{"id":80365,"date":"2026-03-04T22:41:09","date_gmt":"2026-03-04T14:41:09","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/80365.html"},"modified":"2026-03-04T22:41:09","modified_gmt":"2026-03-04T14:41:09","slug":"%e6%95%b0%e5%ad%a6%e7%9a%84%e7%bb%88%e6%9e%81%e9%97%ae%e9%a2%98%ef%bc%9a%e4%b8%8a%e5%b8%9d%e6%98%af%e4%b8%80%e4%bd%8d%e6%95%b0%e5%ad%a6%e5%ae%b6%e5%90%97%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/80365.html","title":{"rendered":"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f"},"content":{"rendered":"

\u6ce8：\u672c\u6587\u4e3a \u201c\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\u201d \u76f8\u5173\u5408\u8f91\u3002 \u82f1\u6587\u5f15\u6587，\u673a\u7ffb\u672a\u6821\u3002 \u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002<\/p>\n

\n

God is a Mathematician<\/h2>\n

\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6<\/h2>\n

Article by Sunil Mehrotra \u4f5c\u8005 \u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9<\/p>\n

Mathematics is abstract, symbolic, structured and precise. It is true everywhere and always and mathematical laws cannot be violated, ever. Math sounds a lot like the attributes of God\u2014eternal, omnipresent and omnipotent. According to theoretical physicist Michio Koku, \u201cThe mind of God we believe is cosmic music, the music of strings resonating through 11\u2011dimensional hyperspace. That is the mind of God.\u201di<\/sup> Vern Poythress, who teaches New Testament in Cambridge University and has two doctorates, a PhD in mathematics from Harvard and a doctorate in Divinity, argues in his book Redeeming Mathematics: A God Centered Approachii<\/sup>, that \u201cthe harmony of abstract mathematics, the physical world of things and our thinking depends on the existence of a Christian God.\u201d Srinivas Ramanujan, on whose life the book and the movie \u201cThe Man who knew Infinity\u201diii<\/sup> are based is known to have said that \u201can equation to me has no meaning unless it represents a thought of God\u201d. \u6570\u5b66\u662f\u62bd\u8c61\u7684\u3001\u7b26\u53f7\u5316\u7684\u3001\u7ed3\u6784\u5316\u7684\u4e14\u7cbe\u786e\u7684\u3002\u5b83\u5728\u4efb\u4f55\u5730\u65b9\u3001\u4efb\u4f55\u65f6\u5019\u90fd\u6210\u7acb，\u6570\u5b66\u89c4\u5f8b\u6c38\u8fdc\u65e0\u6cd5\u88ab\u8fdd\u80cc\u3002\u6570\u5b66\u542c\u8d77\u6765\u4e0e\u4e0a\u5e1d\u7684\u8bf8\u591a\u5c5e\u6027\u5341\u5206\u76f8\u4f3c\u2014\u2014\u6c38\u6052\u3001\u65e0\u6240\u4e0d\u5728\u3001\u65e0\u6240\u4e0d\u80fd\u3002\u7406\u8bba\u7269\u7406\u5b66\u5bb6\u52a0\u6765\u9053\u96c4\u8ba4\u4e3a：\u201c\u6211\u4eec\u6240\u76f8\u4fe1\u7684\u4e0a\u5e1d\u4e4b\u5fc3，\u662f\u5b87\u5b99\u7684\u97f3\u4e50，\u662f\u5f26\u5728 11 \u7ef4\u8d85\u7a7a\u95f4\u4e2d\u5171\u632f\u7684\u4e50\u7ae0\u3002\u8fd9\u4fbf\u662f\u4e0a\u5e1d\u4e4b\u5fc3\u3002\u201d \u5f17\u6069\u30fb\u6ce2\u4f0a\u601d\u96f7\u65af\u5728\u5251\u6865\u5927\u5b66\u6559\u6388\u65b0\u7ea6，\u62e5\u6709\u4e24\u4e2a\u535a\u58eb\u5b66\u4f4d：\u4e00\u4e2a\u662f\u54c8\u4f5b\u5927\u5b66\u7684\u6570\u5b66\u535a\u58eb，\u4e00\u4e2a\u662f\u795e\u5b66\u535a\u58eb，\u4ed6\u5728\u5176\u8457\u4f5c\u300a\u6551\u8d4e\u6570\u5b66：\u4e00\u79cd\u4ee5\u4e0a\u5e1d\u4e3a\u4e2d\u5fc3\u7684\u8fdb\u8def\u300b\u4e2d\u63d0\u51fa：\u201c\u62bd\u8c61\u6570\u5b66\u3001\u7269\u8d28\u4e16\u754c\u4e0e\u4eba\u7c7b\u601d\u7ef4\u4e4b\u95f4\u7684\u548c\u8c10，\u4f9d\u8d56\u4e8e\u57fa\u7763\u6559\u4e4b\u4e0a\u5e1d\u7684\u5b58\u5728\u3002\u201d \u4ee5\u5176\u751f\u5e73\u4e3a\u84dd\u672c\u521b\u4f5c\u4e86\u4e66\u7c4d\u4e0e\u7535\u5f71\u300a\u77e5\u65e0\u6daf\u8005\u300b\u7684\u65af\u91cc\u5c3c\u74e6\u745f\u30fb\u62c9\u9a6c\u52aa\u91d1\u66fe\u8bf4\u8fc7：\u201c\u5bf9\u6211\u800c\u8a00，\u4e00\u4e2a\u65b9\u7a0b\u82e5\u4e0d\u8868\u8fbe\u4e0a\u5e1d\u7684\u601d\u60f3，\u4fbf\u6beb\u65e0\u610f\u4e49\u3002\u201d<\/p>\n

The structures of the universe, from the tiniest (subatomic) to the largest (cosmic), can be visualized as networks or webs of connections. And, these networks are interlocking, pulsating particles, exchanging, sharing and transforming energy from one form to another. Physics today is understood through mathematics. Scientists have long used mathematics to describe the physical properties of the universe, but physicist Max Tegmarkiv<\/sup> goes even further and believes that the universe itself is math. In Tegmark\u2019s view, everything in the universe \u2014 humans included \u2014 is part of a mathematical structure. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical, he says. And space itself has properties such as dimensions, but is still ultimately a mathematical structure. \u4ece\u6700\u5fae\u5c0f\u7684（\u4e9a\u539f\u5b50）\u5230\u6700\u5b8f\u5927\u7684（\u5b87\u5b99），\u5b87\u5b99\u7684\u7ed3\u6784\u90fd\u53ef\u4ee5\u88ab\u89c6\u4f5c\u76f8\u4e92\u8054\u7ed3\u7684\u7f51\u7edc\u6216\u7f51\u72b6\u7cfb\u7edf\u3002\u8fd9\u4e9b\u7f51\u7edc\u7531\u76f8\u4e92\u54ac\u5408\u3001\u8109\u52a8\u7684\u7c92\u5b50\u6784\u6210，\u5b83\u4eec\u8fdb\u884c\u80fd\u91cf\u7684\u4ea4\u6362\u3001\u5171\u4eab\u4e0e\u5f62\u5f0f\u8f6c\u5316\u3002\u5f53\u4eca\u7684\u7269\u7406\u5b66\u901a\u8fc7\u6570\u5b66\u6765\u7406\u89e3\u3002\u957f\u671f\u4ee5\u6765，\u79d1\u5b66\u5bb6\u7528\u6570\u5b66\u63cf\u8ff0\u5b87\u5b99\u7684\u7269\u7406\u6027\u8d28，\u800c\u7269\u7406\u5b66\u5bb6\u9a6c\u514b\u65af\u30fb\u6cf0\u683c\u9a6c\u514b\u5219\u66f4\u8fdb\u4e00\u6b65，\u8ba4\u4e3a\u5b87\u5b99\u672c\u8eab\u5c31\u662f\u6570\u5b66\u3002\u5728\u6cf0\u683c\u9a6c\u514b\u770b\u6765，\u5b87\u5b99\u4e2d\u7684\u4e00\u5207\u2014\u2014\u5305\u62ec\u4eba\u7c7b\u2014\u2014\u90fd\u5c5e\u4e8e\u67d0\u79cd\u6570\u5b66\u7ed3\u6784\u3002\u4ed6\u63d0\u51fa，\u6240\u6709\u7269\u8d28\u90fd\u7531\u7c92\u5b50\u7ec4\u6210，\u7c92\u5b50\u5177\u6709\u7535\u8377\u3001\u81ea\u65cb\u7b49\u6027\u8d28，\u800c\u8fd9\u4e9b\u6027\u8d28\u7eaf\u7cb9\u662f\u6570\u5b66\u6027\u8d28\u3002\u7a7a\u95f4\u672c\u8eab\u5177\u6709\u7ef4\u5ea6\u7b49\u5c5e\u6027，\u4f46\u5f52\u6839\u7ed3\u5e95\u4e5f\u4ecd\u662f\u4e00\u79cd\u6570\u5b66\u7ed3\u6784\u3002<\/p>\n

Mathematics, numbers, symbols, information and energy are different ways by which physicists have attempted to describe the universe. Modern theories in physics are abstract and mystifying to most. For many, faith in the divine origin of the universe provides more certitude than modern physics does. Faith gives one certainty which physics is unable to do, this is the appeal of faith for many. Certainty in an uncertain world is comforting. \u6570\u5b66\u3001\u6570\u5b57\u3001\u7b26\u53f7\u3001\u4fe1\u606f\u4e0e\u80fd\u91cf，\u662f\u7269\u7406\u5b66\u5bb6\u7528\u6765\u63cf\u8ff0\u5b87\u5b99\u7684\u4e0d\u540c\u65b9\u5f0f\u3002\u73b0\u4ee3\u7269\u7406\u7406\u8bba\u5bf9\u5927\u591a\u6570\u4eba\u800c\u8a00\u62bd\u8c61\u800c\u8d39\u89e3\u3002\u5bf9\u8bb8\u591a\u4eba\u6765\u8bf4，\u76f8\u4fe1\u5b87\u5b99\u5177\u6709\u795e\u5723\u8d77\u6e90，\u6bd4\u73b0\u4ee3\u7269\u7406\u80fd\u5e26\u6765\u66f4\u591a\u786e\u5b9a\u6027\u3002\u4fe1\u4ef0\u80fd\u7ed9\u4e88\u4eba\u7269\u7406\u5b66\u65e0\u6cd5\u63d0\u4f9b\u7684\u786e\u5b9a\u6027，\u8fd9\u6b63\u662f\u4fe1\u4ef0\u5bf9\u8bb8\u591a\u4eba\u7684\u5438\u5f15\u529b\u6240\u5728\u3002\u5728\u4e0d\u786e\u5b9a\u7684\u4e16\u754c\u4e2d\u62e5\u6709\u786e\u5b9a\u611f，\u4f1a\u8ba9\u4eba\u611f\u5230\u5b89\u5fc3\u3002<\/p>\n

Scientific knowledge has an asymptotic relationship to truth or Truth. Scientists are getting closer to the Truth but, I suspect, will never reach it. Scientists are like Adam reaching out to touch the hand of God but not making it, as depicted in the frescov<\/sup> on the ceiling of the Sistine Chapel. \u79d1\u5b66\u77e5\u8bc6\u4e0e\u771f\u7406（\u6216\u7edd\u5bf9\u771f\u7406）\u4e4b\u95f4\u5448\u6e10\u8fd1\u5173\u7cfb\u3002\u79d1\u5b66\u5bb6\u6b63\u5728\u4e0d\u65ad\u63a5\u8fd1\u771f\u7406，\u4f46\u6211\u8ba4\u4e3a\u4ed6\u4eec\u6c38\u8fdc\u65e0\u6cd5\u62b5\u8fbe\u3002\u79d1\u5b66\u5bb6\u5c31\u5982\u540c\u897f\u65af\u5ef7\u6559\u5802\u5929\u9876\u58c1\u753b\u4e2d\u6240\u63cf\u7ed8\u7684\u4e9a\u5f53，\u4f38\u624b\u60f3\u8981\u89e6\u78b0\u4e0a\u5e1d\u4e4b\u624b，\u5374\u59cb\u7ec8\u672a\u80fd\u89e6\u53ca\u3002<\/p>\n

