{"id":8858,"date":"2023-11-22T00:13:54","date_gmt":"2023-11-21T16:13:54","guid":{"rendered":"https:\/\/www.andrewsun.net\/panta_rhei\/?p=8858"},"modified":"2023-11-22T00:17:15","modified_gmt":"2023-11-21T16:17:15","slug":"%e5%88%86%e4%ba%ab%e4%b8%80%e4%ba%9b%e4%b9%a6%ef%bc%883%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8858","title":{"rendered":"\u5206\u4eab\u4e00\u4e9b\u4e66\uff083\uff09"},"content":{"rendered":"
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\u867d\u7136\u6211\u8fdc\u6ca1\u6709\u770b\u5b8cPaul Richard Halmos\uff081916~2006\uff09\u7684\u6240\u6709\u6570\u5b66\u6559\u6750\uff0c\u4f46\u6211\u786e\u5b9e\u770b\u4e86\u51e0\u672c\u8bdd\u9898\u6bd4\u8f83\u5927\u4f17\u5316\u7684\u2014\u2014\u56e0\u800c\u540c\u7c7b\u6559\u6750\u4e5f\u6bd4\u8f83\u591a\u7684\u2014\u2014\u4ece\u800c\u5728\u6bd4\u8f83\u7684\u57fa\u7840\u4e0a\u53ef\u4ee5\u652f\u6491\u6211\u5bf9Halmos\u7684\u504f\u7231\u3002\u6240\u4ee5\u6211\u4ecd\u7136\u80fd\u591f\u76f4\u63a5\u8bf4\uff0c\u6211\u7279\u522b\u559c\u6b22Halmos\u7684\u6570\u5b66\u4e66\u3002<\/p>\n\n\n\n

\u6211\u770b\u8fc7\u4ed6\u5199\u7684\u96c6\u5408\u8bba\u3001\u6d4b\u5ea6\u8bba\u3001\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\u548c\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u3002\u8fd9\u4e9b\u4e66\u9664\u4e86\u6d4b\u5ea6\u8bba\u4e4b\u5916\u90fd\u662f\u5f88\u8584\u7684\u3002<\/p>\n\n\n\n

\u81f3\u5c11\u6211\u8fd8\u4e0d\u5b8c\u6574\u5730\u770b\u8fc7\u4ed6\u7684\u81ea\u4f20\u2014\u2014I want to be a mathematician<\/em>\u3002\u8fd9\u4e2a\u81ea\u4f20\u7684\u5f00\u5934\u5c31\u5148\u660e\u786e\u5730\u53cd\u9192\u4e86\u4f5c\u8005\u7684\u4eba\u683c\u7279\u8d28\uff0c\u8bfb\u6765\u53d1\u73b0\u4e0e\u6211\u672c\u4eba\u5341\u5206\u8d34\u8fd1\uff0c\u8fd9\u8db3\u4ee5\u652f\u6301\u6211\u504f\u7231\u4ed6\u7684\u8457\u4f5c\u3002\u8fd9\u6bb5\u8bdd\u5bf9\u5176\u4ed6\u8bfb\u8005\u6765\u8bf4\u4e5f\u53ef\u4ee5\u8bf4\u660eHalmos\u5f88\u53ef\u80fd\u662f\u603b\u80fd\u5199\u51fa\u597d\u6559\u6750\u7684\u6570\u5b66\u5bb6\u3002<\/p>\n\n\n\n

\n

I like words more than numbers, and I always did.<\/p>\n\n\n\n

Then why, you might well ask, am I a mathematician? I don’t know. I can see some of the reasons in the story of my life, and I can see that chance played a role at least as big as choice; I’ll try to tell about all that as we go along. I do know that I wasn’t always sure what I wanted to be.<\/p>\n\n\n\n

The sentence I began with explains the way I feel about a lot of things, and how I got that way. It implies, for instance, or in any event I men for it to imply that in mathematics I like the conceptual more than the computational. To me the definition of a group is far clearer and more important and more beautiful than the Cauchy integral formula. Is it unfair to compare a concept with a fact? Very well, to me the infinite differentiability of a once differentiable complex function is far superior in beauty and depth to the celebrated Campbell–Baker–Hausdorff formula about non-commutative exponentiation.<\/p>\n\n\n\n

