rachnatmakta (ganitache tattvadnyan)

ganitachya tattvadnyanat, rachnatmakta thampane sangate qee eka udaharan astitvat aahe heey siddha karnyasathi ganiteey vastuche vishisht udaharan shodhane (kinwa "rachana") aavashyak aahe. yaulat, shastriy ganitat, ekhadi vyakti ganiteey vastuche astitva spashtapane shodhalyashivay, tiche astitva nahee asey gruheet dharoon aanee nantar tyaa gruhitkapasoon virodhabhas prapt karoon siddha karoo shakate. virodhabhasane asha puravyala gair-rachnatmak mhatale jaau shakate aanee rachanatmakatavadi teey nakaru shakato. rachnatmak drushtikonat astitvavadi parimanakacha padtalanitmak arth lavane samavisht aahe, jeey tyachya shastriy arth lavnyala virodh karate.
rachnatmakteche anek prakar aahet. yamadhye broverne sthapan kelela antardnyanvad, hilbert aanee barnejacha finitism, shannin aanee markovche rachnatmak punaravarti ganeet aanee bishapcha rachnatmak vishleshanacha karyakram yancha samavesh aahe. rachnatmakatemadhye see. zed. eff. sarkhya rachnatmak sanch siddhantancha abhyas aanee topos siddhantacha abhyas dekhil samavisht aahe.
antardnyanvad haah keval eka rachnatmak karyakram asala taree rachnavadala anekada antardnyanvad mhanoon olakhale jaate. antardnyanvad asey sangato qee ganitacha paya vaiyaktik ganitachya antardnyanat asato, jyamule ganitala eka aantarik vyaktiparak kriyakalap banavate. rachnavadache itar prakar antardnyanachya yaa drushtikonavar aadharit naheet aanee ganitachya vastunishtha drushtikonashi susangat aahet.
rachnatmak ganeet.
barech rachnatmak ganeet antardnyani tarkshastra vaparte, jeey moolatah vagalalelya madhyabhagachya niyamashivay shastriy tarkshastra aahe. haah kayada sangato qee, konatyahi prastavasathi, ektar too prastav kharaa aahe kinwa tyacha nishedh aahe. yacha arth assa nahee qee vagalalelya madhyakacha kayada purnapane nakarla gela aahe; kayadyachi vishesh prakarane siddha hotil. heey fakt asey aahe qee samanya niyam haah eka svayamsiddha siddhant mhanoon gruheet dharala jaat nahee. virodhabhasacha kayada (joou sangato qee parasparvirodhi vidhane ekach velee donhi satya asoo shakat naheet) ajunahi vaidh aahe.
udaharanarth, ganitachya uchchankamadhye, konihi heey siddha karoo shakato qee konatyahi prastavasathi "pii" jyamadhye "parimanak nasatat", sootra _ 1 heey eka pramey aahe (jethe "ex", "vaay", "zed"... pii") yaa prastavatil mukt chull aahet. yaa arthane, maryaditateparyant maryadit asalele prastav ajunahi ektar kharey kinwa khote manale jaatat, jase qee teey shastriy ganitat aahet, parantu heey dwait anant sangrahancha sandarbh sandarbh prastavanparyant vistarat nahee.
kharey tarr, ell.eae.jeey. antardnyani shaleche sansthapak, brover yanni vagalalelya madhyakalacha niyam haah maryadit anubhavapasun amoort manla aanee nantar too samarthanashivay anantala lagoo keela. udaharanarth, goldbackcha andaj assa aahe qee pratyek summ sankhya (2 peksha mothee) hii done abhajya sankhyanchi berries aahe. konatyahi vishisht samsankhyesathi tee done avibhajyanchi berries aahe qee nahee yachee chachani karane shakya aahe (udaharanarth sampoorn shodhadvare), tyamule tyapaiki konatihi eka ektar done avibhajyanchi berries aahe kinwa nahee. aanee aataparyant, asha prakare chachani keleli pratyek sankhya hii pratyakshat done avibhajyanchi berries aahe.
