Existence of parallels in axiomatic plane geometries

[lavinia]

TL;DR Summary

Is the existence of parallels provable in Euclidean and Hyperbolic plane geometry?

Parallels exist in both Euclidean and Hyperbolic geometry. Yet each includes a separate postulate that declares the number of parallels to a line in a plane through a given point. But it seems that if both geometries have parallels then their existence - as opposed to how many of them - should be provable without a parallel postulate. One should be able to derive them from the other postulates, the postulates that describe points and lines and their intersections and how they separate the plane.

My question is: Is there a proof of the existence of parallels without Euclid's Fifth Postulate or its equivalents such as Playfair's Axiom? Same question for Hyperbolic geometry.

The following postulates describe simple properties of the plane that it seems one would have to use to prove the existence of parallels without a parallel postulate.

- Given two points in a plane there is a unique line that contains both of them.

- Two lines that intersect intersect in exactly one point

- A point on a line separates the line into two rays - or half lines. The two rays intersect in the point.

- A line separates the plane into two half planes. These two half planes intersect in the line.

- Let two lines $L_1$ and $L_2$ intersect in the point $P$. Let $R_1$ and $R_2$ be the two rays on $L_1$ determined by $P$. Then $R_1$ lies completely in one of the half planes determined by $L_2$ and $R_2$ lies completely in the other.

Note: These postulates exclude Elliptic geometry since, there, two lines intersect in two points. One might try identifying each pair of intersection points to get only one point but then a line will not separate the plane into two disjoint half planes. If one pictures this geometry as a sphere with great circles as the lines then identifying opposite poles creates the projective plane. In the projective plane the projections of the great circles do intersect in a single point but they do not separate the plane into two pieces. This is because the inverse image of the projective plane minus the projection of the circle is two hemispheres. But these are identified in the projective plane. Also on a sphere opposite poles do not determine a unique line. So it seems that an axiomatic version of this geometry would have to say something like if two points are not "opposite" then they determine a unique line but if they are "opposite" they do not. It would be interesting to derive spherical geometry axiomatically.

Geometric intuition says that two perpendicular lines to a given line $L$ must be parallel. Why is this? Suppose they are not. Then they must intersect in one of the half planes and not there other. But which half plane would that be? The lines make right angles to $L$ in both half planes so there is no difference in the way they enter into the half planes. One would think therefore if they intersected in one they would have to intersect in the other. There seems to be a principle of symmetry here that is implicit in the intuition but is not stated in the postulates. Still one would think that this symmetry is intrinsic to the idea of these two geometries and is independent of any parallel postulate.

So is there a proof or is some additional postulate dealing with symmetry needed?

 

[mathman]

Parallel postulate is needed. Without it a geometry with no parallels is possible (spherical).

 

[lavinia]

@mathwonk do you have any thoughts on this?

 

[pbuk]

Note: These postulates exclude Elliptic geometry since, there, two lines intersect in two points.

No they don't, look up the axioms of elliptic geometry.

Geometric intuition says that two perpendicular lines to a given line $L$ must be parallel.

But this is geometric intuition applied to a Euclidean plane - if instead we have a sphere mapped in the conventional way, then any two lines of longitude intersect the equator at right angles, however they meet at the two poles. And here is the crucial point you are missing - in the geometry of a sphere (elliptical geometry), the two poles are actually a single "point". You can see that this is the case by looking at the first three postulates you quote; if antipodes were distinct points then each of these is violated.

 

[lavinia]

No they don't, look up the axioms of elliptic geometry.If you had read mIf

This is from the Wikipedia artice

"Elliptic plane[edit]
The elliptic plane is the real projective plane provided with a metric:"

I explained why the projective plane is excluded in the postulates that I listed. Take a look.

 

[lavinia]

But this is geometric intuition applied to a Euclidean plane -

It is the same or the hyperbolic plane. It is also true for the sphere but not true for the projective plane.

 

[pbuk]

I think we are talking at cross-purposes, and I don't think the Wikipedia articles on this subject are very helpful. As others have said it is not possible to prove the parallel postulate from the first four postulates. We can see this because if you assume the parallel postulate is not true opposite you still have a consistent geometry which obeys the first four postulates.

Bear in mind that when we write "two lines that intersect intersect in exactly one point" we are defining what a point is; the fact that when you look at great circles on a sphere you see two intersections does not mean that this axiom is violated, it means that you need to change your concept of a "point".

A more helpful source:
http://mathworld.wolfram.com/EuclidsPostulates.html
http://mathworld.wolfram.com/ParallelPostulate.html
http://mathworld.wolfram.com/Non-EuclideanGeometry.html

 

[lavinia]

I think we are talking at cross-purposes, and I don't think the Wikipedia articles on this subject are very helpful. As others have said it is not possible to prove the parallel postulate from the first four postulates. We can see this because if you assume the parallel postulate is not true opposite you still have a consistent geometry which obeys the first four postulates.