Physicists are peering into the outer reaches of the cosmos and probing deep into the inner sanctum of atoms, discovering realms that are beyond the reach of our senses. For most of us, these realms are difficult to comprehend, because we cannot see, touch or feel them. No one has seen a quark or been able to visualize Einstein\u2019s four\u2011dimensional space and time. Hence, to some, modern physics is incomprehensible, abstract, hard to relate to, and indistinguishable from a myth. \u7269\u7406\u5b66\u5bb6\u65e2\u773a\u671b\u5b87\u5b99\u7684\u6781\u8fdc\u8fb9\u754c，\u4e5f\u6df1\u5165\u539f\u5b50\u7684\u5185\u90e8\u79d8\u5883，\u53d1\u73b0\u4e86\u8bf8\u591a\u8d85\u51fa\u4eba\u7c7b\u611f\u5b98\u8303\u56f4\u7684\u9886\u57df\u3002\u5bf9\u6211\u4eec\u5927\u591a\u6570\u4eba\u800c\u8a00，\u8fd9\u4e9b\u9886\u57df\u96be\u4ee5\u7406\u89e3，\u56e0\u4e3a\u6211\u4eec\u65e0\u6cd5\u770b\u89c1\u3001\u89e6\u6478\u6216\u611f\u77e5\u5b83\u4eec\u3002\u6ca1\u6709\u4eba\u89c1\u8fc7\u5938\u514b，\u4e5f\u65e0\u6cd5\u76f4\u89c2\u60f3\u8c61\u7231\u56e0\u65af\u5766\u7684\u56db\u7ef4\u65f6\u7a7a\u3002\u56e0\u6b64，\u5bf9\u4e00\u4e9b\u4eba\u6765\u8bf4，\u73b0\u4ee3\u7269\u7406\u5b66\u6666\u6da9\u96be\u61c2\u3001\u62bd\u8c61\u758f\u79bb，\u4e0e\u795e\u8bdd\u51e0\u4e4e\u65e0\u6cd5\u533a\u5206\u3002<\/p>\n

Thankfully, we do not rely just on our senses to understand the universe; if we did, we would still be in the dark ages. \u503c\u5f97\u5e86\u5e78\u7684\u662f，\u6211\u4eec\u5e76\u975e\u53ea\u4f9d\u9760\u611f\u5b98\u6765\u7406\u89e3\u5b87\u5b99\u3002\u5018\u82e5\u5982\u6b64，\u4eba\u7c7b\u4ecd\u5c06\u505c\u7559\u5728\u9ed1\u6697\u65f6\u4ee3\u3002<\/p>\n

Physicists are looking for a single theory or, as Michio Koku states, \u201can equation about six inches long,\u201d which can explain all phenomenon, from the largest (cosmos) to the tiniest (subatomic particles). The holy grail in physics is to find a theory that reconciles general relativity and quantum physics. Science is in search of ultimate unity, the God Particlevi<\/sup>, as Nobel Laureate Leon Lederman wrote in his book of the same name. Particle physicists keep building bigger and bigger particle accelerators, like the one at CERNvii<\/sup>, in search of the God particle. At CERN, in the Large Hadron Collider, energy at the point of collision of the protons approaches the energy moments after the Big Bang, in the hope of finding the God particle. \u7269\u7406\u5b66\u5bb6\u6b63\u5728\u5bfb\u627e\u4e00\u79cd\u5927\u4e00\u7edf\u7406\u8bba，\u6216\u8005\u5982\u52a0\u6765\u9053\u96c4\u6240\u8a00，\u4e00\u4e2a\u201c\u7ea6 6 \u82f1\u5bf8\u957f\u7684\u65b9\u7a0b\u201d，\u5b83\u80fd\u591f\u89e3\u91ca\u4ece\u6700\u5b8f\u5927（\u5b87\u5b99）\u5230\u6700\u5fae\u5c0f（\u4e9a\u539f\u5b50\u7c92\u5b50）\u7684\u4e00\u5207\u73b0\u8c61\u3002\u7269\u7406\u5b66\u7684\u5723\u676f，\u662f\u627e\u5230\u4e00\u79cd\u80fd\u8c03\u548c\u5e7f\u4e49\u76f8\u5bf9\u8bba\u4e0e\u91cf\u5b50\u7269\u7406\u7684\u7406\u8bba\u3002\u79d1\u5b66\u754c\u6b63\u5728\u8ffd\u5bfb\u7ec8\u6781\u7684\u7edf\u4e00，\u5373\u201c\u4e0a\u5e1d\u7c92\u5b50\u201d，\u8bfa\u8d1d\u5c14\u5956\u5f97\u4e3b\u5229\u6602\u30fb\u83b1\u5fb7\u66fc\u5728\u5176\u540c\u540d\u8457\u4f5c\u4e2d\u5982\u6b64\u547d\u540d\u3002\u7c92\u5b50\u7269\u7406\u5b66\u5bb6\u4e0d\u65ad\u5efa\u9020\u8d8a\u6765\u8d8a\u5927\u7684\u7c92\u5b50\u52a0\u901f\u5668，\u4f8b\u5982\u6b27\u6d32\u6838\u5b50\u7814\u7a76\u4e2d\u5fc3（CERN）\u7684\u52a0\u901f\u5668，\u4ee5\u5bfb\u627e\u4e0a\u5e1d\u7c92\u5b50\u3002\u5728\u6b27\u6d32\u6838\u5b50\u7814\u7a76\u4e2d\u5fc3\u7684\u5927\u578b\u5f3a\u5b50\u5bf9\u649e\u673a\u4e2d，\u8d28\u5b50\u78b0\u649e\u77ac\u95f4\u7684\u80fd\u91cf\u63a5\u8fd1\u5b87\u5b99\u5927\u7206\u70b8\u540e\u77ac\u95f4\u7684\u80fd\u91cf，\u4ee5\u671f\u627e\u5230\u4e0a\u5e1d\u7c92\u5b50\u3002<\/p>\n

The truth is that the Truth might not be a particle. Truth might not be a thing. It might be an abstraction, like an \u201cidea in the mind of God\u201d, as some have suggested, or perhaps as Max Tegmark posits, \u201cThere\u2019s something very mathematical about our Universe, and the more carefully we look, the more math we seem to find. \u2026\u2026 So, the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Everything in our world is purely mathematical \u2013 including you.\u201d \u4e8b\u5b9e\u662f，\u7edd\u5bf9\u771f\u7406\u6216\u8bb8\u5e76\u975e\u4e00\u79cd\u7c92\u5b50\u3002\u771f\u7406\u6216\u8bb8\u5e76\u975e\u67d0\u4e2a\u201c\u4e8b\u7269\u201d\u3002\u5b83\u53ef\u80fd\u662f\u4e00\u79cd\u62bd\u8c61\u5b58\u5728，\u6b63\u5982\u4e00\u4e9b\u4eba\u6240\u63d0\u51fa\u7684，\u662f\u201c\u4e0a\u5e1d\u5fc3\u4e2d\u7684\u4e00\u4e2a\u7406\u5ff5\u201d；\u6216\u8005\u5982\u9a6c\u514b\u65af\u30fb\u6cf0\u683c\u9a6c\u514b\u6240\u4e3b\u5f20\u7684：\u201c\u6211\u4eec\u7684\u5b87\u5b99\u5177\u6709\u6781\u5f3a\u7684\u6570\u5b66\u5c5e\u6027，\u6211\u4eec\u89c2\u5bdf\u5f97\u8d8a\u7ec6\u81f4，\u5c31\u4f1a\u53d1\u73b0\u8d8a\u591a\u7684\u6570\u5b66\u89c4\u5f8b\u2026\u2026\u56e0\u6b64，\u7ed3\u8bba\u662f：\u5982\u679c\u4f60\u76f8\u4fe1\u5b58\u5728\u72ec\u7acb\u4e8e\u4eba\u7c7b\u4e4b\u5916\u7684\u5ba2\u89c2\u73b0\u5b9e，\u90a3\u4e48\u4f60\u4e5f\u5fc5\u987b\u76f8\u4fe1，\u6211\u4eec\u7684\u7269\u7406\u73b0\u5b9e\u5c31\u662f\u4e00\u79cd\u6570\u5b66\u7ed3\u6784\u3002\u6211\u4eec\u4e16\u754c\u4e2d\u7684\u4e00\u5207\u2014\u2014\u5305\u62ec\u4f60\u5728\u5185\u2014\u2014\u90fd\u7eaf\u7cb9\u662f\u6570\u5b66\u7684\u3002\u201d<\/p>\n

Theologists, scientists and philosophers seem to agree that Reality, absolute truth, or God, if you will, is an abstract reality. Not a reality that can be detected by our senses or known through our intellect. In this view mathematics are an expression of the mind of God. She is a mathematician! \u795e\u5b66\u5bb6\u3001\u79d1\u5b66\u5bb6\u4e0e\u54f2\u5b66\u5bb6\u4f3c\u4e4e\u8fbe\u6210\u4e86\u5171\u8bc6：\u5b9e\u5728\u3001\u7edd\u5bf9\u771f\u7406，\u6216\u8005\u4f60\u613f\u610f\u79f0\u4e4b\u4e3a\u4e0a\u5e1d，\u662f\u4e00\u79cd\u62bd\u8c61\u7684\u5b9e\u5728\u3002\u5b83\u5e76\u975e\u80fd\u88ab\u4eba\u7c7b\u611f\u5b98\u63a2\u6d4b\u6216\u4ec5\u51ed\u7406\u667a\u7406\u89e3\u7684\u5b9e\u5728\u3002\u5728\u8fd9\u79cd\u89c2\u70b9\u4e0b，\u6570\u5b66\u662f\u4e0a\u5e1d\u601d\u60f3\u7684\u8868\u8fbe\u3002\u4e0a\u5e1d，\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6！<\/p>\n

Further proof is in sacred geometries found in nature. \u66f4\u591a\u7684\u8bc1\u636e，\u5b58\u5728\u4e8e\u81ea\u7136\u754c\u4e2d\u663e\u73b0\u7684\u795e\u5723\u51e0\u4f55\u4e4b\u4e2d\u3002<\/p>\n

Sunil Mehrotra \u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9<\/p>\n

Sunil Mehrotra has an MS in Aeronautical Engineering from Purdue University and an MBA from The University of Chicago. He has held senior executive positions at Fortune 500 companies and has been a CEO and founder of two start\u2011ups. He has also taught at Pepperdine University in Malibu, California. You can read more about his book, Shiva\u2019s Dance: A Scientist Dances with the Sages, at https:\/\/shivasdance.org. *\u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9\u62e5\u6709\u666e\u6e21\u5927\u5b66\u822a\u7a7a\u5de5\u7a0b\u7855\u58eb\u5b66\u4f4d\u4e0e\u829d\u52a0\u54e5\u5927\u5b66\u5de5\u5546\u7ba1\u7406\u7855\u58eb\u5b66\u4f4d\u3002\u4ed6\u66fe\u5728\u8d22\u5bcc 500 \u5f3a\u4f01\u4e1a\u62c5\u4efb\u9ad8\u7ea7\u7ba1\u7406\u804c\u4f4d，\u662f\u4e24\u5bb6\u521d\u521b\u516c\u53f8\u7684\u9996\u5e2d\u6267\u884c\u5b98\u4e0e\u521b\u59cb\u4eba，\u5e76\u66fe\u5728\u52a0\u5229\u798f\u5c3c\u4e9a\u5dde\u9a6c\u91cc\u5e03\u7684\u4f69\u73c0\u4ee3\u56e0\u5927\u5b66\u4efb\u6559\u3002<\/p>\n

