09/03/2022

A survey of the aims and goals of the axiomatic foundation of geometry from Pasch to the present

Abstract

A survey of the history of the axiomatic foundation of geometry, initially a largely German and Italian endeavor, with a limited and rather narrow American involvement, ending in 1942, later with some Polish, Czech and Slovak, Hungarian,  and other involvement (such as the overarching significance of the Danish geometer Johannes Hjelmslev), emphasizing the various aims pursued at various times. The sociology of the endeavor will also be emphasized throughout, given the significant changes in the mathematical enterprise, in the personalities of the protagonists during the long period stretching from 1882 as date of publication to the present.

16/03/2022

The parallel postulate

Abstract

The Euclidean parallel postulate appears to the untrained eye as a closed chapter in the history of the axiomatic foundation of geometry, with interesting things to say about it having been settled during the 19th century. This view is correct only if we regard the Euclidean parallel postulate against the background of what Janos Bolyai called absolute geometry with continuity. If we remove continuity from absolute geometry, then the 19th century offers only the beginning to an understanding of the Euclidean parallel postulate. That understanding was achieved during the 20th century, between 1900 and 2021.

The talk will look at equivalent versions of the parallel postulate, as well as weakenings of it, such as the rectangle axiom, the Lotschnittaxiom, equivalent versions thereof, including one due to Lagrange and another one due to Lebesgue, who are not present in the collective memory of the educated mathematician, as well as the oldest axiom ever mentioned, Aristotle's axiom, together with a recent (2021) surprising splitting of the parallel postulate into two weaker, independent incidence statements. 

24/03/2022

Axiomatizing geometric constructions

Abstract

Geometric constructibility theorems usually state that, given a few geometric objects (points, lines, circles), one can construct all usual constructions in a certain geometry only with a restricted number of drawing instruments. The most famous among these is the Mohr-Mascheroni theorem on constructions with the compass alone. These results can often be rephrased in terms of a certain language, containing only operation symbols corresponding to the ``drawing instruments". It is this particular type of axiomatization, reflecting results in the literature on geometric constructions that will be the focus of this talk. In other words, a talk on axiom systems in which the axioms are quantifier-free statements expressed in languages without relation symbols.

30/03/2022

Intuitionistic axiomatization of geometry and a direct proof of the Steiner-Lehmus theorem

Abstract

A presentation of the various intuitionistic axiomatization of geometry, with special emphasis on a recent one by Michael Beeson, together with its significance for the definitive proof of the existence of a direct proof of the Steiner-Lehmus theorem.

06/04/2022

Into a dark night - the Pasch axiom in ordered geometry

Abstract

A survey of the many things that we know and do not know regarding the Pasch axiom and its weaker forms, introduced by Peano. These will be studied both inside theories of incidence and order, and within geometries with congruence.

07/04/2022

Reverse geometry

Abstract

An analysis of weak axiom systems that are sufficient for the proof of a certain geometric statement of interest. The search for axioms, barely strong enough to prove a certain theorem. For example: What is really needed so one can prove that the medians of a triangle are concurrent? The answer, which has escaped the attention of the educated mathematical public, was found by Johannes Hjelmslev. While a complete proof from that very simple axiom system will be about 7 pages long, it reveals that the apparent reason for the concurrency that one reads off the usual proof in the Euclidean plane is simply misleading. The true reason for the concurrency of the medians of a triangle remained hidden until the Hjelmslev's publications in the 20th century. A wealth of other geometric theorems will be presented with the minimal set of axioms wherefrom they can be proved.

28/04/2022

Simplicity of axiom systems and of proofs

Abstract

A survey of the many possible simplicity criteria for axiom systems, with examples of simplest axiom systems according to those criteria, as well as the question of proof simplicity and the problems with coming up with a meaningful definition thereof (Hilbert's 24th problem).

05/05/2022

The oldest unsolved problem or the trouble with 17

Abstract

A problem raised by Plato's narrative in his Theaetetus, regarding the fact that Theodorus provided proofs of irrationality that stopped at the square root of 17. In Jean Itard's interpretation of the reasons why Theodorus stopped there, we have no proof. This is a problem regarding the impossibility of proving a statement inside a weak arithmetic. It is still unsolved, although it is as elementary as it gets, going back to the Pythagorean arithmetic of the even and the odd.

12/05/2022

Brouwer's intuitionism: mathematics in the being mode of existence

Abstract

An essay on the reasons why Brouwer's intuitionism and his philosophy of mathematics were rejected. An Eastern way of thinking as well as what Erich Fromm referred to as "the being mode of existence" may hold the key. Be prepared to hear about figures not encountered by either philosophers of science or mathematicians: William S Haas, Jiddu Krishnamurti, Buddha, Meister Eckhart, Albert Schweitzer, Humberto Maturana and Francisco Varela, Simone Weil.

18/05/2022

The irreducibility of geometry to algebra

Abstract

Today, all geometry worth studying is permeated by algebra. In fact, geometry appears to be an archaic name for algebraic entities. We will show that this view, suggested by the current research environment cannot be maintained in elementary geometry, and that the reducibility of geometry to algebra runs against certain logical walls.