The discussion about mathematical culture (Cracow 1989)

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Culture is a concept that we rarely associate with any science. More often with nations, communities, civilizations... but there is also personal culture that characterizes every human being. But is it possible to define certain features that would specifying a culture related to mathematics — a mathematical culture? If so, is it possible to go further and define the chemical, technical, engineering culture, etc.? Is mathematical culture something special, or is it a source from which other sciences also derive their culture? Like the workshop that the mathematician gives to the physicist and the physicist to the chemist. Workshop, without which neither of them could do much. Just like a writer who would start writing a novel without knowing the basics of language, grammar, letters.


Almost 30 years ago, the Polish Mathematicians Congress took place in Cracow. One of its parts was the discussion entitled "What is mathematical culture?" The question was not "does it exist ...?" but "what is ...?". Let's do a little flashback and move to Cracow.

Cracow, September 1989

The discussion was open for any comments from auditorium. Polish mathematicians participated in it: R. Duda, S. Hartman, M. Jarnicki, M. Lubański, M. Kordos and J. Waszkiewicz. At the beginning of the discussion there was a question about the sense of learning mathematics: why learn something that is useless in everyday life? Over the centuries, the nature of this discipline of science has changed, as well as the reception of mathematics by society. The general teaching of mathematics at school started only in the 19th century as a result of the demand for specific knowledge from the army and industry. Mathematics then began to supplant Latin and Greek. But let's go back to the title question and use a simple analogy: are poetry, music, literature created for a specified application? If, for example, music was created exclusively for dance, would operas — in which we admire the beauty and harmony of sound, not their ability to revel — exist today? Are we pragmatic enough to also consider other — seemingly unfulfilled activities in everyday life — for completely unnecessary? I think not. And even the opposite. The Triad Truth, Goodness and Beauty is still hip and the claim that mathematics should be deprived of beauty and the sense of doing it for beauty itself is mathematophobic.
But back to the topic: the discussion in Cracow was extremely interesting and full of historical curiosities. Who is able to believe that in the past, humanistic gymnasiums were the best prepared for science studies, and in the XVIII century university mathematics did not differ much from the mathematics used in life? So what has changed over the past few hundred years? I think that the main factors are technical progress and industrialization, which forced the mathematicians to change the course (or rather its periodic disturbances).

Stefan Banach — The Prince of Polish mathematicians — described a mathematician in the following way:

A mathematician is one who can find analogies between statements; better — who sees the analogies between the proofs; even better — who sees analogies between theories; the best — who can see analogies between analogies.


Therefore, the benefit of learning mathematics is quite good: the ability to analyze the situation in life, making decisions, choosing strategies etc. So what is mathematical culture?
In Greek culture, mathematics played an important role among the intellectual elite. The method of practicing mathematics was a model for other sciences. The development of mathematical thought has affected clearly on the development of technology and civilization. But mathematical culture is not only associated with active participation in doing mathematics. Teachers play a great role as well, their own fascination and interest in mathematics has a big impact on students perception of the subject. The discussion about mathematical culture was thrilling, there were various thoughts: from defining the mathematical culture as a way of being, manifesting itself in the elegance of lecture, reliability of work, efficiency in teaching, to extreme opinions questioning mathematical cultures as such.

The discussion was endless, and its echo lasted for a long time. One of the manifestations of such great interest was the extensive comment of the auditor, Ms. Zdzisława Dybiec, who in the context of the whole discussion is very valuable.
The general meaning is: it is very difficult to form a strict definition of a mathematical culture, so maybe let's focus on the attitude that characterizes this term? After all, each of us wants to understand the meaning of the words we use. Mathematical culture is not mathematical knowledge, nor is it a beautiful landscape that, although it makes the whole land famous, is not itself in itself. At the beginning we said that culture at the most elementary level defines a person — then it is called personal culture. The generalization of this concept is also connected with the enlargement of the field of view: looking at the human being as an individual we see only his culture, looking at a wider group of people — we see the culture of a given community, which is always the sum of the components of the culture of its elementary parts. I think it will be useful to consider one particular case and name the features of a human being that testify to having, more or less, a mathematical culture. According to Zdzisława Dybiec, such a person should use basic intellectual techniques, have mental activities typical of mathematics, such as abstraction skills, schematization, comparing, classifying, generalizing, perceiving analogies, organizing, defining, deducing, arguing, reducing, inducing, specifying, algorithmization, optimization, etc. skills.

photo1.jpgAnother article in the cited magazine — about postmodernism in mathematics (in polish)

Admittedly, the above skills can be learned in a different way. You do not need to be friends with math at all. Most of these skills expand, for example, when learning foreign languages. So what is the mathematical nature of these features? I think that mathematics itself contains a comprehensive program that uniquely improves such attributes. So mathematics is in the full sense of the word interdisciplinary science, which is not only a boring calculation and not understood formulas, but it transmits into other areas of life. A simple lesson of the order of doing operations in mathematics teaches us what we should do in order to achieve the result — the correct result. We have a lot of similar situations in our lives: from cooking to driving a car — and although it sounds trivial and funny at that moment — I think that at an early stage of life it brings positive effects in shaping logical thinking and perception.

[1] Dyskusja "Co to jest kultura matematyczna?", Matematyka Społeczeństwo Nauczanie, nr 5, Siedlce 1990.

[1] — Slava Bowman, Unsplash
[2] — Mikito Tateisi, Unsplash

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Culture within the mathematical community has changed quite a bit over time. Especially, in terms of the rigorousness of proofs but also in terms of techniques which are accepted/used.

Mathematical culture and its development within the framework of society is a bit difficult to define, as you also point out. But I think it is possible argue that from a logical perspective we are trying to move more towards consistent logical thought. (Note that consistent logical thought does not mean that the underlying assumptions are true.)

Culture within the mathematical community has changed quite a bit over time. Especially, in terms of the rigorousness of proofs but also in terms of techniques which are accepted/used.
Yeah, I think I know what you mean — proofs using a computer. I heard about it, especially on the occasion of classification of finite simple groups.

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