Lecture 3: Moving Towards Formalized Euclidean Geometry
Formal Euclidean Geometry, Prof. Kontorovich
Rutgers Math Corps, Summer 2025
Homework Problem: A Surprising Result in Hyperbolic Geometry
In our exploration of non-Euclidean geometry, we encountered a remarkable fact: when the parallel postulate fails, seemingly familiar settings result in completely completely unexpected outcomes. Today we'll prove one of the most surprising results in hyperbolic geometry.
The Problem
Assume: The angles in any triangle always add up to less than 180 degrees.
Theorem: If two triangles have the same corresponding angles (AAA), then they are congruent (not just similar)!
In other words, prove that AAA implies congruence in hyperbolic geometry.
This is startling! In Euclidean geometry, triangles with the same angles can have different sizes—think of a small triangle and a large triangle that are similar but not congruent. But in hyperbolic geometry, having the same angles forces the triangles to be exactly the same size.
Problem Setup
Key Insight
The crucial difference in hyperbolic geometry is that angle sums are constrained. When triangles have smaller angle sums, there's less "room" for variation in size while maintaining the same angles.
Given:
- Triangles ABC and DEF with ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F
- Postulate 5': Sum of angles in any triangle is less than 180°
Goal: Prove that AB = DE, BC = EF, and AC = DF (i.e., the triangles are congruent)
Strategy: We'll use proof by contradiction, assuming the sides are different and showing this leads to an impossible situation.
Setup Diagram
Are these triangles necessarily congruent in hyperbolic geometry?
The Proof: AAA Implies Congruence
We'll proceed by contradiction, showing that if the sides are different, we can construct a quadrilateral whose angles sum to exactly 360°—contradicting our assumption about hyperbolic geometry.
QED - Since assuming different side lengths leads to a contradiction, the triangles must be congruent!
Surely there must be something wrong with our argument above, because we all know and can see with our eyes that AAA does not imply congruence, only similarity! But where exactly is the contradiction? People like Gauss, Bolyai, and Lobachevsky tried in vain to find a contradiction but never succeeded. (It turns out that there's nothing wrong with the argument; hyperbolic geometry is just as valid a geometry as Euclid's!)
A Critical Gap in Euclid's First Proposition
Having seen how careful reasoning works in hyperbolic geometry, let's return our attention to Euclid and look again at the proof we gave last time for Proposition I.1.
Euclid's Proposition I.1 (Revisited)
Statement: To construct an equilateral triangle on a given finite straight line.
The Glaring Gap
Do you see the gap in the argument we just gave? It's very difficult to see the first time, and once it's pointed out to you, it's impossible to unsee...
Hilbert's Revolution (1899)
David Hilbert (1862-1943) was one of the greatest mathematicians of the late 19th and early 20th centuries. In 1899, he published "Grundlagen der Geometrie" (Foundations of Geometry), which completely revolutionized how we think about mathematical rigor.
David Hilbert (1862-1943)
Hilbert transformed mathematics by insisting on complete formal rigor
False Positives and False Negatives in Mathematical Reasoning
The examples we've seen reveal a deeper problem with how humans naturally approach mathematical reasoning. We can categorize the errors into two types:
False Positives in Mathematics
Definition: A theorem that appears to be perfectly well-justified but actually contains hidden gaps or unstated assumptions.
Example: Euclid's Proposition I.1 looks completely rigorous—every step seems to follow logically from the postulates. Yet it contains a critical gap: the assumption that circles intersect without any justification for why this must be true.
False Negatives in Mathematics
Definition: A theorem whose proof appears to contain errors or contradictions, when in fact the reasoning is perfectly sound.
Example: The hyperbolic AAA theorem seems impossible—surely triangles with the same angles can have different sizes! Our intuition screams that something must be wrong with the proof. Yet the mathematics is completely valid; it's our Euclidean intuition that misleads us.
These false positives and false negatives reveal a fundamental problem: human intuition and visual perception are unreliable guides to mathematical truth!!! This doesn't mean we should abandon intuition and visualization in our quest to discover universal truths—these remain essential tools for mathematical discovery. But we must also rigorously verify our insights within a completely formal system, since mistakes are all too common and all too easy to make!
The Root Cause
Hilbert realized that what was truly at the core of these issues was simple: when we humans allow our eyes to get involved in mathematical reasoning.
Our visual intuition tells us that circles "obviously" intersect, that parallel lines "clearly" behave a certain way, that triangles with the same angles "must" be similar rather than congruent. But mathematics built on such visual intuitions inevitably contains hidden assumptions and gaps.
