Ngā Ariā Ahurei i roto i te Pāngarau
He maha ngā wā ka tirohia te pāngarau hei kohinga tātai me ngā tataunga, engari kei raro iho he ao whai rawa, ataahua, ā, he mea whakamiharo i ētahi wā o ngā whakaaro. Ko tētahi momo "makutu" i roto i te pāngarau ko te ariā: he kōrero kua whakamātauhia he pono mā te arorau pakari. Heoi, kāore i te rite te hanganga o ngā ariā katoa. Ko ētahi ariā e kiia ana he ahurei nā te mea he ahua ohorere ō rātou hua, he rerekē ō rātou taunakitanga, he whānui rawa rānei ō rātou pānga ka huri i te huarahi e mārama ai tātou ki te tau, te wāhi, me ngā tauira. I roto i tēnei tuhinga, ka matapakihia e tātou te tikanga o te "ariā ahurei", ka tirohia ētahi tauira rongonui, me te take he mea nui aua ariā ki te whakatairanga i te pūtaiao.
He aha te Ariā Ahurei?
He maha ngā take e taea ai te kī he ahurei te ariā. Tuatahi, ka puta pea he whakatau e whakahē ana i ngā whakaaro o ia rā. Tuarua, ka hono pea i ngā wāhanga e rua o te pāngarau e āhua rerekē ana, hei tauira, te hono i te āhuahanga ki te arapū, te ariā tau rānei ki te tātari. Tuatoru, tera pea he tino huatau te taunakitanga o te ariā: he poto engari he kaha, he tino uaua rānei, ā, me tekau, tae atu ki ngā rau whārangi.
Ko te kupu "ahurei" i konei ehara i te mea kotahi anake te ariā nui, engari he āhua motuhake tō te ariā e tū motuhake ai i roto i ngā ariā kē atu. Ko ētahi ariā ka noho hei tohu mō te ahurea pūtaiao nā te mea he maha ngā whakahua, ngā akoranga, me ngā whakamoemiti.
Te Ariā Pythagorean: He mea tawhito engari he mea whakamiharo tonu
Ko tētahi o ngā ariā rongonui ko te ariā Pythagorean e kī ana mō tētahi tapatoru matau me ngā taha poutū \(a\) me \(b\), me te hypotenuse \(c\), ka whai mana tēnei:
\[
a^2 + b^2 = c^2
\]
Ahakoa he ariā tawhito, ko tōna ahurei kei roto i tōna whānui. Ka pā tēnei ariā ki ngā momo horopaki maha: te ine tawhiti, te whakatere, te hangarau ā-iwi, te whakairoiro rorohiko, tae atu ki te ahupūngao. He mea whakamīharo ake, kua kitea ngā rau o ngā taunakitanga mō tēnei ariā puta noa i te hītori, mai i ngā huarahi ā-ira tawhito ki ngā tikanga taupū hou. He onge te maha o ngā taunakitanga o tētahi ariā taketake, ā, e whakaatu ana tēnei i te hohonutanga o tēnei whakaaro māmā.
Te Kaupapa Taketake o te Pāngarau: Ngā Tuakiri o ngā Tau Kāore e Taea te Whakakapi
I roto i te ariā tau, kei reira te Kaupapa Taketake o te Pāngarau: ko ia tauoti nui atu i te 1 ka taea te tuhi hei hua o ngā tau matua i roto i tētahi huarahi ahurei, haunga te raupapa o ngā tauwehe. Hei tauira:
\[
60 = 2^2 \times 3 \times 5
\]
Ko te ahurei o tēnei ariā kei roto i tōna pūtake mō te nuinga o ngā ariā tau. Ki te kore tōna "ahurei" (ahurei i roto i te tikanga o te whakaaturanga tauwehe matua mārama), ka hinga ngā ariā maha atu: te wehewehenga, ngā tau maha, te GCD-LCM, tae noa ki te whakamunatanga hou. I roto i te ao matihiko, ko te haumarutanga o te whakamunatanga pērā i te RSA e whakawhirinaki ana ki ngā āhuatanga o ngā tau matua me te uaua o te tauwehe i ngā tau nui. Nō reira, ehara i te mea he ahurei noa iho tēnei ariā i te taha ariā engari he pānga hangarau nui anō hoki.
Te Ariā Kore-oti a Gödel: He Ahurei i Ru ai te Arorau
Mena i kitea he ariā e "whakaaroaro" ana, e pāngarau ana hoki, he pai te ariā o te kore whakaoti a Gödel. I roto i te poto, i whakamatau a Gödel i roto i tētahi pūnaha whaimana e kaha ana ki te pupuri i ngā pāngarau, he kōrero pono kāore e taea te whakamātau i roto i taua pūnaha. Arā, kāore he pūnaha axiomatic e taea te whakaoti (ka taea te whakamātau i ngā pono katoa) me te ōrite (kāore he whakahē) mena he kaha te pūnaha.
Ko te ahurei o tēnei ariā kei roto i tōna pānga ki te tumanako roa a ngā tohunga pāngarau ki te "whakaoti" i te pāngarau me tētahi pūnaha axiom kotahi, tino tika. I whakaatu a Gödel i ngā rohe ā-roto o te taunakitanga ōkawa. Ahakoa te āhua whakarāpopoto, kua awe tēnei whakaaro i te arorau, te pūtaiao rorohiko ariā, me tō tātou māramatanga ki te pono pāngarau.