The beginning sentence includes also the statement that I like to understand mathematics, and to clarify it for myself and for the world, more even than to discover it. The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me—both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstacy and euphoria of released tension. At the same time, discovering a new truth, similar in subjective pleasure to understanding an old one, is in one way quite different. The difference is the pride, the feeling of victory, the almost malicious satisfaction that comes from being first. “First” implies that someone is second; to want to be first is asking to be “graded on the curve”. I seem to be saying, almost, that clarifying old mathematics is more moral than finding new, and that’s obviously silly—but let me say instead that insight is better without an accompanying gloat than with. Or am I just saying that I am better at polishing than at hunting and I like more what I can do better?<\/p>\nP. Halmos (1985), I want to be a mathematician, Springer<\/cite><\/blockquote>\n\n\n\n

\u4ece\u6211\u8bfb\u8fc7\u7684\u6240\u6709Halmos\u8457\u4f5c\u6765\u770b\uff0c\u4ed6\u4e0d\u6127\u4e8e\u81ea\u4f20\u7684\u7b2c\u4e00\u53e5\u8bdd\u3002\u6211\u5e38\u5e38\u80fd\u6298\u670d\u4e8e\u4ed6\u884c\u6587\u7684\u6587\u7b14\u3002\u770b\u4ed6\u7684\u4e66\uff0c\u7ecf\u5e38\u53ea\u662f\u5ed6\u5ed6\u6570\u884c\u7b80\u5355\u7684\u53e5\u5b50\uff0c\u67d0\u79cd\u601d\u60f3\u5c31\u80fd\u94bb\u8fdb\u4f60\u7684\u8111\u888b\u91cc\u3002<\/p>\n\n\n\n

\u56e0\u6b64\u6211\u63a8\u8350\uff0c\u4e0a\u8ff0\u6211\u6240\u8bfb\u8fc7\u7684\u4ed6\u7684\u4e66\uff0c\u4e3a\u76f8\u5173\u8bdd\u9898\u7684\u4e66\u7684\u9996\u9009\u3001\u5fc5\u8bfb\u3002<\/p>\n\n\n\n

\u6211\u8bfb\u5f88\u591a\u6570\u5b66\u4e66\u90fd\u4e0d\u505a\u4e60\u9898\uff0c\u6216\u8005\u6709\u9650\u5730\u9009\u505a\u4e00\u4e9b\u4e60\u9898\u3002\u4f46\u662fHalmos\u7684\u4e66\u4e0d\u5149\u662f\u5fc5\u987b\u5b8c\u6210\u6bcf\u9053\u4e60\u9898\uff0c\u800c\u4e14\u4ed6\u6b63\u6587\u4e2d\u7684\u6bcf\u53e5\u65ad\u8a00\uff0c\u51e1\u662f\u4f60\u4e0d\u592a\u6e05\u695a\u7406\u7531\u7684\uff0c\u90fd\u503c\u5f97\u7b80\u5355\u8bc1\u660e\u4e00\u904d\u3002\u6240\u5e78\u7684\u662f\u4ed6\u7684\u4e66\u90fd\u5f88\u8584\u3002<\/p>\n\n\n\n

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\u6211\u7ecf\u5e38\u5728\u5404\u7c7b\u65e7\u4e66\u5e02\u573a\u6dd8\u4e00\u4e9b\u539f\u7248\u4e66\u4f5c\u4e3a\u6536\u85cf\u3002<\/figcaption><\/figure>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