\n

\u53e6\u4e00\u4e2a\u89c2\u70b9<\/h2>\n
\n

Mathematics declaring the glory of God<\/h2>\n

\u6570\u5b66\u5f70\u663e\u4e0a\u5e1d\u7684\u8363\u8000<\/h2>\n

Verbum et Ecclesia ISSN: (Online) 2074-7705, (Print) 1609-9982 AOSIS Original Research \u300a\u9053\u4e0e\u6559\u4f1a\u300b\u671f\u520a \u56fd\u9645\u6807\u51c6\u8fde\u7eed\u51fa\u7248\u7269\u53f7：（\u7f51\u7edc\u7248）2074-7705，（\u5370\u5237\u7248）1609-9982 \u975e\u6d32\u5f00\u653e\u79d1\u5b66\u5021\u8bae\u51fa\u7248\u673a\u6784 \u539f\u521b\u7814\u7a76 Author: Volker Kessler\u00b9,\u00b2 \u4f5c\u8005： \u6c83\u5c14\u514b\u00b7\u514b\u65af\u52d2\u00b9，\u00b2 Affiliations: \u00b9Gesellschaft f\u00fcr Bildung und Forschung in Europa, Gummersbach, Germany \u00b2Department of Philosophy, Practical and Systematic Theology, College of Human Sciences, University of South Africa, Pretoria, South Africa \u4f5c\u8005\u5355\u4f4d： \u00b9 \u5fb7\u56fd\u53e4\u9ed8\u65af\u5df4\u8d6b \u6b27\u6d32\u6559\u80b2\u4e0e\u7814\u7a76\u534f\u4f1a \u00b2 \u5357\u975e\u6bd4\u52d2\u9640\u5229\u4e9a \u5357\u975e\u4eba\u6587\u79d1\u5b66\u5927\u5b66 \u54f2\u5b66\u3001\u5b9e\u8df5\u4e0e\u7cfb\u7edf\u795e\u5b66\u7cfb Corresponding author: Volker Kessler, volker.kessler@gbfe.eu \u901a\u8baf\u4f5c\u8005： \u6c83\u5c14\u514b\u00b7\u514b\u65af\u52d2 \u90ae\u7bb1：volker.kessler@gbfe.eu Dates: \u65e5\u671f： Received: 18 Nov. 2021 \/ \u6536\u7a3f\u65e5\u671f： Accepted: 08 Mar. 2022 \/ \u5f55\u7528\u65e5\u671f Published: 19 Apr. 2022 \/ \u51fa\u7248\u65e5\u671f<\/p>\n

How to cite this article: \u672c\u6587\u5f15\u7528\u683c\u5f0f：<\/p>\n

Kessler, V., 2022, \u2018Mathematics declaring the glory of God\u2019, Verbum et Ecclesia 43(1), a2432. https:\/\/doi.org\/10.4102\/ve.v43i1.2432<\/p>\n

\u514b\u65af\u52d2，V.，2022，\u300a\u6570\u5b66\u5f70\u663e\u4e0a\u5e1d\u7684\u8363\u8000\u300b，\u300a\u9053\u4e0e\u6559\u4f1a\u300b，43（1），a2432\u3002 Copyright: \u00a9 2022. The Author. Licensee: AOSIS. This work is licensed under the Creative Commons Attribution License. \u8457\u4f5c\u6743\u8bf4\u660e： \u00a92022 \u672c\u6587\u4f5c\u8005\u3002\u51fa\u7248\u6388\u6743\u65b9：\u975e\u6d32\u5f00\u653e\u79d1\u5b66\u5021\u8bae\u51fa\u7248\u673a\u6784\u3002\u672c\u4f5c\u54c1\u91c7\u7528\u77e5\u8bc6\u5171\u4eab\u7f72\u540d\u8bb8\u53ef\u534f\u8bae\u6388\u6743\u3002<\/p>\n

This article discussed the question \u2018Does God speak through the language of mathematics?\u2019 For centuries, mathematicians with different religious backgrounds would have answered this question in the affirmative. Due to changes in mathematics from the 19th century onwards, this question cannot be answered as easily as it used to be. If one regards mathematical concepts as creations of the human mind, it is difficult to argue that mathematical formulae exist in a divine mind. The article argued that there were traces of the divine in mathematics. Six kinds of traces were explained: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. In practice, traces (1), (2) and (6) are probably the most convincing. \u672c\u6587\u63a2\u8ba8\u4e86\u201c\u4e0a\u5e1d\u662f\u5426\u901a\u8fc7\u6570\u5b66\u7684\u8bed\u8a00\u8a00\u8bf4？\u201d\u8fd9\u4e00\u95ee\u9898\u3002\u6570\u4e2a\u4e16\u7eaa\u4ee5\u6765，\u4e0d\u540c\u5b97\u6559\u80cc\u666f\u7684\u6570\u5b66\u5bb6\u90fd\u4f1a\u5bf9\u8fd9\u4e00\u95ee\u9898\u7ed9\u51fa\u80af\u5b9a\u7684\u7b54\u6848\u300219 \u4e16\u7eaa\u4ee5\u6765，\u6570\u5b66\u9886\u57df\u53d1\u751f\u4e86\u8bf8\u591a\u53d8\u9769，\u8fd9\u4e00\u95ee\u9898\u4e5f\u4e0d\u518d\u50cf\u4ee5\u5f80\u90a3\u6837\u80fd\u8f7b\u6613\u4f5c\u7b54\u3002\u82e5\u5c06\u6570\u5b66\u6982\u5ff5\u89c6\u4f5c\u4eba\u7c7b\u601d\u7ef4\u7684\u521b\u9020\u7269，\u4fbf\u96be\u4ee5\u8bba\u8bc1\u6570\u5b66\u516c\u5f0f\u5b58\u5728\u4e8e\u795e\u5723\u7684\u601d\u7ef4\u4e4b\u4e2d\u3002\u672c\u6587\u63d0\u51fa，\u6570\u5b66\u4e2d\u5b58\u5728\u7740\u795e\u5723\u7684\u8e2a\u8ff9，\u5e76\u9610\u91ca\u4e86\u516d\u79cd\u8e2a\u8ff9：(1)\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406\u7684\u5b58\u5728；(2)\u7f8e\u7684\u5b58\u5728；(3)\u5171\u540c\u4f53\u7684\u91cd\u8981\u6027；(4)\u5bf9\u65e0\u9650\u8fdb\u884c\u7406\u6027\u9610\u91ca；(5)\u53d1\u73b0\u5bf9\u4e0d\u53ef\u89c1\u7684\u62bd\u8c61\u5ba2\u4f53\u8fdb\u884c\u8a00\u8bf4\u5177\u6709\u5408\u7406\u6027；(6)\u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u3002\u5b9e\u9645\u4e0a，\u5176\u4e2d\u7b2c(1)､(2)､(6)\u79cd\u8e2a\u8ff9\u6216\u8bb8\u6700\u5177\u8bf4\u670d\u529b\u3002<\/p>\n

Intradisciplinary and\/or interdisciplinary implications: This article is very much interdisciplinary as it combines mathematics and theology, especially the philosophy of mathematics and systematic theology. \u5b66\u79d1\u5185\u53ca\u8de8\u5b66\u79d1\u610f\u4e49：\u672c\u6587\u5177\u6709\u9ad8\u5ea6\u7684\u8de8\u5b66\u79d1\u6027，\u878d\u5408\u4e86\u6570\u5b66\u4e0e\u795e\u5b66\u4e24\u5927\u9886\u57df，\u5c24\u5176\u7ed3\u5408\u4e86\u6570\u5b66\u54f2\u5b66\u4e0e\u7cfb\u7edf\u795e\u5b66\u7684\u76f8\u5173\u7814\u7a76\u3002<\/p>\n

Keywords: philosophy of mathematics; general revelation; truth; beauty; infinity; unreasonable effectiveness. \u5173\u952e\u8bcd：\u6570\u5b66\u54f2\u5b66；\u666e\u904d\u542f\u793a；\u771f\u7406；\u7f8e；\u65e0\u9650；\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027<\/p>\n

Introduction<\/h3>\n

\u5f15\u8a00<\/h3>\n

This article is about how traces of the divine can be found in the language of mathematics. The title alludes to Psalm 19:1: \u2018The heavens declare the glory of God\u2019, which is part of the Jewish and Christian Bible. \u672c\u6587\u65e8\u5728\u63a2\u8ba8\u5982\u4f55\u5728\u6570\u5b66\u7684\u8bed\u8a00\u4e2d\u53d1\u73b0\u795e\u5723\u7684\u8e2a\u8ff9\u3002\u6587\u7ae0\u6807\u9898\u5316\u7528\u4e86\u72b9\u592a\u6559\u4e0e\u57fa\u7763\u6559\u7ecf\u5178\u300a\u5723\u7ecf\u300b\u4e2d\u300a\u8bd7\u7bc7\u300b19 \u7bc7 1 \u8282\u7684\u5185\u5bb9：\u201c\u8bf8\u5929\u8ff0\u8bf4\u4e0a\u5e1d\u7684\u8363\u8000\u201d\u3002<\/p>\n

Obviously, mathematics is some kind of language (Livio 2009:239\u2013241). Mathematics is often called the \u2018language of nature\u2019, which goes back to a saying by the Italian astronomer Galileo Galilei (1564\u20131642) (Mukunda 2015:347). Isaac Newton called his famous work on physics The Mathematical Principles of Natural Philosophy because he was convinced that natural philosophy has to be expressed in the language of mathematics. \u663e\u7136，\u6570\u5b66\u662f\u4e00\u79cd\u8bed\u8a00（\u5229\u7ef4\u5965，2009：239-241）\u3002\u6570\u5b66\u5e38\u88ab\u79f0\u4f5c\u201c\u81ea\u7136\u7684\u8bed\u8a00\u201d，\u8fd9\u4e00\u8bf4\u6cd5\u53ef\u8ffd\u6eaf\u81f3\u610f\u5927\u5229\u5929\u6587\u5b66\u5bb6\u4f3d\u5229\u7565\u00b7\u4f3d\u5229\u83b1（1564-1642）（\u7a46\u6606\u8fbe，2015：347）\u3002\u827e\u8428\u514b\u00b7\u725b\u987f\u5c06\u5176\u7ecf\u5178\u7684\u7269\u7406\u5b66\u8457\u4f5c\u547d\u540d\u4e3a\u300a\u81ea\u7136\u54f2\u5b66\u7684\u6570\u5b66\u539f\u7406\u300b，\u56e0\u4ed6\u575a\u4fe1，\u81ea\u7136\u54f2\u5b66\u5fc5\u987b\u4ee5\u6570\u5b66\u7684\u8bed\u8a00\u6765\u8868\u8fbe\u3002<\/p>\n

The question to be investigated in this article is as follows: Does God speak through the language of mathematics? Two centuries ago, it would have been easy to argue for a positive answer. Due to changes in mathematics and the growing influence of atheism, today\u2019s answer has to be different from the answers given two centuries ago. In this article, I will first explain the changes in mathematics and their implications for the question to be investigated. Secondly, I will discuss six examples of indications of divine elements in mathematics. I am writing from the perspective of a mathematician and Christian theologian. Thus, I will focus on doctrines of the Christian faith although some implications will also hold for other faith systems. \u672c\u6587\u63a2\u8ba8\u7684\u6838\u5fc3\u95ee\u9898\u4e3a：\u4e0a\u5e1d\u662f\u5426\u901a\u8fc7\u6570\u5b66\u7684\u8bed\u8a00\u8a00\u8bf4？\u4e24\u4e2a\u4e16\u7eaa\u524d，\u4eba\u4eec\u80fd\u8f7b\u6613\u4e3a\u8fd9\u4e00\u95ee\u9898\u7ed9\u51fa\u80af\u5b9a\u7684\u8bba\u8bc1\u3002\u800c\u968f\u7740\u6570\u5b66\u7684\u53d8\u9769\u4e0e\u65e0\u795e\u8bba\u5f71\u54cd\u7684\u4e0d\u65ad\u6269\u5927，\u5982\u4eca\u7684\u7b54\u6848\u5fc5\u7136\u4e0e\u4e24\u4e2a\u4e16\u7eaa\u524d\u6709\u6240\u4e0d\u540c\u3002\u672c\u6587\u4e2d，\u7b14\u8005\u5c06\u9996\u5148\u9610\u91ca\u6570\u5b66\u9886\u57df\u7684\u53d8\u9769\u53ca\u5176\u5bf9\u8fd9\u4e00\u7814\u7a76\u95ee\u9898\u7684\u610f\u4e49；\u5176\u6b21，\u63a2\u8ba8\u6570\u5b66\u4e2d\u5b58\u5728\u795e\u5723\u5143\u7d20\u7684\u516d\u5927\u4f50\u8bc1\u3002\u7b14\u8005\u4ee5\u6570\u5b66\u5bb6\u4e0e\u57fa\u7763\u6559\u795e\u5b66\u5bb6\u7684\u53cc\u91cd\u89c6\u89d2\u5c55\u5f00\u5199\u4f5c，\u56e0\u6b64\u5c06\u805a\u7126\u4e8e\u57fa\u7763\u6559\u7684\u6559\u4e49，\u5c3d\u7ba1\u90e8\u5206\u7814\u7a76\u7ed3\u8bba\u4e5f\u9002\u7528\u4e8e\u5176\u4ed6\u5b97\u6559\u4f53\u7cfb\u3002<\/p>\n

