Solomon Marcus Lectures – August 2008

1) THE LONELINESS OF THE MATHEMATICIAN

This loneliness is manifest in several respects:

1. in education, most children begin to hate math when they are 11, 12, or 13. At this ages. children have to face the big changes related to the first use of the word 'theorem' and the move from natural language to symbolic language (letters such as a, b, c, x, y, z and some artificial signs).
   
   
2. historically, math appeared under the influence of Old Greeks traditions, as a daughter of myths and as a sister of literature; both inherit from myths the symbolic function and the replacement of the real world by a fictional universe. So, with Plato, Kepler, Kant and Goethe math belongs to humanities, while the declared history places math together with the so-called natural sciences and exact sciences. This falsification makes difficult for the common people to understand the psychology of mathematical creativity.
   
   
3. Mathematics is generally considered as a useful tool and sometimes as an adequate language, but its cultural weight and its spiritual dimension remain hidden for most people. This is one more source of the loneliness of the mathematician.
   
   
4. Most problems addressed to mathematicians need a transformation in order to become mathematically meaningful. This fact gives to people working in engineering fields, in economics, medical sciences, linguistics etc, a feeling of frustration and skepticism in respect to mathematician's job; this is one more source of isolation of the mathematician.
   
   
5. Most intellectuals recall nothing meaningful from their school mathematics; the cultural weight of mathematics becomes in this way problematic.
   
   
6. Mathematics cannot be defined in a few words, it is alone by its cognitive status.
   

2)  TO PROVE OR NOT TO PROVE, THAT IS THE QUESTION!

The Hamletian dilemma concerning the history of the great invention made by Old Greeks and called 'mathematical proof' is followed step by step, until its contemporary stage, where the feeling of certainty traditionally associated with math had to leave the scene and to accept to be replaced with the more human feeling of doubt.

3)  SCIENCE TODAY VERTSUS SCIENCE YESTERDAY

Contemporary education does not point out sufficiently the fundamental differences between the science of the XX-th century and the Galileo-Newtonian science. This fact generates much confusion also in the understanding of relations between science and art. We propose a list of basic differences between the two sciences. We also point out some basic similarities and basic differences between science and art, in a contemporary perspective. The relations between science and engineering follow as a corollary.

4)  WE ARE SURROUNDED BY HIDDEN CONFLICTS

Besides the open conflicts, directly visible, there are also conflicts having the face of agreement and compatibility; only a careful analysis reveals them. Examples are given from math, logic, physics, computer science, biology, psychology, sociology, economics, philosophy, literature and art.

5)  INFORMATION: WHAT DOES IT MEAN?

"Information" emerged in the second half of the XIX-th century, from two basic sources: thermodynamics and Darwinian biology. While the first source lead to the quantitative aspects of information, the second one remained consistent with the purely qualitative aspect suggested by the Latin etymology of the word 'information'; in this order of ideas, information is related to 'form'. We follow the evolution of these two lines of the history of information in the XX-th century and make the point of today several tens of meanings of the word 'information'. But the desire to bridge the respective lines of development is still a failure.

6)  MATHEMATICAL MISTAKES AS A SOURCE OF CREATIVITY

Mistakes are generally considered as sins, as negative things. School education and school examinations are based on this idea and children learn from the beginning that mistakes are one of the worst possible things that may happen to them. Against this philosophy, we argue that there are mistakes that should be interpreted as symptoms of creativity. The way to creativity is almost always through mistakes and failures. Most great discoveries and inventions were associated with some mistakes and most great mathematicians are famous not only by their successes but also by their mistakes and failures (one exception: Gauss). Mistakes and failures are the prize we have to pay in order to give a chance to personal thinking and to creativity. A lot of examples, from the elementary to the higher levels, are given to illustrate this claims.

7)  THE BIOLOGICAL CELL IN SPECTACLE

A lot of directions of research, ignoring each other, each of them having its specific jargon, its journals, its meetings, have however a common denominator: each of them begins by claiming that its aim is to understand the functioning of the biological cell. Ten such directions are pointed out.

8) TWO NEIGHBORS IGNORING EACH OTHER: INFINITE WORDS AND FORMAL LANGUAGES

To each infinite word over a finite alphabet we can associate various formal languages and conversely, to each formal language we can associate some infinite words. Despite this situation, these two fields developed to a large extent by ignoring each other. We try to bridge this gap and we sketch some questions and answers in this respect.