Umboni watsopano woti Cantor adalemba Dedekind?

Mpikisano Umene Unapanga Masamu Amakono

M'mbiri ya masamu, maubale ochepa adatsimikizira kuti ndi achonde mwanzeru - kapena ngati amakangana - monga a Georg Cantor ndi Richard Dedekind. Kulemberana kwawo m'zaka zonse za m'ma 1870 ndi 1880 kunatulutsa malingaliro ena osintha kwambiri pamaziko a masamu, kuyambira pakumanga mwamphamvu kwa manambala enieni mpaka vumbulutso lochititsa chidwi loti infinity imabwera mosiyanasiyana. Koma funso lomwe lakhala likufunsidwa pakati pa akatswiri a mbiri ya masamu kwa zaka zopitirira zana layambanso kuwonjezereka: kodi Cantor adalandira ngongole yochulukirapo kuposa momwe amamuyenera, ndipo kodi Dedekind adalandira zochepa kwambiri? Kupenda kwatsopano kwa akatswiri pamakalata awo achinsinsi, zolemba pamanja, ndi nthawi yolondola ya zofalitsa zawo zikukakamiza gulu la masamu kuti liunikenso omwe adatulutsadi malingaliro omwe tsopano tikunena kuti amachokera ku Cantor yekha.

Uku sikungokangana chabe kwamaphunziro pa mawu am'munsi. Funso loti ngati Cantor adaberedwa - kapena sanatchulidwe mokwanira - Dedekind imakhudza mtima wa m'mene timagawira umwini waluntha, momwe mgwirizano umasokonezera kugawidwa, ndi chifukwa chiyani zolemba ndi zolemba zili zofunika m'gawo lililonse, kuyambira masamu enieni mpaka bizinesi yamakono.

Zomwe Mbiri Yakale Imatiuza Kale

Ubale pakati pa Cantor ndi Dedekind umalembedwa bwino kudzera m'makalata angapo omwe anasinthanitsa pakati pa 1872 ndi 1899. Makalata awo, omwe adasindikizidwa koyamba mu kope losonkhanitsidwa ndi Emmy Noether ndi Jean Cavailles mu 1937, akuwonetsa kusinthana kwakukulu kwaluntha. Mu 1872, amuna onsewa adasindikiza paokha zomanga za manambala enieni - Cantor pogwiritsa ntchito zomwe tsopano zimatchedwa kuti Cauchy sequences, ndi Dedekind pogwiritsa ntchito "mabala" ake otchuka. Koma makalatawo akusonyeza kuti Dedekind anakonza zomanga zake zodulidwa kuyambira mu 1858, zaka 14 zathunthu zisanatulutsidwe, pamene ankaphunzitsa masamu ku Polytechnic ku Zürich.

Zomwe akatswiri a mbiri yakale adziwa ndizakuti Cantor adatsamira kwambiri Dedekind m'zaka zoyambira za chiphunzitso chokhazikitsidwa. Munali m'kalata ya 1873 yopita kwa Dedekind kuti Cantor adafunsa koyamba ngati manambala enieni atha kuyikidwa pamakalata amodzi ndi amodzi ndi manambala achilengedwe. Dedekind sanangolimbikitsa kufunsako koma adathandizira kufewetsa kwakukulu ku umboni woyamba wa Cantor kuti zenizeni ndi zosawerengeka. Komabe pamene Cantor adasindikiza chotsatira chodziwika bwino ichi mu Crelle's Journal mu 1874, zomwe Dedekind adapereka sizinatchulidwe.

Kusiya uku sikunangochitika kamodzi. M'mabuku angapo kumapeto kwa zaka za m'ma 1870 ndi 1880, Cantor adapanga malingaliro omwe anali ndi mbiri yodziwika bwino yakusinthana kwake ndi Dedekind - kuphatikiza mafotokozedwe oyambilira a kardinality, lingaliro la kusawerengeka, komanso kapangidwe ka mfundo - popanda kupereka mtundu wa chivomerezo chomwe maphunziro amakono angafune.

