theories geometry - university of cambridge · theories geometry owen gri ths ... geometry euclid...

Introduction to geometry Euclidean Geometry Non-Euclidean Geometry Relative consistency Independence Conclusion

TheoriesGeometry

Owen [email protected]

St John’s College, Cambridge

03/02/14


Previously on Theories

We have seen that, in order for finite beings to grasp infinitetheories, we must use the axiomatic method

An axiom set Σ for a theory Θ is a finitely specifiable set ofwffs from which every theorem of Θ can be deduced

There are two ways of finitely specifying axioms: by listingthem, or by providing schemata

We met Robinson Arithmetic, which is is very weak, not evenbeing able to prove ∀x(0 + x = x). And we met PeanoArithmetic, which is formed by adding an axiom schema ofinduction

Both theories are subject to Godel’s incompleteness theorems


Talk outline

1 Introduction to geometry

2 Euclidean Geometry

3 Non-Euclidean Geometry

4 Relative consistency

5 Independence

6 Conclusion


Geometry

Geometry

Geometry is the theory of points, lines, planes and therelations between them

It predates the ancient Greeks: ancient Egyptians,Babylonians and Chinese all had knowledge of geometry

The Greeks, however, made the discipline more rigourous:before them, geometry was a collection of rule-of-thumbprocedures whose adequacy had to be assessed empirically

The Greeks were the first to attempt an axiomatisation ofgeometry

Euclid was not the first to contribute to this project, but hiscontribution was the most significant. His masterpieceElements attempted to derive 465 geometric theorems from 5axioms


Incidence Geometry 1

Incidence Geometry

Let’s start with a baby theory of geometry: incidencegeometry

This is a fragment of Euclidean geometry that contains as itsonly nonlogical expressions ‘point’, ‘line’ and ‘x is incidentwith y ’ (intuitively, ‘passes through’)

Incidence geometry also contains some defined terms, such as‘x is collinear to y ’, ‘x is concurrent with y ’ and ‘x is parallelto y ’. These are convenient shorthands, but can be defined interms of the nonlogical expressions

This is a very weak theory: it is silent on betweenness andcongruence, which are central concepts of geometry




It has the following 3 axioms:

1 For every point P and for every point Q not identical to Pthere exists a unique line l incident with P and Q.

2 For every line l there exist at least two distinct points incidentwith l .

3 There exist three distinct points with the property that no lineis incident with all three of them.


Proofs in incidence geometry

Proofs in incidence geometry

What sorts of claims can we prove in this simple geometrictheory?

Theorem If l and m are distinct lines that are not parallel, then l and mhave a unique point in common.

Proof To prove this claim, we must appeal to one of thedefined notions of incidence geometry: x is parallel to y just ifthey have no point in common (‘point’ is a nonlogical term).Since we are told that l and m are not parallel, they musthave some P in common. Suppose, for reductio, that l and mhave some other point Q in common, such that P 6= Q.There are, then, 2 distinct lines that are incident with (passthrough) P and Q. But, by Axiom 1 of incidence geometry,for any 2 distinct points, there is a unique line incident withthem. Contradiction. �


Talk outline





5 Independence

6 Conclusion


Euclid’s axioms

Euclid’s axioms

Every theorem of incidence geometry is a theorem ofEuclidean geometry, but not vice versa

Euclid’s axioms are:

1 Given any two points P and Q, exactly one line can be drawnwhich passes through P and Q

2 Any line segment can be indefinitely extended3 A circle can be drawn with any centre and any radius4 All right angles are congruent to each other5 If a line l intersects two distinct lines m and n such that the

sum of the interior angles a and b is less than 180◦, then mand n will intersect at some point


Euclid’s common notions

Euclid’s common notions

Euclid did not attempt to derive his theorems from these 5axioms alone. He also allowed himself 23 definitions and 5‘common notions’

His common notions were close to what we would now calllogical axioms, e.g. laws of logic such as ` P ∨ ¬P

But, if these are in the deductive system of a theory, thenthey do not have to be restated as axioms


Euclid’s definitions

Euclid’s definitions

Euclid’s definitions fall into 2 major categories

The first kind are intended to convey intuitive meanings, suchas:

A point is that of which there is no partA line is a length without breadth

These definitions play no role in formal systems. Indeed, theycannot, since ‘point’ and ‘line’ are primitives in geometry,which cannot be broken down any further

The second kind can be used in the system, such as:

A circle is a plane figure (one bounded by lines) which isbounded by a single line (a circumference) such that all of thelines radiating from a central point to the circumference areequal

Definitions such as this allow us to determine which planefigures are circles


Euclid’s axioms 2

Euclid’s axioms 2

Unlike the arithmetic axioms we encountered last week,Euclid’s axioms are not expressed in a formal language

They are written as permissions on what shapes can be drawn.We can rewrite them so they are not of that form, e.g.