parantu teey serve tase aahet yacha konatahi dnyat purava nahee, kinwa teey serve tase naheet yacha konatahi dnyat purava nahee; kinwa goldbackchya andajala 'purava' kinwa 'apatrata' astitvat asane aavashyak aahe qee nahee heey dekhil mahit nahee (paramparik zdf sanch siddhantat andaj 'anishchit' asoo shakato). tyamule, 'goldbackcha andaj kharaa aahe kinwa nahee' assa dawa karane aamhala yogya vatat nahee."aanee jaree eka divas andaj sodavala jaau shakato, taree yuktivad ashach nirakaran naan jhalelya samasyanna lagoo hoto; ulat, vagalalelya madhyabhagacha niyam haah" pratyek "ganiteey samasyeche nirakaran aahe asey gruheet dharnyaevadha hota.
eka svayamsiddha mhanoon vagalalelya madhyabhagacha niyam vagalalyas, urvarit tarkik pranalimadhye eka astitva gunadharm aahe joou shastriy tarkshastrat nasatoh jevha jevha sootra _ 2 rachnatmakpane siddha hote, tevha pratyakshat sootra _ 3 heey (kiman) eka vishisht sutrasathi rachnatmakpane siddha hote, jyala anekada sakshidar mhanatat. asha prakare ganiteey vastoochya astitvacha purava tyachya nirmitichya shakyateshi jodalela asato.
vastavik vishleshanatun udaharan.
shastriy vastavik vishleshanamadhye, vastavik sankhya paribhashit karnyacha eka marg mhanje parimey sankhyanchya kachee anukramancha samatulya varg.
rachnatmak ganitat, vastavik sankhya tayar karnyacha eka marg mhanje 'f' heey eka asey karya aahe jeey sakaratmak purnank sootra ghete aanee tarksangat 'f' ('n') output dete, ektritpane 'g' heey karya jeey sakaratmak purnank 'n' ghete aanee sakaratmak purnank 'g' ('n') output karate, jase qee
jenekarun "n" jasjase vadhate tastashi "f" ("n") chee moolye jawal yet jaatat. teey darshavit asalelya vastavik sankhyeshi aaplyala jitke aavadate titke tarksangat andaj mojanyasathi aapan "f" aanee "g" ekatra vaparu shakato.
yaa vyakhyeantargat, vastavik sankhya 'eae' chey sadhe pratinidhitva asey aaheh
hii vyakhya rachnatmak valan vagalata, kachee anukram vaparoon shastriy vyakhyeshi sambandhit aaheh shastriy kachee anukramasathi, konatyahi dilelya antarasathi, tyaa kramamadhye eka sadasya asane aavashyak aahe (shastriy arthane) jyanantar serve sadasya tyaa antarapeksha jawal aahet. rachnatmak aavruttimadhye, heey aavashyak aahe qee, konatyahi dilelya antarasathi, heey kuthe ghadate tyaa kramane pratyakshat eka bindu nirdisht karane shakya aahe (yaa aavashyak tapashilala anekada abhisaranache modulus mhanatat). kharey tarr, ganiteey vidhanacha manak rachnatmak arth lavane
heey tantotant abhisaran moduluschi ganana ganana karyache astitva aahe. asha prakare vastavik sankhyanchya done vyakhyanmadhil farak haah 'sarvansathi' yaa vidhanachya vyakhyetil farak manla jaau shakato... astitvat aahe..."
tyanantar heey prashna ughadate qee ganana karnyayogya sanchapasun ganana karnyayogya sanchaparyant, jase qee vareel "f" aanee "g", pratyakshat konatya prakarche karya tayar kele jaau shakate. yaa muddyavar rachanatmakatechya vividh aavrutya bhinn aahet. bandhkamanchi vyakhya vyapakpane mukt nivad anukram mhanoon kelee jaau shakate, jeey antardnyani drushya aahe, kinwa algoridam (kinwa adhik tantrikdrushtya, gananiy karye) kinwa agadi anirdisht mhanoon sodale jaate. udaharanarth, jarr algoridmik drushya ghetale gele tarr yethe tayar kelelya vastavik goshti moolatah shastriyadrushtya gananiy sankhya mhatalya jaatil.
cardinality.
vareel algoridmik arth lavane heey cardinalitychya shastriy kalpananshi visangat vatel. algoridamchi ganana karoon, aapan shastriy paddhatine dakhavu shakato qee gananiy sankhya mojanyayogya aahet. aanee tarihi cantercha karn yuktivad darshavito qee vastavik sankhyanmadhye adhik mukhyata asate. shivay, tirkas yuktivad purnapane rachnatmak disato. gananiy sankhyansah vastavik sankhya olakhane haah nantar virodhabhas tharel.
