Mantiqiylik - Logicism - Wikipedia

In matematika falsafasi, mantiq bu tezislarning bir yoki bir nechtasini o'z ichiga olgan dastur - ba'zi bir izchil ma'noda 'mantiq ' — matematika mantiqning kengaytmasi, matematikaning bir qismi yoki barchasi kamaytirilishi mumkin mantiqqa yoki matematikaning bir qismi yoki barchasi bo'lishi mumkin modellashtirilgan mantiqan.^[1] Bertran Rassel va Alfred Nort Uaytxed tomonidan boshlangan ushbu dasturni qo'llab-quvvatladi Gottlob Frege va keyinchalik tomonidan ishlab chiqilgan Richard Dedekind va Juzeppe Peano.

Umumiy nuqtai

Dedekindning mantiqqa olib boradigan yo'lida burilish nuqtasi bo'lgan, chunki u qoniqtiradigan modelni yaratishga muvaffaq bo'lgan aksiomalar xarakterlovchi haqiqiy raqamlar ning ma'lum to'plamlaridan foydalangan holda ratsional sonlar. Ushbu va shunga o'xshash g'oyalar uni arifmetik, algebra va tahlillarni tabiiy sonlar va sinflarning "mantig'i" bilan kamaytirilishiga ishontirdi. Bundan tashqari, 1872 yilgacha u tabiatshunoslarning o'zi to'plamlar va xaritalarda qisqartirilishi mumkin degan xulosaga keldi. Ehtimol, boshqa mantiqchilar, eng muhimi Frege ham 1872 yilda nashr etilgan haqiqiy sonlarning yangi nazariyalariga amal qilishgan.

Grundlagen der Arithmetik-dan boshlab Frege mantiqiy dasturining asosidagi falsafiy turtki qisman uning noroziligi edi. epistemologik va ontologik o'sha paytgacha mavjud bo'lgan tabiiy sonlar haqidagi hisobotlarning majburiyatlari va Kantning tabiiy sonlar haqidagi haqiqatlarni misol sifatida ishlatganligiga ishonchi sintetik apriori haqiqat noto'g'ri edi.

Bu mantiqiylik uchun kengayish davrini boshladi, Dedekind va Frege uning asosiy namoyandalari bo'lishdi. Biroq, mantiqiy dasturning ushbu boshlang'ich bosqichi klassik paradokslarning kashf etilishi bilan inqirozga yuz tutdi to'plam nazariyasi (Kantor 1896, Zermelo va Rassell 1900-1901). Rassell tan olganidan va muloqot qilganidan keyin Frege loyihadan voz kechdi uning paradoksi Grundgesetze der Arithmetik-da ko'rsatilgan Frege tizimidagi nomuvofiqlikni aniqlash. Yozib oling sodda to'plam nazariyasi bu qiyinchilikdan ham aziyat chekmoqda.

Boshqa tomondan, Rassel yozgan Matematika asoslari ning paradoks va ishlanmalaridan foydalangan holda 1903 yilda Juzeppe Peano geometriya maktabi. U mavzuni davolaganligi sababli ibtidoiy tushunchalar geometriya va to'plamlar nazariyasida ushbu matn mantiqiylikning rivojlanishidagi suv havzasi hisoblanadi. Mantiqiylikni tasdiqlovchi dalillarni Rassel va Uaytxed o'zlarida to'plashdi Matematikaning printsipi.^[2]

Hozirgi kunda mavjud matematikaning asosiy qismi mantiqan oz sonli ekstralogik aksiomalar, masalan, aksiomalaridan kelib chiqadigan deb hisoblanadi. Zermelo-Fraenkel to'plamlari nazariyasi (yoki uning kengaytmasi ZFC ), undan hanuzgacha kelishmovchiliklar kelib chiqmagan. Shunday qilib, mantiqiy dasturlarning elementlari hayotiyligini isbotladi, ammo bu jarayonda sinflar, to'plamlar va xaritalar nazariyalari va yuqori darajadagi mantiqlar Henkin semantikasi, qisman ta'siri ostida ekstralogik sifatida qaraldi Quine Keyinchalik o'yladim.

Kurt Gödel "s to'liqsizlik teoremalari tabiiy sonlar uchun Peano aksiomalaridan kelib chiqadigan biron bir rasmiy tizim, masalan, PMdagi Rassell tizimlari - ushbu tizimning barcha yaxshi shakllangan jumlalarini hal qila olmasligini ko'rsating.^[3] Bu natija Hilbertning matematik asoslari dasturiga zarar etkazdi, bunda "infinitar" nazariyalar, masalan, PM kabi, "infinitar usullar" ga befarq bo'lganlar, ulardan foydalanish shubhasiz bo'lishi kerakligiga ishonch hosil qilish maqsadida, cheklangan nazariyalarga mos kelishi kerak edi. natijada qarama-qarshilik paydo bo'ladi. Gödelning natijasi shuni ko'rsatadiki, klassik matematikani iloji boricha saqlab qolgan holda mantiqiy pozitsiyani saqlab qolish uchun mantiqning bir qismi sifatida ba'zi cheksizlik aksiomalarini qabul qilish kerak. Tashqi tomondan, bu mantiqiy dasturga ham, "infinitar usullar" ga nisbatan shubhali bo'lganlar uchun ham zarar keltiradi. Shunga qaramay, mantiq va Hilbertian finitsizmidan kelib chiqadigan pozitsiyalar Gödelning natijasi nashr etilganidan beri davom etmoqda.

Mantiqiylikdan kelib chiqadigan dasturlarning haqiqiyligini saqlab qoladigan dalillardan biri, to'liqsizlik teoremalarining "boshqa teoremalar singari mantiq bilan isbotlanishi" bo'lishi mumkin. Biroq, bu argument teoremalar orasidagi farqni tan olmasa kerak birinchi darajali mantiq va teoremalari yuqori darajadagi mantiq. Birinchisini finistik usullar yordamida isbotlash mumkin, ikkinchisi esa umuman - mumkin emas. Tarskining aniqlanmaydigan teoremasi Gödel raqamlash sintaktik konstruktsiyalarni isbotlash uchun ishlatilishi mumkinligini ko'rsatadi, ammo semantik tasdiqlarni emas. Shuning uchun mantiqiylik haqiqiy dastur bo'lib qoladi degan da'vo, tabiiy sonlarning mavjudligi va xususiyatlariga asoslangan isbotlash tizimining ba'zi bir rasmiy tizimlarga qaraganda kamroq ishonchli ekanligiga ishontirishga majbur qilishi mumkin.^[4]

Mantiqiylik - ayniqsa, Frejning Rassel va Vitgenshteynga ta'siri orqali^[5] keyinchalik Dummett - rivojlanishiga katta hissa qo'shgan analitik falsafa yigirmanchi asr davomida.

"Mantiq" nomining kelib chiqishi

Ivor Grattan-Ginnes tomonidan "Logistique" frantsuzcha so'zi "tomonidan kiritilgan" Kouturat va boshqalar 1904 yilda Xalqaro falsafa kongressi va shu vaqtdan boshlab Rassel va boshqalar tomonidan turli tillarga mos keladigan versiyalarda ishlatilgan. "(G-G 2000: 501).