Umboni Watsopano: Nthawi Zolemba Pamanja ndi Zosasindikiza Zosasindikiza

Kafukufuku waposachedwa, kutengera zolemba zakale ku Yunivesite ya Göttingen komanso ma marginalia omwe kale anali kunyalanyazidwa ku Dedekind's Nachlass (literary estate), awonjezera kulemera kwakukulu pamlanduwo. Akatswiri a mbiri yakale apeza zolemba pamanja m'manja mwa Dedekind zomwe zimafotokoza mfundo zazikuluzikulu za theoretic - kuphatikiza kumasulira koyambirira kwa zomwe zingakhale chiphunzitso choti seti ilibe malire ngati itayikidwa pagawo lokhalokha - kuyambira nthawi asana Cantor adasindikiza zotsatira zofanana.

Chochititsa chidwi kwambiri ndi zolemba zochokera ku 1874 mpaka 1877 momwe Dedekind amajambula malingaliro okhudza mapu pakati pa magulu a "mphamvu" zosiyanasiyana (zomwe tsopano timazitcha makadinali). Zolemba izi zidachitika zaka zingapo Cantor adasindikiza pamalingaliro omwewo. Ngakhale kuti Dedekind anasankha kusafalitsa - mwa zina chifukwa cha kufunitsitsa kwake kodziwika bwino komanso chifukwa chakuti amawona kuti malingalirowo anali asanakwaniritsidwe - Cantor, yemwe anali ndi mwayi wopeza malingalirowa kudzera m'makalata awo, anasuntha mofulumira kuti asindikize.

Nthawiyi ndiyoyipa kwambiri. Akatswiri apanga mapu osachepera zochitika zisanu ndi ziwiri zosiyana pakati pa 1873 ndi 1885 pomwe lingaliro limawonekera koyamba muzolemba zachinsinsi za Dedekind kapena makalata opita ku Cantor, kenako zimawonekera m'mapepala osindikizidwa a Cantor mkati mwa miyezi 6 mpaka 18 - osatchulapo.

Kubera Kapena Chifunga Chogwirizana?

Musanathamangire kutsutsa Cantor, ndikofunikira kumvetsetsa chikhalidwe chanzeru cha masamu azaka za zana la 19. Miyezo yamatchulidwe ndi mafotokozedwe anali osavomerezeka kwambiri kuposa masiku ano. Panalibe mawonekedwe ovomerezeka, palibe njira zowunikira anzawo momwe timawadziwira, ndipo malire apakati pa "zolimbikitsa zokambirana" ndi "malingaliro obwereka" anali osokonekera kwambiri. Akatswiri a masamu ankagawana maganizo awo m’makalata momveka bwino kuti ufulu wofalitsa ndi wa aliyense amene analemba pepalalo.

" Mzere pakati pa chisonkhezero cha nzeru ndi kuba kwa nzeru umakokedwa osati ndi malingaliro okha, koma ndi zolemba zomwe zimawazungulira. Popanda zolemba zomveka bwino, mikangano yofunika kwambiri imakhala nkhani yotanthauzira - ndipo wofalitsa wolimba mtima nthawi zambiri amapambana mbiri yakale. "

Otsutsa a Cantor amatsutsa kuti adasintha zomwe adaziwona ndi Dedekind kukhala chiphunzitso chokhazikika - kuti Dedekind adapereka mbewu, koma Cantor adamanga mundawo. Pali chowonadi pa izi: Cantor's 1895-1897 Beiträge zur Begründung der transfiniten Mengenlehreikuyimira kaphatikizidwe kokulirapo komwe kumapita kutali kuposa chilichonse chomwe Dedekind adalemba. Koma umboni watsopano ukusonyeza kuti mbewuzo zidapangidwa mokwanira kuposa momwe zimazindikiridwa m'mbuyomu, ndipo kulephera kwa Cantor kuvomereza kunali, pang'onopang'ono, kufooka kwakukulu pamakhalidwe anthawi zonse.