aanee kharey tarr, cantercha karn yuktivad "rachnatmak aahe, yaa arthane qee vastavik sankhya aanee naisargik sankhya yanchyatil dwibhajan dilyas, ekhadi vyakti eka vastavik sankhya tayar karate jee basat nahee aanee tyadvare virodhabhas siddha hoto. aapan kharokharach 'tii' heey karya tayar karnyasathi algoridamchi ganana karoo shakato, jyabaddal aapan suruvatila asey gruheet dharato qee teey naisargik sankhyanpasun vastavik sankhyeparyantche karya aahe. parantu, pratyek algoridammadhe, eka vastavik sankhya asoo shakate kinwa nasoo shakate, kaaran algoridam maryada purn karnyat ayashasvi houu shakate kinwa agadi naan sampanare dekhil asoo shakate ("tii" heey eka aanshik karya aahe), tyamule heey aavashyak bijection tayar karnyat ayashasvi hote. thodakyat, vastavik sankhya (vaiyaktikaritya) prabhavipane gananiy aahet asey mutt asanara vyakti canterchya parinamacha arth assa lavto qee vastavik sankhya (ektritpane) punaravruttiyogyapane gananiy naheet.
tarihi, ekhadi ashee apeksha karoo shakate qee "tii" heey naisargik sankhyanpasun vastavik sankhyanvaril aanshik karya asalyane, tyamule vastavik sankhya "mojanyayogya" peksha jaast naheet. aanee, pratyek naisargik sankhya kshullakpane vastavik sankhya mhanoon darshavili jaau shakate, mhanoon vastavik sankhya mojanyayogya aahet. tyamule teey 'achook' gananiy aahet. tathapi, haah tark rachnatmak nahee, kaaran too ajunahi aavashyak dwibhajan tayar karat nahee. asha paristhitit bijectionche astitva siddha karnare shastriy pramey, mhanje cantor-bernstein-schroeder pramey, heey gair-sanrachanatmak aahe. alikadech asey disun aale aahe qee cantor-bernstein-schroeder pramey vagalalelya madhyabhagacha niyam soochit karato, mhanoon yaa prameyala konatahi rachnatmak purava miloo shakat nahee.
nivadicha swasiddhant.
rachnatmak ganitatil nivadichya svayamsiddhatechi sthitee vegvegalya rachnatmak karyakramanchya vegvegalya drushtikonanmule guntaguntichi aahe. ganitshastradnyanni anaupacharikpane vaparlela 'rachnatmak' yacha eka kshullak arth, 'nivadichya svayamsiddhateshivay zdf sanch siddhantat siddha karata yeto."tathapi, rachnatmak ganitachya adhik maryadit prakaranche samarthak thampane sangatil qee zdf hii swatah eka rachnatmak pranali nahee.
prakar siddhantachya antardnyani siddhantanmadhye (visheshtah uchch-prakarchya ankganitachya), nivadichya svayamsiddhatechya anek prakaranna parvanagi aahe. udaharanarth, vastavik sankhyanchya sanchatil konatyahi sambandhasathi, jarr tumhi heey siddha kele asel qee pratyek vastavik sankhya 'x' sathee ashee vastavik sankhya 'y' aahe jee 'r' ('x', 'y') dharan karate, tarr pratyakshat eka karya 'f' aahe, jeey serve vastavik sankhyansathi 'r' ('x', 'f' ('x') dharan karate. serve maryadit prakaransathi samaan nivad tattve sweekarli jaatat. hii asanrachanatmak tatve swikaranyachi prerana mhanje 'pratyek vastavik sankhya' x 'sathee' r '(' x ',' y ') dharan karnari' y 'ashee vastavik sankhya aahe' yaa puravyachi antardnyani samaj. bii. etch. keey. chya vyakhyenusar, haah purava swatach moolatah ichchit asalele karya "eff" aahe. antardnyani swikartat ashee nivad tattve vagalalelya madhyakacha niyam soochit karat naheet.
tathapi, rachnatmak sanch siddhantasathichya kahee svayamsiddha pranalinmadhye, nivadicha svayamsiddha siddhant, diaconescu-goodman-mile prameyadvare darshavilyapramane, vagalalelya madhyabhagacha (itar svayamsiddha tattvanchya upasthitit) niyam soochit karato. kahee rachnatmak sanch siddhantanmadhye nivadichya svayamsiddhichi kamkuvat rupe samavisht aahet, jase qee mayhilchya sanch siddhantatil aashrit nivadicha swayamsiddhi.