1′ Between any two points there is a line

Euclid’s nonlogical primitives are:

pointlinelie onbetweencongruent


Euclid’s axioms 2

Flaws in Euclid

It turns out that not all of Euclid’s purported theorems followfrom his five axioms

Many of his proofs appeal to diagrams, which do really makeit look like some putative theorem follows from his axioms,when it in fact does not

His axiomatisation can, however, be repaired

David Hilbert, in Die Grundlagen der Geometrie, provides aproper axiomatisation of Euclidean geometry. He uses 20axioms and 9 nonlogical primitives

It will be Euclidean geometry in its repaired, Hilbertian formthat we will now mean by ‘Euclidean geometry’


Talk outline





5 Independence

6 Conclusion


Euclid’s Axiom 5 - 1


Euclid’s first four axioms have always looked acceptable buthis fifth has been the source of great controversy:

5 If a line l intersects two distinct lines m and n such that thesum of the interior angles a and b is less than 180◦, then mand n will intersect at some point

Conider the following illustration:




Why think that Euclid’s fifth axiom (sometimes called theParallel Postulate) is dodgy?

Axioms 1–4 are, in a sense, abstractions from what we canconstruct with a ruler, compass and protractor, but Axiom 5is not like this

E.g. Axiom 5 tells us that lines m and n will intersect at somepoint if they meet certain conditions, but we may have to goan extremely long way down the line to find the intersection

In this way, we may not be able to draw the relevant lines.Indeed, when we make geometric drawings, we only ever drawline segments, not lines in the sense of Axiom 5


Hyperbolic Geometry

Hyperbolic Geometry

Because of this oddity of Axiom 5, mathematicians in the 19th

century began working on geometries without it

The extreme case was geometries that took the negation ofAxiom 5, in some form, as an axiom:

¬5 There exists a line l and point P not on l such that at leasttwo distinct lines parallel to l pass through P

This axiom is always known as the hyperbolic axiom, and thegeometry formed by accepting it is known as hyperbolicgeometry

Hyperbolic geometry is a non-Euclidean geometry. There are,of course, many ways of being a non-Euclidean geometry(reject any Euclidean axiom), but hyperbolic geometry is themost well-known and studied of non-Euclidean geometries


A brief history of hyperbolic geometry 1


Janos Bolyai published a treatise on hyperbolic geometry in1831, as an appendix to a book by his father, WolfgangBolyai, who had spent much of his career attempting to deriveAxiom 5 from Axioms 1–4

Wolfgang Bolyai was so pleased with his son’s work that hesent it to most eminent mathematician of the day, Carl Gauss,who was also a friend of Wolfgang’s

Gauss’s reaction was not what Wolfgang had expected: heclaimed that he had beaten Janos to all of his conclusions inunpublished work. It turns out that Gauss was not lying




The first to actually publish anything on hyperbolic geometrywas Nikolai Lobachevsky in 1829

At first his work was not widely read: it was written inRussian and the few Russian mathematicians to take itseriously were harshly critical

In 1840, Lobachevsky’s research was published in German andwas highly praised by Gauss

Non-Euclidean geometry only gained widespread recognitionafter Lobachevsky’s death in 1855

In 1868 Eugenio Beltrami delivered the result that forcedmathematicians to take non-Euclidean geometry seriously: heshowed that if Euclidean geometry is consistent, then so isnon-Euclidean geometry


Why be afraid of hyperbolic geometry?

Why be afraid of hyperbolic geometry?

Why did it take so long for hyperbolic geometry to be takenseriously?