Aftidan, Rassellning birinchi (va yagona) ishlatishi 1919 yilda paydo bo'lgan: "Rassell bir necha bor Fregega murojaat qildi va uni" matematikani "mantiqiylashtirishda" birinchi o'rinni egallagan kishi "deb tanishtirdi (7-bet). Noto'g'ri bayonotdan tashqari (bu matematikada arifmetikaning o'rni haqidagi o'z nuqtai nazarini tushuntirish bilan qisman Rassell tomonidan tuzatilgan), bu parcha tirnoq qo'ygan so'zi bilan ajralib turadi, ammo ularning mavjudligi asabiylikni anglatadi va u bu so'zni boshqa hech qachon ishlatmadi, shuning uchun " mantiqiylik '1920 yillarning oxiriga qadar paydo bo'lmadi »(GG 2002: 434).^[6]

Taxminan Karnap (1929) bilan bir vaqtning o'zida, ammo aftidan mustaqil ravishda Fraenkel (1928) bu so'zni ishlatgan: "U" Uaytxed / Rassel "pozitsiyasini tavsiflash uchun" mantiq "nomini ishlatgan (244-betdagi bo'lim sarlavhasida). , 263-betdagi tushuntirish) "(GG 2002: 269). Carnap biroz boshqacha "Logistik" so'zini ishlatgan; Behmann uning Carnap qo'lyozmasida ishlatilishidan shikoyat qildi, shuning uchun Carnap 'Logizismus' so'zini taklif qildi, lekin u nihoyat so'z tanlovi 'Logistik' ga sodiq qoldi (G-G 2002: 501). Oxir oqibat "tarqalish asosan 1930 yildan boshlab Carnap tufayli sodir bo'ldi". (G-G 2000: 502).

Mantiqiylikning maqsadi yoki maqsadi

Ramziy mantiq: Mantiqiylikning ochiq maqsadi - barcha matematikani ramziy mantiqdan olish (Frege, Dedekind, Peano, Rassel.) Dan farqli o'laroq. algebraik mantiq (Mantiqiy mantiq ) arifmetik tushunchalarni ishlatadigan, ramziy mantiq juda qisqartirilgan belgilar to'plami (arifmetik bo'lmagan belgilar), "fikr qonunlari" ni o'zida mujassam etgan bir nechta "mantiqiy" aksiomalar va belgilarni qanday yig'ish va boshqarish kerakligini belgilaydigan xulosa qoidalari bilan boshlanadi - masalan, almashtirish va modus ponens (ya'ni [1] dan A moddiy jihatdan B va [2] A ni anglatadi, ulardan B kelib chiqishi mumkin). Mantiqiylik, shuningdek, Frege asoslaridan kelib chiqqan holda, "sub'ekt | predikat" dan tabiiy tilga oid bayonotlarni propozitsion "atomlar" ga yoki "umumlashtirish" ning "argument | funktsiyasi" ga qisqartirishni qabul qiladi - "hamma", "ba'zi", "sinf" ( to'plam, yig'ma) va "munosabat".

Tabiiy sonlar va ularning xususiyatlarini mantiqiy xulosada, hech qanday son "sezgi" aksioma sifatida yoki tasodifan "yashirin" bo'lmasligi kerak. Maqsad, hisoblashning raqamlari va undan keyin haqiqiy sonlardan boshlab, ba'zi tanlangan "fikr qonunlaridan" faqatgina "oldin" va "keyin" yoki "kamroq" va "ko'proq" degan sukutli taxminlarsiz, barcha matematikani olishdir. yoki nuqtaga: "voris" va "salafiy". Gödel 1944 yilda Rassellning intuitivizm va formalizmning ("Hilbert maktabi") asosiy tizimlaridagi "qurilishlar" bilan taqqoslaganda mantiqiy "inshootlari" quyidagicha umumlashtirildi: "Ushbu maktablarning ikkalasi ham o'zlarining qurilishlarini qochish aniq bitta bo'lgan matematik sezgi asosida quradilar. Rassellning asosiy maqsadlaridan biri konstruktivizm "(Gödel 1944 yilda To'plangan asarlar 1990:119).

Tarix: Gödel 1944 yilda Leybnitsning tarixiy tarixi qisqacha bayon qilingan Characteristica universalis, Frege va Peano orqali Rasselga: "Frege asosan fikrni tahlil qilishga qiziqqan va birinchi navbatda o'zining hisobini sof mantiqdan arifmetikani chiqarish uchun ishlatgan", Peano esa "matematikadagi qo'llanmalariga ko'proq qiziqqan". Ammo "Bu faqat [Rassellning] edi Matematikaning printsipi matematikaning katta qismlarini juda oz sonli mantiqiy tushunchalar va aksiomalardan olish uchun yangi usuldan to'liq foydalanilganligi. Bundan tashqari, yosh fan yangi vosita - munosabatlarning mavhum nazariyasi bilan boyitildi "(120-121-betlar).

Kleene 1952 buni quyidagicha ta'kidlaydi: "Leybnits (1666) birinchi navbatda mantiqni boshqa barcha ilmlar asosidagi g'oya va tamoyillarni o'z ichiga olgan fan sifatida tasavvur qildi. Dedekind (1888) va Frege (1884, 1893, 1903) matematik tushunchalarni atamalar bilan aniqlash bilan shug'ullanishdi. mantiqiy bo'lganlar va Peano (1889, 1894-1908) matematik teoremalarni mantiqiy ramziy ma'noda ifodalashda "(43-bet); oldingi xatboshiga u "mantiqiy maktab" ning namunalari sifatida Rassel va Uaytxedni, qolgan ikkita "asos" maktablari intuitiv va "formalistik yoki aksiomatik maktab" (43-bet) sifatida kiritilgan.

Frege 1879 o'zining 1879 yilgi so'zboshisida uning niyatini tasvirlaydi Begriffsschrift: U arifmetikani ko'rib chiqishni boshladi: bu "mantiq" dan kelib chiqdimi yoki "tajriba faktlari" danmi?

"Men birinchi navbatda arifmetikada qanday xulosalar qilish mumkinligi haqida o'ylashim kerak edi, faqatgina barcha tafsilotlardan ustun bo'lgan fikr qonunlarini qo'llab-quvvatlashim kerak edi. Mening dastlabki qadamim buyurtma tushunchasini ketma-ketlikda shu darajaga kamaytirishga urinish edi. ning mantiqiy natija, shuning uchun u erdan raqam tushunchasiga o'tish uchun. Intuitiv narsalarning bu erga sezilmasdan kirib kelishini oldini olish uchun men zanjirni bo'shliqlardan saqlash uchun bor kuchimni sarflashim kerak edi. . . Men tilning etishmovchiligini to'siq deb topdim; qabul qilishga tayyor bo'lgan iboralarim qanchalik qat'iyatli bo'lmasin, munosabatlar tobora murakkablashib, maqsadim talab qiladigan aniqlikka erishish uchun men tobora kamroq imkoniyatga ega bo'ldim. Bu nuqson meni hozirgi ideografiya g'oyasiga olib keldi. Shuning uchun uning birinchi maqsadi bizni xulosalar zanjirining haqiqiyligini eng ishonchli sinovdan o'tkazish va sezilmasdan yashirinishga harakat qiladigan har qanday taxminni ko'rsatishdir "(Frege 1879 van Heijenoort 1967: 5).

