Pojam Infiniti mysteria u Boškovićevim geometrijskim istraživanjima
- Format:
- Journal Article
- Year:
- 2015
- Journal Title:
- Prilozi za istraživanje hrvatske filozofske baštine
- Volume:
- 41
- Issue:
- 1
- Language:
- Abstract:
Ruđer Bošković prvi je put spomenuo pojam ‘tajne beskonačnine’ (Infiniti mysteria) kad je u raspravi De maris aestu (1747) ustvrdio kako je pri istraživanju geometrijskih transformacija potrebno protumačiti upravo tajne beskonačnine. Već je tada, na osnovi dokazā provedenih u raspravi De natura et usu infinitorum et infinite parvorum (1741), imao iskustvo osporavanja aktualne beskonačnine u geometriji, pa je proučavanje tajnā bekonačnine uvjetovao filozofskom pretpostavkom o postojanju beskonačnine.
Dok je u raspravi De transformatione locorum geometricorum (1754) izgrađivao teoriju geometrijskih transformacija, Bošković je izbjegavao definirati ‘tajne beskonačnine’ u duhu euklidske tradicije, ali je odredio široko značenjsko polje tog pojma: sva očitovanja potencijalne i aktualne beskonačnine. Tek je od rasprave De continuitatis lege (1754) počeo strogo lučiti tajnu i apsurd u razumijevanju geometrijske beskonačnine, pa je otada ‘tajnu beskonačnine’ prepoznavao samo u onim geometrijskim veličinama i transformacijama u kojima se očituje potencijalna beskonačnina – uz uvjet da pritom vrijedi princip neprekinutosti.
Dvije potvrde Boškovićeva razumijevanja pojma Infiniti mysteria mogu se pronaći i u njegovoj korespondenciji tijekom 1760–ih: dragocjeno pismo rasprava što ga je od 20. prosinca 1760. do 26. veljače 1761. pisao u Carigradu mladom Giovanu Stefanu Contiju odlikovalo se jasnim govorom u prilog razlikovanju tajna-apsurd, a iscrpan Boškovićev odgovor švicarskom učenjaku Georges-Louisu Le Sageu s nadnevkom 8. svibnja 1765. razmatrao je povezanost između beskonačnine i neprekidnine, odnosno povezanost apsurda i skoka.
Naprotiv, apsurd uvijek slijedi iz pretpostavke o aktualnoj beskonačnini, a ustanovljuje se tijekom dokaznoga postupka u kojem se upotrebljavaju i struktura bijektivnog preslikavanja i sklop ‘dio – cjelina’, dakle oba aspekta koji bitno obilježavaju Bolzanovu paradoksalnu zamisao o odnosu između beskonačnih skupova, a potom Dedekindovu matematičku definiciju beskonačnog sistema.
U svom modelu za utvrđivanje apsurda Bošković se redovito koristi odnosima među geometrijskim veličinama kao reprezentantima odnosā među bekonačnim veličinama. A prekretnica koju je Bolzano pripremio u svojim Paradoxien des Unendlichen (1851), a izveli je Dedekind i Georg Cantor, zbila se u drugom matematičkom području: unutar skupovnog proučavanja realnih brojeva. Ta dva momenta, s jedne strane – uporaba istih matematičkih sadržaja, kao što su struktura bijektivnog preslikavanja i odnos ‘dio – cjelina’, a s druge – bitna razlika između euklidskoga geometrijskog i skupovnog pristupa, određuju mjesto Ruđera Boškovića na povijesnom putu prema egzaktnomu, matematičkom određenju beskonačnine.
When Bošković first mentioned the concept ‘the mysteries of the infinite’ (Infiniti mysteria) in his treatise De maris aestu (1747), he asserted that it was necessary to include the mysteries of the infinite into the investigation of geometric transformations. At that time, on the basis of the demonstration in his early treatise De natura et usu infinitorum et infinite parvorum (1741), he already had some experience in disputing the actual infinite in geometry. Therefore he based his study of the mysteries of the infinite on the philosophical assumption of the existence of the infinite.
While forming the theory of geometric transformations in his treatise De transformatione locorum geometricorum (1754), he gave a large meaning to this concept: all the manifestations of the potential and actual infinite. Only with his treatise De continuitatis lege (1754) did he start to make a strict distinction between mystery and absurdity in the understanding of the geometric infinite, and from that time on he recognized the mysteries of the infinite only in those geometric quantities and transformations in which the potential infinite occurs, on condition that the principle of continuity was preserved.
Two confirmations of Bošković’s understanding of the Infiniti mysteria in this specific way are to be found in his correspondence in the 1760s: his valuable epistolary treatise written in Constantinople from 20 December 1760 to 26 February 1761 for the young Giovan Stefano Conti, and Bošković’s exhaustive reply to the Swiss scholar Le Sage of 8 May 1765.
On the contrary, absurdity always follows from the assumption of the actual infinite, and it is ascertained during the process in which the structure of bijection and relationship ‘part-whole’ are used, that is, both aspects which strongly mark Bernard Bolzano’s paradoxical conception of the relationship between infinite sets, and Richard Dedekind’s mathematical definition of the infinite system.
In his model for ascertaining absurdity, Bošković always uses the relations between geometric quantities as representatives of the relationships between infinite quantities. The turning point which was prepared by Bolzano in his Paradoxien des Unendlichen (1851), and achieved by Dedekind and Georg Cantor, took place in another mathematical field, namely, in the set approach to the real numbers. These two points, the use of the same mathematical contents, such as the structure of bijection and the relationship ‘part-whole,’ on the one hand, and the difference between the Euclidean geometric approach and the set approach on the other hand, determine the place of Ruđer Bošković in the historical process of forming the exact, mathematical definition of the infinite.
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- Page Range:
- 61–90
- ISSN:
- 0350-27911847-4489