{"id":404,"date":"2020-04-16T15:22:27","date_gmt":"2020-04-16T07:22:27","guid":{"rendered":"http:\/\/106.54.217.124\/?p=404"},"modified":"2020-04-16T15:22:27","modified_gmt":"2020-04-16T07:22:27","slug":"dd","status":"publish","type":"post","link":"https:\/\/www.91shuxue.com\/index.php\/2020\/04\/16\/dd\/","title":{"rendered":"\u6570\u5b66\u7684\u771f\u76f8\uff1a\u7269\u7406\u65f6\u7a7a\u7684\u6570\u5b57\u6a21\u578b\u8fd8\u662f\u73b0\u5b9e\u672c\u8eab"},"content":{"rendered":"<p>\u6570\u5b66\u7684\u771f\u76f8\uff1a\u7269\u7406\u65f6\u7a7a\u7684\u6570\u5b57\u6a21\u578b\u8fd8\u662f\u73b0\u5b9e\u672c\u8eab\uff1f<img loading=\"lazy\" decoding=\"async\" class=\"alignright size-full wp-image-351\" src=\"http:\/\/106.54.217.124\/wp-content\/uploads\/2020\/04\/ABUIABACGAAgrbHZyAUo0MSNnwcwnQc4rgM.jpg\" alt=\"\" width=\"925\" height=\"430\" 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\/>\n\u5929\u738b\u661f\u662f\u90a3\u65f6\u5df2\u77e5\u884c\u661f\u4e2d\u79bb\u592a\u9633\u6700\u8fdc\u7684\u4e00\u9897\uff1b\u5b83\u7684\u8fd0\u884c\u8f68\u9053\u4f3c\u4e4e\u603b\u662f\u504f\u79bb\u6839\u636e\u4e07\u6709\u5f15\u529b\u5b9a\u5f8b\u63a8\u7b97\u7684\u7ed3\u679c\u3002<br \/>\n\u5f53\u65f6\u6709\u4e9b\u5b66\u8005\u731c\u6d4b\uff0c\u4e5f\u8bb8\u4e07\u6709\u5f15\u529b\u5b9a\u5f8b\u5bf9\u5982\u6b64\u9065\u8fdc\u7684\u661f\u7403\u53ef\u80fd\u4e0d\u9002\u7528\u3002<br \/>\n\u53e6\u5916\u6709\u4e9b\u79d1\u5b66\u5bb6\u5219\u6c42\u52a9\u4e8e\u6570\u5b66\uff0c\u901a\u8fc7\u8ba1\u7b97\uff0c\u63a8\u6d4b\u5929\u738b\u661f\u9644\u8fd1\u5b58\u5728\u4e00\u56e2\u4f53\u91cf\u5de8\u5927\u7684\u7269\u8d28\uff0c\u5bfc\u81f4\u5b83\u7ed5\u65e5\u516c\u8f6c\u8f68\u9053\u51fa\u73b0\u504f\u79bb\u3002<br \/>\n\u4f26\u6566\u5927\u5b66\u5b66\u9662\u5929\u6587\u7269\u7406\u5b66\u5bb6\u9c81\u897f\u00b7\u683c\u6797\uff08Lucie Green\uff09\u89e3\u91ca\u8bf4\uff0c\u4ed6\u4eec\u901a\u8fc7\u8ba1\u7b97\u7ed3\u679c\u63a8\u6d4b\u8fd9\u56e2\u795e\u79d8\u7269\u8d28\u53ef\u80fd\u5728\u4ec0\u4e48\u65b9\u4f4d\uff0c\u7136\u540e\u628a\u671b\u8fdc\u955c\u76ef\u4f4f\u90a3\u4e2a\u5730\u65b9\uff0c\u7ed3\u679c\u5c31\u53d1\u73b0\u4e86\u6d77\u738b\u661f\u3002\u6d77\u738b\u661f\u7684\u53d1\u73b0\uff0c\u4e3b\u8981\u56e0\u4e3a\u89c2\u5bdf\u5230\u7684\u5929\u738b\u661f\u8fd0\u884c\u8f68\u8ff9\u8ddf\u8ba1\u7b97\u7ed3\u679c\u6709\u504f\u5dee<br \/>\n\u672c\u6765\uff0c\u5929\u738b\u661f\u3001\u6d77\u738b\u661f\u7684\u6545\u4e8b\u5230\u6b64\u53ef\u4ee5\u544a\u4e00\u6bb5\u843d\u3002\u4f46\u7f16\u5199\u5386\u53f2\u7684\u4eba\u628a\u6d77\u738b\u661f\u8ddf\u6570\u5b66\u8fd9\u4e48\u4e00\u5173\u8054\uff0c\u95ee\u9898\u5c31\u6765\u4e86\u3002<br \/>\n\u6570\u5b66\u7a76\u7adf\u662f\u4eba\u4eec\u53d1\u660e\u51fa\u6765\u3001\u7528\u4ee5\u8868\u8ff0\u73b0\u5b9e\u7269\u7406\u65f6\u7a7a\u7684\u62bd\u8c61\u6a21\u578b\u5462\uff0c\u8fd8\u662f\u5b83\u672c\u8eab\u5c31\u662f\u4e00\u79cd\u73b0\u5b9e\u5b58\u5728\uff1f<br \/>\n\u542c\u8d77\u6765\u50cf\u54f2\u5b66\u95ee\u9898\uff0c\u5b9e\u9645\u4e0a\u4e5f\u786e\u5b9e\u662f\u54f2\u5b66\u5bb6\u5343\u767e\u5e74\u6765\u4e00\u76f4\u5728\u7422\u78e8\u7684\u95ee\u9898\u3002<br \/>\n\u800c\u4e14\uff0c\u5bf9\u8fd9\u4e2a\u95ee\u9898\u5373\u4f7f\u5230\u4e86\u4eca\u5929\u4ecd\u6ca1\u6709\u5171\u8bc6\uff0c\u610f\u89c1\u5206\u6b67\u6781\u6df1\u3002\u4f60\u5403\u4e86\u86cb\u7cd5\uff0c\u6216\u8005\u6ca1\u5403\u86cb\u7cd5\u3002\u4f60\u4e0d\u53ef\u80fd\u5403\u4e86\u201c\u8d1f\u86cb\u7cd5\u201d<br \/>\n\u86cb\u7cd5\uff1a\u5403\u5b8c\u5c31\u6ca1\u6709\u4e86<br \/>\n\u6211\u4eec\u8bd5\u7740\u6765\u5543\u4e00\u4e0b\u8fd9\u9053\u5343\u5e74\u96be\u9898\u3002<br \/>\n\u6570\u5b66\u59cb\u4e8e\u4eba\u7c7b\u751f\u5b58\u6d3b\u52a8\u7684\u9700\u8981\uff1a\u8ba1\u6570\u548c\u4e08\u91cf\u3002<br \/>\n\u4ee5\u86cb\u7cd5\u4e3a\u4f8b\u3002\u6570\u5b66\u5bb6\u4eec\u5bf9\u86cb\u7cd5\u6709\u5404\u79cd\u8bf4\u6cd5\uff1a\u5c3a\u5bf8\u3001\u91cd\u91cf\u3001\u5f62\u72b6\u3001\u600e\u4e48\u5207\u5272\u3002\u6240\u6709\u8fd9\u4e9b\u90fd\u662f\u5177\u50cf\u3001\u53ef\u89c1\u7684\u3002<br \/>\n\u6570\u5b66\u8fd8\u53ef\u4ee5\u628a\u6211\u4eec\u5e26\u5230\u62bd\u8c61\u7684\u9886\u57df\uff1a\u5982\u679c\u4f60\u5403\u4e86\u4e09\u5206\u4e4b\u4e00\u7684\u86cb\u7cd5\uff0c\u90a3\u4e48\u5269\u4e0b\u7684\u5c31\u662f\u4e09\u5206\u4e4b\u4e8c\uff0c\u4e24\u5757\u3002\u628a\u8fd9\u4e24\u5757\u5403\u4e86\uff0c\u5c31\u6ca1\u4e86\u3002<br \/>\n\u6570\u5b66\u5bb6\u8d1d\u5a04\u65af\uff08Alex Bellos\uff09\u8bf4\uff0c\u53e4\u4eba\u7528\u6570\u5b57\u8fd0\u7b97\u3001\u4e08\u91cf\u65f6\uff0c\u8fd8\u6ca1\u6709\u8d1f\u6570\u7684\u6982\u5ff5\u3002<br \/>\n\u5982\u679c\u73b0\u5b9e\u610f\u5473\u7740\u6709\u5f62\u6709\u4f53\u7684\u7269\u4ef6\uff0c\u53ef\u4ee5\u8ba1\u6570\u3001\u91cf\u5c3a\u5bf8\u548c\u91cd\u91cf\uff0c\u90a3\u4e48\u6bd4\u96f6\u66f4\u5c0f\u7684\u6982\u5ff5\u662f\u5f88\u96be\u60f3\u8c61\u7684\u3002<br \/>\n\u4e00\u53ea\u86cb\u7cd5\uff0c\u5403\u5b8c\u4e86\u5c31\u5403\u5b8c\u4e86\uff0c\u6ca1\u6709\u201c\u8d1f\u4e09\u5206\u4e4b\u4e00\u201d\u5757\u86cb\u7cd5\u4e4b\u8bf4\u3002\u8d27\u5e01\u548c\u94b1\u8d22\u53ef\u4ee5\u628a\u8d1f\u6570\u8fd9\u4e2a\u6570\u5b66\u6982\u5ff5\u53d8\u5f97\u5f88\u5bb9\u6613\u7406\u89e3\u8d1f\u503a\u548c\u8d1f\u6570<br \/>\n\u4e0d\u8fc7\uff0c\u5728\u9ed1\u4e0e\u767d\u4e2d\u95f4\u7684\u7070\u8272\u5730\u5e26\uff0c\u8d1f\u6570\u8fd9\u4e2a\u6982\u5ff5\u548c\u81ea\u7136\uff0c\u5b8c\u5168\u80fd\u591f\u88ab\u4eba\u7406\u89e3\u7684\uff0c\u90a3\u5c31\u662f\u94b1\u3002<br \/>\n\u94b1\u8fd9\u4e2a\u4e1c\u897f\uff0c\u4f60\u53ef\u4ee5\u6709\uff0c\u4e5f\u53ef\u4ee5\u6b20\u3002<br \/>\n\u8d1f\u6570\u6700\u521d\u7684\u5e94\u7528\u662f\u4ece\u503a\u52a1\u548c\u8bb0\u8d26\u5f00\u59cb\u7684\u3002<br \/>\n\u4f60\u6b20\u522b\u4eba5\u5757\u94b1\uff0c\u6211\u7ed9\u4e86\u4f605\u5757\u8ba9\u4f60\u8fd8\u7ed9\u522b\u4eba\uff0c\u4f60\u624b\u91cc\u5269\u4e0b\u96f6\u3002\u8fd9\u5c31\u662f\u8d1f\u6570\u73b0\u5b9e\u7684\u8d77\u6e90\u3002<br \/>\n\u7136\u540e\u4e8b\u60c5\u5c31\u4e00\u53d1\u4e0d\u53ef\u6536\u62fe\uff0c\u5230\u4eca\u5929\u4eba\u4eec\u5df2\u7ecf\u65e0\u6cd5\u60f3\u50cf\u4e0d\u5b58\u5728\u8d1f\u6570\u7684\u4e16\u754c\u3002<br \/>\n\u6545\u4e8b\u8fd8\u6ca1\u5b8c\u3002\u4e0a\u9762\u8fd9\u4e9b\u90fd\u6839\u690d\u4e8e\u73b0\u5b9e\u4e16\u754c\uff0c\u4f46\u8d1f\u6570\u4f3c\u4e4e\u81ea\u5e26\u9b54\u529b\uff0c\u80fd\u8ba9\u4e0d\u53ef\u601d\u8bae\u7684\u4e8b\u53d1\u751f\u3002<br \/>\n\u5de8\u5927\u7684\u8c1c<br \/>\n\u4e24\u4e2a\u8d1f\u6570\u76f8\u4e58\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f\u6b63\u6570\uff0c\u8d1f\u8d1f\u5f97\u6b63\uff1a(-1) X (-1) = 1\u3002<br \/>\n\u5c31\u8fd9\u4e48\u7b80\u5355\u7684\u4e00\u4e2a\u5b9a\u5f8b\uff0c\u7275\u51fa\u4e86\u4e00\u4e2a\u5b9e\u5b9e\u5728\u5728\u7684\u3001\u5de8\u5927\u7684\u8c1c\u3002<br \/>\n\u8d1d\u5a04\u65af\u89e3\u91ca\u8bf4\uff0c\u5982\u679c\u5728\u6570\u5b66\u7b49\u5f0f\u91cc\u6446\u5f04\u8d1f\u6570\u548c\u6b63\u6570\uff0c\u4f1a\u770b\u5230\u5947\u5999\u7684\u7ed3\u679c\uff1a<img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" src=\"http:\/\/106.54.217.124\/wp-content\/uploads\/2020\/04\/109518369_equation7198084742406886828.jpg\" width=\"643\" height=\"361\" data-wpid=\"16\" \/>\u8fd9\u7ec4\u6570\u5b66\u7b49\u5f0f\u80cc\u540e\u85cf\u7740\u4e00\u4e2a\u8c1c<br \/>\n\u8fd9\u662f\u5565\uff1f\u4e00\u4e2a\u6570\u7684\u5e73\u65b9\uff0c\u5373\u4e00\u4e2a\u6570\u4e58\u4ee5\u81ea\u5df1\uff0c\u7b49\u4e8e-1\uff1b\u90a3\u4e48-1\u7684\u5e73\u65b9\u6839\u5c31\u662f\u8fd9\u4e2a\u6570\uff0c\u4f46\u8fd9\u600e\u4e48\u53ef\u80fd\uff1f<br \/>\n\u6570\u5b66\u5bb6\u8d1d\u5a04\u65af\u8bf4\uff0c\u8fd9\u4e2a\u6570\u4e0d\u53ef\u80fd\u662f\u6b63\u6570\uff0c\u4e5f\u4e0d\u53ef\u80fd\u662f\u8d1f\u6570\uff0c\u56e0\u4e3a\u8d1f\u8d1f\u5f97\u6b63\u3002<br \/>\n\u4e8e\u662f\uff0c\u5076\u7136\u53d1\u73b0\u8fd9\u4e2a\u8c1c\u7684\u4eba\u4eec\u9677\u5165\u56f0\u60d1\u3002\u8fd9\u592a\u8352\u8bde\u4e86\u3002<br \/>\n\u540e\u6765\uff0c\u6570\u5b66\u5bb6\u4eec\u9010\u6e10\u5f00\u59cb\u610f\u8bc6\u5230\uff0c\u8fd9\u786e\u5b9e\u4e0d\u5bfb\u5e38\uff0c\u4f46\u5b9e\u9645\u5e94\u7528\u4e2d\u5374\u80fd\u63a8\u5bfc\u51fa\u6b63\u786e\u7b54\u6848\u3002\u6240\u4ee5\uff0c\u8fd9\u4e2a\u8c1c\u5c31\u7559\u7ed9\u54f2\u5b66\u5bb6\u4eec\u53bb\u7ede\u8111\u6c41\u5427\u3002\u6211\u4eec\u6570\u5b66\u5bb6\u53ea\u8981\u6570\u5b57\uff0c\u53ea\u8981\u7b54\u6848\u3002<br \/>\n\u8fd9\u65f6\uff0c\u4e00\u6247\u4ece\u73b0\u5b9e\u901a\u5411\u975e\u73b0\u5b9e\u7684\u95e8\u6084\u6084\u6253\u5f00\u4e86\u3002<br \/>\n\u865a\u6570\u3001\u5b9e\u6570\u3001\u590d\u6570<br \/>\n\u5f53\u7136\uff0c\u5168\u90e8\u6254\u7ed9\u54f2\u5b66\u5bb6\u53bb\u89e3\u91ca\u662f\u4e0d\u53ef\u80fd\u7684\uff1b\u6570\u5b66\u4ecd\u65e7\u53ef\u4ee5\u7528\u6765\u89e3\u91ca\u8fd9\u4e2a\u73b0\u8c61\u3002<br \/>\n-1\u7684\u5e73\u65b9\u6839\u6709\u4e00\u4e2a\u540d\u5b57\uff0c\u53eb\u505a\u201c\u865a\u6570\u201d\u3002<br \/>\n\u8d1d\u5a04\u65af\u8bf4\uff0c\u8fd9\u4e2a\u540d\u5b57\u4e0d\u662f\u5f88\u597d\u542c\uff0c\u56e0\u4e3a\u6570\u5b66\u672c\u8eab\u662f\u771f\u5b9e\u7684\uff0c\u4f46\u201c\u865a\u201d\u5374\u7ed9\u4eba\u4e0d\u771f\u5b9e\u7684\u5370\u8c61\u3002<br \/>\n\u4f46\u5176\u5b9e\u6570\u5b66\u4ece\u4e00\u5f00\u59cb\u5c31\u662f\u865a\u7684\uff0c\u662f\u62bd\u8c61\u7684\u3002\u6bd4\u5982\u4e09\u5757\u86cb\u7cd5\uff0c\u6211\u4eec\u770b\u89c1\u7684\u662f\u86cb\u7cd5\uff0c\u800c\u201c\u4e09\u201d\u5219\u662f\u4e2a\u62bd\u8c61\u6982\u5ff5\u3002I\u53c1\u3001\u4e09\u30013\uff0c\u65e0\u8bba\u662f\u5b57\u8fd8\u662f\u6570\u5b57\uff0c\u90fd\u662f\u4e00\u4e2a\u6570\u7684\u7b26\u53f7\uff0c\u800c\u8fd9\u4e2a\u6570\u662f\u4e2a\u62bd\u8c61\u7684\u6982\u5ff5<br \/>\n\u865a\u6570\u4e5f\u4e00\u6837\uff0c\u770b\u8d77\u6765\u5f88\u8352\u8bde\uff0c\u4f46\u4ed4\u7ec6\u7422\u78e8\uff0c\u5c31\u4f1a\u53d1\u73b0\u5b83\u5f88\u7b26\u5408\u903b\u8f91\u3002<br \/>\n\u8d1d\u5a04\u65af\u89e3\u91ca\u8bf4\uff0c\u6211\u4eec\u6240\u8bf4\u7684\u5b9e\u6570\uff0c\u52a0\u4e0a\u865a\u6570\uff0c\u8fd8\u6709\u590d\u6570\uff0c\u5171\u540c\u6784\u6210\u4e86\u4e00\u79cd\u903b\u8f91\u8bed\u8a00\uff0c\u80fd\u591f\u7528\u6765\u5b8c\u7f8e\u8be0\u91ca\u8bb8\u591a\u73b0\u8c61\uff0c\u6bd4\u5982\u5faa\u73af\u65cb\u8f6c\u3002<br \/>\n\u4ed6\u8bf4\uff1a\u201c\u4eca\u5929\uff0c-1\u7684\u5e73\u65b9\u6839\u5728\u6211\u4eec\u770b\u6765\u5c31\u548c-1\u4e00\u6837\u771f\u5b9e\uff0c\u5373\u4f7f\u5b83\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5f88\u96be\u7406\u89e3\uff0c\u5c31\u50cf-1\u8fd9\u4e2a\u6570\u5bf9\u6211\u4eec\u7684\u8001\u7956\u5b97\u6765\u8bf4\u65e0\u6cd5\u7406\u55bb\u4e00\u6837\u3002\u201d<br \/>\n\u518d\u63a5\u518d\u5389<br \/>\n\u73b0\u5728\u8f6e\u5230\u590d\u6570\uff08complex number) \u3002\u6709\u4e9b\u7528\u5b9e\u6570\u65e0\u89e3\u7684\u65b9\u7a0b\u5f0f\uff0c\u7528\u590d\u6570\u5c31\u53ef\u4ee5\u89e3\u7b54\u3002<br \/>\n\u5b83\u4eec\u5c31\u662f\u90a3\u4e48\u9b54\u6027\uff0c\u5bf9\u6211\u4eec\u7406\u89e3\u73b0\u5b9e\u63d0\u4f9b\u4e86\u5b9e\u9645\u5e2e\u52a9\uff0c\u51e0\u4e4e\u6240\u6709\u6d89\u53ca\u5faa\u73af\u65cb\u8f6c\u548c\u6ce2\u6d6a\u7684\u6570\u5b66\u9898\u79bb\u4e0d\u5f00\u590d\u6570\u8fd9\u4e2a\u5de5\u5177\u3002<br \/>\n\u590d\u6570\u5728\u7535\u673a\u5de5\u7a0b\u3001\u96f7\u8fbe\u3001\u533b\u7597\u5f71\u50cf\u9886\u57df\u4e0d\u53ef\u6216\u7f3a\uff0c\u5bf9\u89e3\u91ca\u6b21\u539f\u5b50\u7c92\u5b50\u7684\u884c\u4e3a\u4e5f\u81f3\u5173\u91cd\u8981\u3002<br \/>\n\u90a3\u4e48\u95ee\u9898\u53c8\u6765\u4e86\uff1a\u4e3a\u4ec0\u4e48\u53ea\u5b58\u5728\u4e8e\u6570\u5b66\u5047\u8bbe\u4e2d\u7684\u4e1c\u897f\uff0c\u5728\u73b0\u5b9e\u4e16\u754c\u7528\u9014\u5982\u6b64\u4e4b\u5927\u5462\uff1f<br \/>\n\u5308\u7259\u522920\u4e16\u7eaa\u7269\u7406\u5b66\u5bb6\u5c24\u91d1\u00b7\u7ef4\u683c\u7eb3\uff08Eugene Wigner\uff09\u8ba4\u4e3a\u8fd9\u5c31\u662f\u4e2a\u5947\u8ff9\u3002\u8fd8\u6709\u4e0d\u5c11\u4eba\u8ddf\u4ed6\u770b\u6cd5\u4e00\u81f4\u3002<br \/>\n\u7ef4\u683c\u7eb31960\u5e74\u53d1\u8868\u4e86\u4e00\u7bc7\u91cd\u8981\u8bba\u6587\uff0c\u5176\u4e2d\u628a\u590d\u6570\u53eb\u505a\u201c\u6570\u5b66\u5728\u7269\u7406\u79d1\u5b66\u4e2d\u4e0d\u5408\u7406\u7684\u6709\u6548\u6027\u201d\uff08unreasonable effectiveness\uff09\u3002\u5982\u679c\u6570\u5b66\u672c\u6765\u5c31\u662f\u4eba\u7c7b\u4e3a\u4e86\u7406\u89e3\u73b0\u5b9e\u4e16\u754c\u53d1\u660e\u7684\u5de5\u5177\uff0c\u90a3\u4e48\u6570\u5b66\u8868\u8ff0\u73b0\u5b9e\u4e16\u754c\u5c31\u4e0d\u8db3\u4e3a\u5947<br \/>\n\u4e0d\u5408\u7406\u6709\u6548\u6027<br \/>\n\u5982\u679c\u6570\u5b66\u662f\u4eba\u7c7b\u53d1\u660e\u8bbe\u8ba1\u51fa\u6765\uff0c\u7528\u4f5c\u8868\u8ff0\u73b0\u5b9e\u7684\u5de5\u5177\uff0c\u90a3\u4e48\u5b83\u5177\u6709\u8fd9\u79cd\u529f\u80fd\u96be\u9053\u4e0d\u662f\u5408\u60c5\u5408\u7406\uff0c\u5b8c\u5168\u7b26\u5408\u903b\u8f91\u5417\uff1f<br \/>\n\u6709\u4ec0\u4e48\u201c\u4e0d\u5408\u7406\u201d\u7684\u5462\uff1f<br \/>\n\u8fd9\u4e2a\u95ee\u9898\u6216\u8bb8\u8981\u627e\u6570\u5b66\u3001\u7269\u7406\u4e24\u6816\u7684\u4e13\u5bb6\u6765\u89e3\u91ca\uff0c\u6bd4\u5982\u57c3\u8389\u8bfa\u00b7\u8bfa\u514b\u65af\uff08Eleanor Knox\uff09\uff1b\u5979\u7684\u4e13\u4e1a\u9886\u57df\u662f\u7269\u7406\u4e2d\u7684\u54f2\u5b66\u3002<br \/>\n\u5979\u8bf4\uff0c\u786e\u5b9e\uff0c\u5982\u679c\u53d1\u660e\u6570\u5b66\u662f\u4e3a\u4e86\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u7269\u7406\u7cfb\u7edf\uff0c\u90a3\u4e48\u5b83\u80dc\u4efb\u8fd9\u4e2a\u4efb\u52a1\u5b8c\u5168\u5408\u4e4e\u903b\u8f91\u3002\u4f46\u6570\u5b66\u7684\u6f14\u53d8\u8fc7\u7a0b\u4f3c\u4e4e\u8ddf\u8fd9\u79cd\u89e3\u91ca\u4e0d\u76f8\u7b26\u3002<br \/>\n\u5386\u53f2\u4e0a\u4e0d\u4e4f\u8fd9\u6837\u7684\u5148\u4f8b\uff1a\u6570\u5b66\u5bb6\u7eaf\u7cb9\u51fa\u4e8e\u4e2a\u4eba\u5174\u8da3\u505a\u51fa\u4e86\u4e00\u4e9b\u53d1\u660e\uff0c \u8fc7\u4e86\u6bb5\u65f6\u95f4\u8fd9\u4e9b\u53d1\u660e\u6070\u597d\u7528\u6765\u4fc3\u6210\u67d0\u9879\u91cd\u5927\u7684\u7269\u7406\u5b66\u53d1\u73b0\u6216\u7a81\u7834\u3002<br \/>\n\u6570\u5b66\u4e2d\u7684\u975e\u6b27\u51e0\u4f55\uff08non-Euclidean geometry\uff09\u5c31\u662f\u4e00\u4e2a\u8457\u540d\u7684\u4f8b\u5b50\u3002<br \/>\n\u572819\u4e16\u7eaa\u672b\u7684\u65f6\u5019\uff0c\u8bb8\u591a\u6570\u5b66\u5bb6\u90fd\u8ff7\u4e0a\u4e86\u975e\u6b27\u51e0\u4f55\uff0c\u5c31\u56e0\u4e3a\u4ed6\u4eec\u89c9\u5f97\u8fd9\u4e2a\u547d\u9898\u975e\u5e38\u6709\u610f\u601d\uff0c\u5f15\u4eba\u5165\u80dc\u3002<br \/>\n\u5f53\u65f6\u516c\u8ba4\u6574\u4e2a\u4e16\u754c\u90fd\u53ef\u4ee5\u7528\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u5b9a\u5f8b\u6765\u8868\u8ff0\uff0c\u5b66\u6821\u6559\u7684\u4e5f\u662f\u6b27\u51e0\u91cc\u5fb7\uff0c\u6bd4\u5982\u4e09\u89d2\u5f62\u7684\u4e09\u4e2a\u5185\u89d2\u4e4b\u548c\u662f180\u5ea6\u3002<br \/>\n1800\u5e74\u4ee3\u672b\u671f\u7684\u6570\u5b66\u5bb6\u5e76\u4e0d\u60f3\u63a8\u7ffb\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u5b66\uff0c\u53ea\u4e0d\u8fc7\u662f\u5728\u6570\u5b66\u63a2\u7d22\u8fc7\u7a0b\u4e2d\u53d1\u73b0\u4e86\u5999\u4e0d\u53ef\u8a00\u7684\u65b0\u7684\u6570\u5b66\u7ed3\u6784\u3002\u975e\u6b27\u51e0\u4f55\u628a\u66fe\u7ecf\u53ea\u5b58\u5728\u4e8e\u6570\u5b66\u5bb6\u60f3\u8c61\u4e2d\u7684\u5f62\u72b6\u53d8\u6210\u4e86\u5e38\u4eba\u770b\u5f97\u89c1\u6478\u5f97\u7740\u7684\u591a\u7ef4\u5ea6\u7269\u4f53<br \/>\n\u8bfa\u514b\u65af\u8bf4\uff0c\u5230\u4e8620\u4e16\u7eaa\uff0c\u7231\u56e0\u65af\u5766\u5728\u5e7f\u4e49\u76f8\u5bf9\u8bba\u7814\u7a76\u4e2d\u53d1\u73b0\u81ea\u5df1\u9700\u8981\u4e00\u79cd\u65b0\u7684\u7406\u8bba\u7528\u6765\u8868\u8ff0\u76f8\u5e94\u7684\u65f6\u7a7a\u5b9a\u5f8b\uff0c\u800c\u975e\u6b27\u51e0\u4f55\u6b63\u597d\u6ee1\u8db3\u4e86\u4ed6\u7684\u9700\u6c42\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u6ca1\u6709\u975e\u6b27\u51e0\u4f55\uff0c\u7231\u56e0\u65af\u5766\u7684\u76f8\u5bf9\u8bba\u5c31\u4e0d\u4f1a\u95ee\u4e16\u3002<br \/>\n\u4f46\u662f\uff0c19\u4e16\u7eaa\u6c89\u6d78\u5728\u975e\u6b27\u51e0\u4f55\u7814\u7a76\u4e2d\u7684\u6570\u5b66\u5bb6\u5f88\u663e\u7136\u6ca1\u6709\u9884\u89c1\u5230\u591a\u5e74\u540e\u7269\u7406\u5b66\u754c\u4f1a\u51fa\u73b0\u5e7f\u4e49\u76f8\u5bf9\u8bba\u3002<br \/>\n\u8bf8\u5982\u6b64\u7c7b\u7684\u53f2\u5b9e\u96be\u514d\u4ee4\u4eba\u4ea7\u751f\u5404\u79cd\u8054\u60f3\uff1a\u6570\u5b66\u548c\u73b0\u5b9e\u4e16\u754c\u7684\u5173\u7cfb\u5982\u679c\u4e0d\u662f\u5947\u8ff9\uff0c\u81f3\u5c11\u79f0\u5f97\u4e0a\u4ee4\u4eba\u60ca\u8be7\u3002<br \/>\n\u73b0\u5b9e\u672c\u8d28<br \/>\n\u73b0\u4ee3\u7269\u7406\u5b66\u4e0d\u65ad\u53d1\u5c55\uff0c\u5b66\u95ee\u8d8a\u6765\u8d8a\u7cbe\u6df1\uff0c\u51e1\u592b\u4fd7\u5b50\u5982\u4f60\u6211\u79bb\u5176\u4e2d\u8574\u542b\u7684\u590d\u6742\u7684\u6570\u5b66\u8d8a\u6765\u8d8a\u8fdc\uff0c\u5bf9\u6570\u5b66\u6240\u8868\u793a\u7684\u53e4\u602a\u73b0\u5b9e\u4e5f\u8d8a\u6765\u8d8a\u96be\u4ee5\u7406\u89e3\u3002<br \/>\n\u5f53\u7136\uff0c\u8fd9\u4e5f\u4e0d\u5947\u602a\u3002\u6ca1\u6709\u4efb\u4f55\u7406\u7531\u8ba4\u4e3a\u6211\u4eec\u51ed\u611f\u5b98\u63a5\u89e6\u5230\u7684\u65e5\u5e38\u73b0\u5b9e\u5c31\u662f\u5b87\u5b99\u73b0\u5b9e\u4e4b\u6839\u672c\u3002<br \/>\n\u771f\u6b63\u4ee4\u4eba\u60ca\u5947\u7684\u662f\u6570\u5b66\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u5728\u4eba\u7c7b\u611f\u5b98\u65e0\u6cd5\u4f01\u53ca\u7684\u65f6\u7a7a\u5c55\u5f00\u63a2\u7d22\u3002<br \/>\n\u63a5\u4e0b\u6765\u7684\u95ee\u9898\u662f\uff0c\u5728\u63a2\u7d22\u73b0\u5b9e\u672c\u8d28\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6570\u5b66\u7684\u8868\u8ff0\u80fd\u529b\u662f\u4e0d\u662f\u6709\u6781\u9650\uff1f<br \/>\n\u8bfa\u514b\u65af\u8bf4\uff0c20\u4e16\u7eaa\u8bde\u751f\u4e86\u4e24\u4e2a\u4f1f\u5927\u7684\u7269\u7406\u7406\u8bba\uff1a\u91cf\u5b50\u529b\u5b66\u548c\u5e7f\u4e49\u76f8\u5bf9\u8bba\u3002<br \/>\n\u4f46\u662f\uff0c\u8fd9\u4e24\u5927\u7406\u8bba\u80cc\u540e\u7684\u6570\u5b66\u67b6\u6784\u5374\u50cf\u4e00\u5bf9\u8001\u6b7b\u4e0d\u76f8\u5f80\u6765\u7684\u5919\u654c\uff0c\u8981\u628a\u4e24\u8005\u6574\u5408\u8d77\u6765\uff0c\u5176\u590d\u6742\u7a0b\u5ea6\u96be\u4ee5\u590d\u52a0\u3002\u501f\u52a9\u6570\u5b66\u7684\u9b54\u529b\uff0c\u4eba\u7c7b\u5f97\u4ee5\u63a2\u7d22\u611f\u5b98\u65e0\u6cd5\u4f01\u53ca\u7684\u65f6\u7a7a\u3002\u56fe\u4e3a\u7269\u7406\u5b66\u4e2d\u7684\u5f26\u7406\u8bba\u56fe\u793a<br \/>\n\u8bfa\u514b\u65af\u89e3\u91ca\uff1a\u201c\u6ca1\u6709\u4e00\u4e2a\u8fde\u8d2f\u3001\u8026\u5408\u7684\u6846\u67b6\u6765\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u8fd9\u4e24\u5927\u7406\u8bba\u662f\u5982\u4f55\u5728\u540c\u4e00\u4e2a\u4e16\u754c\u5e76\u5b58\u7684\uff0c\u5b83\u4eec\u662f\u600e\u6837\u8868\u8ff0\u540c\u4e00\u4e2a\u73b0\u5b9e\u672c\u4f53\u7684\u3002\u201d<br \/>\n\u8fd9\u65b9\u9762\u7684\u63a2\u7d22\u96be\u5ea6\u4e4b\u5927\u5e38\u4eba\u65e0\u6cd5\u60f3\u50cf\uff0c\u79d1\u5b66\u5b9e\u8df5\u4e2d\u6700\u57fa\u672c\u7684\u7ebd\u5e26 &#8211; \u601d\u60f3\u548c\u5b9e\u9a8c\u4e4b\u95f4\u7684\u7ebd\u5e26 &#8211; \u5728\u8fd9\u91cc\u4f3c\u4e4e\u65ad\u4e86\u3002<br \/>\n\u8fd9\u96be\u9053\u5c31\u662f\u6570\u5b66\u7684\u6781\u9650\u5417\uff1f<br \/>\n\u8bfa\u514b\u65af\u8ba4\u4e3a\uff0c\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u51e0\u4e2a\u7ed3\u8bba\uff1a<br \/>\n\u6570\u5b66\u4ecd\u80dc\u4efb\u8868\u8ff0\u6211\u4eec\u7684\u5b87\u5b99\u7684\u4efb\u52a1\uff0c\u8fd9\u662f\u6211\u4eec\u7684\u5927\u5e78\uff1b<br \/>\n\u6570\u5b66\u8868\u8ff0\u4e86\u5b87\u5b99\u7684\u4e00\u4e9b\u652f\u79bb\u7247\u65ad\u7684\u90e8\u5206\uff0c\u800c\u4e0d\u662f\u5168\u90e8\uff1b<br \/>\n\u5bf9\u5b87\u5b99\u6574\u4f53\u52a0\u4ee5\u8868\u8ff0\u8d85\u7ea7\u590d\u6742\u8d85\u7ea7\u96be\uff1b<br \/>\n\u6570\u5b66\u592a\u590d\u6742\uff0c\u6211\u4eec\u65e0\u6cd5\u627f\u53d7\uff1b<br \/>\n\u6211\u4eec\u8fd8\u4e0d\u7406\u89e3\u6570\u5b66\uff0c\u4f46\u5c06\u6765\u4f1a\u9010\u6e10\u61c2\u5b83\u30021 + 1 \u7b49\u4e8e 2\uff0c\u6709\u4ec0\u4e48\u95ee\u9898\uff1f<br \/>\n\u5de8\u5927\u5dee\u522b<br \/>\n\u4e5f\u8bb8\u6211\u4eec\u5bf9\u81ea\u5df1\u592a\u82db\u523b\uff0c\u6216\u8005\u592a\u72c2\u5984\uff0c\u60f3\u8ba9\u6570\u5b66\u5b9a\u5f8b\u548c\u7269\u7406\u5b9a\u5f8b\u624b\u62c9\u624b\u6b65\u5165\u540c\u4e00\u4e2a\u6bbf\u5802 &#8211; \u8fd9\u4ef6\u4e8b\u6bd4\u767b\u5929\u8fd8\u96be\uff0c\u5c31\u56e0\u4e3a\u5b83\u4eec\u662f\u4e24\u56de\u4e8b\u3002<br \/>\n\u7231\u56e0\u65af\u5766\u66fe\u7ecf\u8bf4\u8fc7\uff0c\u6570\u5b66\u5b9a\u5f8b\u79bb\u73b0\u5b9e\u8d8a\u8fd1\u5c31\u8d8a\u4e0d\u786e\u5b9a\uff1b\u6570\u5b66\u5b9a\u5f8b\u8d8a\u786e\u5b9a\uff0c\u79bb\u73b0\u5b9e\u5c31\u8d8a\u8fdc\u3002<br \/>\n\u8bfa\u514b\u65af\u8fd9\u4e48\u89e3\u91ca\uff1a\u6570\u5b66\u6709\u4e00\u4e2a\u7279\u70b9\uff0c\u975e\u9519\u5373\u5bf9\u3002\u6211\u7528\u6570\u5b66\u6765\u9a8c\u8bc1\u4e00\u4ef6\u4e8b\uff0c\u8c01\u90fd\u4e0d\u4f1a\u6709\u7591\u95ee\u3002\u4f46\u7269\u7406\u5c31\u4e0d\u662f\u8fd9\u6837\u3002<br \/>\n\u8fd9\u6b63\u662f\u6570\u5b66\u548c\u7269\u7406\u7684\u6700\u5927\u533a\u522b\u4e4b\u4e00\u3002<br \/>\n\u7269\u7406\u5b9a\u5f8b\u548c\u7406\u8bba\u7ecf\u5e38\u4f1a\u9519\u3002\u725b\u987f\u7684\u4e07\u6709\u5f15\u529b\u5b9a\u5f8b\u5f88\u7f8e\u3001\u5f88\u9ad8\u96c5\uff0c\u800c\u4e14\u7ecf\u5e38\u662f\u6709\u610f\u4e49\u7684\uff0c\u4f46\u5b83\u4eec\u4e0d\u4ee3\u8868\u5168\u90e8\u7684\u771f\u7406\u3002<br \/>\n\u8bfa\u514b\u65af\u9884\u8a00\uff0c\u6beb\u65e0\u7591\u95ee\uff0c\u672a\u6765\u7684\u79d1\u5b66\u5bb6\u5c06\u80fd\u591f\u8bc1\u660e\u7231\u56e0\u65af\u5766\u7684\u7406\u8bba\u4e5f\u53ea\u662f\u201c\u5927\u81f4\u6b63\u786e\u201d\u3002\u53e4\u57c3\u53ca\u8bf8\u795e\u4e2d\u6709\u4e00\u4e2a\u6570\u5b66\u5973\u795e\uff0c\u540d\u53eb\u585e\u838e\u7279\uff08Seshat\uff09<br \/>\n\u53d1\u73b0\u8fd8\u662f\u53d1\u660e?<br \/>\n\u6570\u5b66\u4ece\u54ea\u91cc\u6765\uff1f\u8fd9\u4e2a\u95ee\u9898\u5e94\u8be5\u95ee\u6570\u5b66\u5bb6\u3002<br \/>\n\u90d1\u4e50\u96bd\uff08Eugenia Cheng\uff09\u662f\u829d\u52a0\u54e5\u827a\u672f\u5b66\u9662\u9a7b\u6821\u79d1\u5b66\u5bb6\uff0c\u51fa\u751f\u5728\u82f1\u56fd\uff0c\u5251\u6865\u5927\u5b66\u6bd5\u4e1a\u3002\u903b\u8f91\u548c\u827a\u672f\u662f\u5979\u611f\u5174\u8da3\u7684\u8bdd\u9898\u4e4b\u4e00\u3002<br \/>\n\u5979\u7684\u4e13\u4e1a\u662f\u8303\u7574\u8bba\uff08Category Theory\uff09\uff0c\u4e5f\u88ab\u79f0\u4e3a\u6570\u5b66\u7684\u6570\u5b66\uff0c\u7814\u7a76\u6570\u5b66\u738b\u56fd\u4e2d\u5404\u4e2a\u4e0d\u540c\u9886\u57df\u4e4b\u95f4\u5982\u4f55\u5efa\u7acb\u8054\u7cfb\uff0c\u67b6\u6865\u94fa\u8def\u3002<br \/>\n\u5979\u8bf4\uff1a\u201c\u6211\u786e\u5b9e\u611f\u89c9\u81ea\u5df1\u662f\u5728\u53d1\u73b0\u6982\u5ff5\uff0c\u7136\u540e\u53d1\u660e\u601d\u8003\u5b83\u4eec\u7684\u65b9\u6cd5\u3002\u6211\u505a\u62bd\u8c61\u7814\u7a76\u65f6\uff0c\u5c31\u89c9\u5f97\u81ea\u5df1\u50cf\u5728\u62bd\u8c61\u7684\u4e1b\u6797\u91cc\u6f2b\u6e38\u3001\u63a2\u7d22\uff0c\u7136\u540e\u53d1\u660e\u4e00\u79cd\u65b9\u5f0f\u6765\u8c08\u8bba\u5b83\u4eec\uff0c\u5c06\u5b83\u4eec\u7406\u8bba\u5316\u3001\u7cfb\u7edf\u5316\uff0c\u8fd9\u6837\u6211\u5c31\u53ef\u4ee5\u68b3\u7406\u81ea\u5df1\u7684\u601d\u60f3\u5e76\u4e0e\u4eba\u4ea4\u6d41\u3002\u201d\u62bd\u8c61\u7684\u4e1c\u897f\u672a\u5fc5\u4e0d\u771f\u5b9e\uff01<br \/>\n\u771f\u5b9e\u600e\u4e48\u5b9a\u4e49\uff1f<br \/>\n\u786e\u5b9e\u5f88\u96be\u60f3\u8c61\u8fd8\u6709\u4ec0\u4e48\u6bd4\u8fd9\u66f4\u62bd\u8c61\u7684\u5de5\u4f5c\u3002<br \/>\n\u90a3\u4e48\uff0c\u90d1\u4e50\u96bd\u7814\u7a76\u8fc7\u7684\u90a3\u4e9b\u6570\u5b66\u5bb6\u662f\u4e0d\u662f\u8ddf\u73b0\u5b9e\u6709\u5173\u8054\u5462\uff1f<br \/>\n\u201c\u522b\u4eba\u8ddf\u6211\u63d0\u73b0\u5b9e\uff0c\u6211\u5c31\u60f3\u53cd\u95ee\uff1a\u5230\u5e95\u4ec0\u4e48\u662f\u771f\u5b9e\uff1f<br \/>\n\u6211\u4eec\u79f0\u4e4b\u4e3a\u2018\u73b0\u5b9e\u2019\u7684\u53ea\u4e0d\u8fc7\u662f\u6211\u4eec\u5047\u8bbe\u771f\u5b9e\u4e8b\u7269\u7684\u5e7b\u8c61\uff0c\u56e0\u4e3a\u6211\u4eec\u5927\u5bb6\u90fd\u503e\u5411\u4e8e\u7528\u540c\u6837\u7684\u65b9\u5f0f\u53bb\u770b\u5f85\u5b83\u3002<br \/>\n\u4eba\u4eec\u8bf4\u6570\u5b57\u4e0d\u662f\u771f\u5b9e\u7684\uff0c\u56e0\u4e3a\u4f60\u6478\u4e0d\u7740\u5b83\u4eec\u3002\u53ef\u662f\u6709\u8bb8\u591a\u4e1c\u897f\u662f\u771f\u5b9e\u7684\u73b0\u5b9e\u4f46\u6211\u6478\u4e0d\u7740\uff0c\u6bd4\u5982\u9965\u997f\u3002\u201d<br \/>\n\u5979\u8bf4\uff1a\u201c\u6240\u4ee5\u6211\u66f4\u613f\u610f\u8c08\u8bba\u5177\u4f53\u3001\u5b9e\u5728\u7684\u4e1c\u897f\uff0c\u6211\u4eec\u53ef\u4ee5\u6478\u5f97\u5230\u3001\u53ef\u4ee5\u76f4\u63a5\u4e0e\u4e4b\u4e92\u52a8\u7684\u4e1c\u897f\uff1b\u800c\u62bd\u8c61\u7684\u4e8b\u7269\uff0c\u6211\u4eec\u53ea\u80fd\u5728\u8111\u5b50\u91cc\u8ddf\u5b83\u4eec\u4e92\u52a8\u3002\u201d<br \/>\n\u6570\u5b66\u662f\u62bd\u8c61\u7684\uff0c\u4f46\u62bd\u8c61\u6982\u5ff5\u4e5f\u53ef\u4ee5\u65e0\u6bd4\u771f\u5b9e\uff0c\u8ddf\u5176\u4ed6\u771f\u5b9e\u7684\u4e8b\u7269\u4e00\u6837\u3002\u6211\u4eec\u751f\u6d3b\u4e8e\u5176\u4e2d\u7684\u65f6\u7a7a\u5230\u5904\u90fd\u85cf\u7740\u6570\u5b57<br \/>\n\u73b0\u5b9e\u662f\u4ec0\u4e48\uff1f<br \/>\n\u4e00\u65b9\u9762\uff0c\u53ef\u4ee5\u8bf4\u6570\u5b66\u5c31\u662f\u73b0\u5b9e\u3002\u6bd4\u5982\u751f\u7269\u5b66\u57fa\u4e8e\u5316\u5b66\uff0c\u800c\u5316\u5b66\u672c\u8d28\u4e0a\u670d\u4ece\u7269\u7406\u5b66\u5b9a\u5f8b\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u81ea\u7136\u8bf4\u5230\u6570\u5b57\u548c\u6570\u5b66\u4e86\u3002\u518d\u60f3\u60f3\u851a\u84dd\u7684\u5929\u7a7a\uff0c\u53ef\u4ee5\u7528\u6298\u5c04\u5149\u7684\u6ce2\u957f\u7406\u8bba\u6765\u89e3\u91ca\u2026\u2026\u90a3\u5168\u90fd\u662f\u6570\u5b57\u3002<br \/>\n\u770b\u6765\uff0c\u5728\u66f4\u6df1\u7684\u5c42\u6b21\u4e0a\uff0c\u7269\u8d28\u7684\u73b0\u5b9e\u5177\u6709\u6570\u5b66\u5c5e\u6027\u3002<br \/>\n\u4f46\u540c\u65f6\u6570\u5b66\u4f3c\u4e4e\u65e0\u6cd5\u89e3\u7b54\u4eba\u7c7b\u751f\u6d3b\u4e2d\u4e00\u4e9b\u81f3\u5173\u91cd\u8981\u7684\u4e8b\uff0c\u6bd4\u5982\u7231\u60c5\u3001\u9965\u997f\u548c\u751f\u547d\u3002<br 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