1887 yil o'zining birinchi nashrining 1887 yil muqaddimasida o'zining niyatini tasvirlaydi Raqamlarning mohiyati va ma'nosi. U "eng sodda ilm-fan asoslarida; ya'ni mantiqning raqamlar nazariyasi bilan shug'ullanadigan qismi" da'vo qilinmagan - "isbotlashga qodir bo'lgan hech narsa isbotsiz qabul qilinmasligi kerak" deb hisoblagan:

Arifmetik (algebra, tahlil) mantiqning bir qismi sifatida gapirganda, men raqamlar kontseptsiyasini makon va vaqt sezgi tushunchalaridan mutlaqo mustakil deb bilishni, shuni fikrlash qonunlarining bevosita natijasi deb bilishni nazarda tutmoqchiman. . . raqamlar inson ongining bepul ijodidir. . . [va] faqat raqamlar haqidagi fanni shakllantirishning mantiqiy jarayoni orqali. . . Bizning kosmos va vaqt haqidagi tushunchalarimizni ularni ongimizda yaratilgan ushbu raqam-domen bilan bog'lash orqali tekshirishga aniq tayyormiz "(Dedekind 1887 Dover respublikasi 1963: 31).

Peano 1889 yil uning niyatini o'zining 1889 yilgi muqaddimasida aytadi Arifmetikaning asoslari:

Matematikaning asoslariga taalluqli savollar, so'nggi paytlarda ko'pchilik tomonidan ko'rib chiqilgan bo'lsa-da, hali ham qoniqarli echim yo'q. Qiyinchilik tilning noaniqligidan asosiy manbaga ega. ¶ Shuning uchun biz ishlatadigan so'zlarni diqqat bilan o'rganish juda muhimdir. Mening maqsadim bu imtihondan o'tish edi "(Peano 1889, van Heijenoort 1967: 85).

Rassell 1903 yil o'zining 1903 yilgi so'zboshisida uning niyatini tasvirlaydi Matematika tamoyillari:

"Hozirgi ishda ikkita asosiy ob'ekt mavjud. Ulardan biri, ya'ni dalil barcha sof matematikalar juda oz sonli mantiqiy tushunchalar nuqtai nazaridan aniqlanadigan tushunchalar bilan shug'ullanishi va uning barcha takliflari juda oz sonli mantiqiy printsiplardan kelib chiqishi mumkin "(Prezdum 1903: vi).

"Hozirgi ishning kelib chiqishi to'g'risida bir nechta so'zlar muhokama qilingan savollarning muhimligini ko'rsatishga xizmat qilishi mumkin. Taxminan olti yil oldin men Dinamika falsafasini o'rganishni boshladim ... [Ikkala savoldan - tezlashtirish va mutloq harakat "kosmosning relyatsion nazariyasida"] meni Geometriya tamoyillarini, shu sababli davomiylik va cheksizlik falsafasini qayta ko'rib chiqishga, so'ngra so'zning ma'nosini ochib berishga undashdi. har qanday, ramziy mantiqqa "(1903-yilgi kirish so'zi: vi-vii).

Epistemologiya, ontologiya va mantiqiylik

Dedekind va Frege: Dedekind va Frege epistemologiyalari Rasselnikiga qaraganda unchalik aniq emas, ammo ikkalasi ham shunday qabul qilgandek apriori oddiy propozitsion bayonotlarga oid odatiy "fikr qonunlari" (odatda e'tiqod); agar bu qonunlar sinflar va munosabatlar nazariyasi bilan kengaytirilsa (masalan, masalan) x R y) jismoniy shaxslar o'rtasida x va y umumlashtirish R. bilan bog'langan.

Dedekindning "inson ongining erkin shakllanishi" Kronekerning "qat'iy" laridan farqli o'laroq: Dedekindning argumenti "1. bilan boshlanadi, men bundan keyin nimani tushunaman narsa fikrimizning har qanday ob'ekti "; biz odamlar ongimizdagi ushbu" narsalarni "muhokama qilish uchun ramzlardan foydalanamiz;" Biror narsa to'liq tasdiqlanishi yoki u haqida o'ylanishi mumkin bo'lgan narsalar bilan belgilanadi "(44-bet). Keyingi paragrafda Dedekind muhokama qiladi nima "tizim S bu: bu agregat, ko'p qirrali, bog'liq elementlarning (narsalarning) umumiyligi a, b, v"; u bunday tizim" deb ta'kidlaydi S . . . bizning fikrimiz ob'ekti sifatida ham xuddi shunday narsa (1); u har bir narsaga nisbatan uning elementi ekanligi aniqlanganda to'liq aniqlanadi S yoki yo'q. * "(45-bet, kursiv qo'shilgan). * izohga quyidagicha ishora qiladi:

"Kronecker yaqinda (Krelning jurnali, Jild 99, 334-336-betlar) matematikada tushunchalarni erkin shakllanishida men cheklashlar kiritishga intildi, men ularni asosli deb bilmayman "(45-bet).

Darhaqiqat, u Kroneckerni "ushbu cheklovlarning zarurligi yoki shunchaki maqsadga muvofiqligi sabablarini e'lon qilishini" kutmoqda (45-bet).

Leopold Kronecker, "Xudo butun sonlarni yaratdi, qolganlari insonning ishidir" degan fikri bilan mashhur^[7] uning dushmanlari bor edi, ular orasida Xilbert ham bor edi. Hilbert Kronekerni "dogmatist, u butun sonni o'zining muhim xususiyatlari bilan dogma sifatida qabul qiladigan darajada va orqaga qaramaydi "^[8] va uning o'ta konstruktivistik pozitsiyasini Brouwer bilan tenglashtirdi sezgi, ikkalasini ham "sub'ektivizm" da ayblagan holda: "Bizni o'zboshimchalik, his-tuyg'ular va odatlardan ozod qilish va bizni allaqachon Kroneckerning qarashlarida his qilgan sub'ektivizmdan himoya qilish fan vazifasining bir qismidir. intuitivizmdagi kulminatsiya ".^[9] Keyin Xilbert "matematika oldindan taxmin qilinmaydigan fan. Buni topish uchun menga Kroneker kabi Xudo kerak emas ..." deb ta'kidlaydi. (479-bet).

Rassel realist sifatida: Rassellniki Realizm unga inglizlarga qarshi vosita sifatida xizmat qilgan Idealizm,^[10] Evropadan olingan qismlar bilan Ratsionalizm va inglizlar empiriklik.^[11] Birinchidan, "Rassel ikkita asosiy masala bo'yicha realist edi: universal va moddiy narsalar" (Rassell 1912: xi). Rassel uchun jadvallar kuzatuvchi Rasseldan mustaqil ravishda mavjud bo'lgan haqiqiy narsalardir. Ratsionalizm tushunchasiga yordam beradi apriori bilim,^[12] empirikizm tajriba bilimlarining rolini (tajribadan kelib chiqish) hissa qo'shadi.^[13] Rassell Kantga "apriori" bilim g'oyasi bilan ishonar edi, ammo u Kantga "o'limga olib keladigan" deb e'tiroz bildirdi: "Faktlar [dunyo] har doim mantiq va arifmetikaga mos kelishi kerak. Mantiq va arifmetik bizning hissamiz bu bilan hisoblanmaydi "(1912: 87); Rassel shunday degan xulosaga keladi apriori biz egallagan bilim "narsalar haqida, nafaqat fikrlar haqida" (1912: 89). Va bu Rasselning epistemologiyasida Dedekindning "raqamlar inson ongining erkin ijodi" (Dedekind 1887: 31) degan e'tiqodidan farq qiladi.^[14]

Ammo uning tug'ma haqidagi epistemologiyasi (u so'zni afzal ko'radi apriori mantiqiy tamoyillarga qo'llanganda, qarang. 1912: 74) juda murakkab. U qat'iy va aniq so'zlarni qo'llab-quvvatlashini bildiradi Platonik "universallar" (qarang: 1912: 91-118) va u haqiqat va yolg'on "u erda" degan xulosaga keladi; aql yaratadi e'tiqodlar va ishonchni haqiqatga aylantiradigan narsa haqiqatdir, "va bu haqiqat (istisno holatlar bundan mustasno) e'tiqodga ega bo'lgan odamning ongini o'z ichiga olmaydi" (1912: 130).