Bear in mind that when we write "two lines that intersect intersect in exactly one point" we are defining what a point is; the fact that when you look at great circles on a sphere you see two intersections does not mean that this axiom is violated, it means that you need to change your concept of a "point".

A more helpful source:
http://mathworld.wolfram.com/EuclidsPostulates.html
http://mathworld.wolfram.com/ParallelPostulate.html
http://mathworld.wolfram.com/Non-EuclideanGeometry.html

The problem with the projective plane is that it is not separable by a line. One of the postulates is that a line separates the plane into two half planes.

On the other hand if you accept the separation postulate for a moment then what about the question of symmetry that ended my post? That is what I am really talking about.

BTW: If you do not use the separation postulate then the argument about symmetry that I suggested does not hold since there are not two half planes.

 

[Infrared]

This doesn't fully answer your question, but I think it covers the case of Euclidean, elliptic/projective, hyperbolic geometry. Let's suppose that our plane is a Riemannian 2-manifold, and the lines are geodesics.

Suppose for contradiction that your axioms are met, but there are no parallels, so every two lines meet in exactly one point. Let $L$ be a line, and $p$ be a point not on $L$. Then, for every nonzero $v\in T_pM$, the geodesic through $p$ with initial tangent vector $v$ meets $L$ exactly once by assumption. So, we have a map $T_pM-\{0\}\to L$. Scaling $v$ does not affect the geodesic, and since $M$ is $2$-dimensional, this gives a map $\mathbb{RP}^1=S^1\to L$. This map is onto (since any two points are connected by a geodesic) and one-to-one (since if two lines pass through $p$ and the same point on $L$, they must agree by your second axiom). I think this map is also continuous, so we have a continuous bijection $S^1\to L$. Since $S^1$ is compact and $L$ is Hausdorff, it is a homeomorphism, so $L\cong S^1$.

Now let $P_1$ and $P_2$ be points in one of the regions bounded by $L$. Let $L'$ be the line passing through $P_1$ and $P_2$. Since $L$ and $L'$ are homeomorphic to circles and $P_1$ and $P_2$ are on the same side of $L$, in order for $L$ and $L'$ to meet exactly once, they must be tangent at some point. But two geodesics that are tangent at a point are equal. Contradiction.

 

[pbuk]

The problem with the projective plane is that it is not separable by a line. One of the postulates is that a line separates the plane into two half planes.

I think it would be better to stop talking about projective plane/elliptic geometry because you are restricting your interest to Euclidean and hyperbolic spaces. Some of the statements you have made about elliptic geometry are not correct and this is adding unnecessary confusion.

On the other hand if you accept the separation postulate for a moment then what about the question of symmetry that ended my post? That is what I am really talking about.

Are you suggesting that we can just assume symmetry? I don't think we can, in mathematics we can't just make an assumption because it looks intuitive, we either have to prove it as a theorem or we have to introduce it as another postulate/axiom (e.g. "lines that meet in one half plane must meet in the other"*). And we already know that we can replace the parallel postulate with e.g. the triangle rule or anyone of a number of equivalents so where has this got us?

* EDIT - I am not suggesting that this is a sufficient condition to be equivalent to the parallel postulate, just asserting the existence of such conditions.

 

[lavinia]

I think it would be better to stop talking about projective plane/elliptic geometry because you are restricting your interest to Euclidean and hyperbolic spaces. Some of the statements you have made about elliptic geometry are not correct and this is adding unnecessary confusion.

I agree there is confusion. But what specifically did I say that was incorrect?

 

[lavinia]

This doesn't fully answer your question, but I think it covers the case of Euclidean, elliptic/projective, hyperbolic geometry. Let's suppose that our plane is a Riemannian 2-manifold, and the lines are geodesics.

Suppose for contradiction that your axioms are met, but there are no parallels, so every two lines meet in exactly one point. Let $L$ be a line, and $p$ be a point not on $L$. Then, for every nonzero $v\in T_pM$, the geodesic through $p$ with initial tangent vector $v$ meets $L$ exactly once by assumption. So, we have a map $T_pM-\{0\}\to L$. Scaling $v$ does not affect the geodesic, and since $M$ is $2$-dimensional, this gives a map $\mathbb{RP}^1=S^1\to L$. This map is onto (since any two points are connected by a geodesic) and one-to-one (since if two lines pass through $p$ and the same point on $L$, they must agree by your second axiom). I think this map is also continuous, so we have a continuous bijection $S^1\to L$. Since $S^1$ is compact and $L$ is Hausdorff, it is a homeomorphism, so $L\cong S^1$.

