Proofs as Constructions

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Proofs as Constructions

Evidence in Euclidean Plane Geometry

Ariel Kellison

September 11, 2018

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Euclid’s Propositions

Proposition 1 : To construct an equilateral triangle on a finite straight line.

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Euclid’s Propositions

Proposition 2 : To place a straight line equal to a given straight line with one end at a given point.

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Euclid’s Propositions

Proposition 4 : If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

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Euclid’s Propositions

Proposition 6 : If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

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Euclid’s Propositions

Proposition 9: To bisect a given rectilinear angle.

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Euclid’s Propositions

How can we explain the meaning of these propositions?

I Proposition 1 : To construct an equilateral triangle on a finite straight line.

I Proposition 2 : To place a straight line equal to a given straight line with one end at a given point.

I Proposition 4 : If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

I Proposition 6 : If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

I Proposition 9: To bisect a given rectilinear angle.

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Euclid’s Propositions

Problems and Theorems

I Proposition 1 : To construct an equilateral triangle on a finite straight line.

I Proposition 2 : To place a straight line equal to a given straight line with one end at a given point.

I Proposition 4 : If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

I Proposition 6 : If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

I Proposition 9: To bisect a given rectilinear angle.

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Euclid’s Propositions

I The evidence for a problem is the explicit construction of a geometric object and the demonstration that the object has the properties specified.

I The evidence for a theorem is the demonstration that certain relations hold for a given geometric object.

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Euclid’s Propositions

I The evidence for a problem is the explicit construction of a geometric object and the demonstration that the object has the properties specified.

I The evidence for a theorem is the demonstration that certain relations hold for a given geometric object.

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Euclid’s Postulates

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Euclid’s Postulates

1. To draw a straight line form any point to any point

2. To produce a finite straight line continuously in a straight line 3. To describe a circle with any center and distance

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Euclid’s Postulates

2. To produce a finite straight line continuously in a straight line

3. To describe a circle with any center and distance

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Euclid’s Postulates

2. To produce a finite straight line continuously in a straight line 3. To describe a circle with any center and distance

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Euclid’s Postulates

· specifies a function with two points as inputs and a line as output

· specifies a function with a straight line as input and the extension of the straight line as output

· specifies a function with a point and finite straight line (distance) as input and a circle as output

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Euclid’s Postulates

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Euclid’s Postulates

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Euclid’s Postulates

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Euclid’s Postulates

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Euclid’s Postulates

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Evidence Semantics

I A is some arbitrary proposition

I a is some specific evidence for A

I [A] is the type of evidences for A

I a : [A] (or a ∈ [A]) ⇐⇒ “a is evidence for A”

Proposition Evidence Type Evidence Term

A ⇒ B function space: [A] → [B] λa.b

A ∧ B Cartesian product: [A] × [B] (a, b) A ∨ B disjoint union: [A] + [B] inl (a), inr (b)

∀x : A.B(x) dependent fun space: x : [A] → [B(x )] λa.b

∃x : A.B(x) dependent product: x : [A] × [B(x )] (a, b)

⊥ empty set : {} 0

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Evidence Semantics

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Evidence Semantics

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Evidence Semantics

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Evidence Semantics

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What is a logical form for Euclid’s propositions?

Think of it in terms of evidence:

the evidence for this proposition is a function taking a straight line into a pair consisting of a geometric object and a proof that the

geometric object is an equilateral triangle.

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What is a logical form for Euclid’s propositions?

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What is a logical form for Euclid’s propositions?

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What is a logical form for Euclid’s propositions?

l : line → (t : ∆(l ) × R(line, ∆(l )))

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General Forms for Euclid’s Propositions

I Problems

∀x : A ∃y : B(x).C (x, y )

I Theorems

∀x : A.B(x)

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General Forms for Euclid’s Propositions

I Problems

∀x : A ∃y : B(x).C (x, y )

I Theorems

∀x : A.B(x)

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General Forms for Euclid’s Propositions

I Problems

∀x : A ∃y : B(x).C (x, y )

I Theorems

∀x : A.B(x)

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Formalizing the Straightedge and Compass

Primitive Objects and Relations Definition

A Euclidean Plane structure has a primitive type Point together with the following relations for any a, b, c, d ∈ Point.

