Nā Manaʻo Kūikawā ma ka Makemakika
Ua manaʻo pinepine ʻia ka makemakika he hōʻiliʻili o nā ʻano hana a me nā helu ʻana, akā ma lalo ona aia kahi honua waiwai, nani, a i kekahi manawa he honua kupanaha o nā manaʻo. ʻO kekahi ʻano o ka "kilokilo" i ka makemakika ʻo ia ka theorem: kahi ʻōlelo i hōʻoia ʻia he ʻoiaʻiʻo ma o ka logic paʻa. Eia naʻe, ʻaʻole i hana like ʻia nā theorem āpau. Ua manaʻo ʻia kekahi mau theorem he kū hoʻokahi no ka mea me he mea lā ʻaʻole i manaʻo ʻia kā lākou mau hopena, he mea ʻē kā lākou mau hōʻoia, a i ʻole ua lōʻihi loa ko lākou hopena a hoʻololi lākou i ke ʻano o ko mākou hoʻomaopopo ʻana i nā helu, ka hakahaka, a me nā ʻano. Ma kēia ʻatikala, e kūkākūkā mākou i ke ʻano o "theorem kū hoʻokahi," e nānā i kekahi mau hiʻohiʻona kaulana, a no ke aha he mea nui ia mau theorem i ka holomua o ka ʻepekema.
He aha ke Kumumanaʻo Kūikawā?
Hiki ke manaʻo ʻia he kū hoʻokahi kahi theorem no kekahi mau kumu. ʻO ka mea mua, hiki iā ia ke hāʻawi i kahi hopena e kūʻē i ka naʻau o kēlā me kēia lā. ʻO ka lua, hiki iā ia ke hoʻohui i ʻelua mau wahi like ʻole o ka makemakika, no ka laʻana, e hoʻopili ana i ka geometry me ka algebra a i ʻole ke kumumanaʻo helu me ka nānā ʻana. ʻO ke kolu, hiki i ka theorem ke loaʻa kahi hōʻike nani loa: pōkole akā ikaika, a paʻakikī paha e pono ai nā ʻaoʻao he ʻumi a i ʻole nā haneli.
ʻO ka huaʻōlelo "kū hoʻokahi" ma ʻaneʻi, ʻaʻole ia he manaʻo he hoʻokahi wale nō theorem koʻikoʻi, akā, he ʻano kūikawā ko ka theorem e hoʻokaʻawale ai iā ia i waena o nā theorem ʻē aʻe. Ua lilo kekahi mau theorem i mau kiʻi o ka moʻomeheu ʻepekema no ka mea ua ʻōlelo pinepine ʻia, aʻo ʻia, a mahalo ʻia.
ʻO ke Kumumanaʻo Pythagorean: Kahiko akā Kupaianaha Mau
ʻO kekahi o nā theorem kaulana loa ʻo ia ka Pythagorean Theorem e ʻōlelo ana no kahi huinakolu ʻākau me nā ʻaoʻao perpendicular \(a\) a me \(b\), a me ka hypotenuse \(c\), penei ka paʻa:
\[
a^2 + b^2 = c^2
\]
ʻOiai he theorem kuʻuna, aia kona ʻano kū hoʻokahi i kona ʻano honua. Pili kēia theorem i nā ʻano pōʻaiapili like ʻole: ke ana ʻana i ka mamao, ka hoʻokele ʻana, ka ʻenekinia kīwila, nā kiʻi kamepiula, a me ke kinoea. ʻOi aku ka hoihoi, ua ʻike ʻia nā haneli o nā hōʻoia o kēia theorem ma o ka mōʻaukala, mai nā ʻano geometric kahiko a hiki i nā ʻano algebraic hou. Kākaʻikahi ka loaʻa ʻana o nā hōʻoia he nui i kahi theorem kumu, a hōʻike kēia i ka hohonu hohonu o kēia manaʻo maʻalahi.