\u867d\u7136\u6211\u8fdc\u6ca1\u6709\u770b\u5b8cPaul Richard Halmos\uff081916~2006\uff09\u7684\u6240\u6709\u6570\u5b66\u6559\u6750\uff0c\u4f46\u6211\u786e\u5b9e\u770b\u4e86\u51e0\u672c\u8bdd\u9898\u6bd4\u8f83\u5927\u4f17\u5316\u7684\u2014\u2014\u56e0\u800c\u540c\u7c7b\u6559\u6750\u4e5f\u6bd4\u8f83\u591a\u7684\u2014\u2014\u4ece\u800c\u5728\u6bd4\u8f83\u7684\u57fa\u7840\u4e0a\u53ef\u4ee5\u652f\u6491\u6211\u5bf9Halmos\u7684\u504f\u7231\u3002\u6240\u4ee5\u6211\u4ecd\u7136\u80fd\u591f\u76f4\u63a5\u8bf4\uff0c\u6211\u7279\u522b\u559c\u6b22Halmos\u7684\u6570\u5b66\u4e66\u3002 \u6211\u770b\u8fc7\u4ed6\u5199\u7684\u96c6\u5408\u8bba\u3001\u6d4b\u5ea6\u8bba\u3001\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\u548c\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u3002\u8fd9\u4e9b\u4e66\u9664\u4e86\u6d4b\u5ea6\u8bba\u4e4b\u5916\u90fd\u662f\u5f88\u8584\u7684\u3002 \u81f3\u5c11\u6211\u8fd8\u4e0d\u5b8c\u6574\u5730\u770b\u8fc7\u4ed6\u7684\u81ea\u4f20\u2014\u2014I want to be a mathematician\u3002\u8fd9\u4e2a\u81ea\u4f20\u7684\u5f00\u5934\u5c31\u5148\u660e\u786e\u5730\u53cd\u9192\u4e86\u4f5c\u8005\u7684\u4eba\u683c\u7279\u8d28\uff0c\u8bfb\u6765\u53d1\u73b0\u4e0e\u6211\u672c\u4eba\u5341\u5206\u8d34\u8fd1\uff0c\u8fd9\u8db3\u4ee5\u652f\u6301\u6211\u504f\u7231\u4ed6\u7684\u8457\u4f5c\u3002\u8fd9\u6bb5\u8bdd\u5bf9\u5176\u4ed6\u8bfb\u8005\u6765\u8bf4\u4e5f\u53ef\u4ee5\u8bf4\u660eHalmos\u5f88\u53ef\u80fd\u662f\u603b\u80fd\u5199\u51fa\u597d\u6559\u6750\u7684\u6570\u5b66\u5bb6\u3002 I like words more than numbers, and I always did. Then why, you might well ask, am I a mathematician? I don’t know. I can see some of the reasons in the story of my life, and I can see that chance played a role […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[279,303,275],"tags":[],"class_list":["post-8858","post","type-post","status-publish","format-standard","hentry","category-279","category-rheo","category-275"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-2iS","jetpack-related-posts":[{"id":8728,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8728","url_meta":{"origin":8858,"position":0},"title":"\u5173\u4e8e\u7ebf\u6027\u4ee3\u6570\u6559\u6750","author":"Andrew","date":"2022\u5e7412\u67087\u65e5","format":false,"excerpt":"\u6211\u8c03\u67e5\u4e86\u90e8\u5206\u4ee5\u77e9\u9635\u548c\u884c\u5217\u5f0f\u8fd0\u7b97\u4e3a\u4e3b\u8981\u5185\u5bb9\u7684\u7ebf\u6027\u4ee3\u6570\u6559\u6750\uff0c\u5728\u8fd9\u91cc\u603b\u7ed3\u4e00\u4e0b\u3002\u505a\u8fd9\u4ef6\u4e8b\u7684\u521d\u59cb\u52a8\u673a\u6765\u7f18\u662f\u77e5\u4e4e\u4e0a\u5f88\u591a\u5b66\u751f\u5bf9\u8fd9\u7c7b\u7ebf\u6027\u4ee3\u6570\u6559\u6750\u7684\u6028\u8a00\u3002\u4f46\u672c\u6587\u8fd8\u662f\u4e3b\u9898\u6bd4\u8f83\u51cc\u6563\u7684\uff0c\u672a\u5fc5\u662f\u5bf9\u8fd9\u4e9b\u6028\u8a00\u7684\u56de\u5e94\u3002 