One reason is that it has some fairly unintuitive consequences:

For every line l and point P not on l there are at least twodistinct lines parallel to l which pass through PFor any triangle ABC , the sum of the interior angles of ABC isstrictly less than 180◦

Rectangles do not existAll similar triangles are congruent (i.e. there are no triangles ofthe same shape but different sizes)


Possible objection

Possible objection

You might claim that the hyperbolic axiom and itsconsequences are plainly false

Why think this? Well, geometry is about lines we can draw,and we can’t draw the lines of hyperbolic geometry

But geometry is not about lines we can draw: that is appliedgeometry, which is just a part of geometry

Pure geometry is about ideal lines, and the only experimentswe can perform on these are thought experiments

The question, then, is not whether we can draw the lines ofhyperbolic geometry but whether we can conceive of them.And we can conceive of them if hyperbolic geometry isconsistent


Talk outline





5 Independence

6 Conclusion


Relative consistency 1


Hyperbolic geometry is, then, deeply weird and unintuitive

We may therefore think that it is inconsistent

Recall from last week that a theory Θ is consistent iff there isno φ such that Θ ` φ and Θ ` ¬φThere will of course be a corresponding notion of semanticconsistency: Θ is consistent iff there is no φ such that Θ |= φand Θ |= ¬φIt will be consistency in the latter sense that we mean today,since it will be much easier to reason semantically

Of course, semantic consistency implies syntactic consistencyif the deductive system is complete. Alfred Tarski showed thatEuclidean geometry can be formalised in first-order logic,which we know is complete, so we know there will be acorresponding syntactic proof




How can we prove that hyperbolic geometry is consistent?

It is quite a daunting task to prove that a theory is consistentonce and for all

Instead, we frequently settle for proof of relative consistency

This involves proving conditionals such as

If Θ1 is consistent, then Θ2 is consistent

When this conditional has been proved, we say that Θ2 isconsistent relative to Θ1




Generally, we show theory Θ2 to be consistent relative to Θ1

in two steps

First, we give an interpretation of the nonlogical primitives ofΘ2 in the language of Θ1

Second, we show that so interpreted, the axioms of Θ2 are alltheorems of Θ1

If Θ1 is consistent, it follows that Θ2 is consistent whenunderstood in this new way

The Beltrami-Klein model provides a proof of the relativeconsistency of hyperbolc geometry to Euclidean geometry


Strategy

Strategy

We can use the Beltrami-Klein model to provide a relativeconsistency proof of hyperbolic geometry

Here’s the strategy:

(1) Give an interpretation of the undefined nonlogical primitives ofhyperbolic geometry in the language of Euclidean geometry

(2) Show that, so interpreted, the axioms of hyperbolic geometryare all theorems of Euclidean geometry

Possible confusion: but aren’t the terms involved in Euclideanand hyperbolic geometries just the same, i.e. ‘line’, ‘point’?No: the two geometries feature terms that are syntacticallyidentical but they have different intended interpretations, e.g.‘line’ and ‘parallel’ have different properties in hyperbolicgeometry


The hyperbolic axiom

The hyperbolic axiom

First consider the hyperbolic axiom:

H There exists a line l and point P not on l such that at leasttwo distinct lines parallel to l pass through P

There are two undefined nonlogical primitives in H: ‘line’ and‘point’

‘x is parallel to y ’ is also nonlogical, but it is not undefined. Itcan be defined in terms of points and lines, e.g. two lines areparallel just if they have no points in common

Let’s rewrite H to reflect this:

H′ There exists a line l and point P such that at least two distinctlines that have no points in common with l pass through P

Our first job, then, is to reinterpret hyperbolic geometry’s‘point’ and ‘line’ in the language of Euclidean geometry


‘Point’

‘Point’

Let’s fix on a particular circle γ with centre O and say that a‘point’ is any position X in the interior of γ such thatOX < OP


‘Line’

‘Line’

Let’s say that a ‘line’ is any open chord of circle γ. A chord isa line segment joining two points of the circumference of acircle. An open chord is a chord without its ends


‘x is parallel to y ’

‘x is parallel to y ’

Now we have:

H There exists a line l and point P not on l such that at leasttwo distinct lines parallel to l pass through P

H′ There exists a line l and point P such that at least two distinctlines that have no points in common with l pass through P

H′′ There exists an open chord l and a point in circle P not on lsuch that at least two distinct open chords with no points inthe circle in common with l pass through P

Let’s see if H′′ is a theorem of Euclidean geometry


H′′

H′′

There exists an open chord l and a point in circle P not on lsuch that at least two distinct open chords with no points inthe circle in common with l pass through P

Points outside the circle are irrelevant! They do not representpoints in the hyperbolic plane


Axiom 1

Axiom 1

We must now reinterpret the other axioms of hyperbolicgeometry (the Euclidean axioms excluding Axiom 5) and showthat so interpreted they are theorems of Euclidean geometry

Consider Axiom 1:

1 Given any two points P and Q, exactly one line can be drawnwhich passes through P and Q

On our interpretation, this becomes

1′ Given any two distinct points P and Q in the circle, thereexists exactly one open chord l such that P and Q both lie on l


Axiom 1′

Axiom 1′

Given p and q as points in the circle, let pq be the Euclideanline through them. pq is a segment of open chord mn. Pointsp and q lie on the open chord mn and, by Euclid’s Axiom 1,there must be only one open chord on which they lie.