ʻO ke Kumumanaʻo Kumu o ka Arithmetic: Nā ʻIke o nā Helu Hiki ʻole ke Hoʻololi
Ma ke kumumanaʻo helu, aia ke Kumumanaʻo Kumu o ka Arithmetic: hiki ke kākau ʻia kēlā me kēia helu piha ma mua o 1 ma ke ʻano he huahana o nā helu mua ma ke ʻano kū hoʻokahi, koe wale nō ke kaʻina o nā kumu. Eia kekahi laʻana:
\[
60 = 2^2 \times 3 \times 5
\]
ʻO ke ʻano kū hoʻokahi o kēia theorem aia i kona kumu no ka aneane o nā kumumanaʻo helu āpau. Me ka ʻole o kona "ʻano kū hoʻokahi" (kū hoʻokahi ma ke ʻano o kahi hōʻike hoʻohālikelike mua ʻole), e hiolo nā manaʻo ʻē aʻe he nui: ka mahele ʻana, nā multiples, GCD-LCM, a me ka cryptography hou. I loko o ke ao kikohoʻe, hilinaʻi ka palekana o ka hoʻopāʻālua e like me RSA i nā waiwai o nā helu mua a me ka paʻakikī o ka hoʻohālikelike ʻana i nā helu nui. No laila, ʻaʻole wale kēia theorem he mea kū hoʻokahi ma ke ʻano kumumanaʻo akā he hopena ʻenehana koʻikoʻi hoʻi.
ʻO ke Kumumanaʻo Hoʻopau ʻole o Gödel: He ʻano kū hoʻokahi i hoʻoluliluli i ka Logic
Inā loaʻa kekahi theorem i manaʻo ʻia he "akamai" a me ka makemakika, a laila he moho ikaika ka Gödel's Incompleteness Theorem. I ka pōkole, ua hōʻoia ʻo Gödel i loko o kekahi ʻōnaehana kūhelu i lawa ka mana e hoʻopaʻa i ka helu helu, aia nā ʻōlelo ʻoiaʻiʻo i hiki ʻole ke hōʻoia ʻia i loko o kēlā ʻōnaehana. ʻO ia hoʻi, ʻaʻohe ʻōnaehana axiomatic hiki ke piha (hiki ke hōʻoia ʻia nā ʻoiaʻiʻo āpau) a kūlike (ʻaʻohe kūʻē) inā lawa ka mana o ka ʻōnaehana.
ʻO ke ʻano kū hoʻokahi o kēia theorem aia i kona hopena i ka manaʻolana lōʻihi o ka poʻe makemakika e "hoʻoponopono" i ka makemakika me kahi ʻōnaehana axiom hoʻokahi a kūpono. Ua hōʻike ʻo Gödel i nā palena kūlohelohe o ka hōʻoia kūhelu. ʻOiai ke ʻano he abstract, ua hoʻohuli kēia manaʻo i ka logic, ka ʻepekema kamepiula theoretical, a me ko mākou ʻike ʻana i ka ʻoiaʻiʻo makemakika.
ʻO ke kumumanaʻo hope loa a Fermat: ʻŌlelo maʻalahi, hōʻoia nui
ʻŌlelo ka Fermat's Last Theorem ʻaʻohe helu piha maikaʻi \(a\), \(b\), a me \(c\) e hoʻokō ana i:
\[
a^n + b^n = c^n
\]
no \(n > 2\). He maʻalahi loa ka ʻōlelo, ua like ia me kahi ʻano like ʻole o ka Pythagorean Theorem. Eia nō naʻe, ua "hoʻowalewale" kēia theorem i nā makemakika no nā makahiki he 350 a ʻoi. ʻAʻole a hiki i ka makahiki 1990 i holomua ai ʻo Andrew Wiles i ka hōʻoia ʻana me ka hoʻohana ʻana i nā mea hana makemakika hou i mamao loa mai ka pilikia mua, me ke kumumanaʻo o nā piʻo elliptic a me nā ʻano modular.
ʻO ke ʻano kū hoʻokahi o kēia theorem ʻaʻole wale ma kona mōʻaukala lōʻihi, akā ma kona hōʻoia ʻana hoʻi, e hōʻike ana i kahi ʻoiaʻiʻo koʻikoʻi: ulu pinepine ka makemakika ma nā ʻano like ʻole. ʻO ka hōʻoia ʻana i kahi ʻōlelo maʻalahi i kekahi manawa e pono ai kahi kumumanaʻo paʻakikī loa.
