Mantiqiya

MANTIQIYa (ba zaboni russi «Matematicheskaya logika») – in jarayon dar bunyodi matematika va falsafai matematika, ki tezisi asosii on tasdiqi «muvofiqati matematika ba mantiqi, yane imkoniyati (va zarurati) muayyan kardani tamomi mafhumhoi ibtidoii matematiki (ki dar doirai khudi matematika muayyan nameshavand) dar istilohhoi mantiqi «kholis» va isboti hamai jumlahoi matematiki (az on jumla aksioma) bo vositahoi mantiqi meboshad. Goyahoi Mantiqiya avvalin bor az tarafi Gotfrid Vilgelm Leybnis peshnihod shuda, bad dar talimoti G. Frege purra tashakkul yoftaand.

Gotfrid Vilgelm Leybnis mafhumhoi asosii matematika — mafhumi adadi naturaliro nisbat ba khajmi mafhumho istifoda namuda, sistemai mantiqiero kor karda baromad, ki ba vositai on tamomi teoremahoi arifmetikaro isbot kardan mumkin bud. Bo hamin ba aqidai Frege programmai Mantiqiya ijro gardid, zero to in davra dar matematika masalai payvastani mafhumhoi asosii tahlili matematiki, geometriya va algebra bo arifmetika aslan anjom yofta bud.

Vale to nashri jildi 2-yumi asari Frege «Qonunhoi asosii arifmetika» (1893-—1903) B. Rassel dar sistemai Frege ziddiyatro (odatan onro paradoksi Rassel menomand, nig. Xopija) oshkor kard. Vale khudi Rassel tezishoi asosii programmai Mantiqiyaro taqsim namud; u kushish kard, ki sistemai Fregero «isloh» namuda, az ziddiyat «najot» dihad. Natijai in kor kitobi sejildai Rassel (yakjoya bo A. N. Uaytkhed) «Principia Mathematica» (1910—13) bud. Navigarii asosii sistemai Rassel-Uaytkhed (minbad — RM) tartibi mantiq dar shakli «hisobi tadriji» yo «nazariyai navho» meboshad. Obekthoi shaklii in nazariyai ba navho (zinaho) judo meshavad va in «silsilai navho» baroi az khamai paradokshoi mamul raho yoftan imkoiiyat dod. Ammo baroi tanzimp matematikai klassiki tavassuti RM ba in sistema baze aksiomahoero ki mazmunan khosiyathoi muhimi «olami matematikai» muayyanro tavsif menamoyad, hamro kardan lozim omad.

Hamin tavr na hamai matematikai Rassel az mantiq cap mezavad. Chunonki K. Gyodel (1951) nishon dod sistemahoi navi RM komilan nopurraand. Hamin tariq programmai Mantiqiya— isboti «sof mantiqi»-i matematika ijronashavanda bud. Bo mururi zamon natijahoi Rassel va korhoi olimoni digare, ki badtar baroi takmili sistemai RM peshnhod shudaand, (masalan korhoi matematiki amerikoi U. Van , O. Kuayn) ba inkishofi mantiqi matematiki va umuman ilm tasiri kalon rasondaand.