Hilbert's Radical Solution
The Problem: Human perception and intuition systematically mislead mathematical reasoning
The Solution: Remove humanity from mathematics entirely
If we want mathematics to be truly precise and reliable, we must eliminate human judgment, visual intuition, and "obvious" assumptions. Mathematics must become a purely formal, mechanical process that could be carried out by a machine with no understanding of what the symbols "mean."
This insight was revolutionary and deeply unsettling. Hilbert was essentially arguing that to achieve mathematical certainty, we must strip away everything that makes mathematics feel human, intuitive, and meaningful. The price of absolute rigor is the elimination of human insight from the reasoning process.
The Problem of "Meaningless" Definitions
This viewpoint already reveals other problems with Euclid's approach. Look back at Euclid's very first definitions from Lecture 1:
Euclid's Definitions, Book I:
- Point: "That which has no part"
- Line: "Breadthless length"
- Straight line: "A line which lies evenly with the points on itself"
What is a "part"? What does it mean to have or not have a part? What does it mean for something to be "breadthless"? What is a "length"? How can we tell if a line "lies evenly with the points on itself"? This is all abject nonsense! These are poetic descriptions, not mathematical definitions; they are essentially meaningless!
There's nothing wrong with the English dictionary defining words in terms of other English words—this circular referencing is actually an essential feature of language and one of the reasons Large Language Models exist! But mathematics cannot tolerate such infinite recursion and circular logic. Just as we must start with unjustified axioms to avoid infinite recursion in logic, we must also start with UNDEFINED terms and specify only their logical relationships.
The Formal Approach to Geometry
Hilbert's Famous Quote
Something like: "Take my entire reworking of Euclid's geometry, and replace every instance of 'point', 'line', and 'plane' with 'table', 'chair', and 'beer bottle', and all the LOGIC should remain valid."
This quote (even if apocryphal) captures the essence of Hilbert's approach: geometry should be about relationships and logical structure, not about the intuitive meaning of geometric objects.
In Hilbert's system:
- Points, lines, and circles are completely undefined objects
- Incidence (when a point lies on a line or circle; when a circle and line intersect, etc, etc) are undefined relations
- Axioms specify exactly how these objects and relations behave with respect to one-another
- Theorems follow by pure logical deduction from the axioms
There are (at least) two benefits of formalization. The first is that any potential hidden assumptions or logical gaps are immediately revealed and dealt with (as was our primary rationale for working formally). The second is the beautiful generality of the arguments: the same logical structure can potentially apply to many different "models". Let's elaborate on this last point.
This generality has profound implications for our understanding of geometry. Whatever theorems we prove using only the first four postulates (or their formal analogs) will automatically be valid in any geometry that satisfies those postulates—including hyperbolic geometry; since we never specified exactly what we mean by "point" or "line," our proofs work for any interpretation that satisfies the logical relationships! It's only when we add the fifth postulate (or its negation) that the logical conclusions between Euclidean and non-Euclidean geometries begin to diverge. This reveals why Hilbert's abstract approach is so powerful: by avoiding concrete definitions, we discover which geometric truths are truly universal and which depend on specific assumptions about the nature of space.
Modern Formalization:
Here's how we might begin to formalize Euclidean plane geometry as a class containing objects and relations:
Pt : Type
Line : Type
Circ : Type
That is, we begin by declaring that our EuclideanPlane will contain undefined things (Types) called Pt (Point), Line, and Circ (Circle).
Let's add some primitive relations:
That is, Pt_on_Line is a function which takes a Pt and then a Line, and then returns True or False; obviously this is meant to represent incidence, determining whether the given point lies on the line or not.
Lets add a few more:
Center_of_Circ : Pt → Circ → Prop
Of course these functions are meant to represent whether a given point is on a circle, and whether the point is the center of the circle. (But as they currently stand, they are completely meaningless, just abstract functions! We'll need more axioms to disambiguate how they behave...)
Here's the formal statement of Postulate 1, that given any two points, there exists a line passing through both. The mathematical symbol ∀ means "for all", and the symbol ∃ means "there exists".
(Pt_on_Line a L) ∧ (Pt_on_Line b L)
In natural language, this says that, for any pair of points a and b, there exists some line L so that both a and b lie on it.
Euclid's second Postulate was that, given a finite line segment (drawn on a finite piece of paper), it can be extended (by straightedge and compass) as much as necessary, in either direction. Since we are doing purely abstract formal geometry, we need not confine ourselves to finite pieces of paper. So let's do away with segments altogether (notice we didn't make a primitive for them! All our "Line"s are already infinite in both directions). Then we can likewise do away with Postulate 2.
Euclid's Postulate 3 says that, given two points, one can create a line which has the first as its center and passing through the second. Can you already see how to write this formally?
Looking Forward
In the next Lecture, we will continue fleshing out the axioms, until we've successfully and purely formally proved Euclid's Proposition I.1.