Chifukwa chiyani Dedekind Adakhala Chete

Chimodzi mwazinthu zopatsa chidwi kwambiri m'nkhaniyi ndikuyankha kwa Dedekind - kapena m'malo mwake, kusowa kwake. Ngakhale anali ndi umboni wokwanira wosonyeza kuti iyeyo ndi wofunika kwambiri, Dedekind sananenepo poyera Cantor kuti ndi wachinyengo. Zinthu zingapo zimathandizira kufotokozera kudziletsa uku:

• Kusiyana kwanyengo: Dedekind anali wosungidwa, wosamala, komanso wachinsinsi kwambiri. Cantor anali wofuna kutchuka, wochulukirachulukira, ndi wofunitsitsa kuzindikiridwa ndi bungwe la masamu lomwe kaŵirikaŵiri limakana ntchito yake.
• Chiwopsezo cha akatswiri: Cantor adathera nthawi yayitali ku yunivesite ya Halle, bungwe lachiwiri, ndipo adatsutsidwa kwambiri ndi Leopold Kronecker. Dedekind, yemwe adakhazikitsidwa bwino ku Brunswick Polytechnic, atha kuganiza kuti mkangano ungakhale pansi pake.
• Kudalirana: Ngakhale kuti pali kusiyana pakati pa ngongole, amuna onsewa amayamikira ubalewo. Ntchito yaukadaulo ya Dedekind ya 1888 Kodi sind und was sollen die Zahlen? idamangidwa pamalingaliro omwe adapanga limodzi, ndipo mkangano wapagulu ukadayipitsa zolowa zonse ziwiri.
• Nzeru zofalitsa: Dedekind ankakhulupirira kuti malingaliro ayenera kusindikizidwa pamene afika pachinthu chomveka bwino komanso chokwanira. Anasankha mosapita m'mbali kuti asasindikize zotsatira zambiri, zomwe adaziwona ngati zosakhalitsa. M’malingaliro ake, lingaliro losasindikizidwa linali lisanakonzekere dziko.

Nthawi yomalizayi mwina ndiyovuta kwambiri. Kufuna kuti Dedekind kukhale kwangwiro kudapangitsa kuti Cantor adadzaza. Umboni watsopanowu sumawulula kwambiri munthu woyipa chifukwa umawunikira vuto la kapangidwe kake: ngati palibe njira zowonetsera zowonekera, wofalitsa wochulukira amalanda ngongoleyo, mosasamala kanthu za yemwe anali ndi lingaliro loyamba.

Kodi Izi Zikutanthauza Chiyani pa Luntha la Luntha Lero

Mlandu wa Cantor-Dedekind wakhudza kwambiri mbiri ya masamu. M'gawo lililonse lachigwirizano - kuyambira kafukufuku wasayansi kupita ku chitukuko cha mapulogalamu mpaka njira zamabizinesi - funso la yemwe adayambitsa lingaliro ndi amene adangolipereka limakhala lovuta kulithetsa. Maphunziro amakono ayankha ndi machitidwe okhwima okhudzana ndi mawu, olemba anzawo, komanso zolemba zowonekera. Koma m’dziko lazamalonda, kumene magulu amagwirira ntchito limodzi tsiku lililonse, vuto likupitirirabe.

Ganizirani kuchuluka kwa zisankho zovuta zamabizinesi, zopanga zatsopano, ndi njira zoyambira zomwe zimatuluka kuchokera pazokambirana zamwambo - uthenga wa Slack pano, gawo la boardboard pamenepo, ndemanga yapamsonkhano. Popanda zolembedwa mwadongosolo, munthu amene amalemba lipoti lomaliza kapena kupereka ulaliki womaliza nthawi zambiri amalandira chiwongolero, pomwe amene wayambitsa lingalirolo amazimiririka kumbuyo. Ili ndiye vuto la Dedekind mumgwirizano wamakampani.