Now let $P_1$ and $P_2$ be points in one of the regions bounded by $L$. Let $L'$ be the line passing through $P_1$ and $P_2$. Since $L$ and $L'$ are homeomorphic to circles and $P_1$ and $P_2$ are on the same side of $L$, in order for $L$ and $L'$ to meet exactly once, they must be tangent at some point. But two geodesics that are tangent at a point are equal. Contradiction.

I love this proof.

Technically all of this differential topology is not included in the axioms. Still the proof seems to show that existence of parallels depends only on the separation axioms. Really very nice.

Interestingly in elliptic geometry the lines are circles.

 

[lavinia]

Are you suggesting that we can just assume symmetry? I don't think we can, in mathematics we can't just make an assumption because it looks intuitive, we either have to prove it as a theorem or we have to introduce it as another postulate/axiom (e.g. "lines that meet in one half plane must meet in the other"*). And we already know that we can replace the parallel postulate with e.g. the triangle rule or anyone of a number of equivalents so where has this got us?

I am suggesting that there is an underlying principle of symmetry that depends only on the separation axioms.

Symmetry is not equivalent to the parallel postulate. It merely guarantees the existence of parallels. It does not say how many.

BTW: The symmetry principle also works on the sphere. A great circle splits the sphere into two disjoint hemispheres. However, once one takes the quotient topology by identifying antipodal points the projections of the circles do not separate the the quotient space i.e. the projective plane.

If you read through @Infrared 's post #9 it seems to say that a model for the separation axioms is a smooth two dimensional manifold whose geodesics are smooth 1 dimensional submanifolds. In this model no parallel postulate is assumed nor any specific metric. Yet the existence of parallels is proved. It does not say how many parallels. I am not sure yet if there is a symmetry principle used in the proof. Will have to think about that. He uses the theorem that the only connected one manifolds without boundary are the line and the circle.

 

[Infrared]

- A point on a line separates the line into two rays - or half lines. The two rays intersect in the point.

I realized that I don't use that axiom, and in fact it is inconsistent with having lines that are homeomorphic to circles (and adjusting this would also mean changing your last axiom). The axiom that lines can't meet more than once already rules out spherical geometry, so maybe this unnecessary?

He uses the theorem that the only connected one manifolds without boundary are the line and the circle.

I'm probably missing something, but where exactly did I use this?

 

[lavinia]

I realized that I don't use that axiom, and in fact it is inconsistent with having lines that are homeomorphic to circles (and adjusting this would also mean changing your last axiom). The axiom that lines can't meet more than once already rules out spherical geometry, so maybe this unnecessary?I'm probably missing something, but where exactly did I use this?

Right. So it seems either that the ray postulate is not necessary or the circles contradict it.

 

[lavinia]

I'm probably missing something, but where exactly did I use this?

You are right. You did not use it.

 

[TeethWhitener]

This doesn't fully answer your question, but I think it covers the case of Euclidean, elliptic/projective, hyperbolic geometry. Let's suppose that our plane is a Riemannian 2-manifold, and the lines are geodesics.

Suppose for contradiction that your axioms are met, but there are no parallels, so every two lines meet in exactly one point. Let $L$ be a line, and $p$ be a point not on $L$. Then, for every nonzero $v\in T_pM$, the geodesic through $p$ with initial tangent vector $v$ meets $L$ exactly once by assumption. So, we have a map $T_pM-\{0\}\to L$. Scaling $v$ does not affect the geodesic, and since $M$ is $2$-dimensional, this gives a map $\mathbb{RP}^1=S^1\to L$. This map is onto (since any two points are connected by a geodesic) and one-to-one (since if two lines pass through $p$ and the same point on $L$, they must agree by your second axiom). I think this map is also continuous, so we have a continuous bijection $S^1\to L$. Since $S^1$ is compact and $L$ is Hausdorff, it is a homeomorphism, so $L\cong S^1$.

Now let $P_1$ and $P_2$ be points in one of the regions bounded by $L$. Let $L'$ be the line passing through $P_1$ and $P_2$. Since $L$ and $L'$ are homeomorphic to circles and $P_1$ and $P_2$ are on the same side of $L$, in order for $L$ and $L'$ to meet exactly once, they must be tangent at some point. But two geodesics that are tangent at a point are equal. Contradiction.

This is going to be a stupid question. I'm not a mathematician, so the language is going to be rough, at the very least; I'm just trying to wrap my head around the argument. At least intuitiviely, isn't the geodesic through $p$ with initial tangent vector $v$ equivalent to the geodesic through $p$ with initial tangent vector $-v$ (I don't know what else to call it)? Wouldn't that mean that the map is onto but not one-to-one?