In this article, I will not treat mathematical theistic proofs such as G\u00f6del \u00b9 or Meyer (1987). I do not think that mathematics alone can say anything about God\u2019s existence. Mathematics is a formal construct, and as such it does not say anything about the real nature of things.\u00b2 Instead I will argue that mathematics provides an opportunity to discover some properties that are usually associated with God. I will list several aspects of mathematics that I regard as traces or hints leading one to reflect upon God and divine attributes. \u672c\u6587\u4e0d\u63a2\u8ba8\u8bf8\u5982\u54e5\u5fb7\u5c14 \u00b9 \u6216\u8fc8\u8036（1987）\u6240\u63d0\u51fa\u7684\u6570\u5b66\u795e\u5b58\u5728\u8bc1\u660e\u3002\u7b14\u8005\u8ba4\u4e3a，\u4ec5\u51ed\u6570\u5b66\u65e0\u6cd5\u5bf9\u4e0a\u5e1d\u7684\u5b58\u5728\u4f5c\u51fa\u4efb\u4f55\u8bba\u65ad\u3002\u6570\u5b66\u662f\u4e00\u79cd\u5f62\u5f0f\u5316\u7684\u5efa\u6784，\u5176\u672c\u8eab\u65e0\u6cd5\u63ed\u793a\u4e8b\u7269\u7684\u771f\u5b9e\u672c\u8d28 \u00b2\u3002\u7b14\u8005\u5c06\u8bba\u8bc1，\u6570\u5b66\u4e3a\u4eba\u4eec\u53d1\u73b0\u90a3\u4e9b\u901a\u5e38\u4e0e\u4e0a\u5e1d\u76f8\u5173\u7684\u5c5e\u6027\u63d0\u4f9b\u4e86\u5951\u673a，\u5e76\u5217\u4e3e\u6570\u5b66\u4e2d\u82e5\u5e72\u53ef\u88ab\u89c6\u4f5c\u8e2a\u8ff9\u6216\u7ebf\u7d22\u7684\u65b9\u9762，\u5f15\u5bfc\u4eba\u4eec\u601d\u8003\u4e0a\u5e1d\u4e0e\u795e\u5723\u7684\u5c5e\u6027\u3002<\/p>\n

  • G\u00f6del noted his proof on two pages in 1970 but kept it secret until his death, see Bromand and Kreis (2013:483\u2013491). \u54e5\u5fb7\u5c14\u4e8e 1970 \u5e74\u5728\u4e24\u9875\u7eb8\u4e2d\u5199\u4e0b\u4e86\u5176\u795e\u5b58\u5728\u8bc1\u660e，\u5374\u5c06\u5176\u79d8\u800c\u4e0d\u5ba3，\u76f4\u81f3\u79bb\u4e16\u3002\u53c2\u89c1\u5e03\u7f57\u66fc\u5fb7\u4e0e\u514b\u96f7\u65af（2013：483-491）\u3002 \u00a0<\/li>\n
  • See Poincar\u00e9\u2019s statement: \u2018The object of mathematical theories is not to reveal to us the real nature of things\u2019 (Poincar\u00e9 1905:211). \u53c2\u89c1\u5e9e\u52a0\u83b1\u7684\u8bba\u8ff0：\u201c\u6570\u5b66\u7406\u8bba\u7684\u76ee\u7684\u5e76\u975e\u5411\u6211\u4eec\u63ed\u793a\u4e8b\u7269\u7684\u771f\u5b9e\u672c\u8d28\u3002\u201d（\u5e9e\u52a0\u83b1，1905：211）\u3002<\/li>\n

    This article develops some ideas that I presented in two previous articles (Kessler 2018, 2019). In particular, I take into account some reactions to these articles.\u00b3 \u672c\u6587\u8fdb\u4e00\u6b65\u53d1\u5c55\u4e86\u7b14\u8005\u5728\u4e24\u7bc7\u524d\u671f\u8bba\u6587\u4e2d\u63d0\u51fa\u7684\u89c2\u70b9（\u514b\u65af\u52d2，2018\u30012019），\u5c24\u5176\u7ed3\u5408\u4e86\u5b66\u754c\u5bf9\u8fd9\u4e24\u7bc7\u8bba\u6587\u7684\u76f8\u5173\u56de\u5e94\u00b3\u3002<\/p>\n

  • In Kessler (2018), I argued that \u2018God becomes beautiful \u2026 in mathematics\u2019. This article received a response by Smit and De Heide (2021), in which they criticise that I did \u2018not discuss in detail the current state of research in the field of philosophy of mathematics\u2019 (p. 3). Their observation is correct. Because I wanted the article to be readable for non-mathematicians, my presentation of the mathematical foundations was a bit (too) simplistic. In this article, I am making up for what I deliberately omitted in Kessler (2018). \u5728\u514b\u65af\u52d2（2018）\u7684\u8bba\u6587\u4e2d，\u7b14\u8005\u63d0\u51fa\u201c\u4e0a\u5e1d\u5728\u6570\u5b66\u4e2d\u5f70\u663e\u5176\u7f8e\u2026\u2026\u201d\u3002\u65af\u5bc6\u7279\u4e0e\u5fb7\u00b7\u6d77\u5fb7（2021）\u5bf9\u8be5\u6587\u4f5c\u51fa\u56de\u5e94，\u6279\u8bc4\u7b14\u8005\u201c\u672a\u8be6\u7ec6\u63a2\u8ba8\u6570\u5b66\u54f2\u5b66\u9886\u57df\u7684\u5f53\u524d\u7814\u7a76\u73b0\u72b6\u201d（\u7b2c 3 \u9875）\u3002\u8fd9\u4e00\u6279\u8bc4\u5341\u5206\u4e2d\u80af\u3002\u56e0\u7b14\u8005\u5e0c\u671b\u8be5\u6587\u80fd\u4e3a\u975e\u6570\u5b66\u9886\u57df\u7684\u8bfb\u8005\u6240\u7406\u89e3，\u5bf9\u6570\u5b66\u57fa\u7840\u7684\u9610\u8ff0\u7565\u663e（\u8fc7\u4e8e）\u7b80\u7565\u3002\u672c\u6587\u5c06\u5f25\u8865\u7b14\u8005\u5728\u514b\u65af\u52d2（2018）\u4e00\u6587\u4e2d\u523b\u610f\u7701\u7565\u7684\u5185\u5bb9\u3002<\/li>\n

    When the answer seemed to be straightforward<\/h3>\n

    \u7b54\u6848\u66fe\u4e00\u76ee\u4e86\u7136\u7684\u65f6\u4ee3<\/h3>\n

    Mathematics and the mind of God \u2013 before the big changes<\/h4>\n

    \u6570\u5b66\u4e0e\u4e0a\u5e1d\u7684\u601d\u7ef4\u2014\u2014\u91cd\u5927\u53d8\u9769\u4e4b\u524d<\/h4>\n

    In the past, strong links between mathematics and the divine have always been assumed; see, for example, the historical study Koetsier and Bergmann (eds. 2005). For many, mathematical knowledge used to be \u2018an aspect of spiritual knowledge, knowledge in the mind of God\u2019 (Hersh 1997:236). Jonas (1966) and Livio (2009) raised the same question: \u2018Is God a mathematician?\u2019 \u2074 \u5728\u8fc7\u53bb，\u4eba\u4eec\u59cb\u7ec8\u8ba4\u4e3a\u6570\u5b66\u4e0e\u795e\u5723\u4e4b\u95f4\u5b58\u5728\u7740\u7d27\u5bc6\u7684\u8054\u7cfb，\u4f8b\u5982\u53ef\u53c2\u89c1\u79d1\u9f50\u5c14\u4e0e\u4f2f\u683c\u66fc\u4e3b\u7f16\u7684\u76f8\u5173\u5386\u53f2\u7814\u7a76（2005）\u3002\u5bf9\u8bb8\u591a\u4eba\u800c\u8a00，\u6570\u5b66\u77e5\u8bc6\u66fe\u662f\u201c\u7075\u6027\u77e5\u8bc6\u7684\u4e00\u90e8\u5206，\u662f\u5b58\u5728\u4e8e\u4e0a\u5e1d\u601d\u7ef4\u4e2d\u7684\u77e5\u8bc6\u201d（\u8d6b\u4ec0，1997：236）\u3002\u7ea6\u7eb3\u65af（1966）\u4e0e\u5229\u7ef4\u5965（2009）\u90fd\u63d0\u51fa\u8fc7\u540c\u4e00\u4e2a\u95ee\u9898：\u201c\u4e0a\u5e1d\u662f\u6570\u5b66\u5bb6\u5417？\u201d \u2074<\/p>\n

  • Jonas\u2019 (1966:92) answer is a distinct \u2018no\u2019. Livio (2009:252) leaves the question open. \u7ea6\u7eb3\u65af\u5bf9\u8fd9\u4e00\u95ee\u9898\u7ed9\u51fa\u4e86\u660e\u786e\u7684\u5426\u5b9a\u7b54\u6848（1966：92），\u800c\u5229\u7ef4\u5965\u5219\u5bf9\u6b64\u95ee\u9898\u6301\u5f00\u653e\u6001\u5ea6（2009：252）\u3002<\/li>\n

    In the Classic Greek tradition, theology and mathematics were close to each other (Phillips 2009). Pythagoras (ca. 570\u2013510 BC) was convinced that numbers and formulae had an inherent spiritual meaning, and his school at Croton was \u2018more a religious brotherhood than an academy\u2019 (Phillips 2009:5). Pythagoras and Plato both regarded mathematics as a spiritual path leading to the divine, but they came to different conclusions. For Plato (428\u2013348 BC), mathematical studies were a preparation for the contemplation of the divine principles, whereas for the Pythagoreans mathematics actually was God (Livio 2009:28). \u5728\u53e4\u5e0c\u814a\u4f20\u7edf\u4e2d，\u795e\u5b66\u4e0e\u6570\u5b66\u8054\u7cfb\u7d27\u5bc6（\u83f2\u5229\u666e\u65af，2009）\u3002\u6bd5\u8fbe\u54e5\u62c9\u65af（\u7ea6\u516c\u5143\u524d 570-\u524d 510 \u5e74）\u575a\u4fe1，\u6570\u5b57\u4e0e\u516c\u5f0f\u8574\u542b\u7740\u5185\u5728\u7684\u7075\u6027\u610f\u4e49，\u4ed6\u5728\u514b\u7f57\u6258\u5185\u521b\u529e\u7684\u5b66\u56ed\u201c\u4e0e\u5176\u8bf4\u662f\u4e00\u6240\u5b66\u672f\u673a\u6784，\u4e0d\u5982\u8bf4\u662f\u4e00\u4e2a\u5b97\u6559\u5144\u5f1f\u4f1a\u201d（\u83f2\u5229\u666e\u65af，2009：5）\u3002\u6bd5\u8fbe\u54e5\u62c9\u65af\u4e0e\u67cf\u62c9\u56fe\u5747\u5c06\u6570\u5b66\u89c6\u4f5c\u901a\u5f80\u795e\u5723\u7684\u7075\u6027\u4e4b\u8def，\u5374\u5f97\u51fa\u4e86\u4e0d\u540c\u7684\u7ed3\u8bba\u3002\u5bf9\u67cf\u62c9\u56fe（\u516c\u5143\u524d 428-\u524d 348 \u5e74）\u800c\u8a00，\u6570\u5b66\u7814\u7a76\u662f\u9ed8\u60f3\u795e\u5723\u6cd5\u5219\u7684\u51c6\u5907；\u800c\u5bf9\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u800c\u8a00，\u6570\u5b66\u672c\u8eab\u5c31\u662f\u4e0a\u5e1d（\u5229\u7ef4\u5965，2009：28）\u3002<\/p>\n