Rassel bu epistemik tushunchalarni qaerdan olgan? U bizga o'zining 1903 yilgi muqaddimasida aytadi Matematika tamoyillari. E'tibor bering, u: "Emili - bu quyon" degan e'tiqod mavjud emas, ammo bu mavjud bo'lmagan taklifning haqiqati har qanday biladigan aqlga bog'liq emas; agar Emili chindan ham quyon bo'lsa, bu haqiqat Rassel yoki boshqa biron bir tirikmi yoki o'likmi yoki yo'qligidan qat'i nazar mavjud va Emilining quyon qalpoqchasiga munosabati "nihoyatda":

"Falsafaning asosiy masalalari bo'yicha mening pozitsiyam o'zining barcha asosiy xususiyatlari bilan janob G.E. Murdan kelib chiqadi. Men undan takliflarning mavjud bo'lmagan tabiatini (mavjudlikni tasdiqlash kabi holatlar bundan mustasno) va har qanday bilimga ega bo'lishdan mustaqilligini qabul qildim aql; shuningdek, o'zaro bog'liq bo'lgan cheksiz sonli o'zaro mustaqil mavjudotlardan tashkil topgan dunyoni, mavjudotlarni ham, mavjudotlarni ham, ularning atamalari yoki ular atamalarining sifatlari bilan kamaytirmaydigan munosabatlar bilan yakunlovchi plyuralizm. tuzish ... Yuqorida aytib o'tilgan ta'limotlar, mening fikrimcha, matematikaning har qanday toqatli darajada qoniqarli falsafasi uchun juda zarurdir, chunki umid qilamanki, keyingi sahifalar ko'rsatadiki ... Rasmiy ravishda mening taxminlarim shunchaki taxmin qilingan; ammo haqiqat ular matematikaning haqiqat bo'lishiga imkon beradi, hozirgi zamon falsafalarining aksariyati bu haqiqatan ham ularning foydasiga kuchli dalil. " (Muqaddima 1903: viii)

Rassellning paradoksi: 1902 yilda Rassel "ayanchli doirani" kashf etdi (Rassellning paradoksi ) Frege's-da Grundgesetze der Arithmetik, Frejning V asosiy qonunidan kelib chiqqan va u 1903 yilda uni takrorlamaslikka qaror qilgan Matematika tamoyillari. So'nggi daqiqada qo'shilgan ikkita Ilovada u 28 sahifani Frege nazariyasini qarama-qarshi bo'lganini batafsil tahlil qilishga va paradoksni tuzatishga bag'ishladi. Ammo u natija haqida optimistik bo'lmagan:

"Sinflarga kelsak, men tan olishim kerak, men sinf tushunchasi uchun zarur bo'lgan shartlarni bajaradigan biron bir kontseptsiyani sezmadim. Va x bobda muhokama qilingan qarama-qarshilik biron bir narsaning noto'g'ri ekanligini isbotlaydi, ammo men shu paytgacha muvaffaqiyatsizlikka uchraganman kashf qilish. (Rassellning 1903 yilgi so'zboshisi: vi) "

"Fantizm" va Rasselning sinfsiz nazariyasi: Gödel 1944 yilda 1903 yildagi yosh Rassel bilan rozi bo'lmas edi ("[mening premissiyalarim] matematikaning haqiqat bo'lishiga imkon beradi"), lekin, ehtimol Rassellning yuqorida keltirilgan bayonotiga qo'shilishi mumkin ("biron bir narsa noto'g'ri"); Rassell nazariyasi matematikaning qoniqarli asosiga kela olmadi: natija "mohiyatan salbiy; ya'ni shu tarzda kiritilgan sinflar va tushunchalar matematikadan foydalanish uchun zarur bo'lgan barcha xususiyatlarga ega emas" (Gödel 1944: 132).

Rassel bu vaziyatga qanday etib keldi? Gödel Rassellning ajablantiradigan "realist" ekanligini ta'kidlaydi: u Rasselning 1919: 169-yilgi "Mantiq haqiqiy dunyo bilan ham xuddi zoologiya kabi" (Gödel 1944: 120). Ammo u "aniq bir muammo bilan shug'ullana boshlaganda, tez orada tahlil qilinadigan ob'ektlar (masalan, sinflar yoki takliflar)" mantiqiy uydirmalar "ga aylanib ketganini [faqat ma'no] biz to'g'ridan-to'g'ri idrok etmasligimizni ta'kidlaydi. ularni. " (Gödel 1944: 120)

Rassellning mantiqiy brendiga tegishli bo'lgan kuzatuvda, Perri Rassell realizmning uch bosqichidan o'tganligini ta'kidladi: o'ta, o'rtacha va konstruktiv (Perry 1997: xxv). 1903 yilda u o'zining o'ta bosqichida edi; 1905 yilga kelib u mo''tadil bosqichida bo'ladi. Bir necha yil ichida u "jismoniy yoki moddiy narsalardan dunyoning asosiy mebellari sifatida voz kechadi. U ularni aql-idrok asosida qurishga harakat qiladi" keyingi kitobida Tashqi dunyo haqidagi bilimlarimiz [1914] "(Perry 1997: xxvi).

1944 yil Gödel deb ataydigan ushbu qurilishlar "nominalistik konstruktivizm. . . deb nomlanishi mumkin xayoliylik "Rassellning" yanada radikal g'oyasi, sinfsiz nazariya "dan olingan (125-bet):

"qaysi sinflarga yoki tushunchalarga muvofiq hech qachon haqiqiy ob'ektlar sifatida mavjud bo'lib, ushbu atamalarni o'z ichiga olgan jumlalar faqat ular sifatida talqin qilinishi mumkin bo'lgan ma'noga ega. . . boshqa narsalar haqida gapirish uslubi "(125-bet).

Quyida keltirilgan Tanqid bo'limlarida ko'proq ma'lumot oling.