 

[Infrared]

My map is from $\mathbb{RP}^1=(T_pM-\{0\})/\mathbb{R^\times}$ to $L$. Note that $v$ and $-v$ represent the same element of $\mathbb{RP}^1.$

Edit: If you're not familiar with projective space, here's a way to restate the argument without it. Choose coordinates so that $\{e^{i\theta}\}$ is the unit circle in $T_pM$, and consider the arc $C=\{e^{i\theta}:0\leq\theta\leq\pi\}.$ Every geodesic based at $p$ is equivalent (can be reparameterized) to one with initial tangent vector in $C$, and the only two tangent vectors in $C$ that give equivalent geodesics are $1=e^{0}$ and $-1=e^{i\pi}$. A map from an interval (arc) to a space where the two endpoints of the arc are mapped to the same point is equivalent to a map from a circle to the space. This map (from the circle) is bijective because now we've counted all the directions exactly once.

 

[mathman]

On a sphere every pair of lines meet at exactly two points. Where did exactly one point come from?

 

[pbuk]

One of our definitions is that two lines meet at exactly one point. We therefore interpret a "point" on a sphere as the pair of antipodal points where two great circles cross; the resulting geometry is consistent.

See post #7 and http://mathworld.wolfram.com/SphericalGeometry.html

 

[lavinia]

On a sphere every pair of lines meet at exactly two points. Where did exactly one point come from?

The projective space its the quotient space of the sphere obtained by identifying each pair of its antipodal points to a single point. Since two great circles intersect in two antipodal points, their projections in the projective plane intersect in one point.

When you identify antipodal points on a circle you still end up with a circle. Think of twisting a rubber band so that it overlaps itself twice. So in the projective plane great circles project to circles. You get a geometry where lines are circles and each pair of lines intersect in exactly one point.

 

[lavinia]

Note: Why a line does not separate the projective plane into two half planes.

Choose a line in the projective plane. Its preimage in the sphere is a great circle. Think of this great circle as the equator. It is surrounded by a cylindrical strip bounded by the Tropics of Cancer and Capricorn.

Since this strip is symmetrical around the equator all of its antipodal points lie inside of it. A point in the Southern half of the strip is antipodal to a point in the Northern half. It is identified to itself in the projective plane where identification turns it into a Mobius stip. So every line in the projective plane is surrounded by an embedded Mobius strip. One knows that when one removes the equator of a Mobius strip it does not fall into two pieces but just into one connected cylindrical band.

More simply the equator on the sphere splits the sphere into two hemispheres but these become one connected region in the projective plane..

 

[mathwonk]

Euclid proves in Prop. I. 31 the existence of parallel lines, without the use of the parallel postulate, i.e. in any "neutral" geometry (also called a Hilbert plane), in particular also in hyperbolic geometry. The proof uses only Prop. I. 28 (equivalent to I.27), which do not use the parallel postulate, which is first used in Prop. I.29.

 

[jim mcnamara]

See: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI31.html
I did.

 

[lavinia]

Following on @Infrared's proof, I thought a sketch of a proof that does not use calculus might go like this. ( Sorry, I do not know how to include diagrams.)

In a plane, given a line $L$, a point $O$ on it, and a line $L_{OA}$ perpendicular to it, draw a line perpendicular to $L_{OA}$ through $A$ and assume that there are no parallels. This line intersects $L$ in a point $B$.

$B$ lies in one of the half planes $H_1$ determined by $L_{OA}$. In the other half plane $H_2$ a point $P$ on $L$ determines a ray that makes some angle $∠OAP$ with $L_{OA}$. Let $α$ be the least upper bound of all of these angles.

Now draw a line through $A$ that makes the angle $α$ with $L_{OA}$ in the same half plane $H_2$. Call this line $L_{AX}$ where $X$ is the point where the line intersects the original line $L$. $X$ can not be in the half plane $H_2$ since then a larger angle than $α$ could be found. One does this by selecting a point on the ray in $L$ whose base point is $X$ and which does not contain the point $O$.

So $X$ must lie in the half plane $H_1$, the same half plane as the intersection point $B$ of $L_{AB}$ with $L$. Further $X$ can not lie on the line segment between $O$ and $B$ but must be in the half plane determine by $L_{AB}$ that does not contain $O$. (For the moment the case that $X$ and $B$ are the same will be excluded.)

Now consider the angles formed by the constructed lines at the point $A$. One has the 90 degree angle $∠OAB$; the angle $α$ , and a residual angle $∠BAX$ . These angles all together add up to 180 degrees.

* The lines through $A$ that are inside $α$ all intersect $L$ in the half plane $H_2$. The lines inside $∠OAB$ all intersect $L$ on the line segment $OB$. The lines through $∠BAX$ all intersect $L$ on the line segment $BX$.

No line intersects $L$ at a point on the ray that starts at $X$ and does not contain $B$.

But one can always draw such a line.

Note:

- If $X$ and $B$ are the same then one gets the same situation as in * but with the residual angle $∠BAX$ equal to zero.

To flesh this out, one needs a few ideas and facts. All of these - I think - can be defined in terms of the separation axioms.