Mapulatifomu amakono ngati Mewayz amathetsa vutoli poyika mgwirizano wamagulu pakati, zolemba zamapulojekiti, ndi kufufuza kachitidwe ka ntchito mu dongosolo limodzi. Ndi ma module ophatikizika 207 omwe amatenga CRM, kasamalidwe ka projekiti, kulumikizana ndi gulu, ndi kusanthula, zopereka zilizonse zimalowetsedwa, zolembedwa nthawi, komanso zopezeka. Pamene membala wa gulu apanga njira muzolemba za polojekiti, mbiriyo imapitilira. Ntchito ikasinthidwa, mbiri yosintha ikuwonetsa yemwe adakonza komanso liti. Mtundu wa kusiyana komwe kwavutitsa Dedekind kwazaka zopitilira zana kumakhala kosatheka mwamapangidwe pomwe zolembedwa zimamangidwa papulatifomu yokha.

Kuwunikanso Cholowa cha Cantor

Palibe chilichonse mwa izi chomwe chikuchepetsa nzeru zenizeni za Georg Cantor. Mtsutso wake wa diagonal wa 1891, kukula kwake kwa manambala osinthika ndi ma cardinal, ndi malingaliro ake opitilira muyeso akadali opambana kwambiri omwe ali ndi siginecha yake yodabwitsa. Funso lomwe ladzutsidwa ndi umboni watsopano siliri ngati Cantor anali katswiri wamasamu - mosakayikira anali - koma ngati mbiri yakale idasokonekera mopanda chilungamo.

Zothandizira za Dedekind pamaziko a masamu zikuzindikirika mochulukira ngati maziko m'lingaliro lenileni. Kupanga kwake manambala enieni kudzera m'madula kumakhalabe njira yokhazikika m'mabuku amakono osanthula. Chiphunzitso chake cha manambala a algebra chinakhudza mibadwo ya akatswiri a masamu kuyambira Emmy Noether mpaka André Weil. Ndipo zidziwitso zake zamalingaliro, zomwe tsopano zalembedwa mokwanira kudzera muumboni zakale, zimawulula woganiza yemwe sanali mtolankhani wa Cantor chabe koma waluntha wofanana - ndipo, nthawi zina, wolowa m'malo mwake.

Kuwunikanso sikukhudza kugwetsa cholowa chimodzi kuti mupange china. Ndiko kukwanilitsa kumvetsetsa bwino momwe malingaliro osintha zinthu amakulirakulira: osati mu nthawi yokhayokha ya luso, koma mwa kukambirana kokhazikika, kukopana, ndi kukonzanso pang'onopang'ono kwa malingaliro ogawana. Chomvetsa chisoni n'chakuti zolembedwazo zinali zochepa kwambiri, ndipo zofalitsa zomwe zimasindikizidwa ndizochepa kwambiri, kuti zitha kufotokoza zenizeni zenizenizi mu nthawi yeniyeni.

Maphunziro a Zolemba-Dziko Loyamba

Kukangana kwa Cantor-Dedekind kumapereka phunziro lamphamvu lomwe limapitilira maphunziro. M'nthawi yomwe mikangano yazinthu zanzeru imatha kudziwa tsogolo lamakampani ndi ntchito, kufunikira kwa zolembedwa zolimba, zenizeni sikungapitirire. Kugwirizana kulikonse kumapanga malingaliro, ndipo lingaliro lirilonse liri ndi chiyambi. Mabungwe omwe adzachite bwino adzakhala omwe amatengera chiyambi chimenecho monga momwe ziliri - osati monga kuganiza mozama, koma monga gawo lophatikizidwa la momwe ntchito imagwiridwira.

Kwa mabizinesi a 138,000 omwe akugwiritsa ntchito kale Mewayz kuyang'anira ntchito zawo, mfundoyi imamangidwa mumayendedwe a tsiku ndi tsiku. Kulumikizana kulikonse kwamakasitomala komwe kumalowa mu CRM, ma invoice aliwonse opangidwa, projekiti iliyonse yomwe yatsatiridwa imapanga mbiri yosasinthika ya yemwe adapereka chiyani komanso liti. Mwanjira ina, ndi zomangamanga zomwe Dedekind sanakhalepo nazo - dongosolo lomwe limawonetsetsa kuti zopereka zanzeru sizithawike m'mabuku achinsinsi, kudikirira zaka zana kuti zizindikirike.

