ИСТИНА

CATHEDRAL BUILDERS: THE SUBLIME IN MATHEMATICS 1. The topic of mathematical aims and goals is not so widely discussed as the kindred one on scientific aims and goals. Nevertheless, there is a wide range of specific aims pursued by mathematicians as mathematicians. Moreover external goals differ from multilevel internal ones. External goals are immediate objectives of mathematician’s efforts. They are equation solving, classification, generalization, proving and so on. Among internal goals one can find systematization, unification, explanation and justification. On the next level one can ask why we are looking for say justification, what kind of justification is appropriate for our purposes and what are these purposes. Finally one can try to uncover the ultimate internal goals of the mathematical activity. It is sensible to stop just before we lose the specificity of mathematical activity and get into the field of general human goals. 2. The same external goal may be motivated by diverse internal goals. One may recall an old parable, the parable of the three stonecutters working at the construction of a cathedral. Their external goal was exactly the same: to cut stones giving them the shape required; but being asked what they were doing they gave different answers. The first one said: “I am making a living”. The second one: “I am doing the best job of stonecutting in the entire county”. The third one: “I am building a cathedral!” Their internal goals turned out to be quite diverse. I would like to interpret the parable as representing three attitudes to doing mathematics: pragmatic, aesthetic and theological. According to the first attitude mathematics is something very useful in science and everyday life, no wonder it helps mathematicians to earn their living. According to the second one a pure mathematician inhabits a sort of ivory tower and practices a kind of art for art’s sake. These two attitudes are widely recognized, while theological one is less known. I dare interpret the metaphor of ‘building a cathedral’ as the theological attitude to doing mathematics according to another famous metaphor describing a cathedral as ‘theology in stone’. The third attitude is possible owing to the phenomenon of the mathematical sublime. 3. The sublime and beauty are closely connected and often confused. The sublime forms a religious stratum within aesthetic values. Expression ‘scientific sublime’ usually refers to the image of natural science in Romanticism. The ‘sublime’ in that context means a feeling on its way from terror to awe. If we try to apply it to contemporary science it will require something at the frontiers of knowledge. There are some contemporary attempts at working with ‘microbiological sublime’ as well as ‘computational’ or ‘digital sublime’. ‘Mathematical sublime’ has been used almost exclusively in the Kant’s sense. I use this expression in the sense of the experience of the sublime evoked especially by doing mathematics. It does not require something at the forefront of mathematics, and is almost free from straightforward sense of horror, but still is constituted by “the informing of the infinite into the finite” (as Schelling defined the sublimity). I am going to discuss different variants and examples of the mathematical sublime: mathematical perfection; mathematical infinite; absolute certainty of proofs, etc. My thesis is that the experience of mathematical sublime serves as one of the irreducible ultimate goals of mathematical activity.