- An idea of betweeness for three points on a line. This by itself excludes circles as candidates for lines.
- An idea of a line passing through the interior of an angle.
- The assumption that rays always contain at least two points.

I think this line of proof differs from Infrared's only in not using the continuity for the mapping of the projective line at $A$ onto the line $L$.

 

[mathwonk]

One of the most rewarding mathematical experiences I had in my career was finally teaching geometry from Euclid, in my 60's, guided in that endeavor by the excellent book of Hartshorne: Geometry, Euclid and beyond. I found that the reputation of Euclid for being unrigorous is highly overstated, and it in fact provides the background for the basic modern theories of mathematics, i.e. both algebra and analysis, in elementary geometric terms.

There are two deep concepts that are in fact equivalent, something I had not before realized, namely area and proportion, or similarity. Euclid begins with a fairly careful development of area, culminating in the Pythagorean theorem, and derives the concept of similarity from it, in a way that gives a glimpse of the Dedekind definition of real numbers. He then shows how conversely one could derive the Pythagorean theorem from the principle of similarity.

The opposite development, postulating similarity and deriving the theory of area, is nowadays done in the Birkhoff approach to geometry. This backwards derivation is "easier" but at the cost of assuming a much more sophisticated concept as opposed to an elementary one in my opinion, and hence very unnatural. Today's math student may prefer it since we are steeped in the theory of real numbers and proportion from early on. But to understand the origin of the idea of real numbers I greatly recommend studying Euclid.

He also presages the theory of algebraic operations by means of elementary geometry, giving simple geometric interpretations of the laws of multiplication and its properties, distributivity for instance. He even shows how to solve quadratic equations geometrically. (To see this interpretation one only has to consider the product of two segments as the rectangle with them as sides.) After relating the study of circles to the study of triangles, the construction of a regular pentagon is a tour de force in Euclid.

To approach Euclid I highly recommend beginning with Hartshorne, as I had avoided Euclid all my life because of being put off by the beginning, the vague definitions for example. Once I got into the theorems I began to see the beauty and clarity of it. I think micromass had a similarly positive experience with Euclid, according to his comments here.

After teaching from Hartshorne and Euclid to college students, I had the privilege of teaching a shorter version of the same course to a brilliant group of 10 year olds, who also enjoyed it, and it helped me to really grasp the material while trying to make it completely accessible to bright but naive students. In fact, asked to introduce similarity but without time to reach the fundamental result, Proposition VI.2, I realized the same idea was already contained in Prop. III.35 and I presented it that way. At the end of the course I understood how Archimedes had refined the treatment of area to give a foundation to the ideas of integral calculus, for example he clearly knew the so called Cavalieri principle, and used it to compute the volume of a ball.

I.e. while the fundamental theorem of calculus allows us to calculate area and volume formulas from height and area formulas, the more basic Cavalieri principle allows one to compare areas and volumes of different figures, and hence to deduce new such formulas from old. I.e. although Archimedes could not go directly from a formula for the slice areas to a volume formula, he knew that two figures with the same slice areas have the same volume. So even without the FTC, Archimedes deduced the volume of a 3-ball by comparing it to the difference of the volumes of a circular cylinder and a cone, which he deduced in turn from comparisons with cubes and polygonal cylinders.

I ended the course for the 10 year olds with this result, as well as the field theoretic sketch of the impossibility of certain ruler and compass constructions, and the calculation of the volume of a "bicylinder", a result missing from the partially recovered "palimpsest" of Archimedes, but which can be reconstructed from his ideas. The volume of a bicylinder is an example of a calculation which is often posed as a challenge problem in calc books, but which is as easy by Archimedes' method as that of a 3-ball. One can even use his methods to calculate the volume of a 4-ball and hence also its "surface area" i.e. that of the 3 dimensional sphere S^3, and I added this result in an afterword to the course. my notes from this course are here:
http://alpha.math.uga.edu/~roy/camp2011/10.pdf

 

[lavinia]

One of the most rewarding mathematical experiences I had in my career was finally teaching geometry from Euclid, in my 60's, guided in that endeavor by the excellent book of Hartshorne: Geometry, Euclid and beyond. I found that the reputation of Euclid for being unrigorous is highly overstated, and it in fact provides the background for the basic modern theories of mathematics, i.e. both algebra and analysis, in elementary geometric terms.

There are two deep concepts that are in fact equivalent, something I had not before realized, namely area and proportion, or similarity. Euclid begins with a fairly careful development of area, culminating in the Pythagorean theorem, and derives the concept of similarity from it, in a way that gives a glimpse of the Dedekind definition of real numbers. He then shows how conversely one could derive the Pythagorean theorem from the principle of similarity.

The opposite development, postulating similarity and deriving the theory of area, is nowadays done in the Birkhoff approach to geometry. This backwards derivation is "easier" but at the cost of assuming a much more sophisticated concept as opposed to an elementary one in my opinion, and hence very unnatural. Today's math student may prefer it since we are steeped in the theory of real numbers and proportion from early on. But to understand the origin of the idea of real numbers I greatly recommend studying Euclid.

