The Philosophy of Geometry
– On the deadly wrestling of Certainty and Intuition –

Zoltán Győrfi (K, 18:00-19:30 i/224)
BMA-LOTD-361.01 BMI-LOTD-361E.01 
BBN-FIL-402.50 BMA-FILD-402.50

Abstract

Geometry is the most ancient art. To be sure, Geometry is at least a three hundred million year old science; rampaging animals were its first practitioners. One of the survival tools of these creatures turned out to be their urinal geometry by which they designated―less and less clumsily―their territories.   At the same time the flickering consciousness, however, demanded more and more certainty and allowed less bravery and less thirst for blood:

The creating body created for itself spirit, as a hand to its will. 

The battle then began between the quick and superficial brown fox of intuition and the lazy dog of certainty. Once upon a time our already spiritual animals, reached the critical point when one of them could not withhold the
roaring question:

Is my understanding only blindness to my own lack of understanding? 

I will argue that―paradoxically enough―this blindness of the spirit is nothing else but the intuition―the spirit’s only sense organ for vision… 

   This is me, posing for myself

The history of geometry best exemplifies this ever lasting wrestling between intuition and certainty. My aim is to present a colorful series of case studies smelled, sensed or even seen in the arena of this struggle with myself being in love with intuition and certainty at the same time. This romantic triangle is unbearable.
A few words about my method: I will physically exemplify the existence of a plane geometry whose structure is apparently different from that of the planes of our “mother space”―the Euclidean one. We are going to talk about the geometry of those creatures that live and exercising their urinal geometry in this non-Euclidean world, and will follow them in discovering a (dangerously heretic) way of thinking about space: the Euclidean way.

A less romantic Syllabus

General remarks: This course is about the role of intuition and philosophy in geometries and it is not about proofs. Only a few formal proofs will be presented. We are going to focus on modeling geometries in each other. As far as intuition: our goal is to try to illustrate that the essence of intuition is always a kind of conscious or unconscious translation between models of something ideal an sich.  As a side product of these investigations is a (hopefully new) approach to Plato’s idea theory.

   The Bolyai lamp creating straight the lines for the hyperbolic crow.

The success of an intuitive approach depends on the formal perfection of the models between which the intuitive translation takes place. This is the chief paradox of being intuitive: Our intuition is as good as its formal basis is, even if we don’t know anything about that. 
My method of being intuitive is very counterintuitive in many ways. The name of the game is: falling in love with our image shown in the distorting mirror. 

The thematic order of the presentation

1.	On the intuition: its dangers and its advantages. From physics to mathematics. Card tricks of philosophical interest. The concept of order.
2.	On the order – continued. The Veblen axioms of order. Sylvester’s theorem. Some further examples of formal systems. Thinking without intuition. Turning on the light of intuition.
3.	On the historical geometries: Euclidean geometry and hyperbolic geometry. Introducing a physical model, the Bolyai lamp. Basic constructions with the help of the physical model.
4.	Basic constructions with the help of the physical model – continued. The life of the hyperbolic creatures. Hyperbolic parallelism.
5.	Basic constructions with the help of the physical model – continued.  Introduction to a computer program modeling the Bolyai lamp. The core concept of the historical geometries: Euclidean and hyperbolic congruence. Absolute geometry. Perpendicularity and the circle. The theorem of Thales.
6.	On the history of the conceptual thinking. From Aristotle to Hilbert. The fall of the great Saccheri.
7.	The role of the axiom of parallelism. Our physical model is a model of hyperbolic geometry. The Hjelmslev model: Modeling hyperbolic geometry in itself. 
8.	Kant’s negative role in the development of geometry. Why did not publish Gauss his results?
9.	Discovering the affine plane on the hyperbolic plane. The pantograph. On the conics.
10.	How to get to the Euclidean geometry from the hyperbolic one. Modeling the Euclidean geometry on the hyperbolic plane.
11.	Modeling the hyperbolic plane on the Euclidean plane while it is modeled on the hyperbolic one. Translation between the different models of the Euclidean and the hyperbolic geometry.  Models embedded in each other: an intuition theory.
12.	The Klein model and the cross ratio. On the absolute units of the hyperbolic plane.
13.	Using the Euclidean geometry in the discovery of the hyperbolic plane. The circumference of the hyperbolic circle.
14.	Hyperbolic cosmology. Hyperbolic planes of different parameter.