    To give an example of the close link between mathematics and theology some centuries ago, we look at the beliefs of some important mathematicians of the 17th century, the start of the scientific age and of modern mathematics. In the Europe of the 17th century, it was quite common to believe in the Christian God, and this is also true for the most brilliant European mathematicians of the 17th century. It is, of course, debatable that mathematician was most important at a given time; for example, Stewart (2017) selects 25 mathematicians throughout history. Nevertheless there are good reasons to regard the following six persons as the most important mathematicians of the 17th century: (1) Ren\u00e9 Descartes, France (1596\u20131650), who invented analytic geometry; his compatriots (2) Pierre de Fermat (1607\u20131665) and (3) Blaise de Pascal (1623\u20131662), both of whom (among others) laid the foundation of probability theory; (4) Sir Isaac Newton, England (1642\u20131726) and (5) Gottfried Wilhelm Leibniz, Germany (1646\u20131716), both of whom invented calculus independently of each other and (6) Jacob Bernoulli, Switzerland (1654\u20131705), after whom the \u2018law of large numbers\u2019 is named. It is interesting to note that five of them saw a strong connection between their mathematics and their Christian faith. (We do not have a statement from Fermat on this.) \u4e3a\u4f50\u8bc1\u6570\u4e2a\u4e16\u7eaa\u524d\u6570\u5b66\u4e0e\u795e\u5b66\u7684\u7d27\u5bc6\u8054\u7cfb，\u6211\u4eec\u6765\u770b\u770b 17 \u4e16\u7eaa\u4e00\u4f17\u91cd\u8981\u6570\u5b66\u5bb6\u7684\u4fe1\u4ef0\u300217 \u4e16\u7eaa\u662f\u79d1\u5b66\u65f6\u4ee3\u4e0e\u73b0\u4ee3\u6570\u5b66\u7684\u5f00\u7aef，\u5f7c\u65f6\u7684\u6b27\u6d32，\u4fe1\u4ef0\u57fa\u7763\u6559\u4e0a\u5e1d\u662f\u4e00\u79cd\u666e\u904d\u73b0\u8c61，17 \u4e16\u7eaa\u6b27\u6d32\u6700\u6770\u51fa\u7684\u6570\u5b66\u5bb6\u4eec\u4ea6\u4e0d\u4f8b\u5916\u3002\u5f53\u7136，\u5173\u4e8e\u67d0\u4e00\u65f6\u671f\u54ea\u4f4d\u6570\u5b66\u5bb6\u6700\u4e3a\u91cd\u8981，\u5386\u6765\u4f17\u8bf4\u7eb7\u7ead，\u4f8b\u5982\u65af\u56fe\u5c14\u7279（2017）\u4fbf\u9074\u9009\u51fa\u4e86\u5386\u53f2\u4e0a\u7684 25 \u4f4d\u6770\u51fa\u6570\u5b66\u5bb6\u3002\u5c3d\u7ba1\u5982\u6b64，\u6211\u4eec\u4ecd\u6709\u5145\u5206\u7684\u7406\u7531\u5c06\u4ee5\u4e0b\u516d\u4f4d\u6570\u5b66\u5bb6\u89c6\u4f5c 17 \u4e16\u7eaa\u6700\u5177\u5f71\u54cd\u529b\u7684\u6570\u5b66\u5bb6：（1）\u6cd5\u56fd\u7684\u52d2\u5185\u00b7\u7b1b\u5361\u5c14（1596-1650），\u89e3\u6790\u51e0\u4f55\u7684\u53d1\u660e\u8005；\u5176\u6cd5\u56fd\u540c\u80de（2）\u76ae\u57c3\u5c14\u00b7\u5fb7\u00b7\u8d39\u9a6c（1607-1665）\u4e0e（3）\u5e03\u83b1\u58eb\u00b7\u5e15\u65af\u5361（1623-1662），\u4e8c\u4eba\u5171\u540c\u4e3a\u6982\u7387\u8bba\u5960\u5b9a\u4e86\u57fa\u7840；（4）\u82f1\u56fd\u7684\u827e\u8428\u514b\u00b7\u725b\u987f\u7235\u58eb（1642-1726）\u4e0e（5）\u5fb7\u56fd\u7684\u6208\u7279\u5f17\u91cc\u5fb7\u00b7\u5a01\u5ec9\u00b7\u83b1\u5e03\u5c3c\u8328（1646-1716），\u4e8c\u4eba\u5404\u81ea\u72ec\u7acb\u53d1\u660e\u4e86\u5fae\u79ef\u5206；（6）\u745e\u58eb\u7684\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229（1654-1705），\u201c\u5927\u6570\u5b9a\u5f8b\u201d\u4fbf\u662f\u4ee5\u4ed6\u7684\u540d\u5b57\u547d\u540d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f，\u8fd9\u516d\u4f4d\u6570\u5b66\u5bb6\u4e2d，\u6709\u4e94\u4f4d\u90fd\u8ba4\u4e3a\u81ea\u5df1\u7684\u6570\u5b66\u7814\u7a76\u4e0e\u57fa\u7763\u6559\u4fe1\u4ef0\u4e4b\u95f4\u5b58\u5728\u7740\u7d27\u5bc6\u7684\u8054\u7cfb（\u76ee\u524d\u5c1a\u672a\u53d1\u73b0\u8d39\u9a6c\u5bf9\u6b64\u7684\u76f8\u5173\u8bba\u8ff0）\u3002<\/p>\n

    Descartes provided two theistic proofs in his Meditations (Descartes 1976). Pascal provided a famous \u2018wager\u2019 in which he gives a probabilistic argument for choosing to believe in God (discussed in Heller 2018:42\u201344). Newton himself believed strongly in a Designer who worked through mathematical laws (Davies 1992:76). \u2018The most beautiful system of sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being\u2019 (Newton, quoted in Livio 2009:114). According to the Leibniz specialist Martin (1960), Leibniz regarded mathematical theorems as \u2018primarily and continuously thought by God\u2019, and when a mathematician discovers them, \u2018this knowing is a repetition of the primary divine knowing\u2019 (quoted in Hersh 1997:126). Bernoulli was a strong Calvinist and reflected on the theological implications of his discoveries in probability theory (Heller 2018:48\u201355). \u7b1b\u5361\u5c14\u5728\u5176\u300a\u7b2c\u4e00\u54f2\u5b66\u6c89\u601d\u96c6\u300b\u4e2d\u63d0\u51fa\u4e86\u4e24\u79cd\u795e\u5b58\u5728\u8bc1\u660e（\u7b1b\u5361\u5c14，1976）\u3002\u5e15\u65af\u5361\u63d0\u51fa\u4e86\u8457\u540d\u7684\u201c\u5e15\u65af\u5361\u8d4c\u6ce8\u201d，\u4ee5\u6982\u7387\u8bba\u7684\u89c6\u89d2\u8bba\u8bc1\u4e86\u4fe1\u4ef0\u4e0a\u5e1d\u7684\u5408\u7406\u6027（\u6d77\u52d2\u5bf9\u6b64\u6709\u76f8\u5173\u63a2\u8ba8，2018：42-44）\u3002\u725b\u987f\u672c\u4eba\u575a\u4fe1，\u5b58\u5728\u4e00\u4f4d\u901a\u8fc7\u6570\u5b66\u6cd5\u5219\u65bd\u5c55\u5176\u521b\u9020\u7684\u9020\u7269\u4e3b（\u6234\u7ef4\u65af，1992：76），\u4ed6\u66fe\u8a00：\u201c\u592a\u9633\u3001\u884c\u661f\u4e0e\u5f57\u661f\u6784\u6210\u7684\u8fd9\u4e00\u6700\u7f8e\u5999\u7684\u4f53\u7cfb，\u53ea\u80fd\u8bde\u751f\u4e8e\u4e00\u4f4d\u667a\u6167\u800c\u5168\u80fd\u7684\u5b58\u5728\u7684\u8c0b\u5212\u4e0e\u4e3b\u5bb0\u3002\u201d（\u725b\u987f，\u8f6c\u5f15\u81ea\u5229\u7ef4\u5965，2009：114）\u3002\u83b1\u5e03\u5c3c\u8328\u7814\u7a76\u4e13\u5bb6\u9a6c\u4e01（1960）\u6307\u51fa，\u83b1\u5e03\u5c3c\u8328\u5c06\u6570\u5b66\u5b9a\u7406\u89c6\u4f5c\u201c\u4e0a\u5e1d\u9996\u8981\u4e14\u6301\u7eed\u7684\u601d\u8003\u4e4b\u7269\u201d，\u800c\u6570\u5b66\u5bb6\u5bf9\u5b9a\u7406\u7684\u53d1\u73b0，\u201c\u8fd9\u79cd\u8ba4\u77e5\u662f\u5bf9\u795e\u5723\u672c\u6e90\u8ba4\u77e5\u7684\u91cd\u73b0\u201d（\u8f6c\u5f15\u81ea\u8d6b\u4ec0，1997：126）\u3002\u4f2f\u52aa\u5229\u662f\u575a\u5b9a\u7684\u52a0\u5c14\u6587\u6559\u5f92，\u4ed6\u6df1\u5165\u601d\u8003\u4e86\u81ea\u5df1\u5728\u6982\u7387\u8bba\u9886\u57df\u7684\u53d1\u73b0\u6240\u8574\u542b\u7684\u795e\u5b66\u610f\u4e49（\u6d77\u52d2，2018：48-55）\u3002<\/p>\n

    In a world where mathematical truths were seen as part of the divine mind, it was easy to argue that God reveals himself through mathematics. The quotation from Edward Everett (1794\u20131865) gives a poetic description of this viewpoint: \u5728\u90a3\u4e2a\u5c06\u6570\u5b66\u771f\u7406\u89c6\u4f5c\u795e\u5723\u601d\u7ef4\u4e00\u90e8\u5206\u7684\u65f6\u4ee3，\u4eba\u4eec\u5f88\u5bb9\u6613\u8bba\u8bc1\u4e0a\u5e1d\u901a\u8fc7\u6570\u5b66\u5f70\u663e\u81ea\u8eab\u3002\u7231\u5fb7\u534e\u00b7\u57c3\u5f17\u96f7\u7279（1794-1865）\u7684\u4e00\u6bb5\u8bdd\u8bd7\u610f\u5730\u8be0\u91ca\u4e86\u8fd9\u4e00\u89c2\u70b9：<\/p>\n

    In the pure mathematics, we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven. (quoted in Hersh 1997:9) \u5728\u7eaf\u7cb9\u7684\u6570\u5b66\u4e2d，\u6211\u4eec\u51dd\u89c6\u7740\u7edd\u5bf9\u7684\u771f\u7406，\u8fd9\u4e9b\u771f\u7406\u5728\u6668\u661f\u4e00\u540c\u6b4c\u5531\u4e4b\u524d，\u4fbf\u5df2\u5b58\u5728\u4e8e\u795e\u5723\u7684\u601d\u7ef4\u4e4b\u4e2d；\u5373\u4fbf\u5f53\u6700\u540e\u4e00\u9897\u661f\u8fb0\u4ece\u5929\u9645\u9668\u843d，\u5b83\u4eec\u4e5f\u5c06\u6c38\u5b58\u4e8e\u90a3\u795e\u5723\u7684\u601d\u7ef4\u91cc\u3002（\u8f6c\u5f15\u81ea\u8d6b\u4ec0，1997：9）<\/p>\n

    Similar statements are known from mathematicians with a different religious background. For example, the remarkable Indian mathematician S. Ramanujan (1887\u20131920) once noted: \u2018An equation means nothing to me unless it expresses a thought of God\u2019 (quoted in Kanigel 1991: pos 221). \u4e0d\u540c\u5b97\u6559\u80cc\u666f\u7684\u6570\u5b66\u5bb6\u4e5f\u6709\u8fc7\u7c7b\u4f3c\u7684\u8868\u8ff0\u3002\u4f8b\u5982，\u5370\u5ea6\u6770\u51fa\u7684\u6570\u5b66\u5bb6 S\u00b7\u62c9\u9a6c\u52aa\u91d1（1887-1920）\u66fe\u8bf4：\u201c\u82e5\u4e00\u4e2a\u65b9\u7a0b\u65e0\u6cd5\u8868\u8fbe\u4e0a\u5e1d\u7684\u601d\u60f3，\u90a3\u5b83\u5bf9\u6211\u800c\u8a00\u4fbf\u6beb\u65e0\u610f\u4e49\u3002\u201d（\u8f6c\u5f15\u81ea\u5361\u5c3c\u683c\u5c14，1991：\u7b2c 221 \u6bb5）\u3002<\/p>\n