Te Ariā Whakamutunga a Fermat: Te Tauākī Māmā, Te Whakamātautau Nui
E ai ki te ariā whakamutunga a Fermat, kāore he tauoti pai \(a\), \(b\), me \(c\) e tutuki ana i:
\[
a^n + b^n = c^n
\]
mō \(n > 2\). He tino māmā te kōrero, ā, he rite tonu ki te momo rerekētanga o te Ariā Pythagorean. Heoi, kua "whakamātau" tēnei ariā i ngā tohunga pāngarau mō ngā tau neke atu i te 350. Tae noa ki te tekau tau atu i 1990 ka angitu a Andrew Wiles ki te whakamatau i tēnei mā te whakamahi i ngā taputapu pāngarau hou i tawhiti atu i te raruraru taketake, tae atu ki te ariā o ngā pihi porowhita me ngā āhua modular.
Ko te ahurei o tēnei ariā ehara i te mea kei roto i tōna hītori roa anake, engari kei roto hoki i tōna taunakitanga, e whakaatu ana i tētahi meka nui: he maha ngā wā ka whanake te pāngarau puta noa i ngā mara. I ētahi wā, me whai ariā tino matatau hei whakamana i tētahi kōrero māmā.
Te Ariā a Bayes: He Ahurei Nā te mea ka Huri i te Huarahi o Tā Tātou Whakaaro
I roto i ngā tatauranga me te tūponotanga, ka whakaratohia e te Ariā a Bayes tētahi huarahi hei whakahou i ngā whakapono i runga i ngā mōhiohio hou. Ko tōna āhua whānui ko:
\[
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
\]
He ahurei tēnei ariā nā te mea ehara i te mea he taputapu tatau anake engari he anga whakaaro anō hoki. I tēnei ao hou, e whakamahia ana te ariā a Bayes i roto i te tātaritanga hauora, te tātari pāme īmēra, ngā pūnaha taunakitanga, tae noa ki te ako mīhini. E whakaako ana i te "tūponotanga" ehara i te mea he tūponotanga matapōkere noa iho engari ka taea hoki te tohu i tētahi taumata whakapono ka pai ake i te kohikohinga o ngā raraunga.
Te ariā mō te mutunga kore: Te kaiwaiata me te ine i te mutunga kore
I whakaurua mai e Georg Cantor tētahi whakaaro whakamiharo: kāore ngā mutunga kore katoa i te rite te rahi. Hei tauira, ka taea te tatau i ngā huinga o ngā tauoti me ngā tau whaitake, engari kāore e taea te tatau i te huinga o ngā tau tūturu. Ko tētahi o ngā hua rongonui a Cantor ko te tautohe whakarara, e whakaatu ana kāore e taea te whakarārangi katoa i ngā tau tūturu.
Ko te ahurei o ngā ariā a Cantor kei roto i tōna āhua rerekē. He tokomaha te hunga e whakaaro ana "ko te mutunga kore te mutunga kore," engari e whakaatu ana te pāngarau i te noho o ngā taumata mutunga kore. Ehara i te mea he mea nui tēnei whakaaro i roto i te ariā huinga anake engari he awe anō hoki ki ngā turanga o te pāngarau o ēnei rā.
He aha te Hiranga o te Ariā Ahurei?
He mea nui te mahi a ngā ariā ahurei ki te whanaketanga o te pāngarau me te pūtaiao. Tuatahi, ka whakakitea e rātou ngā hanganga huna e kore e kitea e te tirohanga noa. Tuarua, he maha ngā wā ka whakatuwherahia e rātou he mara hou, ka whakapakari rānei i ngā hononga i waenganui i a rātou. Tuatoru, ka whakangungu ngā ariā ahurei i te whakaaroaro arohaehae: ka ako tātou he hē te māramatanga i ētahi wā, ā, ko te taunakitanga pakari anake te mea tino whakatau i te pono.
Heoi anō, he mea whakahihiri ngā ariā ahurei. Ina taea e tētahi whakaaro iti te whakaputa hua whānui, ka kite tātou ehara te pāngarau i te taputapu noa iho, engari he toi whakaaro hohonu.
Te Katinga
Kī tonu te pāngarau i ngā ariā, engari ko ētahi o aua ariā he mea ahurei nā te ataahua, te ohorere, te awe whakamiharo rānei. Mai i te ariā Pythagorean matarohia ki te ariā taketake o te pāngarau, ki te whakakorikori a Gödel rāua ko Cantor i tō tātou māramatanga ki te arorau me te mutunga kore, kotahi te mea e whakaatu ana i tēnei katoa: ka tupu te pāngarau mā te whakakotahitanga o te auahatanga me te pakari. Ehara i te mea ka whakawhānui noa i tō tātou mātauranga ngā ariā ahurei, engari ka huri anō hoki i te huarahi e kite ai tātou i te ao—arā, kei muri i ngā tau me ngā tohu, he kōrero mō ngā whakaaro whakamiharo.
Ki te hiahia koe, ka taea e au te whakarerekē i tēnei tuhinga kia arotahi ake ki tētahi kaupapa kotahi (hei tauira, "te ariā tino rerekē," "te ariā i whakarerekē i te hangarau," "he ariā huatau me tētahi taunakitanga poto"), tāpirihia rānei he rārangi pukapuka me ngā tohutoro.