Natural sonlarning mantiqiy qurilishiga misol: Rasselning in Printsipiya

Frege va Dedekindning mantiqiyligi Rasselnikiga o'xshaydi, lekin tafsilotlaridagi farqlar bilan (quyida Tanqidlarga qarang). Umuman olganda, tabiiy sonlarning mantiqiy kelib chiqishi, masalan, Zermelo to'plamlari ('Z') aksiomalaridan farq qiladi. Holbuki, Z dan kelib chiqqan holda, "raqam" ning bitta ta'rifi ushbu tizimning aksiomasidan foydalanadi juftlashtirish aksiomasi - bu "buyurtma qilingan juftlik" ta'rifiga olib keladi - yo'q ochiq son aksiomasi tabiiy sonlarni chiqarishga imkon beradigan turli mantiqiy aksiomalar tizimlarida mavjud. E'tibor bering, sonning ta'rifini olish uchun zarur bo'lgan aksiomalar har qanday holatda ham to'plam nazariyasi uchun aksioma tizimlari o'rtasida farq qilishi mumkin. Masalan, ZF va ZFC da juftlashish aksiomasi va shuning uchun oxir-oqibat tartiblangan juft tushunchasi Infinity Axiom va Almashtirish Aksiomasidan kelib chiqadi va Von Neumann raqamlari ta'rifida talab qilinadi (lekin Zermelo emas) raqamlar), Holbuki NFUda Frege raqamlari Grundgesetsedagi hosil bo'lishiga o'xshash tarzda olinishi mumkin.

The Printsipiya, uning kashshofi kabi Grundgesetze, raqamlarni qurishni "sinf", "propozitsion funktsiya" kabi ibtidoiy takliflardan, xususan, "o'xshashlik" munosabatlaridan ("tenglik": kollektsiyalar elementlarini birma-bir yozishmalarga joylashtirish) va " buyurtma berish "(" tengdosh sinflar kollektsiyalarini buyurtma qilish bilan bog'liqlik "ning" vorisidan "foydalanish").^[15] Mantiqiy kelib chiqish tenglamani tenglashtiradi asosiy raqamlar qurilgan bu raqamlar tabiiy sonlarga va shu sonlarning barchasi bir xil "tip" ga, ya'ni sinflar sinfiga o'xshab ketadi, ba'zi bir nazariy konstruktsiyalarda, masalan, fon Neumman va Zermelo raqamlari - har bir raqam avvalgisiga ega . Kleene quyidagilarni kuzatadi. (Kleinning taxminlari (1) va (2) 0 ning xossaga ega ekanligini bildiradi P va n+1 mulkka ega P har doim n mulkka ega P.)

"Bu erda nuqtai nazar [Kronecker] ning" Xudo butun sonlarni yaratdi "degan fikridan juda farq qiladi Peano aksiomalari sonlar va matematik induktsiya], bu erda biz tabiiy sonlar ketma-ketligining intuitiv tushunchasini taxmin qildik va undan printsipga asoslanib, har doim ma'lum bir xususiyatga ega bo'ldik. P natural sonlar shunday berilganki (1) va (2), u holda har qanday berilgan natural son xususiyatga ega bo'lishi kerak P"(Kleene 1952: 44).

Natural sonlarni yasash mantiqiy dasturining ahamiyati Rasselning "Barcha an'anaviy sof matematikaning tabiiy sonlardan kelib chiqishi mumkinligi, bu uzoq vaqtdan beri gumon qilingan bo'lsa-da, juda yaqinda topilgan kashfiyotdir" (1919: 4) degan fikridan kelib chiqadi. Ning bir hosilasi haqiqiy sonlar nazariyasidan kelib chiqadi Dedekind kesadi ratsional sonlarda ratsional sonlar o'z navbatida tabiatdan kelib chiqadi. Buning qanday amalga oshirilganligi haqidagi misol foydali bo'lsa-da, u avvalo tabiiy sonlarning hosil bo'lishiga tayanadi. Shunday qilib, agar tabiiy sonlarning mantiqiy xulosasida falsafiy qiyinchiliklar paydo bo'lsa, bu muammolar echimini topguncha dasturni to'xtatish uchun etarli bo'lishi kerak (Quyidagi Tanqidlarga qarang).

Natural sonlarni yasashga qaratilgan bir urinish 1930-1931 yillarda Bernays tomonidan umumlashtirildi.^[16] Ammo ba'zi tafsilotlarda to'liq bo'lmagan Bernaysning prezisasidan foydalanish o'rniga, ba'zi bir cheklangan rasmlarni o'z ichiga olgan Rassel qurilishining parafraziga urinish quyida keltirilgan:

Dastlabki bosqichlar

Rassel uchun to'plamlar (sinflar) - bu takliflar (narsa yoki narsalar to'g'risida dalillarni tasdiqlash) natijasida paydo bo'lgan, maxsus nomlar bilan ko'rsatilgan "narsalar" ning yig'indisi. Rassel ushbu umumiy tushunchani tahlil qildi. U jumlalarda "atamalar" bilan boshlanadi va ularni quyidagicha tahlil qiladi:

Shartlar: Rassel uchun "atamalar" "narsalar" yoki "tushunchalar" dir: "Nima bo'lishidan qat'iy nazar fikr ob'ekti bo'lishi mumkin yoki har qanday to'g'ri yoki yolg'on taklifda bo'lishi mumkin yoki bitta deb hisoblash mumkin bo'lsa, men uni muddat. Demak, bu falsafiy lug'at tarkibidagi eng keng so'z. Men unga sinonim sifatida birlik, birlik va shaxs so'zlarini ishlataman. Birinchi ikkitasi har bir atama bitta ekanligini, uchinchisi har bir atama borligi, ya'ni ma'lum ma'noda ekanligidan kelib chiqadi. Odam, bir lahza, son, sinf, munosabat, ximaera yoki boshqa har qanday narsani aytib o'tish mumkin, albatta, bu atama; va falon narsaning atamasi ekanligini inkor etish har doim yolg'on bo'lishi kerak "(Rassell 1903: 43)

Narsalar tegishli nomlar bilan ko'rsatilgan; tushunchalar sifat yoki fe'l bilan ko'rsatiladi: "Atamalar orasida ikkita turni ajratish mumkin, men ularni navbati bilan chaqiraman narsalar va tushunchalar; birinchisi - xususiy ismlar bilan ko'rsatilgan atamalar, ikkinchisi - barcha boshqa so'zlar bilan ko'rsatilgan. . . Tushunchalar orasida yana hech bo'lmaganda ikkita turni ajratish kerak, ya'ni sifatlar va fe'llar bilan belgilanadi "(1903: 44).

Kontseptsiya-sifatlar "predikatlar"; tushuncha-fe'llar "munosabatlar": "Avvalgi tur ko'pincha predikatlar yoki sinf tushunchalari deb ataladi; ikkinchisi har doim yoki deyarli doimo munosabatlardir." (1903: 44)

Taklifda paydo bo'ladigan "o'zgaruvchan" mavzu tushunchasi: "Men haqida gapiraman shartlar Taklifda ushbu atamalar mavjud bo'lsa-da, ammo bu taklifda uchraydigan ko'plab mavzular sifatida ko'rib chiqilishi mumkin. Taklif shartlarining o'ziga xos xususiyati shundan iboratki, ularning birortasini bizning taklifimiz bo'lmasdan turib, boshqa biron bir shaxs o'zgartirishi mumkin. Shunday qilib, biz "Suqrot - bu inson" degan taklif faqat bitta muddatga ega; taklifning qolgan komponentidan biri fe'l, ikkinchisi predikat ... . Demak, taxminlar faqat bitta atama yoki predmetga ega bo'lgan takliflarda yuzaga keladigan fe'llardan tashqari tushunchalardir. "(1903: 45)

Haqiqat va yolg'on: Deylik, kimdir biron bir narsaga ishora qilib: "Mening oldimdagi" Emili "degan narsa ayol kishi", deb aytishi kerak edi. Bu tashqi dunyo "dalillari" ga qarshi sinovdan o'tkazilishi kerak bo'lgan ma'ruzachining e'tiqodiga oid taklif, da'vo: "Aql yo'q yaratmoq haqiqat yoki yolg'on. Ular e'tiqodlarni yaratadilar. . . ishonchni haqiqatga aylantiradigan narsa bu haqiqatVa bu haqiqat (istisno holatlar bundan mustasno) hech qanday tarzda e'tiqodga ega bo'lgan odamning ongiga taalluqli emas "(1912: 130). Agar aytilgan so'zlarni va" haqiqat "bilan yozishmalarni tekshirish orqali Rassel aniqlasa, Emili agar quyon bo'lsa, demak uning gapi "yolg'on" deb hisoblanadi; agar Emili ayol odam bo'lsa (ayol "tuksiz oyoq"), chunki Rassel odamlarga qo'ng'iroq qilishni yaxshi ko'radi Diogenes Laërtius Aflotun haqidagi anekdot), keyin uning so'zlari "to'g'ri" deb hisoblanadi.