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analysis\uff09\u3002\u8fd9\u4e5f\u662f\u6211\u8c03\u67e5\u7684\u4e66\u7684\u8303\u56f4\u9650\u5b9a\u3002\u57fa\u4e8e\u6211\u5f88\u6709\u9650\u7684\u8c03\u67e5\u7684\u7ed3\u679c\u9996\u5148\u53ef\u4ee5\u660e\u786e\u7684\u81f3\u5c11\u662f\uff0c\u5728\u4e8c\u5341\u4e16\u7eaa\u7684\u5927\u5b66\u672c\u79d1\uff0c\u4eba\u4eec\u672c\u6765\u5c31\u662f\u6253\u7b97\u53ea\u6559\u7ed9\u5b66\u751f\u77e9\u9635\u7684\u90a3\u4e9b\u201c\u8fd0\u7b97\u201d\uff0c\u800c\u4e0d\u5173\u5fc3\u62bd\u8c61\u4ee3\u6570\u610f\u4e49\u7684\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u3002\u6211\u4eec\u5173\u5fc3\u7684\u662f\uff0c\u8fd9\u4e9b\u6559\u6750\u7684\u4f5c\u8005\u662f\u5426\u89c9\u5f97\u201c\u77e9\u9635\u4ee3\u6570\u201d\u548c\u201c\u77e9\u9635\u5206\u6790\u201d\u5c31\u662f\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u7684\u540c\u4e49\u8bcd\uff1b\u540c\u65f6\uff0c\u4e3a\u4f55\u8fd9\u4e9b\u4e13\u95e8\u5173\u4e8e\u77e9\u9635\u7684\u3001\u72ec\u7acb\u4e8e\u5411\u91cf\u7a7a\u95f4\u7684\u62bd\u8c61\u4ee3\u6570\u601d\u60f3\u7684\u8fd0\u7b97\u6cd5\u5219\u77e5\u8bc6\u90a3\u4e48\u91cd\u8981\u3002 [1] Maxime B\u00f4cher (1915), Introduction to higher algebra, The MacMillan Company \u8fd9\u672c\u4e66\u7684\u5185\u5bb9\u7f16\u6392\uff0c\u9c9c\u660e\u5730\u4f53\u73b0\u4e86\u201c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7406\u8bba\u201d\u7684\u7279\u8272\u3002\u56e0\u6b64\uff0c\u8be5\u4e66\u6807\u9898\u7528algebra\u4e00\u8bcd\uff0c\u4ee3\u8868\u4e86\u4e0a\u4e00\u8282\u6240\u8ff0\u201c\u4ee3\u6570\u5c31\u662f\u89e3\u65b9\u7a0b\u201d\u7684\u201c\u524d20\u4e16\u7eaa\u7406\u89e3\u201d\u3002 [2] Roger Horn, Charles Johnson (1985), Matrix analysis,\u2026","rel":"","context":"In "\u4e2a\u4eba\u6587\u96c6"","block_context":{"text":"\u4e2a\u4eba\u6587\u96c6","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/featured"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":8305,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8305","url_meta":{"origin":8858,"position":1},"title":"\u6211\u7684\u201c\u54c1\u5473\u201d","author":"Andrew","date":"2021\u5e7412\u670820\u65e5","format":false,"excerpt":"\u5728\u957f\u671f\u7684\u5b66\u4e60\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5bf9\u4e00\u4ef6\u4e8b\u7684\u786e\u8ba4\u4e0d\u65ad\u5730\u5f3a\u5316\uff0c\u90a3\u5c31\u662f\u6211\u8ddfClifford Truesdell\u548cWalter Noll\u6709\u7740\u76f8\u540c\u7684\u201c\u54c1\u5473\u201d\u3002 \u6211\u8d5e\u540c\u5e76\u6b23\u8d4f\u5df2\u6709\u7269\u7406\u7406\u8bba\u7684\u516c\u7406\u5316\u52aa\u529b\u3002\u6211\u8ba4\u4e3a\u8fd9\u662f\u4e00\u4e2a\u6709\u610f\u4e49\u7684\u7814\u7a76\u9886\u57df\u3002\u6211\u5e0c\u671b\u6709\u4e00\u5929\u6211\u4eec\u7684\u672c\u79d1\u6559\u6750\u80fd\u66ff\u6362\u6210\u8fd9\u4e00\u9886\u57df\u7684\u7814\u7a76\u6210\u679c\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6211\u5728\u6d41\u53d8\u5b66\u8bb2\u4e49\u4e2d\u5c31\u5c1d\u8bd5\u9075\u5faa\u4e00\u4e9b\u5df2\u7ecf\u5efa\u7acb\u7684\u516c\u7406\u5316\u5f15\u5165\u3002 \u6211\u7b2c\u4e00\u6b21\u770b\u5230Water Noll\u7684neo-classical space-time\u6784\u5efa\u662f\u5728\u4e0b\u9762\u8fd9\u4e2a\u5730\u65b9\uff1a W. Noll (1966), The Foundations of Mechanics, In: G. Grioli, C. Truesdell (eds), Non-linear Continuum Theories, Springer-Verlag \u8fd9\u4e66\u7684\u5370\u5237\u662f\u7528\u6253\u5b57\u673a\u7684\uff0c\u6709\u5f88\u82b1\u4f53notation\u662f\u540e\u9762\u518d\u7528\u624b\u5199\u8865\u4e0a\u7684\u3002\u6211\u4e0d\u77e5\u9053\u8fd9\u4e9b\u7b14\u8ff9\u662f\u4e0d\u662fNoll\u672c\u4eba\u7684\u3002\u4f46\u4ece\u6392\u7248\u4e0a\u53ef\u4ee5\u8bf4\u662f\u975e\u5e38\u4e0d\u53cb\u597d\u3002\u6211\u5bf9\u65f6\u7a7a\u6784\u5efa\u7684\u5185\u5bb9\u4e5f\u5b8c\u5168\u964c\u751f\u3002\u4f46\u662f\u4e0d\u77e5\u9053\u4e3a\u4ec0\u4e48\uff0c\u6211\u4e00\u5f00\u59cb\u5c31\u611f\u53d7\u5230\u8fd9\u4e2a\u8bdd\u9898\u662f\u5f88\u53ef\u80fd\u611f\u5174\u8da3\u7684\uff0c\u9b3c\u4f7f\u795e\u5dee\u5730\u628a\u5b83\u786c\u5543\u5b8c\u4e86\uff0c\u679c\u7136\u611f\u53d7\u5230\u4e86\u65e0\u6cd5\u7528\u8a00\u8bed\u5f62\u5bb9\u7684\u7f8e\u5b66\u4eab\u53d7\u3002\u9664\u4e86\u7f8e\u7684\u4eab\u53d7\u4e4b\u5916\uff0c\u65f6\u7a7a\u6784\u5efa\u7684\u4e25\u5bc6\u6027\uff0c\u53ef\u4ee5\u8ba9\u540e\u7eed\u7684\u5f88\u591a\u8fde\u7eed\u4ecb\u8d28\u529b\u5b66\u5f53\u4e0b\u6559\u6750\u4e2d\u5bb9\u6613\u8ba9\u5b66\u751f\u641e\u6df7\u7684\u8bdd\u9898\u53d8\u5f97\u6839\u672c\u4e0d\u53ef\u80fd\u641e\u6df7\u3002 \u65e9\u671f\uff0c\u6211\u8865\u4e60\u6570\u5b66\u7684\u552f\u4e00\u53c2\u8003\u5c31\u662f S. Hassani (1999), Mathematical Physics: A Modern Introduction to Its Foundations, Springer \u6211\u77e5\u9053\u8fd9\u672c\u4e66\uff0c\u4e5f\u662f\u4ece\u65e9\u5e74\u6d41\u4f20\u7684\u201cFang\u7684\u4e66\u5355\u201d\u4e2d\u77e5\u9053\u7684\u3002\u8fd9\u672c\u4e66\u5bf9Fang\u800c\u8a00\u662f\u65b0\u4e66\uff0c\u4ed6\u53ea\u662f\u8bf4\u597d\u50cf\u8fd8\u53ef\u4ee5\u3002\u73b0\u5728\u56de\u770b\uff0c\u8fd9\u672c\u4e66\u7ed9\u4e86\u6211\u4e24\u4e2a\u5f71\u54cd\uff1a1\uff09\u662f\u8fd9\u672c\u4e66\u5185\u5bb9\u5f88\u5168\uff0c\u4e14\u5728\u4e00\u672c\u4e66\u4e4b\u5185Notation\u548c\u6982\u5ff5\u4f53\u7cfb\u662f\u7edf\u4e00\u7684\u3002\u8ba9\u6211\u5728\u521a\u5b66\u6570\u5b66\u5c31\u611f\u53d7\u5230\u4e86\u4e16\u4e0a\u53ea\u6709\u4e00\u4e2a\u6570\u5b66\uff0c\u800c\u4e0d\u662f\u5272\u88c2\u770b\u4f3c\u65e0\u5173\u7684\u4e0d\u540c\u8bfe\u7a0b\u3002\u540e\u8005\u53ef\u80fd\u662f\u5f88\u591a\u6570\u5b66\u6216\u7269\u7406\u4e13\u4e1a\u7684\u672c\u79d1\u751f\u5728\u73b0\u5728\u7684\u6559\u80b2\u4f53\u7cfb\u4e2d\u83b7\u5f97\u7684\u5370\u8c61\u30022\uff09\u7531\u4e8e\u8981\u505a\u52301\uff09\u8fd9\u4e66\u4e0d\u53ef\u907f\u514d\u5730\u8981\u5982\u5176\u526f\u6807\u9898\u6240\u8a00\u6bd4\u8f83\u73b0\u4ee3\u3002\u4f8b\u5982\u6240\u6709\u6784\u5efa\u90fd\u4ece\u96c6\u5408\u51fa\u53d1\uff0ccoordinate-free\u7684\u7ebf\u6027\u4ee3\u6570\uff0c\u628a\u5f20\u91cf\u5f15\u5165\u4e3a\u591a\u7ebf\u6027\u5f62\u5f0f\u7b49\u3002\u4e4b\u524d\u4e00\u5ea6\u6709\u4e00\u70b9\u8ba9\u6211\u5f88\u8be7\u5f02\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u6211\u770b\u8fc7\u4e00\u4e9b\u4e66\uff0c\u