    Discovery of non-Euclidean geometry and its implications<\/h4>\n

    \u975e\u6b27\u51e0\u4f55\u7684\u53d1\u73b0\u53ca\u5176\u610f\u4e49<\/h4>\n

    As the above examples show, the validity of Euclidean geometry used to be regarded as indisputable and was often taken as an analogy for certainty in theological issues. \u2018Geometry served from the time of Plato as proof that certainty is possible in human knowledge \u2013 including religious certainty\u2019 (Hersh 1997:137). This reasoning was challenged by the discovery of non-Euclidean geometry in the 19th century by Gauss (Germany), Lobachevsky (Russia), Bolyai (Hungary) and finally Riemann (Germany). \u5982\u4e0a\u8ff0\u4f8b\u5b50\u6240\u793a，\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u7684\u6709\u6548\u6027\u66fe\u88ab\u89c6\u4f5c\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406，\u4e14\u5e38\u88ab\u7528\u4f5c\u7c7b\u6bd4，\u8bba\u8bc1\u795e\u5b66\u95ee\u9898\u4e2d\u5b58\u5728\u786e\u5b9a\u7684\u7b54\u6848\u3002\u201c\u81ea\u67cf\u62c9\u56fe\u65f6\u4ee3\u8d77，\u51e0\u4f55\u5b66\u4fbf\u6210\u4e3a\u8bc1\u660e\u4eba\u7c7b\u7684\u8ba4\u77e5\u4e2d\u5b58\u5728\u786e\u5b9a\u6027\u7684\u4f9d\u636e\u2014\u2014\u8fd9\u5176\u4e2d\u4e5f\u5305\u62ec\u5b97\u6559\u8ba4\u77e5\u7684\u786e\u5b9a\u6027\u3002\u201d（\u8d6b\u4ec0，1997：137）\u3002\u800c 19 \u4e16\u7eaa，\u9ad8\u65af（\u5fb7\u56fd）\u3001\u7f57\u5df4\u5207\u592b\u65af\u57fa（\u4fc4\u7f57\u65af）\u3001\u9c8d\u8036（\u5308\u7259\u5229）\u4ee5\u53ca\u9ece\u66fc（\u5fb7\u56fd）\u76f8\u7ee7\u53d1\u73b0\u4e86\u975e\u6b27\u51e0\u4f55，\u8fd9\u4e00\u8bba\u8bc1\u903b\u8f91\u53d7\u5230\u4e86\u6311\u6218\u3002<\/p>\n

    The angle-sum theorem teaches that the angles in a triangle add up to 180\u00b0. This theorem was often used as an example of an absolutely certain statement. For example, Descartes (1976:59) claimed in the fifth meditation that the existence of God is as certain as the fact that three angles of a triangle are equal to two right angles. Spinoza regarded this theorem as indubitable (Hersh 1997:121). David Hume noted that Euclidean geometry was as solid as the rock of Gibraltar (Livio 2009:151), and for Kant, who regarded the concept of space as a priori, the space was indisputably Euclidean (Kant 1966:88; Livio 2009:152). \u4e09\u89d2\u5f62\u5185\u89d2\u548c\u5b9a\u7406\u6307\u51fa，\u4e09\u89d2\u5f62\u7684\u5185\u89d2\u548c\u4e3a<\/p>\n

    180<\/p>\n

    \u2218<\/p>\n

    180^\\\\circ<\/p>\n

    <\/span><\/span>18<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u8fd9\u4e00\u5b9a\u7406\u5e38\u88ab\u7528\u4f5c\u7edd\u5bf9\u786e\u5b9a\u547d\u9898\u7684\u8303\u4f8b\u3002\u4f8b\u5982，\u7b1b\u5361\u5c14\u5728\u5176\u300a\u7b2c\u4e00\u54f2\u5b66\u6c89\u601d\u96c6\u300b\u7684\u7b2c\u4e94\u4e2a\u6c89\u601d\u4e2d\u63d0\u51fa，\u4e0a\u5e1d\u7684\u5b58\u5728\u4e0e\u4e09\u89d2\u5f62\u7684\u5185\u89d2\u548c\u7b49\u4e8e\u4e24\u4e2a\u76f4\u89d2\u8fd9\u4e00\u4e8b\u5b9e\u540c\u6837\u786e\u5b9a（\u7b1b\u5361\u5c14，1976：59）\u3002\u65af\u5bbe\u8bfa\u838e\u5c06\u8fd9\u4e00\u5b9a\u7406\u89c6\u4f5c\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406（\u8d6b\u4ec0，1997：121）\u3002\u5927\u536b\u00b7\u4f11\u8c1f\u8ba4\u4e3a，\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u5982\u540c\u76f4\u5e03\u7f57\u9640\u5de8\u5ca9\u4e00\u822c\u575a\u4e0d\u53ef\u6467（\u5229\u7ef4\u5965，2009：151）；\u800c\u5eb7\u5fb7\u5c06\u7a7a\u95f4\u6982\u5ff5\u89c6\u4f5c\u5148\u9a8c\u7684\u5b58\u5728，\u5728\u4ed6\u770b\u6765，\u7a7a\u95f4\u5fc5\u7136\u662f\u6b27\u51e0\u91cc\u5f97\u5f0f\u7684（\u5eb7\u5fb7，1966：88；\u5229\u7ef4\u5965，2009：152）\u3002<\/p>\n

    But this theorem does not hold in non-Euclidean geometry. The sum of the angles in a triangle can be more than 180\u00b0 (elliptic or Riemannian geometry) or less than 180\u00b0 (hyperbolic geometry).\u2075 \u4f46\u8fd9\u4e00\u5b9a\u7406\u5728\u975e\u6b27\u51e0\u4f55\u4e2d\u5e76\u4e0d\u6210\u7acb\u3002\u5728\u975e\u6b27\u51e0\u4f55\u4e2d，\u4e09\u89d2\u5f62\u7684\u5185\u89d2\u548c\u53ef\u5927\u4e8e<\/p>\n

    180<\/p>\n

    \u2218<\/p>\n

    180^\\\\circ<\/p>\n

    <\/span><\/span>18<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>（\u692d\u5706\u51e0\u4f55\u6216\u9ece\u66fc\u51e0\u4f55），\u4e5f\u53ef\u5c0f\u4e8e<\/p>\n

    180<\/p>\n

    \u2218<\/p>\n

    180^\\\\circ<\/p>\n

    <\/span><\/span>18<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>（\u53cc\u66f2\u51e0\u4f55）\u2075\u3002<\/p>\n

  • See Figure 41 in Livio 2009:155. \u53c2\u89c1\u5229\u7ef4\u5965（2009：155）\u7684\u56fe 41\u3002<\/li>\n

    This discovery led to a re-thinking of the foundations of mathematics. The famous French mathematician and physicist Henri Poincar\u00e9 (1854\u20131912) made the consequences very clear: The geometrical axioms are therefore neither synthetic \u00e0 priori intuitions\u2076 nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free \u2026 (Poincar\u00e9 1905:50) \u8fd9\u4e00\u53d1\u73b0\u4fc3\u4f7f\u4eba\u4eec\u91cd\u65b0\u601d\u8003\u6570\u5b66\u7684\u57fa\u7840\u3002\u6cd5\u56fd\u8457\u540d\u6570\u5b66\u5bb6\u3001\u7269\u7406\u5b66\u5bb6\u4ea8\u5229\u00b7\u5e9e\u52a0\u83b1（1854-1912）\u6e05\u6670\u5730\u9610\u660e\u4e86\u8fd9\u4e00\u53d1\u73b0\u7684\u5f71\u54cd： \u56e0\u6b64，\u51e0\u4f55\u516c\u7406\u65e2\u975e\u5148\u5929\u7efc\u5408\u76f4\u89c2\u2076，\u4ea6\u975e\u5b9e\u9a8c\u4e8b\u5b9e，\u800c\u662f\u4e00\u79cd\u7ea6\u5b9a\u3002\u6211\u4eec\u5728\u8bf8\u591a\u53ef\u80fd\u7684\u7ea6\u5b9a\u4e2d\u4f5c\u51fa\u9009\u62e9\u65f6，\u4f1a\u4ee5\u5b9e\u9a8c\u4e8b\u5b9e\u4e3a\u6307\u5f15，\u4f46\u8fd9\u79cd\u9009\u62e9\u59cb\u7ec8\u662f\u81ea\u7531\u7684\u2026\u2026（\u5e9e\u52a0\u83b1，1905：50）<\/p>\n

    The new situation was that the mathematicians would offer different sorts of geometry from which the physicists had to choose the geometry, which would be most appropriate for studying the physical world. This new situation challenged the understanding of mathematics as a divine science. \u6b64\u540e，\u6570\u5b66\u5bb6\u63d0\u51fa\u4e86\u591a\u79cd\u51e0\u4f55\u4f53\u7cfb，\u7269\u7406\u5b66\u5bb6\u5219\u9700\u8981\u4ece\u4e2d\u9009\u62e9\u6700\u9002\u5408\u7814\u7a76\u7269\u7406\u4e16\u754c\u7684\u51e0\u4f55\u4f53\u7cfb\u3002\u8fd9\u4e00\u65b0\u7684\u5c40\u9762，\u5bf9\u5c06\u6570\u5b66\u89c6\u4f5c\u795e\u5723\u79d1\u5b66\u7684\u4f20\u7edf\u8ba4\u77e5\u63d0\u51fa\u4e86\u6311\u6218\u3002<\/p>\n

    The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. (Livio 2009:159) \u4eba\u4eec\u53ef\u4ee5\u9009\u62e9\u4e0d\u540c\u7684\u516c\u7406\u4f53\u7cfb，\u6784\u5efa\u51fa\u4e0d\u540c\u7684\u51e0\u4f55\u4f53\u7cfb，\u8fd9\u4e00\u4e8b\u5b9e\u9996\u6b21\u8ba9\u4eba\u4eec\u4ea7\u751f\u4e86\u8fd9\u6837\u7684\u8d28\u7591：\u6570\u5b66\u7ec8\u7a76\u662f\u4eba\u7c7b\u7684\u53d1\u660e，\u800c\u975e\u5bf9\u72ec\u7acb\u4e8e\u4eba\u7c7b\u601d\u7ef4\u4e4b\u5916\u7684\u771f\u7406\u7684\u53d1\u73b0（\u5229\u7ef4\u5965，2009：159）\u3002<\/p>\n

    At this point, it became obvious that intuition alone is not sufficient for mathematical certainty. Cantor\u2019s set theory was a candidate for the foundation of mathematics. But then paradoxes in the set theory were discovered by Zermelo in 1899 and by Russell in 1901 independently of each other. This finally led to three different schools of thought on how to lay a good foundation for mathematics: logicism, formalism and intuitionism (Shapiro 2000:107\u2013197). It turned out that each of these three major schools has its deficiencies; none can fulfil the promises of their inventors. Intuitionism is almost dead, and G\u00f6del destroyed \u2018all hope for a consistent and complete axiomatization of mathematics\u2019 (Mendelson 1987:175). \u81f3\u6b64，\u4eba\u4eec\u660e\u786e\u8ba4\u8bc6\u5230，\u4ec5\u51ed\u76f4\u89c2\u65e0\u6cd5\u5b9e\u73b0\u6570\u5b66\u7684\u786e\u5b9a\u6027\u3002\u5eb7\u6258\u5c14\u7684\u96c6\u5408\u8bba\u66fe\u88ab\u89c6\u4f5c\u5960\u5b9a\u6570\u5b66\u57fa\u7840\u7684\u5019\u9009\u7406\u8bba，\u7136\u800c\u7b56\u6885\u6d1b\u4e8e 1899 \u5e74\u3001\u7f57\u7d20\u4e8e 1901 \u5e74\u5404\u81ea\u72ec\u7acb\u53d1\u73b0\u4e86\u96c6\u5408\u8bba\u4e2d\u7684\u6096\u8bba\u3002\u8fd9\u4e00\u53d1\u73b0\u6700\u7ec8\u50ac\u751f\u4e86\u4e09\u5927\u6570\u5b66\u57fa\u7840\u5b66\u6d3e：\u903b\u8f91\u4e3b\u4e49\u3001\u5f62\u5f0f\u4e3b\u4e49\u4e0e\u76f4\u89c9\u4e3b\u4e49（\u590f\u76ae\u7f57，2000：107-197）\u3002\u4e8b\u5b9e\u8bc1\u660e，\u8fd9\u4e09\u5927\u4e3b\u6d41\u5b66\u6d3e\u5747\u5b58\u5728\u7f3a\u9677，\u6ca1\u6709\u4e00\u4e2a\u5b66\u6d3e\u80fd\u5b9e\u73b0\u5176\u521b\u7acb\u8005\u7684\u521d\u8877\u3002\u76f4\u89c9\u4e3b\u4e49\u5982\u4eca\u51e0\u8fd1\u6d88\u4ea1，\u800c\u54e5\u5fb7\u5c14\u7684\u7406\u8bba\u5219\u5f7b\u5e95\u6253\u7834\u4e86\u201c\u4e3a\u6570\u5b66\u5efa\u7acb\u4e00\u81f4\u4e14\u5b8c\u5907\u7684\u516c\u7406\u5316\u4f53\u7cfb\u7684\u6240\u6709\u5e0c\u671b\u201d（\u95e8\u5fb7\u5c14\u677e，1987：175）\u3002<\/p>\n