He also presages the theory of algebraic operations by means of elementary geometry, giving simple geometric interpretations of the laws of multiplication and its properties, distributivity for instance. He even shows how to solve quadratic equations geometrically. (To see this interpretation one only has to consider the product of two segments as the rectangle with them as sides.) After relating the study of circles to the study of triangles, the construction of a regular pentagon is a tour de force in Euclid.

To approach Euclid I highly recommend beginning with Hartshorne, as I had avoided Euclid all my life because of being put off by the beginning, the vague definitions for example. Once I got into the theorems I began to see the beauty and clarity of it. I think micromass had a similarly positive experience with Euclid, according to his comments here.

After teaching from Hartshorne and Euclid to college students, I had the privilege of teaching a shorter version of the same course to a brilliant group of 10 year olds, who also enjoyed it, and it helped me to really grasp the material while trying to make it completely accessible to bright but naive students. In fact, asked to introduce similarity but without time to reach the fundamental result, Proposition VI.2, I realized the same idea was already contained in Prop. III.35 and I presented it that way. At the end of the course I understood how Archimedes had refined the treatment of area to give a foundation to the ideas of integral calculus, for example he clearly knew the so called Cavalieri principle, and used it to compute the volume of a ball.

I.e. while the fundamental theorem of calculus allows us to calculate area and volume formulas from height and area formulas, the more basic Cavalieri principle allows one to compare areas and volumes of different figures, and hence to deduce new such formulas from old. I.e. although Archimedes could not go directly from a formula for the slice areas to a volume formula, he knew that two figures with the same slice areas have the same volume. So even without the FTC, Archimedes deduced the volume of a 3-ball by comparing it to the difference of the volumes of a circular cylinder and a cone, which he deduced in turn from comparisons with cubes and polygonal cylinders.

I ended the course for the 10 year olds with this result, as well as the field theoretic sketch of the impossibility of certain ruler and compass constructions, and the calculation of the volume of a "bicylinder", a result missing from the partially recovered "palimpsest" of Archimedes, but which can be reconstructed from his ideas. The volume of a bicylinder is an example of a calculation which is often posed as a challenge problem in calc books, but which is as easy by Archimedes' method as that of a 3-ball. One can even use his methods to calculate the volume of a 4-ball and hence also its "surface area" i.e. that of the 3 dimensional sphere S^3, and I added this result in an afterword to the course. my notes from this course are here:
http://alpha.math.uga.edu/~roy/camp2011/10.pdf

@mathwonk

How did the Greeks think about numbers? It seems that they spent a lot of time on ruler and compass constructions almost as if they thought that the continuum was of the points defined by intersections of circles and lines in these constructions.

It also seems that they may have thought that all numbers are rational until they found the Pythagorean Theorem.

What kind of number did they think did they think π is?

 

[mathwonk]

I'll try to remember my thoughts from some years ago. And anything I say is just my own impression after reading and thinking about their works. Reading Galileo, one sees that he represents a real number as a pair of line segments. So to him a real number is the ratio of two geometric lengths. Now this concept seems to me to be the one in the Greek approach in Euclid.

I don’t think the Greeks really had a concept of real number. (In fact Euclid defines a "number" as an integer, in perhaps Book VII) Nonetheless they dealt with an equivalent concept, namely they had an equivalence relation between pairs of segments, which amounted to the pairs being in the same ratio with respect to length, and they knew how to calculate with this relation. If they had given a name to the equivalence classes, they would have had the set of real numbers.

Since the treatment of these equivalence classes is so detailed and extensive, it seems instructive to me to study it in their terms, to acquire a sense of what went before the modern theory of real numbers, in order to realize just how sophisticated was the early study, and understand how the concept of real numbers developed historically. So even if you do not believe the Greeks had a notion of real number, it still interests me to ask just what part of it did they have, and how does it relate to the modern theory of real numbers.

My first good acquaintance with real numbers was from the treatment in Courant, where he starts from the notion of the ratio of two segments on the real line, and then measures this by sequences of integers between 0 and 9, i.e. decimals. The upshot is that if one thinks of a real number as a pair of segments, then once one chooses a unit segment, a real number is determined and can be described by an infinite decimal. Then one can go backward and define a real number as an infinite decimal.

But one could also go back further and define a real number as a pair of segments, provided one tells how to operate “algebraically” on those pairs, and gives a robust equivalence relation on those pairs, e.g. how to multiply two pairs. Note that even using decimals one still cannot avoid equivalence relations, since one must always deal with that bugabear of the young student, the fact that two different decimals such as 1.0000…. and .9999…. sometimes define the same real number.