Sinflar (agregatlar, komplekslar): "Sinf, sinf tushunchasidan farqli o'laroq, ushbu predikatga ega bo'lgan barcha atamalarning yig'indisi yoki bog'lanishidir" (1903 y. 55-bet). Sinflarni kengaytma (a'zolarini ro'yxati) yoki intentsiya bilan, ya'ni "x a u" yoki "x is v" kabi "propozitsion funktsiya" bilan belgilanishi mumkin. Ammo "agar biz kengaytmani toza deb qabul qiladigan bo'lsak, bizning sinfimiz uning atamalarini sanab chiqishi bilan belgilanadi va bu usul Symbolic Logic singari cheksiz sinflar bilan muomala qilishga imkon bermaydi. Shunday qilib, bizning sinflarimiz umuman tushunchalar bilan belgilangan ob'ektlar sifatida qaralishi kerak. va shu darajaga intilish nuqtai nazari juda muhimdir. " (1909 bet 66)

Taklif funktsiyalari: "Odatda atamalardan ajralib turadigan sinf kontseptsiyasining o'ziga xos xususiyati shundaki," x - u u "propozitsion funktsiya bo'lib, u qachonki u u sinf tushunchasi bo'lsa." (1903: 56)

Sinfning kengaytirilgan va intensiv ta'rifi: "71. Sinf kengaytirilgan yoki intensiv ravishda belgilanishi mumkin. Ya'ni, biz sinfga tegishli bo'lgan ob'ekt turini yoki sinfni anglatadigan tushunchaning turini aniqlashimiz mumkin: bu kengaytmaning qarama-qarshiligining aniq ma'nosi Va shunga bog'liq ravishda intentlik. Ammo umumiy tushunchani ushbu ikki qavatli usulda aniqlash mumkin bo'lsa-da, ayrim sinflar, agar ular cheklangan bo'lishi mumkin bo'lgan holatlar bundan mustasno, faqat intensiv ravishda, ya'ni falon kontseptsiyalar bilan belgilanadigan ob'ektlar sifatida belgilanishi mumkin. . mantiqan; kengayish ta'rifi cheksiz sinflarga bir xil darajada taalluqli bo'lib tuyuladi, ammo amalda, agar biz buni sinab ko'rmoqchi bo'lsak, o'lim o'z maqsadiga erishguncha bizning maqtovga sazovor harakatlarimizni qisqartiradi. "(1903: 69)

Natural sonlarning ta'rifi

Prinipsiyada natural sonlar quyidagilardan kelib chiqadi barchasi tasdiqlanishi mumkin bo'lgan takliflar har qanday sub'ektlar to'plami. Rassel buni quyida keltirilgan ikkinchi (kursiv) jumla bilan aniq ko'rsatib beradi.

"Birinchi navbatda raqamlarning o'zi cheksiz to'plamni tashkil qiladi va shuning uchun ularni sanab chiqish bilan aniqlash mumkin emas. Ikkinchi o'rinda, ma'lum miqdordagi atamalarga ega bo'lgan to'plamlarning o'zi, ehtimol, cheksiz to'plamni tashkil qiladi: masalan, dunyoda cheksiz trioslar to'plami mavjud deb taxmin qilish kerak., chunki agar bunday bo'lmasa, dunyodagi narsalarning umumiy soni cheklangan bo'lar edi, bu mumkin bo'lsa ham, ehtimol dargumon. Uchinchi o'rinda biz "raqam" ni cheksiz sonlar mumkin bo'ladigan tarzda aniqlamoqchimiz; shuning uchun biz cheksiz to'plamdagi atamalar soni to'g'risida gaplasha olishimiz kerak va bunday to'plam intentlik bilan, ya'ni uning barcha a'zolari uchun umumiy va ularga xos bo'lgan xususiyat bilan belgilanishi kerak. "(1919: 13)

Buni tasavvur qilish uchun quyidagi so'nggi misolni ko'rib chiqing: bir ko'chada 12 ta oila bor deylik. Ba'zilarning bolalari bor, ba'zilarida yo'q. Ushbu xonadonlardagi bolalarning ismlarini muhokama qilish uchun 12 ta taklifni tasdiqlash kerak "bola nomi bu Fn oilasidagi bolaning ismi "F1, F2,. ismlari bo'lgan oilalarning ma'lum bir ko'chasida joylashgan ushbu uy xo'jaliklari to'plamiga nisbatan qo'llaniladi. 12 ta taklifning har biri" argument "bo'ladimi yoki yo'qmi bilan bog'liq. bola nomi ma'lum bir uydagi bolaga tegishli. Bolalarning ismlari (bola nomi) f (x) propozitsiya funktsiyasidagi x sifatida qaralishi mumkin, bu erda funktsiya "Fn ismli oiladagi bolaning ismi" dir.^{[17][asl tadqiqotmi? ]}

1-qadam: Barcha sinflarni yig'ing: Holbuki oldingi misol cheklangan taklif funktsiyasiga nisbatan cheklangan "bolalar ismlari aftidan sonli sonli oilalarning cheklangan ko'chasida joylashgan Fn '"oilasidagi bolalardan Rassel, aftidan, barcha raqamlarni yaratishga imkon beradigan cheksiz domen bo'ylab kengaytirilgan barcha taklif funktsiyalariga amal qilishni maqsad qilgan.

Kleen, Rassell bir yo'lni belgilagan deb hisoblaydi ishonchli u hal qilishi kerak bo'lgan ta'rif yoki shunga o'xshash narsalarni keltirib chiqarishi mumkin Rassel paradoksi. "Buning o'rniga biz tabiiy sonlar ketma-ketligi ta'rifidan oldin mantiqan mavjud bo'lgan asosiy sonlarning barcha xususiyatlarining umumiyligini taxmin qilamiz" (Kleene 1952: 44). Muammo, hatto Rassell birlik sinfi bilan shug'ullanganida, bu erda keltirilgan cheklangan misolda ham paydo bo'ladi (qarang: Rassel 1903: 517).