    Actually, the question of the foundation of mathematics has no clear answer to this day (Hersh 1997, Shapiro 2000). As a consequence, the vast majority of working mathematicians do not spend too much time on the question of foundations. \u4e8b\u5b9e\u4e0a，\u6570\u5b66\u57fa\u7840\u95ee\u9898\u81f3\u4eca\u4ecd\u65e0\u660e\u786e\u7b54\u6848（\u8d6b\u4ec0，1997；\u590f\u76ae\u7f57，2000）\u3002\u56e0\u6b64，\u7edd\u5927\u591a\u6570\u4e00\u7ebf\u6570\u5b66\u5bb6\u5e76\u4e0d\u4f1a\u5728\u6570\u5b66\u57fa\u7840\u95ee\u9898\u4e0a\u8017\u8d39\u8fc7\u591a\u7cbe\u529b\u3002<\/p>\n

    Pragmatic Platonism afterwards<\/h4>\n

    \u6b64\u540e\u7684\u5b9e\u7528\u67cf\u62c9\u56fe\u4e3b\u4e49<\/h4>\n

    Although Platonism is no longer considered an up-to-date epistemology or philosophy of science, it is still alive among working mathematicians. A crucial question to mathematicians is \u2018is mathematics created or discovered?\u2019 In 1940, the renowned British mathematician G.F. Hardy (1877\u20131947) wrote about the \u2018mathematical reality\u2019, admitting that there is no sort of agreement on the nature of this reality (Hardy 2001:123). Hardy (p. 123) believed \u2018that the mathematical reality lies outside us, that our function it to discover or observe it\u2019. Similarly, the German-Austrian logician Kurt G\u00f6del (1906\u20131978) argued in 1944: Classes and concepts may \u2026 be considered as real objects \u2026 It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. (quoted in Shapiro 2000:10) \u5c3d\u7ba1\u67cf\u62c9\u56fe\u4e3b\u4e49\u5df2\u4e0d\u518d\u88ab\u89c6\u4f5c\u524d\u6cbf\u7684\u8ba4\u8bc6\u8bba\u6216\u79d1\u5b66\u54f2\u5b66\u7406\u8bba，\u4f46\u5728\u4e00\u7ebf\u6570\u5b66\u5bb6\u4e2d，\u8fd9\u4e00\u601d\u60f3\u4ecd\u6709\u4e00\u5e2d\u4e4b\u5730\u3002\u5bf9\u6570\u5b66\u5bb6\u800c\u8a00，\u4e00\u4e2a\u6838\u5fc3\u95ee\u9898\u662f：\u201c\u6570\u5b66\u662f\u88ab\u521b\u9020\u7684，\u8fd8\u662f\u88ab\u53d1\u73b0\u7684？\u201d1940 \u5e74，\u82f1\u56fd\u8457\u540d\u6570\u5b66\u5bb6 G\u00b7F\u00b7\u54c8\u4ee3（1877-1947）\u5728\u63a2\u8ba8\u201c\u6570\u5b66\u5b9e\u5728\u201d\u65f6\u5766\u8a00，\u5b66\u754c\u5bf9\u8fd9\u4e00\u5b9e\u5728\u7684\u672c\u8d28\u59cb\u7ec8\u672a\u80fd\u8fbe\u6210\u5171\u8bc6（\u54c8\u4ee3，2001：123）\u3002\u54c8\u4ee3\u8ba4\u4e3a，\u201c\u6570\u5b66\u5b9e\u5728\u5b58\u5728\u4e8e\u6211\u4eec\u4e4b\u5916，\u6211\u4eec\u7684\u4efb\u52a1\u662f\u53bb\u53d1\u73b0\u6216\u89c2\u5bdf\u5b83\u201d（\u540c\u4e0a，\u7b2c 123 \u9875）\u3002\u65e0\u72ec\u6709\u5076，\u5fb7\u88d4\u5965\u5730\u5229\u903b\u8f91\u5b66\u5bb6\u5e93\u5c14\u7279\u00b7\u54e5\u5fb7\u5c14（1906-1978）\u5728 1944 \u5e74\u63d0\u51fa： \u7c7b\u4e0e\u6982\u5ff5\u53ef\u4ee5\u88ab\u89c6\u4f5c\u771f\u5b9e\u7684\u5ba2\u4f53\u2026\u2026\u5728\u6211\u770b\u6765，\u5047\u5b9a\u8fd9\u7c7b\u5ba2\u4f53\u7684\u5b58\u5728，\u4e0e\u5047\u5b9a\u7269\u7406\u5b9e\u4f53\u7684\u5b58\u5728\u540c\u6837\u5408\u7406，\u6211\u4eec\u4e5f\u6709\u540c\u6837\u5145\u5206\u7684\u7406\u7531\u76f8\u4fe1\u5b83\u4eec\u7684\u5b58\u5728\u3002（\u8f6c\u5f15\u81ea\u590f\u76ae\u7f57，2000：10）<\/p>\n

  • This was Kant\u2019s position presented in 1781 in his Kritik der reinen Vernunft. \u2018Geometrie ist eine Wissenschaft, welche die Eigenschaften des Raums synthetisch und doch a priori bestimmt\u2019 (Kant 1966:88). \u8fd9\u662f\u5eb7\u5fb7\u5728 1781 \u5e74\u51fa\u7248\u7684\u300a\u7eaf\u7cb9\u7406\u6027\u6279\u5224\u300b\u4e2d\u63d0\u51fa\u7684\u89c2\u70b9，\u5176\u539f\u6587\u4e3a：\u201c\u51e0\u4f55\u5b66\u662f\u4e00\u95e8\u7efc\u5408\u5730\u4e14\u5148\u5929\u5730\u89c4\u5b9a\u7a7a\u95f4\u5c5e\u6027\u7684\u79d1\u5b66\u3002\u201d（\u5eb7\u5fb7，1966：88）\u3002<\/li>\n

    A small survey done by Pamela Aschbacher, the spouse of an American mathematician shows that this is probably the thinking of most working mathematicians. Aschbacher (2015:17) asked the colleagues of her husband whether math was something like an unseen star, out there to be discovered, or something created by us. Except in one case, she received the answer: \u2018Oh, it\u2019s definitely the \u201ctruths\u201d of the world waiting to be discovered\u2019 (p. 17). This attitude is also implicit in mathematical parlance, because most mathematicians would say: \u2018I discovered this mathematical law\u2019. The underlying assumption is that it was already there. \u7f8e\u56fd\u4e00\u4f4d\u6570\u5b66\u5bb6\u7684\u914d\u5076\u5e15\u6885\u62c9\u00b7\u963f\u5e0c\u5df4\u8d6b\u5f00\u5c55\u7684\u4e00\u9879\u5c0f\u578b\u8c03\u67e5\u8868\u660e，\u8fd9\u4e00\u89c2\u70b9\u6216\u8bb8\u4ee3\u8868\u4e86\u7edd\u5927\u591a\u6570\u4e00\u7ebf\u6570\u5b66\u5bb6\u7684\u60f3\u6cd5\u3002\u963f\u5e0c\u5df4\u8d6b\u8be2\u95ee\u5176\u4e08\u592b\u7684\u540c\u4e8b，\u6570\u5b66\u7a76\u7adf\u662f\u5982\u540c\u9065\u8fdc\u5929\u9645\u4e2d\u672a\u88ab\u53d1\u73b0\u7684\u661f\u8fb0，\u7b49\u5f85\u4eba\u4eec\u53bb\u63a2\u5bfb，\u8fd8\u662f\u7531\u4eba\u7c7b\u521b\u9020\u7684\u4ea7\u7269（2015：17）\u3002\u9664\u4e00\u4eba\u4e4b\u5916，\u6240\u6709\u4eba\u90fd\u7ed9\u51fa\u4e86\u8fd9\u6837\u7684\u7b54\u6848：\u201c\u663e\u7136，\u6570\u5b66\u662f\u4e16\u95f4\u7b49\u5f85\u88ab\u53d1\u73b0\u7684\u771f\u7406\u3002\u201d（\u540c\u4e0a，\u7b2c 17 \u9875）\u3002\u8fd9\u79cd\u6001\u5ea6\u4e5f\u9690\u542b\u5728\u6570\u5b66\u7684\u4e13\u4e1a\u8868\u8ff0\u4e2d，\u591a\u6570\u6570\u5b66\u5bb6\u90fd\u4f1a\u8bf4\u201c\u6211\u53d1\u73b0\u4e86\u8fd9\u4e00\u6570\u5b66\u5b9a\u5f8b\u201d，\u5176\u80cc\u540e\u7684\u9884\u8bbe\u662f，\u8fd9\u4e00\u5b9a\u5f8b\u672c\u5c31\u5b58\u5728\u3002<\/p>\n

    Harrison (2017:481) writes that \u2018virtually everyone in (the mathematical) community assumes a realist interpretation of ontology\u2019. This statement \u2018virtually everyone\u2019 might be a bit too strong. For example, the American mathematician Reuben Hersh (1927\u20132020) does not share this realist interpretation and quotes several mathematicians on his side (Hersh 1997:182\u2013232). Actually Hersh strongly objects to the fact that mainstream mathematicians still stick to Platonism: \u2018Why do mathematicians believe something so unscientific, so far-fetched as an independent immaterial timeless world of mathematical truth?\u2019 (Hersh 1997:11).\u2077 \u2018The trouble with today\u2019s Platonism is that it gives up God, but wants to keep mathematics a thought in the mind of God\u2019 (p. 135). \u54c8\u91cc\u68ee（2017：481）\u5199\u9053：\u201c\u6570\u5b66\u754c\u51e0\u4e4e\u6240\u6709\u4eba\u90fd\u79c9\u6301\u672c\u4f53\u8bba\u7684\u5b9e\u5728\u8bba\u89e3\u91ca\u3002\u201d\u201c\u51e0\u4e4e\u6240\u6709\u4eba\u201d\u8fd9\u4e00\u8868\u8ff0\u6216\u8bb8\u7565\u663e\u7edd\u5bf9\u3002\u4f8b\u5982，\u7f8e\u56fd\u6570\u5b66\u5bb6\u9c81\u672c\u00b7\u8d6b\u4ec0（1927-2020）\u4fbf\u4e0d\u8ba4\u540c\u8fd9\u79cd\u5b9e\u5728\u8bba\u89e3\u91ca，\u8fd8\u63f4\u5f15\u4e86\u591a\u4f4d\u6301\u76f8\u540c\u89c2\u70b9\u7684\u6570\u5b66\u5bb6\u7684\u8bba\u8ff0（\u8d6b\u4ec0，1997：182-232）\u3002\u4e8b\u5b9e\u4e0a，\u8d6b\u4ec0\u5f3a\u70c8\u53cd\u5bf9\u4e3b\u6d41\u6570\u5b66\u5bb6\u5bf9\u67cf\u62c9\u56fe\u4e3b\u4e49\u7684\u575a\u5b88，\u4ed6\u63d0\u51fa\u8d28\u7591：\u201c\u4e3a\u4f55\u6570\u5b66\u5bb6\u4f1a\u76f8\u4fe1\u6570\u5b66\u771f\u7406\u5b58\u5728\u4e8e\u4e00\u4e2a\u72ec\u7acb\u3001\u975e\u7269\u8d28\u3001\u6c38\u6052\u7684\u4e16\u754c\u4e4b\u4e2d，\u8fd9\u4e00\u89c2\u70b9\u65e2\u4e0d\u79d1\u5b66，\u53c8\u8352\u8bde\u4e0d\u7ecf\u3002\u201d（\u8d6b\u4ec0，1997：11）\u2077\u3002\u4ed6\u8fd8\u6307\u51fa：\u201c\u5982\u4eca\u7684\u67cf\u62c9\u56fe\u4e3b\u4e49\u7684\u95ee\u9898\u5728\u4e8e，\u5b83\u629b\u5f03\u4e86\u4e0a\u5e1d，\u5374\u4ecd\u60f3\u5c06\u6570\u5b66\u89c6\u4f5c\u5b58\u5728\u4e8e\u4e0a\u5e1d\u601d\u7ef4\u4e2d\u7684\u4e8b\u7269\u3002\u201d（\u540c\u4e0a，\u7b2c 135 \u9875）\u3002<\/p>\n