If one takes as a charge to describe in detail how to do algebraic operations on pairs of segments, and how to determine when two pairs of segments determine the same real number, I believe one will find that Euclid discusses all these matters in detail. So he has done all the work that would be needed to define a real number as an equivalence class of segments, even if he does not do this.

Now let me say right away that the theory in Euclid does not lead to the field of real numbers, since there is no axiom of completeness there for the “numbers” arising from his geometry. If one carries through the theory suggested here and makes pairs of segments into a set of numbers, one gets a field that Hartshorne calls a ”Euclidean” field, which is a certain type of ordered field having positive square roots. If one assumes the Archimedean axiom, which is described in Euclid, one gets a Euclidean subfield of the real numbers.

So when does one actually obtain the real numbers from Euclid’s theory? Recall that Hilbert has clarified the need for certain betweenness axioms that are implicit in Euclid, including the idea that each point on a line divides that line into two disjoint rays. If one were to state, as Euclid does not to my knowledge, that conversely every division of a line into two disjoint rays must occur at a point of that line, then one would have the Dedekind completeness axiom, and the resulting field defined by segment arithmetic would indeed be isomorphic to the field of real numbers. All this is very carefully treated in Hartshorne’s wonderful book Geometry: Euclid and beyond, especially chapter 3: geometry over fields, and chapter 4: segment arithmetic.

Briefly, my own more naive approach was to think of a segment as a number, and define the product of two segments x,y as the rectangle (xy) they define as its sides. After choosing a unit segment, one makes this product area (xy) into a length by constructing a rectangle with the same area but having one side equal to the given unit, i.e. then the other side of that rectangle has length equal to the given area xy.

I.e. it is easy to define the product of two segments as a rectangle, but to define it as a segment needs choosing a unit segment.

Various geometric constructions in Euclid’s chapter II amount to showing that multiplication in this sense is distributive and commutative.

Since Euclid shows how to compare areas, one can say when two products xy and ab are equal, by comparing the rectangles they define. This then let's us define the equality of ratios of lengths, since x/a = b/y precisely when xy = ab.

The fundamental similarity result for triangles then becomes the Prop. III. 35 in Euclid, since an arbitrary pair of similar triangles can be represented in this way by two secants in a circle passing through a common interior point. Note this approach to similarity does not require the Archimedes axiom. I made up this approach myself when I needed it for my class of 10 year olds.

Euclid’s approach to similarity introduces the Archimedean axiom, and the concept of rational approximations to irrational ratios. I.e. two pairs of line segments, both of whose ratios are rational, can be determined by a finite subdivision. Euclid extends this to arbitrary ones by saying they define the same ratio when every finite subdivision shows either that both ratios are greater than the corresponding rational number, or else both are less than it. As we would put it, two real numbers are equal if they are both greater than the same set of rational numbers.

Anyway, I highly recommend Hartshorne, and readily admit that I myself have not fully read all the relevant sections I have cited.

In regard to the specific number π, Euclid proves in Prop. XII.2 that the ratio of the area of a circle to that of the square on its diameter, hence also to that on its radius, is a constant. Hence he seems to have regarded π as the ratio of two areas, i.e. as defined by the formula area(circle of radius r) = πr^2. Archimedes, in his work Measurement of a circle, specifically Props 1 and 3, proves this ratio π equals also that of the circumference of a circle to its diameter, and that this ratio lies strictly between 22/7 and 223/71.

Curiously, Prop. 2 of that work apparently makes the erroneous and contradictory claim that the ratio of the area of a circle to the area of the square on its diameter is 11/14, i.e. that π = 22/7. The editor T.L. Heath notes that "the text of this proposition is not satisfactory, and Archimedes cannot have placed it before Prop. 3, as the approximation depends upon the result of that proposition".

 

[lavinia]

I'll try to remember my thoughts from some years ago. And anything I say is just my own impression after reading and thinking about their works. Reading Galileo, one sees that he represents a real number as a pair of line segments. So to him a real number is the ratio of two geometric lengths. Now this concept seems to me to be the one in the Greek approach in Euclid.

I don’t think the Greeks really had a concept of real number. (In fact Euclid defines a "number" as an integer, in perhaps Book VII) Nonetheless they dealt with an equivalent concept, namely they had an equivalence relation between pairs of segments, which amounted to the pairs being in the same ratio with respect to length, and they knew how to calculate with this relation. If they had given a name to the equivalence classes, they would have had the set of real numbers.
...

@mathwonk
A couple questions: Why weren't geometric lengths themselves considered to be numbers? Why were numbers ratios of pairs of geometric lengths?

How were ratios defined without having numbers first?

It almost seems like ratios were the key idea of interest.

 

[mathwonk]

What do you mean by geometric lengths? just knowing two segments are congruent does not assign a numerical length to them. numbers are intrinsically dependent on an introduction of a unit. numbers only arise when you want to compare two segments. this is true of all quantities.