Congruence, written ab ∼= cd , says that segments ab and cd have the same length.

Betweenness, written a b c, says that the point b lies between a and c. This relation is not strict, so b could be equivalent to either a or c.

Apartness is a binary relation, signified by #, on points. If a#b we say that a is separated from b.

Leftness is a ternary relation on points, written a left of bc, and says that the point a is to the left of the line bc (by bc we mean the directed line from b to c).

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The Postulates

Straightedge-Straightedge“SS”

a

b x

y z

Figure:The Straightedge-Straightedge constructor for z = SS (a, b, x , y ).

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The Postulates

Straightedge Compass “SC”

a b

d

v u

Figure:The Straightedge Compass constructor: u = SCO(a, b, c, d ) and v = SCS(a, b, c, d )

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The Postulates

Compass-Compass “CC”

a

b

c

d u

v

Figure:The Compass-Compass constructor for the points u and v such that u left of ac and v right of ac:

u = CCL(a, b, c, d ) and v = CCR(a, b, c, d ).

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The Postulates

The Magnifying glass

M(a, b, c) is the function that, by magnification, decides whether c#a or c#b. Thus M(a, b, c) ∈ c#a + c#b.

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The Postulates

Orientation

For any a, b, c ∈ Point, the function LeftOrRight(a, b, c) takes evidence for a#bc into evidence for orientation:

LeftOrRight(a, b, c) ∈ (a left of bc + a right of bc).

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The Postulates

Non-triviality

guarantees that there exist two separated points, O and X such that O#X .

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Euclid’s Proposition 1

To construct an equilateral triangle on a given finite straight line.

A B

CCL(A, B, B, A)

∀A : Point. ∀B : {Point|B#A}. ∃C : {Point| Cong3(A, B, C )∧C left of AB} where Cong3(A, B, C ) := AB ∼= BC ∧ BC ∼= CA ∧ CA ∼= AB

Our Nuprl extract reflects the simplicity of the construction: λA.λB. CCL(A, B, B, A)

So we can define

4(A, B) = CCL(A, B, B, A) as the program for Euclid’s Proposition 1.

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Euclid’s Proposition 1

A B

CCL(A, B, B, A)

∀A : Point. ∀B : {Point|B#A}. ∃C : {Point| Cong3(A, B, C )∧C left of AB}

where Cong3(A, B, C ) := AB ∼= BC ∧ BC ∼= CA ∧ CA ∼= AB

Our Nuprl extract reflects the simplicity of the construction: λA.λB. CCL(A, B, B, A)

So we can define

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Euclid’s Proposition 1

A B

CCL(A, B, B, A)

where Cong3(A, B, C ) := AB ∼= BC ∧ BC ∼= CA ∧ CA ∼= AB Our Nuprl extract reflects the

simplicity of the construction:

λA.λB. CCL(A, B, B, A)

So we can define

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Euclid’s Proposition 1

A B

CCL(A, B, B, A)

where Cong3(A, B, C ) := AB ∼= BC ∧ BC ∼= CA ∧ CA ∼= AB Our Nuprl extract reflects the

simplicity of the construction:

λA.λB. CCL(A, B, B, A)

So we can define

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Euclid’s Proposition 2 (Lemma)

To place a straight line equal to a given straight line with one end at a given point.

∀A : Point. ∀B : {Point|B#A}. ∀C : Point. ∃D : {Point|AD ∼= BC }.