5728\u77e5\u4e4e\u4e0a\u5c31\u4ea7\u751f\u4e86\u4e00\u4e2a\u611f\u89c9\u5c31\u662f\u6211\u770b\u5f97\u61c2\u90a3\u4e9b\u79d1\u73ed\u51fa\u8eab\u7684\u4eba\u8bf4\u4ec0\u4e48\uff0c\u8fd8\u597d\u50cf\u80fd\u5bf9\u4e00\u4e9b\u8d85\u8d8a\u672c\u79d1\u7684\u8bdd\u9898\u8fdb\u884c\u8bc4\u8bba\u3002\u5982\u679c\u4e0d\u5b8c\u5168\u662f\u9519\u89c9\uff0c\u90a3\u53ef\u80fd\u662f\u56e0\u4e3a\u6211\u4e00\u5f00\u59cb\u5c31\u63a5\u89e6\u73b0\u4ee3\u7684\u6570\u5b66\u6587\u672c\u3002 \u540e\u6765\u6211\u4eceHassani\u7684\u4e66\u540e\u53c2\u8003\u4e66\u76ee\u53c8\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u77e5\u9053\u4e86\u5f88\u591a\u4e66\u548c\u4f5c\u8005\u3002\u4f8b\u5982Halmos\u548c Hoffman and Kunze\u7684\u7ebf\u6027\u4ee3\u6570\u3002\u540e\u8005\u6210\u4e3a\u4e86\u6211\u7b2c\u4e00\u672c\u4e3b\u8981\u5b66\u4e60\u7684\u4e66\u3002Hoffman and Kunze\u5176\u5b9e\u662f\u4e00\u672c\u5904\u5904\u900f\u7740\u62bd\u8c61\u4ee3\u6570\u7684\u666e\u901a\u7ebf\u6027\u4ee3\u6570\uff0ccoordinate-free\u7684\u8272\u5f69\u66f4\u6d53\u4e86\u3002\u4e00\u4e2a\u8ba9\u6211\u66fe\u7ecf\u56f0\u60d1\uff0c\u5374\u53cd\u800c\u662f\u6781\u5176\u5177\u6709\u542f\u53d1\u6027\u7684\u7279\u70b9\u662f\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u7ebf\u6027\u4ee3\u6570\uff0c\u5168\u4e66\u6ca1\u6709\u201ccross product\u201d\u8fd9\u4e2a\u8bcd\u3002\u7ffb\u904d\u5168\u4e66\u53ea\u6709tensor product\u6ca1\u6709tensor\uff0c\u8fdb\u884ctensor product\u7684\u5bf9\u8c61\u53eb\u591a\u7ebf\u6027\u5f62\u5f0f\u3002\u73b0\u5728\u56de\u60f3\u8d77\u6765\uff0c\u6211\u6562\u8bf4\u5f88\u591a\u529b\u5b66\u8bfe\u7a0b\u7684\u8bb2\u8ff0\u8005\u4e0e\u8bb8\u672a\u5fc5\u80fd\u89e3\u7b54\u5b66\u751f\u5728\u4e00\u5207\u5176\u4ed6\u4e0d\u540c\u5730\u65b9\u770b\u5230\u7528\u201c\u5f20\u91cf\u201d\u4e8c\u5b57\u79f0\u547c\u7684\u5bf9\u8c61\u662f\u4e0d\u662f\u540c\u4e00\u4e2a\u4e1c\u897f\uff0c\u6709\u4ec0\u4e48\u8054\u7cfb\u3002 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\u4e94\u3001 \u5f88\u591a\u9762\u5411\u7269\u7406\u7cfb\u5b66\u751f\u7684\u201c\u6570\u5b66\u7269\u7406\u65b9\u6cd5\u201d\u8bfe\u672c\uff0c\u662fanti-mathematics\u7684\u3002\u5b83\u4eec\u4e0d\u8fc7\u662f\u4e00\u4e2a\u7a0d\u5fae\u6269\u5927\u4e86\u7684\u201c\u6570\u5b66\u51c6\u5907\u201d\uff0c\u662f\u201c\u4f1a\u7b97\u5c31\u884c\u201d\u90aa\u6559\u4ea7\u7269\u3002\u56e0\u6b64\u6211\u4e0d\u5efa\u8bae\u7eaf\u6d01\u7684\u5316\u5b66\u4e13\u4e1a\u5b66\u751f\u627e\u8fd9\u7c7b\u4e66\u6765\u5b66\u4e60\u6570\u5b66\u3002\u4f46\u9664\u4e86\u4e00\u672c\uff1a Sadri Hassani (2003), Mathematical Physics: A Modern Introduction to Its Foundations, 2nd ed., Springer 