Mbiri silingapereke chigamulo chotsimikizika ngati Cantor adalemba Dedekind. Umboni watsopanowu umapendekeka, koma chowonadi chonse chabisika muzobisika zaubwenzi wazaka za m'ma 1800 womwe udachitika kudzera m'makalata olembedwa pamanja ndikulankhulana pamasom'pamaso zomwe palibe zosunga zakale zomwe zingapangenso. Zomwe tingaphunzire, komabe, ndizosamvetsetseka: lembani chilichonse, perekani ngongole mowolowa manja, ndikumanga machitidwe omwe amapangitsa kuti kubwerekedwe zisachitike. Dedekind wotsatira akuyenera kuchita bwino.

Mafunso Ofunsidwa Kawirikawiri

Kodi ndi umboni wanji ukusonyeza kuti Cantor mwina adalemba Dedekind?

Maphunziro aposachedwa amawunika makalata awo ochulukirapo kuyambira 1870s ndi 1880s, kuwulula kuti malingaliro ambiri oyambira a Cantor pa chiphunzitso chokhazikitsidwa ndi chikhalidwe cha malingaliro osawerengeka agalasi omwe Dedekind adagawana nawo kale. Akatswiri a mbiri yakale amatchula kusiyana kwa nthawi pakati pa mipukutu yosasindikizidwa ya Dedekind ndi zofalitsa zotsatila za Cantor, pamodzi ndi ndime za m'makalata awo pamene Dedekind anafotokoza mfundo zazikulu zomwe pambuyo pake zinawonekera mu ntchito ya Cantor popanda kutchulidwa koyenera.

Kodi ubale wa Cantor-Dedekind unakhudza bwanji masamu amakono?

Kugwirizana kwawo ndi kupikisana kwawo kunapanga maziko a masamu amakono. Kumanga mosamalitsa kwa Dedekind kwa manambala enieni kudzera m'madula ndi chitukuko cha Cantor cha chiphunzitso chosinthika pamodzi chinakhazikitsa dongosolo lomwe pafupifupi masamu onse amakono ali. Kusinthana kwawo pa lingaliro la kusatha, kupitiriza, ndi mtundu wa zinthu za masamu kunayambitsa mikangano yomwe ikupitiriza kuyendetsa kafukufuku wa logic, filosofi ya masamu, ndi maphunziro oyambira lero.

Nchifukwa chiyani mkangano wakuba ukuyambanso tsopano?

Zosungidwa zatsopano zosungidwa pakompyuta, kuphatikiza zilembo zosafikirika m'mbuyomu ndi zolembedwa pamanja, zalola akatswiri a mbiri yakale kukonzanso nthawi yolondola ya kakulidwe ka malingaliro. Zida zamakono zowunikira malemba ndi njira zowunikiranso zapangitsa kuti zikhale zosavuta kufufuza momwe malingaliro akuyendera pakati pa akatswiri a masamu awiriwa. Zomwe atulukirazi zatsitsimutsanso chidwi cha maphunziro ndipo zapangitsa kuti mabuku angapo owunikiridwa ndi anzawo awonenso momwe Cantor adathandizira.

Kodi ndingapeze kuti zolembedwa zozama za masamu ndi mbiri yaukadaulo?

Majenali amaphunziro, zakale zamayunivesite, ndi malaibulale osungidwa a digito ndi poyambira zabwino kwambiri pakufufuza mozama. Kwa akatswiri ndi opanga zinthu omwe akufuna kufalitsa ndi kuyang'anira maphunziro awo moyenera, Mewayz imapereka bizinesi ya 207-module OS yoyambira pa $19/mo yomwe imaphatikizapo kulemba mabulogu, zida za SEO, ndi kasamalidwe ka omvera - chilichonse chofunikira kuti mupange chidziwitso chovomerezeka.