Recall the words of euler in his introduction to algebra, Elements of algebra, pages 1-2,

"Whatever is capable of increase or diminution, is called a magnitude or quantity."...Mathematics, in general, is the science of quantity. Now we cannot measure or determine any quantity except by considering some other quantity of the same kind as known, and pointing out their mutual relation...So that the determination or the measure of magnitudes of all kinds is reduced to this: fix at pleasure upon anyone known magnitude of the same species with that which is to be determined, and consider it as the measure or unit; then determine the proportion of the proposed magnitude to this known measure. This proportion is always expressed by numbers; so that this number is nothing but the proportion of one magnitude to another, arbitrarily assumed as the unit."

This makes clear that a number is determined by any pair of magnitudes of the same sort, but to me the most usual sort of magnitudes to use in analysis and geometry is a pair of segments, i.e. "lengths", but length in an abstract sense, not associated with a number, just a concept arising from the phenomenon of one dimensional extension, i.e. intrinsic in a segment. Euclid of course also compares pairs of areas, and relates the comparison of two areas to the comparison of two lengths, by expressing both areas as rectangles on the same base, or rather congruent bases, so that the areas are equal when, having arranged equal bases, the heights are also equal as segments.

We are so used to numbers we have trouble separating quantity from its measure in terms of numbers. Go back to the Odyssey of Homer wherein we have a blinded Cyclops who knows how to tell if his sheep have all returned to the cave at night without assigning a number to them. Namely he piles up stones, one for each sheep, as they go out, and removes the stones from the pile as they come back in. He knows thereby that there is the same number of sheep and of stones, but he does not know what that number is. He has no sense of number in the abstract. The notion of whether two numbers are the same is different from the notion of what that number is.

The modern leap of abstraction is to take the definition of equivalence of sets via bijection, i.e. the knowledge of when two sets have the same number of elements, and make that into definition of a cardinal number, i.e. a number is an equivalence class of objects of the same kind, with respect to an appropriate equivalence relation.

Of course it depemnds what you mean by a number, since one can add segments in Euclid, without a unit being chosen, as Hartshorne emphasizes it is when you want to multiply them that you need a unit. Of course this is what we said above, since the product of two segments is naturally an area, i.e. a rectangle, and to equate a rectangle with a segment you must choose a unit segment, and express all your rectangles as having that given unit length base.

does this scan?

of course we have a long evolution of these concepts arriving at the idea that a cardinal number is one of the objects constructed by genius wherein zero IS the empty set, and 1 IS the set whose only element is the empty set, and so on...i.e. n is the set whose elements are 0,1.2,...n-1. In this instance we have chosen one distinguished representative of the equivalence class and chosen that as the name of the whole class, i.e. we have decided that n is a certain particular set of n elements, which is thus in bijection with every set of n elements. In this tradition, my hand is the number 5.

 

[mathwonk]

I agree with you that ratios preceded numbers. I.e. the ability to compare things of the same kind is the fundamental idea, i.e. equivalence relations, or when are two things to be considered the same in some sense. e.g. if you consider two plane vectors as the same if they have the same length and direction, you have an equivalence relation that can be used to define complex numbers, i.e. to add them you use vector addition and to multiply them you add their angles and multiply their lengths.
It is interesting that again to define multiplication you need a unit vector, i.e. to define addition of angles, as well as multiplication of lengths.

numbers arise when you try to capture the essence of an equvalence class by some one thing. but "numbers" also imply to me some way to have computational operations on the classes that obey some of the usual laws.

 

[bhobba]

Bear in mind that when we write "two lines that intersect intersect in exactly one point" we are defining what a point is;

I thought point was a primitive of the axioms rather than a definition in it based on other primitives:
https://en.wikipedia.org/wiki/Hilbert's_axioms

Or am I missing something?

Thanks
Bill

 

[pbuk]

I thought point was a primitive of the axioms rather than a definition in it based on other primitives:
https://en.wikipedia.org/wiki/Hilbert's_axioms

Maybe (this thread is not using Hilbert's axiomatisation), but I don't think it is wrong to say "we are defining what a point is" rather than "we are adopting a primitive notion of what a point is".

 

[lavinia]

@mathwonk

So starting with two geometrically determined line segments what does it mean to compare them with no unit as a guide?

Is it modular arithmetic?

 

[mathwonk]

Part of the axiom system is the ability to compare two segments and say whether they are equal and if not, which is larger. Euclid's axioms are full of conditions on the notion of "equal", which he seems to apply to lengths, areas, volumes and angles. His theorem statements also include statements about angles being equal, greater than or less than another. See his postulates 4,5 and all 5 common notions, as well as many propositions, e.g. all of the first 20 or so that I have quickly scanned, contain one of these comparison words. Hilbert has a set of axioms for congruence, and one can define less than and greater than in terms of congruence to a subset.