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\u6211\u57282009\u5e74\u7684\u90a3\u7bc7\u6587\u7ae0\u4e2d\u66fe\u8868\u793a\u5bf9\u4e2d\u5ddd\u81ea\u8ff0\u7684\u642d\u5efa\u4eea\u5668\u7684\u7ecf\u5386\u7684\u656c\u4f69\uff08\u8be6\u89c1\u539f\u6587\uff09\u3002\u73b0\u5728\u56de\u8fc7\u6765\u770b\uff0c\u6211\u672c\u4eba\u5728\u79d1\u5b66\u7814\u7a76\u4e0a\u7684\u6210\u957f\u4e5f\u8d70\u5411\u4e86\u4e00\u4e2a\u5b9e\u9a8c\u6d4b\u91cf\u8005\u3002\u5728\u539f\u7530\u91cd\u6cbb\u7684\u56de\u5fc6\u4e2d\u4e5f\u8bb2\u5230\uff0c\u8fd9\u786e\u5b9e\u662f\u4e2d\u5ddd\u9e64\u592a\u90ce\u7684\u5178\u578b\u7814\u7a76\u98ce\u683c\uff0c\u5e76\u63d0\u4f9b\u4e86\u66f4\u591a\u7684\uff0c\u5728\u4e2d\u5ddd\u7684\u6307\u5bfc\u4e0b\u653b\u8bfb\u7814\u7a76\u751f\u671f\u95f4\u7684\u4eea\u5668\u642d\u5efa\u7ecf\u5386\u3002\u65e5\u672c\u5728\u6218\u540e\u91cd\u5efa\u65f6\u671f\u7269\u8d44\u5f88\u7d27\u7f3a\uff0c\u5404\u65b9\u9762\u90fd\u5341\u5206\u56f0\u96be\uff0c\u5f88\u50cf\u6211\u56fd\u5728\u6587\u9769\u7ed3\u675f\u4e4b\u540e\u7684\u60c5\u5f62\u3002\u4e0d\u7ba1\u4f60\u8981\u505a\u4ec0\u4e48\u7814\u7a76\uff0c\u4f60\u90fd\u4e0d\u5f97\u4e0d\u719f\u77e5\u5404\u7c7b\u4e94\u91d1\u7535\u8def\u77e5\u8bc6\u548c\u5b9e\u8df5\u6280\u80fd\u3002\u6211\u57282009\u5e74\u7684\u6587\u7ae0\u4e2d\u5f3a\u8c03\u4e86\u81ea\u5df1\u642d\u5efa\u4eea\u5668\u7684\u91cd\u8981\u6027\u3002\u4e8b\u5b9e\u4e0a\uff0c\u4eca\u5929\u7684\u5316\u5b66\u4e13\u4e1a\u5b66\u751f\u5728\u8fd9\u65b9\u9762\u57fa\u672c\u77e5\u8bc6\u90fd\u662f\u7a7a\u767d\u7684\u3002\u4e0d\u77e5\u9053\u4e3a\u4ec0\u4e48\uff0c\u793e\u4f1a\u751f\u4ea7\u7684\u5206\u5de5\u7ec6\u5316\uff0c\u4e0d\u4ec5\u8ba9\u4e13\u4e1a\u5206\u5f97\u66f4\u7ec6\uff0c\u8fd8\u8ba9\u201c\u516c\u6c11\u5e38\u8bc6\u201d\u4e5f\u53d1\u751f\u4e86\u538b\u7f29\u3002\u66fe\u7ecf\u662f\u4e0d\u5206\u4e13\u4e1a\u7684\u5e38\u8bc6\u6216\u5fc5\u5907\u6280\u80fd\uff0c\u4e5f\u56e0\u5206\u5de5\u7ec6\u5316\u800c\u53d8\u6210\u4e86\u201c\u7531\u4e13\u4e1a\u4eba\u58eb\u5305\u529e\u201d\u7684\u4e8b\u60c5\u3002\u6bcf\u4e2a\u4e13\u4e1a\u7684\u4eba\u90fd\u56e0\u5206\u5de5\u7ec6\u5316\u800c\u53d8\u5f97\u66f4\u52a0\u201c\u5de5\u5177\u5316\u201d\uff0c\u540c\u65f6\u4e5f\u56e0\u5e38\u8bc6\u6b20\u7f3a\u800c\u53d8\u5f97\u9ebb\u6728\uff0c\u4e0d\u540c\u4e13\u4e1a\u7684\u4eba\u65e0\u6cd5\u6c9f\u901a\u3002\u8fd9\u662f\u4e0d\u662f\u8d44\u672c\u4e3b\u4e49\u7684\u9634\u8c0b\u5462\uff1f \u201c\u4e45\u5c71\u6c0f\u306e\u65b9\u6cd5\u201d \u6211\u57282009\u5e74\u7684\u90a3\u7bc7\u6587\u7ae0\u4e2d\u66fe\u5f15\u7528\uff0c\u4e2d\u5ddd\u63d0\u5230\u8fc7\u4e00\u4e2a\u53eb\u201c\u4e45\u5c71\u591a\u7f8e\u7537\u201d\u53d1\u8868\u5728\u201c\u79d1\u5b66\u201d\u6742\u5fd7\u4e0a\u7684\u4e00\u4e2a\u6d4b\u91cf\u7c98\u5f39\u6027\u7684\u65b9\u6cd5\u3002\u8fd9\u4e2a\u300a\u79d1\u5b66\u300b\u6742\u5fd7\u4e0d\u662f\u7f8e\u56fd\u7684\u90a3\u4e2a\uff0c\u800c\u662f\u65e5\u672c\u7684\u4e00\u4e2a\u671f\u520a\u300aKagaku\u300b\u3002\u73b0\u5728\u6211\u53ef\u4ee5\u544a\u8bc9\u5f53\u5e74\u9976\u6709\u5174\u8da3\u7684\u90a3\u4e2a\u81ea\u5df1\uff1a\u4e2d\u5ddd\u5728\u53e6\u4e00\u4e2a\u8bba\u6587\u91cc\u63d0\u5230\u4e86\u8fd9\u4e2a\u201c\u4e45\u5c71\u6c0f\u306e\u65b9\u6cd5\u201d\uff0c\u4ee5\u53ca\u5176\u4ed6\u7684\u51e0\u79cd\u6d4b\u91cf\u7c98\u5f39\u6027\u7684\u88c5\u7f6e\u8bbe\u8ba1\u3002\u5176\u4e2d\u4e45\u5c71\u7684\u4eea\u5668\u793a\u610f\u56fe\u5982\u4e0b\uff1a \u4e45\u5c71\u591a\u7f8e\u7537\u7684\u88c5\u7f6e \u5b9e\u9645\u4e0a\u5f88\u63a5\u8fd1\u4eca\u5929\u7684\u5e94\u529b\u63a7\u5236\u578b\u6d41\u53d8\u4eea\u3002 \u540c\u6837\u5730\uff0c2009\u5e74\u6211\u5f15\u7528\u7684\uff0c\u4e2d\u5ddd\u5728\u8bfb\u7814\u65f6\uff0c\u653e\u5f03\u7814\u7a76\u201c\u6ce1\u201d\u7684\u95ee\u9898\uff0c\u9009\u62e9\u7814\u7a76\u201c\u62c9\u4e1d\u201d\u7684\u95ee\u9898\uff0c\u6700\u540e\u6709\u6210\u679c\u5417\uff1f\u4eca\u5929\u6211\u4e5f\u53ef\u4ee5\u544a\u8bc9\u5f53\u5e74\u90a3\u4e2a\u81ea\u5df1\uff1a\u4ed6\u5728BCSJ\u4e0a\u53d1\u8868\u4e86\u4e24\u7bc7\u7cfb\u5217\u8bba\u6587\uff1aSpinnability of Liquid. A visco-elastic State. I & II\u3002\u5728\u300a\u6d41\u52a8\u7684\u56fa\u4f53\u300b\u4e2d\u663e\u793a\u7684\u201c\u6d4b\u91cf\u62c9\u4e1d\u6027\u7684\u88c5\u7f6e\u201d\u63d2\u56fe\uff0c\u5c31\u5728\u8fd9\u4e2a\u8bba\u6587I\u4e2d\u7684Fig. 3\u3002\u800c\u5bf9\u4e45\u5c71\u591a\u7f8e\u7537\u88c5\u7f6e\u7684\u501f\u9274\uff0c\u5219\u53ef\u89c1\u4e8e\u8bba\u6587II\u4e2d\u7684Fig. 2\u3002 \u9c81\u8fc5 \u4e2d\u5ddd\u5728\u4e00\u7bc7\u6587\u7ae0\u4e2d\u63d0\u5230\u8fc7\u9c81\u8fc5\u3002 1960\u5e74\u4ee3\u7684\u65e5\u672c\u662f\u6bd4\u8f83\u52a8\u8361\u7684\u3002\u4e2d\u5ddd\u57281967\u5e743\u6708\u53f7\u7684\u300a\u9ad8\u5206\u5b50\u300b\u6742\u5fd7\u4e2d\u53d1\u8868\u4e86\u4e00\u7bc7\u9898\u4e3a\u300a\u5e94\u8be5\u600e\u4e48\u505a\uff08\u4f55\u3092\u306a\u3059\u3079\u304d\u304b\uff09\u300b\u7684\u6587\u7ae0\uff0c\u6292\u53d1\u4e86\u5bf9\u65e5\u672c\u73b0\u72b6\u7684\u4e0d\u6ee1\u3002\u4ed6\u5c31\u662f\u5728\u8fd9\u7bc7\u8bba\u6587\u4e2d\u63d0\u5230\u4e86\u9c81\u8fc5\u3002\u800c\u57281970\u5e74\u53c8\u5728\u540c\u4e00\u672c\u6742\u5fd7\u53d1\u8868\u4e86\u8ddf\u653f\u6cbb\u76f8\u5173\u7684\u6587\u7ae0\u300a\u65b0\u7684\u975e\u7406\u6027\u4e3b\u4e49\u300b\uff0c\u66f4\u52a0\u76f4\u63a5\u5730\u6279\u8bc4\u4e86\u5f53\u65f6\u60f3\u8981\u6d17\u8131\u65e5\u672c\u5728\u4e8c\u6218\u4e2d\u7684\u7f6a\u884c\u7684\u601d\u60f3\u3002\u8fd9\u79cd\u601d\u60f3\u53ef\u80fd\u662f1960\u5e74\u4ee3\u5bf9\u7f8e\u56fd\u4fb5\u72af\u65e5\u672c\u6743\u76ca\u548c\u5c0a\u4e25\u7684\u6297\u8bae\u7684\u8fc7\u5ea6\u53cd\u5e94\u3002\u8fd9\u7bc7\u8bba\u6587\u6700\u540e\u4e00\u53e5\u5f88\u7cbe\u5f69\uff1a \u610f\u8bc6\u5f62\u6001\u7684\u201c\u9524\u5b50\u201d\u53ef\u4ee5\u81ea\u7531\u6325\u821e\u610f\u8bc6\u5f62\u6001\uff0c\u4f46\u4e0d\u80fd\u81ea\u7531\u6325\u821e\u73b0\u5b9e\u3002 \u6211\u4ece\u4e2d\u5ddd\u8fd9\u4e9b\u96f6\u661f\u7684\u653f\u6cbb\u5c0f\u6587\u7ae0\uff0c\u4f3c\u4e4e\u770b\u5230\u4e86\u9c81\u8fc5\u7b14\u4e0b\u7684\u5bf9\u4e2d\u56fd\u53cb\u597d\u7684\u90a3\u4e2a\u85e4\u91ce\u5148\u751f\u3002 \u5728\u7985\u5bfa\u7684\u95e8\u524d\u5f98\u5f8a 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