Savol tug'iladi, aynan qaysi "sinf" bu yoki bo'lishi kerak. Dedekind va Frej uchun sinf - bu o'ziga xos alohida birlik, ya'ni ba'zi bir propozitsion funktsiyani qondiradigan barcha xlar bilan aniqlanishi mumkin bo'lgan "birlik" (bu ramz Rasselda paydo bo'lib, u erda Fregega tegishli: " Funktsiyaning mohiyati - qachon qolgan bo'lsa x olib tashlanadi, ya'ni yuqoridagi holatda, 2 ()³ + (). Bahs x funktsiyaga tegishli emas, lekin ikkalasi bir butunlikni hosil qiladi (ib. 6-bet [yani Frejning 1891 y.) Funktsiya va Begriff] "(Rassell 1903: 505).) Masalan, ma'lum bir" birlik "ga nom berilishi mumkin edi; masalan, Fa oilasida Enni, Barbi va Charlz ismli bolalar bor:

{a, b, c}_Fa

This notion of collection or class as object, when used without restriction, results in Rassellning paradoksi; see more below about impredicative definitions. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments x do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic" (First edition of Matematikaning printsipi 1927:24).

Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions":^{[asl tadqiqotmi? ]}

"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)

And in the second edition of Bosh vazir (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.

Step 2: Collect "similar" classes into 'bundles' : These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by ≈, i.e. one-one correspondence of the elements,^[18] and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes" (Russell 1919:14).

Step 3: Define the null class: Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection.

The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? Yilda Bosh vazir Russell says that "A class is said to mavjud when it has at least one member . . . the class which has no members is called the "null class" . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in Russell's 1903 work.^[19] After he discovered the paradox in Frege's Grundgesetze he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types"; see more about the unit class, the problem of impredicative definitions and Russell's "vicious circle principle" below.^[19]

Step 4: Assign a "numeral" to each bundle: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary.

Step 5: Define "0" Following Frege, Russell picked the empty or bekor class of classes as the appropriate class to fill this role, this being the class of classes having no members. This null class of classes may be labelled "0"

Step 6: Define the notion of "successor": Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, the successor of n". (1903:21). He asserts that "the natural numbers are the avlodlar — the "children", the inheritors of the "successor" — of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).

Note Russell has used a few words here without definition, in particular "number series", "number n", and "successor". He will define these in due course. Observe in particular that Russell does not use the unit class of classes "1" to construct the successor. The reason is that, in Russell's detailed analysis,^[20] if a unit class becomes an entity in its own right, then it too can be an element in its own proposition; this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, unit classes must be defined so as not to assume that we know what is meant by bitta (1919:181).

For his definition of successor, Russell will use for his "unit" a single entity or "term" as follows:

"It remains to define "successor". Given any number n ruxsat bering a be a class which has n members, and let x be a term which is not a member of a. Then the class consisting of a bilan x added on will have +1 a'zolar. Thus we have the following definition:

the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class." (1919:23)

Russell's definition requires a new "term" which is "added into" the collections inside the bundles.

Step 7: Construct the successor of the null class.

Step 8: For every class of equinumerous classes, create its successor.

Step 9: Order the numbers: The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "S", between the various "numerals". "We must now consider the ketma-ket character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the sinf of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)

Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of "asymmetry" i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, he defines the notion of "transitivity" for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of "connected": "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)

He concludes: ". . . [natural] number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers]." (1919:35)

Tanqid

The presumption of an 'extralogical' notion of iteration: Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)

Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, "a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be an elementary structural concept . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of "logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).

Hilbert 1931:266-7, like Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive apriori mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The apriori is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain extra-logical concrete objects that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).

In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an apriori notion that lies outside symbolic logic.

Hilbert dismissed logicism as a "false path": "Some tried to define the numbers purely logically; others simply took the usual number-theoretic modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267). The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.

Mancosu states that Brouwer concluded that: "the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation]; they are derived from the post factum record of mathematical constructions . . . Theoretical logic . . . [is] an empirical science and an application of mathematics" (Brouwer quoted by Mancosu 1998:9).

Gödel 1944 yil: With respect to the texnik aspects of Russellian logicism as it appears in Matematikaning printsipi (either edition), Gödel was disappointed:

"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foundations (contained in *1–*21 of Printsipiya) that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944 To'plangan asarlar 1990:120).

In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their ta'riflar" (Russell 1944:120)

With respect to the philosophy that might underlie these foundations, Gödel considered Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) — to be faulty. See more in "Gödel's criticism and suggestions" below.

Grattan-Ginnes: A complicated theory of relations continued to strangle Russell's explanatory 1919 Matematik falsafaga kirish and his 1927 second edition of Printsipiya. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets. Grattan-Guinness observes that in the second edition of Printsipiya Russell ignored this reduction that had been achieved by his own student Norbert Wiener (1914). Perhaps because of "residual annoyance, Russell did not react at all".^[21] By 1914 Hausdorff would provide another, equivalent definition, and Kuratowski in 1921 would provide the one in use today.^[22]

The unit class, impredicativity, and the vicious circle principle

A benign impredicative definition: Suppose a librarian wants to index her collection into a single book (call it Ι for "index"). Her index will list all the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has four books: I, Ά, β, and Γ. Her task is not difficult. When completed, the contents of her index I are 4 pages, each with a unique title and unique location (each entry abbreviated as Title.Location_T):

I ← { I.L_Men, Ά.L_Ά, β.L_β, Γ.L_Γ}.

This sort of definition of I was deemed by Poincaré to be "impredicative ". He seems to have considered that only predicative definitions can be allowed in mathematics:

"a definition is 'predicative' and logically admissible only if it chiqarib tashlaydi aniqlangan tushunchaga bog'liq bo'lgan barcha ob'ektlar, ya'ni uni har qanday yo'l bilan aniqlash mumkin ".^[23]

By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle":

"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in To'plangan asarlar jildi II 1990:125).^[24]

A pernicious impredicativity: α = NOT-α: To illustrate what a pernicious instance of impredicativity might be, consider the consequence of inputting argument α into the funktsiya f with output ω = 1 – α. This may be seen as the equivalent 'algebraic-logic' expression to the 'symbolic-logic' expression ω = NOT-α, with truth values 1 and 0. When input α = 0, output ω = 1; when input α = 1, output ω = 0.

To make the function "impredicative", identify the input with the output, yielding α = 1-α

Within the algebra of, say, rational numbers the equation is satisfied when α = 0.5. But within, for instance, a Boolean algebra, where only "truth values" 0 and 1 are permitted, then the equality qila olmaydi be satisfied.

Fatal impredicativity in the definition of the unit class: Some of the difficulties in the logicist programme may derive from the α = NOT-α paradox^[25] Russell discovered in Frege's 1879 Begriffsschrift^[26] that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but also from the function's own output.^[27]

As described above, Both Frege's and Russell's constructions of the natural numbers begin with the formation of equinumerous classes of classes ("bundles"), followed by an assignment of a unique "numeral" to each bundle, and then by the placing of the bundles into an order via a relation S that is asymmetric: x S y ≠ y S x. But Frege, unlike Russell, allowed the class of unit classes to be identified as a unit itself:

But, since the class with numeral 1 is a single object or unit in its own right, it too must be included in the class of unit classes. This inclusion results in an "infinite regress" (as Gödel called it) of increasing "type" and increasing content.

Russell avoided this problem by declaring a class to be more or a "fiction". By this he meant that a class could designate only those elements that satisfied its propositional function and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". Bu yig'ish but is not in Russell's view "worthy of thing-hood":

"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus x ε siz will mean " x biri siz 's." This must not be taken as a relation of two terms, x va siz, chunki siz as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's; its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents.[etc]" (1903:516).