mapan siddhant.
shastriy mojamapacha siddhant haah moolbhootpane gair-rachnatmak aahe, kaaran lebsgue mojamapachi shastriy vyakhya sanchache mojmap kinwa karyache avibhajya kase mojayache yache varnan karat nahee. kharey tarr, jarr ekhadyane ekhadya karyacha keval eka niyam mhanoon vichar keela qee "vastavik sankhya pravisht karate aanee vastavik sankhya output karate" tarr karyachya avibhajyatechi ganana karnyasathi konatahi algoridam asoo shakat nahee, kaaran konatahi algoridam ekach velee karyachya keval maryaditpane anek moolyanna coll karoo shakel aanee konatyahi kshullak nasalelya achukatesathi avibhajya ganana karnyasathi maryaditpane anek moolye pureshi nasatat. yaa samasyevar pratham upay mhanje, abhisaran darachya mahitisah, satatchya karyanchi (satatya dnyat moduluses) binduvrutti maryada mhanoon lihileli keval karye vicharat ghene. rachnatmak mapan siddhantacha eka fayda assa aahe qee jarr ekhada sanch rachnatmakpane purn mojamapacha aahe heey siddha karoo shakat asel tarr tyaa sanchamadhye eka bindu shodhanyasathi eka algoridam aahe (punha pahaa). udaharanarth, yaa drushtikonacha vapar pratyek besasathi normal asaleli vastavik sankhya tayar karnyasathi keela jaau shakato.
ganitamadhye rachnatmakteche sthan.
paramparikpane, kahee ganitshastradnya ganiteey rachanatmakatebaddal, virodhi nasale taree, sanshayaspad rahile aahet, mukhytah teey rachnatmak vishleshanasathi tayar karatat asey tyanna vatat asalelya maryadanmule.
1928 madhye, jevha tyanni 'grundlagen dair mathematic' madhye lihile, tevha david hilbert yanni hii matey baljabarine vyakt kelee, 'ganitshastratun vagalalele madhya tattva ghene, udaharanarth, khagolshastradnyala kinwa mushtiyoddhyala durbinicha vapar karnyas bandee ghalanyasarkhech asel.
erret bishop yanni tyanchya 1967 chya 'rachnatmak vishleshanacha paya' yaa pustakat, rachnatmak chaukatit mothya pramanat paramparik vishleshan vikasit karoon yaa bheeti door karnyasathi kaam kele.
jaree bahutek ganitshastradnya rachnatmak paddhatinvar aadharit kelele ganitach yogya aahe haah rachanatmakvadicha prabandh swikarat nasale taree, rachnatmak paddhati gair-saiddhantik aadharanvar adhikadhik swarasyapoorn aahet. udaharanarth, vishleshanatil rachnatmak purave sakshidar nishkarshanachi khatri asha prakare karoo shakatat qee rachnatmak paddhatinchya maryadanmadhye kaam kelyane shastriy paddhati vaparnyapeksha siddhantanche sakshidar shodhane sope houu shakate. rachnatmak ganitache anuprayog lambda calculi, topos siddhant aanee shrenibaddha tarkshastra yaa tankit prakaranmadhye dekhil aadhalale aahet, jeey moolbhut ganeet aanee sanganak vidnyanatil ullekhaneey vishay aahet. bijganitemadhye, topoi aanee hoff algebra sarkhya ghatkansathi, rachana antargat bhashela samarthan dete jee eka rachnatmak siddhant aahe; tyaa bhashechya maryadanmadhye kaam karane heey bahudha sambhavya thos algebrachya sanchabaddal aanee tyanchya samarupatebaddal tark karnyachya madhyamandvare bahyapane kaam karnyapeksha adhik antardnyani aanee lavchik asate.
bhautikshastradnya lii smollin yanni 'quantum gurutvakarshanache teen marg' yaa pustakat lihile aahe qee topos siddhant haah 'vishvashastrasathi tarkshastracha yogya prakar' aahe (prushth 30) aanee 'tyachya pahilya swarupat tyala' antardnyani tarkshastra '(prushth 31) asey mhatale gele. "asha prakarchya tarkshastrat, vishvabaddal nirikshak karoo shakanari vidhane kiman teen gatanmadhye vibhagali jaatatah jyanna aapan kharey tharvu shakato, jyanna aapan khote tharvu shakato aanee jyanna satya aahe heey aapan sadhya tharvu shakat nahee" (prushth 28).