So 1′ is a theorem of Euclidean geometry


The other Euclidean axioms

The other Euclidean axioms

The same procedure can be used to show that all of the otheraxioms of hyperbolic geometry can be interpreted as axiomsof Euclidean geometry

See Chapter 7 of Greenberg’s Euclidean and Non-EuclideanGeometries for the details

To repeat, we have taken the axioms of hyperbolic geometry(Euclid’s axioms with the hyperbolic axiom instead of Axiom5) and reinterpreted all of their undefined terms

We have then shown that, so reinterpreted, the axioms are alltheorems of Euclidean geometry

If Euclidean geometry is consistent, therefore, hyperbolicgeometry is also consistent


Models

Models

On the assumption that Euclidean geometry is consistent, theBeltrami-Klein model is a model of hyperbolic geometry

A model M of a theory Θ is an interpretation that satisfies allof the theorems of Θ

One way of showing that a theory is consistent is to provide amodel

We have shown that, if Euclidean geometry is (semantically)consistent, then there is a model of hyperbolic geometry, i.e.hyperbolic geometry is also (semantically) consistent


Talk outline





5 Independence

6 Conclusion


Independence

Independence

A sentence φ is independent of a set of sentences Γ iff Γ 0 φIndependence is a desirable property in axioms, i.e. we wouldlike each axiom α in an axiom set Σ to be such that Σ−α 0 αWhy is this desirable? We want our axiom sets to be as smallas possible, and if an axiom set contains an axiom that isdependent on the others, then that axiom is in a senseredundant: it is doing no real work, since anything provablewith it is provable without it


Euclid’s Axiom 5

Euclid’s Axiom 5

We have proved

Theorem If Euclidean geometry is consistent, then so is hyperbolicgeometry

This theorem has the following corollary:

Corollary If Euclidean geometry is consistent, then Euclid’s Axiom 5 isindependent of the other axioms


Proof of Corollary

Proof of Corollary

Corollary If Euclidean geometry is consistent, then Euclid’sfifth axiom is independent of the other axioms

Proof Assume for reductio that Corollary is false. It is aconditional, so if it is false Euclidean geometry is consistentbut Euclid’s Axiom 5 is not independent. If it is notindependent, then there is a proof of Axiom 5 from the otheraxioms. Euclidean geometry is consistent, so it does not provethe negation of Axiom 5, i.e. it does not prove the hyperbolicaxiom. But then hyperbolic geometry would be inconsistent,since it too would prove Axiom 5, but it would also prove thehyperbolic axiom (since it has the hyperbolic axiom as anaxiom!). But hyperbolic geometry is consistent if Euclideangeometry is consistent, so hyperbolic geometry is consistent.Contradiction �


Talk outline





5 Independence

6 Conclusion


Conclusion

Conclusion

We have seen that it is difficult to show that a theory isabsolutely consistent, and that frequently we must give proofsof relative consistency

We proved that hyperbolic geometry is consistent relative toEuclidean geometry

We saw that it is a corollary of this result that Euclid’scontroversial Axiom 5 is independent of the other axioms:there is no proof of Axiom 5 from the other axioms

Generally, independence is a desirable property of axioms: itensures that our axiom sets are as lean as possible


Conclusion

Next week

We may be concerned about relative consistency proofs: wewant our theories to have a definite subject matter, so whenwe reinterpret them, aren’t we talking about a differenttheory?

Frege thinks so. Axioms must be true, so they must be fullyinterpreted sentences, not partially interpreted sentencescapable of reinterpretation

Hilbert disagrees. Axioms need not be true, so they can becapable of reinterpretation and relative consistency proofs areperfectly legitimate

Next week, we’ll think about who wins

theories geometry - university of cambridge · theories geometry owen gri ths ... geometry euclid...

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