This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.

A solution to impredicativity: a hierarchy of types

Classes as non-objects, as useful fictions: Gödel 1944:131 observes that "Russell adduces two reasons against the extensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and (2) the unit classes, which would have to be identical with their single elements." He suggests that Russell should have regarded these as fictitious, but not derive the further conclusion that barchasi classes (such as the class-of-classes that define the numbers 2, 3, etc) are fictions.

But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Frege in his 1903, Russell concludes:

"The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms; this is the case with all propositions asserting numbers other than 0 and 1" (1903:516).

In the following notice the wording "the class as many"—a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a thing-in-itself:

"Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI; that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes" (1903:518).

It is as if a rancher were to round up all his livestock (sheep, cows and horses) into three fictitious corrals (one for the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually exist are the sheep, the cows and the horses (the extensions), but not the fictitious "concepts" corrals and ranch.^{[asl tadqiqotmi? ]}

Ramified theory of types: function-orders and argument-types, predicative functions: When Russell proclaimed barchasi classes are useful fictions he solved the problem of the "unit" class, but the umuman olganda problem did not go away; rather, it arrived in a new form: "It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on reklama infinitum; we shall have to hold that no member of one set is a member of any other set, and that x ε siz shuni talab qiladi x should be of a set of a degree lower by one than the set to which siz tegishli. Shunday qilib x ε x will become a meaningless proposition; and in this way the contradiction is avoided" (1903:517).

This is Russell's "doctrine of types". To guarantee that impredicative expressions such as x ε x can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions. This supposition requires the notions of function-"orders" and argument-"types". First, functions (and their classes-as-extensions, i.e. "matrices") are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the "type" of a function's arguments (the function's "inputs") to be their "range of significance", i.e. what are those inputs a (individuals? classes? classes-of-classes? etc.) that, when plugged into f(x), yield a meaningful output ω. Note that this means that a "type" can be of mixed order, as the following example shows:

"Joe DiMaggio and the Yankees won the 1947 World Series".

This sentence can be decomposed into two clauses: "x won the 1947 World Series" + "y won the 1947 World Series". The first sentence takes for x an individual "Joe DiMaggio" as its input, the other takes for y an aggregate "Yankees" as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).

By "predicative", Russell meant that the function must be of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth. But these types can be mixed (for example, for this sentence to be (sort of) true: " z won the 1947 World Series " could accept the individual (type 0) "Joe DiMaggio" and/or the names of his other teammates, va it could accept the class (type 1) of individual players "The Yankees".

The axiom of reducibility: The kamaytirilishi aksiomasi is the hypothesis that har qanday funktsiyasi har qanday order can be reduced to (or replaced by) an equivalent predikativ function of the appropriate order.^[28] A careful reading of the first edition indicates that an n^th order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the nisbiy types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that nazariy jihatdan a reduction "all the way down" is possible.

Russell 1927 abandons the axiom of reducibility: By the 2nd edition of Bosh vazir of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:

"All propositions, of whatever order, are derived from a matrix composed of elementary propositions combined by means of the stroke" (Bosh vazir 1927 Appendix A, p. 385)

(The "stroke" is Sheffer's stroke — adopted for the 2nd edition of PM — a single two argument logical function from which all other logical functions may be defined.)

The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom" (Bosh vazir 1927:xiv).

Gödel 1944 agrees that Russell's logicist project was stymied; he seems to disagree that even the integers survived:

"[In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) of higher orders and types is to make it possible to assert more complicated truth-functions of atomic propositions" (Gödel 1944 in To'plangan asarlar:134).

Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 134-135); the proof in Russell 1927 Bosh vazir Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plus types] must be considered as unsolved at the present time". Gödel concluded that it wouldn't matter anyway because propositional functions of order n (har qanday n) must be described by finite combinations of symbols (all quotes and content derived from page 135).

Gödel's criticism and suggestions

Gödel, in his 1944 work, identifies the place where he considers Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).

Russell's no-class theory is the root of the problem: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his "constructivistic (or nominalistic"^[29]) standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p. 128).

Indeed, Russell's "no class" theory, Gödel concludes:

"is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data³³. The "data" are to understand in a relative sense here; i.e. in our case as logic without the assumption of the existence of classes and concepts]. The result has been in this case essentially negative; i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away" (p. 132)

He concludes his essay with the following suggestions and observations:

"One should take a more conservative course, such as would consist in trying to make the meaning of terms "class" and "concept" clearer, and to set up a consistent theory of classes and concepts as objectively existing entities. This is the course which the actual development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major among the attempts in this direction . . . are the simple theory of types . . . and axiomatic set theory, both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others . . .." (p. 140)

Neo-mantiq

Neo-mantiq describes a range of views considered by their proponents to be successors of the original logicist program.^[30] More narrowly, neo-logicism may be seen as the attempt to salvage some or all elements of Frege programme through the use of a modified version of Frege's system in the Grundgesetze (which may be seen as a kind of ikkinchi darajali mantiq ).

For instance, one might replace Asosiy qonun V (ga o'xshash cheklanmagan tushunish aksiomasi sxemasi yilda sodda to'plam nazariyasi ) with some 'safer' axiom so as to prevent the derivation of the known paradoxes. The most cited candidate to replace BLV is Xyumning printsipi, the contextual definition of '#' given by '#F = #G if and only if there is a bijection between F and G'.^[31] This kind of neo-logicism is often referred to as neo-Fregeanism.^[32] Proponents of neo-Fregeanism include Krispin Rayt va Bob Xeyl, ba'zan ham Shotlandiya maktabi yoki abstractionist Platonism,^[33] who espouse a form of epistemik asoschilik.^[34]

Other major proponents of neo-logicism include Bernard Linsky va Edvard N. Zalta, ba'zan Stenford-Edmonton maktabi, abstract structuralism yoki modal neo-logicism who espouse a form of aksiomatik metafizika.^[34][32] Modal neo-logicism derives the Peano aksiomalari ichida ikkinchi darajali modali ob'ekt nazariyasi.^[35][36]

Another quasi-neo-logicist approach has been suggested by M. Randall Holmes. In this kind of amendment to the Grundgesetze, BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's NF va tegishli tizimlar. Essentially all of the Grundgesetze then 'goes through'. The resulting system has the same consistency strength as Jensen 's NFU + Rosser 's Axiom of Counting.^[37]

Izohlar

Bibliografiya

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  • Amit Hojar 2005 yil Kirish Bertran Rasselga, 1919 yil, Matematik falsafaga kirish, Barnes & Noble, Inc, Nyu-York, NY, ISBN  978-1-4114-2942-0.

• Alfred Nayt Uaydxed va Bertran Rassel, 1927 yil 2-nashr, (birinchi nashr 1910-1913), Mathematica Principia-dan * 56,1962 gacha nashr, University Press-da Kembrij, Buyuk Britaniyaning Kembrijda, ISBN yo'q. * 56 ga qisqartirilgan ikkinchi nashr, bilan Ikkinchi nashrga kirish sahifalar Xiii-xlvi va yangi Ilova (* 8) Ko'rinadigan o'zgaruvchilarni o'z ichiga olgan takliflar) o'rniga * 9 Ko'rinadigan o'zgaruvchilar nazariyasiva C ilova Haqiqat funktsiyalari va boshqalar.

Tashqi havolalar

• Matematika entsiklopediyasida "mantiqiylik"