying geometry should know that there's this decision procedure okay but meanwhile i think the criticisms of the enthusiasm especially are quite strong and so i can't be so enthusiastic now because the algorithm is useless it just takes way too long it's double exponential time we might as well not have an algorithm at all if it takes so long that we can't use it to answer any particular question not even a very tiny question uh the algorithm simply isn't going to give us an answer and so in a sense we don't have a useful answer it's a theoretical algorithm only um and so of course there's plenty still to do in geometry and open questions and the geometers are quite busy but also there's another respect in which that algorithm isn't any good and that is it's only working in the very restricted language that tauroski had set up where you can quantify over points and you can refer to between this and and to equi equidistance and using that you can you can make questions like well is every uh conic section that you arise by intersecting this plane and this cone and so on does it for any such thing is there a point such that for the sphere that you construct blah blah blah like this so you can make such kind of statements in that language but what you can't say in that language is any statement that's referring to the integers or to arithmetic or to the possibility of a construction you can't ask in toroski's formal language is there a construction that will achieve this or does this construction eventually terminate at a point with a certain property um those questions are not formalizable in tarsus language and they're not subject to the decision procedure and we know by girdle's incompleteness theorem which we're going to talk about in week 7 i think that it's impossible to have a decision procedure that answers questions about arithmetic um precisely because the the incompleteness theorem implies that arithmetic is not desolatable um so that's a sense so those are all senses in which um of course geometry isn't finished and there's plenty of work to do and plenty of open questions and so on and there's several ways in which one can object to that decision procedure and so maybe the existence of a decision procedure doesn't really matter so much in terms of uh mathematicians leaving the subject certainly they're not leaving the subject because of choski's you know okay are there other questions right is the geometric theory that tosca proved completeness for a first order theory um yes his theory is complete it's that's what he proved so the theory is called the element of hero geometry and it's basically the geometry that you get um in in cartesian geometry because it's based on this analysis of the real close field so it's the geometry that you would be led to if you were thinking of geometric points as represented by real numbers in in some fixed dimension um uh you know by tubals of real numbers in that dimension so he has he has a completeness axiom and then these between his things and so on and and uh and his maximization is complete and decidable i mean it's a consequence that is decidable because of the completeness of course whenever you have a complete theory with a computable accidental theory it will be decidable because given any question you can just systematically look through all proofs and either you're going to find a proof of the statement or you're going to find a proof of the negation and therefore you will eventually be able to answer yes or no whether that statement is true or not so decidability is generally a consequence of completeness and that's how tasky proved it okay any other comments or questions all right so i guess uh so that's it for this week then and i'll see you all next week if you want to come and join us um uh so see you next week bye

Channel: Joel David Hamkins

Views: 6,623

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Id: V4Jo4VGRq5M

Length: 94min 22sec (5662 seconds)

Published: Wed Nov 04 2020