ʻO ke Kumumanaʻo o Bayes: Kū hoʻokahi no ka mea e hoʻololi ia i ke ala a mākou e noʻonoʻo ai
Ma nā helu helu a me ka hiki ke loaʻa, hāʻawi ka Theorem a Bayes i kahi ala e hōʻano hou ai i nā manaʻoʻiʻo e pili ana i ka ʻike hou. ʻO kona ʻano maʻamau:
\[
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
\]
He mea kū hoʻokahi kēia theorem no ka mea, ʻaʻole ia he mea hana helu wale nō akā he ʻōnaehana no ka noʻonoʻo ʻana. I kēia au hou, hoʻohana ʻia ka Theorem a Bayes i ka ʻike lapaʻau, ka kānana ʻana i ka spam leka uila, nā ʻōnaehana ʻōlelo paipai, a me ke aʻo ʻana i ka mīkini. Ke aʻo nei ia ʻaʻole wale ka "probability" he mea kūpono wale nō akā hiki ke hōʻike i kahi kekelē o ka hilinaʻi e hoʻomaikaʻi ana i ka hōʻiliʻili ʻana o ka ʻikepili.
ʻO ke kumumanaʻo ma ka palena ʻole: ʻO Cantor a me ke ana o ka palena ʻole
Ua hoʻolauna ʻo Georg Cantor i kahi manaʻo kupanaha: ʻaʻole like ka nui o nā infinities āpau. No ka laʻana, hiki ke helu ʻia nā set o nā helu helu a me nā helu rational, akā ʻaʻole hiki ke helu ʻia ka set o nā helu maoli. ʻO kekahi o nā hopena kaulana a Cantor ʻo ia ka hoʻopaʻapaʻa diagonal, e hōʻike ana he hiki ʻole ke papa inoa piha i nā helu maoli āpau.
ʻO ke ʻano kū hoʻokahi o nā theorems a Cantor aia i ko lākou ʻano kūʻē i ka manaʻo maʻamau. Manaʻo ka poʻe he nui "ʻo ka palena ʻole ka palena ʻole," akā hōʻike ka makemakika i ke ola ʻana o nā pae palena ʻole. ʻAʻole wale kēia manaʻo he mea nui i ke kumumanaʻo hoʻonohonoho akā hoʻohuli pū kekahi i nā kumu o ka makemakika hou.
No ke aha he mea nui ka Theorem Unique?
He kuleana koʻikoʻi ko nā theorems kū hoʻokahi i ka hoʻomohala ʻana o ka makemakika a me ka ʻepekema. ʻO ka mea mua, hōʻike lākou i nā ʻano huna i ʻike ʻole ʻia e ka nānā maʻamau. ʻO ka lua, wehe pinepine lākou i nā kahua hou a hoʻoikaika paha i nā pilina ma waena o lākou. ʻO ke kolu, hoʻomaʻamaʻa nā theorems kū hoʻokahi i ka noʻonoʻo koʻikoʻi: ʻike mākou he hewa ka naʻau i kekahi manawa, a ʻo ka hōʻike paʻa wale nō ka mea hoʻoholo hope loa o ka ʻoiaʻiʻo.
Eia kekahi, he mea hoʻoikaika nā kumumanaʻo kūikawā. Ke hiki i kahi manaʻo liʻiliʻi ke hoʻopuka i nā hopena ākea, ʻike mākou ʻaʻole wale ka makemakika he mea hana, akā he hana noʻeau hohonu o ka noʻonoʻo.
Pani
Ua piha ka makemakika i nā theorems, akā kū kekahi o lākou i mea kū hoʻokahi ma muli o ko lākou nani, kahaha, a i ʻole ka mana kupaianaha. Mai ka Pythagorean Theorem maʻamau a hiki i ka Fundamental Theorem of Arithmetic, a hiki i ka hoʻoluliluli ʻana o Gödel lāua ʻo Cantor i ko mākou ʻike i ka logic a me ka palena ʻole, hōʻike lākou a pau i hoʻokahi mea: ulu ka makemakika ma o ka hui pū ʻana o ka noʻeau a me ka paʻakikī. ʻAʻole wale nā theorems kū hoʻokahi e hoʻonui i ko mākou ʻike, akā hoʻololi pū i ke ʻano o kā mākou ʻike ʻana i ka honua—ma hope o nā helu a me nā hōʻailona, aia kahi moʻolelo o nā manaʻo kupaianaha.
Inā makemake ʻoe, hiki iaʻu ke hoʻoponopono i kēia ʻatikala e ʻoi aku ka nānā ʻana ma kahi kumuhana hoʻokahi (e.g., "ka theorem kūʻē loa i ka manaʻo maʻamau," "ka theorem i hoʻololi i ka ʻenehana," a i ʻole "kahi theorem nani me kahi hōʻoia pōkole") a i ʻole e hoʻohui i kahi papa inoa a me nā kuhikuhi.