L2(A,B,C) =

let X = 4(A, B)

in

let U = SCO(X , B, B, C ) in SCS(A, X , X , U)

A B

X C

D U

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Euclid’s Proposition 2 (Lemma)

L2(A,B,C) =

let X = 4(A, B)

in

A B

X C

D U

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Euclid’s Proposition 2 (Lemma)

L2(A,B,C) =

let X = 4(A, B)

in

A B

X C

D U

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Euclid’s Proposition 2 (Lemma)

L2(A,B,C) =

let X = 4(A, B) in let U = SCO(X , B, B, C )

in SCS(A, X , X , U)

A B

X C

U

D

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Euclid’s Proposition 2 (Lemma)

L2(A,B,C) =

let X = 4(A, B) in

A B

X C

D U

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Euclid’s Proposition 2 (Unrestricted)

∀A, B, C : Point. ∃D : {Point|AD ∼= BC } Proof.

We must construct L without assuming A#B. Non-triviality guarantees O and X in the plane such that O#X . Apply the magnifying glass to O#X and such that A#X or A#O; without loss of generality we have A#A⁰. Apply the magnifying glass again to A#A⁰ such that A#B or A⁰#B:

case 1 A#B: Apply L2(A, B, C ) to get L.

case 2 A⁰#B: Apply L2(A⁰, B, C ) to get L⁰ then apply L2(A, A⁰, L⁰) to get L.

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Euclid’s Proposition 2 (Unrestricted)

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Euclid’s Proposition 2 (Unrestricted)

We must construct L without assuming A#B.

Non-triviality guarantees O and X in the plane such that O#X . Apply the magnifying glass to O#X and such that A#X or A#O; without loss of generality we have A#A⁰. Apply the magnifying glass again to A#A⁰ such that A#B or A⁰#B:

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Euclid’s Proposition 2 (Unrestricted)

We must construct L without assuming A#B. Non-triviality guarantees O and X in the plane such that O#X .

Apply the magnifying glass to O#X and such that A#X or A#O; without loss of generality we have A#A⁰. Apply the magnifying glass again to A#A⁰ such that A#B or A⁰#B:

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Euclid’s Proposition 2 (Unrestricted)

We must construct L without assuming A#B. Non-triviality guarantees O and X in the plane such that O#X . Apply the magnifying glass to O#X and such that A#X or A#O; without loss of generality we have A#A⁰.

Apply the magnifying glass again to A#A⁰ such that A#B or A⁰#B:

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Euclid’s Proposition 2 (Unrestricted)

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Euclid’s Proposition 2 (Unrestricted)

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Euclid’s Proposition 2 (Unrestricted)

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Euclid’s Proposition 2 (Unrestricted)

∀A, B, C : Point. ∃D : {Point|AD ∼= BC } Prop2(A,B,C) = if M(O, X , A)

then if M(A, O, B) then L2(A, B, C ) elseL2(A, O, L2(O, B, C )) else if M(A, X , B) then L2(A, B, C )

elseL2(A, X , L2(X , B, C )).

B C

A⁰

4(A⁰, B) L2(A⁰, B, C )

A

4(A, A⁰)

G D

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Euclid’s Proposition 2 (Unrestricted)

elseL2(A, X , L2(X , B, C )).

B C

A⁰

4(A⁰, B) L2(A⁰, B, C )

A

4(A, A⁰)

G D

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Euclid’s Proposition 2 (Unrestricted)

elseL2(A, X , L2(X , B, C )).

B C

A⁰

4(A⁰, B) L2(A⁰, B, C )

A

4(A, A⁰)

G

D

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Euclid’s Proposition 2 (Unrestricted)

elseL2(A, X , L2(X , B, C )).

B C

A⁰

4(A⁰, B) L2(A⁰, B, C )

A

4(A, A⁰)

G D

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Closing Statements

I Interpreting the meaning of logical formulas by describing the evidence that can be given for them gives a faithful

interpretation of Euclid’s methodology of geometric constructions.

I This interpretation gives novel insights into Euclid’s geometry (Prop 2).

I Both geometric construction problems and programming problems require the construction of a (computable) function that transforms an object of the input type into an object of the output type that satisfies some property.

I Thus, the semantics introduced here are generally suitable for expressing constructions in mathematics and computer science.