  • See similar remarks on Hersh (1997:13, 236), etc. \u4ea6\u53ef\u53c2\u89c1\u8d6b\u4ec0\u7684\u76f8\u5173\u8bba\u8ff0（1997：13\u3001236）\u7b49\u3002<\/li>\n

    As rightly noted by Hersh (1997:42), the working mathematician usually oscillates between Platonism and formalism: \u8d6b\u4ec0（1997：42）\u7684\u89c2\u70b9\u5341\u5206\u4e2d\u80af，\u4ed6\u6307\u51fa\u4e00\u7ebf\u6570\u5b66\u5bb6\u7684\u601d\u60f3\u5f80\u5f80\u5728\u67cf\u62c9\u56fe\u4e3b\u4e49\u4e0e\u5f62\u5f0f\u4e3b\u4e49\u4e4b\u95f4\u6447\u6446：<\/p>\n

    The working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he\u2019s a Platonist, convinced he\u2019s dealing with an objective reality whose properties he\u2019s trying to determine. On weekends, if challenged to give a philosophical account of this reality, it\u2019s easiest to pretend he doesn\u2019t believe in it. He plays formalist, and pretends mathematics is a meaningless game. (Hersh 1997:39f.)\u2078 \u4e00\u7ebf\u6570\u5b66\u5bb6\u5728\u5de5\u4f5c\u65e5\u662f\u67cf\u62c9\u56fe\u4e3b\u4e49\u8005，\u5728\u5468\u672b\u5219\u6210\u4e86\u5f62\u5f0f\u4e3b\u4e49\u8005\u3002\u5de5\u4f5c\u65e5\u505a\u7814\u7a76\u65f6，\u4ed6\u575a\u4fe1\u81ea\u5df1\u6240\u7814\u7a76\u7684\u662f\u5ba2\u89c2\u7684\u5b9e\u5728，\u81f4\u529b\u4e8e\u63a2\u5bfb\u5176\u5c5e\u6027，\u662f\u575a\u5b9a\u7684\u67cf\u62c9\u56fe\u4e3b\u4e49\u8005；\u800c\u5230\u4e86\u5468\u672b，\u82e5\u88ab\u8981\u6c42\u4ece\u54f2\u5b66\u89d2\u5ea6\u9610\u91ca\u8fd9\u4e00\u5b9e\u5728，\u6700\u7701\u4e8b\u7684\u505a\u6cd5\u4fbf\u662f\u4f6f\u88c5\u81ea\u5df1\u5e76\u4e0d\u76f8\u4fe1\u5b83\u7684\u5b58\u5728，\u8f6c\u800c\u7ad9\u5728\u5f62\u5f0f\u4e3b\u4e49\u7684\u7acb\u573a，\u5c06\u6570\u5b66\u89c6\u4f5c\u4e00\u573a\u65e0\u610f\u4e49\u7684\u6e38\u620f\u3002（\u8d6b\u4ec0，1997：39-40）\u2078<\/p>\n

  • This illustrative description is also quoted in Livio (2009:225). \u8fd9\u4e00\u5f62\u8c61\u7684\u63cf\u8ff0\u4e5f\u88ab\u5229\u7ef4\u5965\u5f15\u7528（2009：225）\u3002 This attitude could be called \u2018pragmatic Platonism\u2019 (Kessler 2019:51). \u8fd9\u79cd\u6001\u5ea6\u53ef\u88ab\u79f0\u4f5c\u201c\u5b9e\u7528\u67cf\u62c9\u56fe\u4e3b\u4e49\u201d（\u514b\u65af\u52d2，2019：51）\u3002<\/li>\n

    Hersh has a different view in mathematics. He sees mathematics as part of human culture and history and regards mathematics as a \u2018socio-historical reality\u2019 (Hersh 1997:17), comparable, for example, with institutions like the Supreme Court. \u8d6b\u4ec0\u5219\u5bf9\u6570\u5b66\u6709\u7740\u4e0d\u540c\u7684\u7406\u89e3\u3002\u4ed6\u5c06\u6570\u5b66\u89c6\u4f5c\u4eba\u7c7b\u6587\u5316\u4e0e\u5386\u53f2\u7684\u4e00\u90e8\u5206，\u8ba4\u4e3a\u6570\u5b66\u662f\u4e00\u79cd\u201c\u793e\u4f1a\u5386\u53f2\u5b9e\u5728\u201d（\u8d6b\u4ec0，1997：17），\u53ef\u4e0e\u6700\u9ad8\u6cd5\u9662\u8fd9\u7c7b\u793e\u4f1a\u673a\u6784\u76f8\u7c7b\u6bd4\u3002<\/p>\n

    My own position<\/h4>\n

    \u7b14\u8005\u7684\u89c2\u70b9<\/h4>\n

    I think that Livio presents a well-balanced answer to the question of whether mathematics is created or discovered. Our mathematics is a combination of inventions and discoveries\u2026 Humans commonly invent mathematical concepts and discover the relations among those concepts. (Livio 2009:238, 242) \u7b14\u8005\u8ba4\u4e3a，\u5229\u7ef4\u5965\u5bf9\u201c\u6570\u5b66\u662f\u88ab\u521b\u9020\u8fd8\u662f\u88ab\u53d1\u73b0\u201d\u8fd9\u4e00\u95ee\u9898\u7ed9\u51fa\u4e86\u6700\u4e3a\u516c\u5141\u7684\u7b54\u6848： \u6211\u4eec\u7684\u6570\u5b66，\u662f\u53d1\u660e\u4e0e\u53d1\u73b0\u7684\u7ed3\u5408\u2026\u2026\u4eba\u7c7b\u521b\u9020\u6570\u5b66\u6982\u5ff5，\u800c\u540e\u53d1\u73b0\u8fd9\u4e9b\u6982\u5ff5\u4e4b\u95f4\u7684\u5173\u8054\u3002（\u5229\u7ef4\u5965，2009：238\u3001242）<\/p>\n

    A mathematician might invent a mathematical concept like a \u2018prime ideal\u2019or a \u2018crossed-product order\u2019 (Kessler 1994). \u6570\u5b66\u5bb6\u53ef\u80fd\u4f1a\u521b\u9020\u51fa\u201c\u7d20\u7406\u60f3\u201d\u6216\u201c\u4ea4\u53c9\u79ef\u5e8f\u201d\u8fd9\u7c7b\u6570\u5b66\u6982\u5ff5（\u514b\u65af\u52d2，1994）\u3002<\/p>\n

    But once the concept and its axioms are in place, the mathematician is bound by this chosen concept. It is a bit comparable to the writing of a novel. An author can freely choose a character for a book, but once the character is chosen the author has to stick to it. The axioms of a mathematical object can be freely chosen, invented, but then the theorems about this object are discovered. \u4f46\u4e00\u65e6\u6570\u5b66\u6982\u5ff5\u53ca\u5176\u516c\u7406\u4f53\u7cfb\u786e\u7acb，\u6570\u5b66\u5bb6\u7684\u7814\u7a76\u4fbf\u4f1a\u53d7\u9650\u4e8e\u8fd9\u4e00\u65e2\u5b9a\u6982\u5ff5\u3002\u8fd9\u4e00\u70b9\u4e0e\u5c0f\u8bf4\u521b\u4f5c\u9887\u6709\u76f8\u4f3c\u4e4b\u5904：\u4f5c\u8005\u53ef\u4ee5\u81ea\u7531\u5851\u9020\u5c0f\u8bf4\u4e2d\u7684\u4eba\u7269，\u4f46\u4eba\u7269\u5f62\u8c61\u4e00\u65e6\u786e\u7acb，\u4f5c\u8005\u7684\u521b\u4f5c\u4fbf\u9700\u7b26\u5408\u5176\u6027\u683c\u8bbe\u5b9a\u3002\u6570\u5b66\u5ba2\u4f53\u7684\u516c\u7406\u53ef\u4ee5\u7531\u4eba\u7c7b\u81ea\u7531\u9009\u62e9\u3001\u521b\u9020，\u4f46\u5173\u4e8e\u8be5\u5ba2\u4f53\u7684\u5b9a\u7406，\u5219\u662f\u4eba\u4eec\u53bb\u53d1\u73b0\u7684\u7ed3\u679c\u3002<\/p>\n

    Without digging deeper into the ontology of mathematics, I just want to state that I have much sympathy for Popper\u2019s model of the three worlds. There is the physical world (world 1), the mental or psychological world (world 2), which is our inner world, inaccessible from outside and there is a third world (world 3), consisting of \u2018the products of the human mind\u2019 (Popper 1980:44) such as concepts, languages and theorems. At the moment when Fermat made public the formula, which was in his mind (world 2) and became later known as \u2018Fermat\u2019s Last Theorem\u2019, this formula could be discussed and analysed by the mathematical community as part of world 3. Singh (1997) tells the story of these discussions. \u7b14\u8005\u4e0d\u6253\u7b97\u6df1\u5165\u63a2\u8ba8\u6570\u5b66\u7684\u672c\u4f53\u8bba\u95ee\u9898，\u4ec5\u60f3\u8868\u660e，\u7b14\u8005\u5341\u5206\u8ba4\u540c\u6ce2\u666e\u5c14\u7684\u4e09\u4e2a\u4e16\u754c\u7406\u8bba\u3002\u6ce2\u666e\u5c14\u63d0\u51fa，\u4e16\u754c\u5206\u4e3a\u4e09\u4e2a\u5c42\u6b21：\u7269\u7406\u4e16\u754c（\u4e16\u754c 1）；\u7cbe\u795e\u6216\u5fc3\u7406\u4e16\u754c（\u4e16\u754c 2），\u5373\u4eba\u7c7b\u7684\u5185\u5728\u4e16\u754c，\u5177\u6709\u4e0d\u53ef\u88ab\u5916\u90e8\u7aa5\u63a2\u7684\u7279\u6027；\u4ee5\u53ca\u7531\u201c\u4eba\u7c7b\u601d\u7ef4\u7684\u4ea7\u7269\u201d（\u6ce2\u666e\u5c14，1980：44）\u6784\u6210\u7684\u7b2c\u4e09\u4e16\u754c（\u4e16\u754c 3），\u6982\u5ff5\u3001\u8bed\u8a00\u3001\u5b9a\u7406\u7686\u5c5e\u4e8e\u8fd9\u4e00\u4e16\u754c\u3002\u5f53\u8d39\u9a6c\u5c06\u5176\u8111\u6d77\u4e2d（\u4e16\u754c 2）\u7684\u516c\u5f0f\u516c\u4e4b\u4e8e\u4f17\u65f6，\u8fd9\u4e00\u540e\u6765\u88ab\u79f0\u4f5c\u201c\u8d39\u9a6c\u5927\u5b9a\u7406\u201d\u7684\u516c\u5f0f\u4fbf\u6210\u4e3a\u4e86\u4e16\u754c 3 \u7684\u4e00\u90e8\u5206，\u4f9b\u6570\u5b66\u754c\u63a2\u8ba8\u4e0e\u5206\u6790\u3002\u8f9b\u683c（1997）\u8be6\u7ec6\u8bb0\u8ff0\u4e86\u5b66\u754c\u5bf9\u8fd9\u4e00\u5b9a\u7406\u7684\u63a2\u8ba8\u5386\u7a0b\u3002<\/p>\n

    Already in 1918, the German logician Gottlob Frege (1848\u20131925) published a similar idea in his essay Der Gedanke: That seems to be the result: Thoughts are neither things of the external world nor conceptions. A third realm must be recognised. (Frege 1986:43)\u2079 \u65e9\u5728 1918 \u5e74，\u5fb7\u56fd\u903b\u8f91\u5b66\u5bb6\u6208\u7279\u6d1b\u5e03\u00b7\u5f17\u96f7\u683c（1848-1925）\u4fbf\u5728\u5176\u8bba\u6587\u300a\u601d\u60f3\u300b\u4e2d\u63d0\u51fa\u4e86\u7c7b\u4f3c\u7684\u89c2\u70b9： \u7ed3\u8bba\u4f3c\u4e4e\u662f：\u601d\u60f3\u65e2\u975e\u5916\u90e8\u4e16\u754c\u7684\u4e8b\u7269，\u4ea6\u975e\u4e3b\u89c2\u7684\u8868\u8c61，\u6211\u4eec\u5fc5\u987b\u627f\u8ba4\u5b58\u5728\u7b2c\u4e09\u4e2a\u9886\u57df\u3002（\u5f17\u96f7\u683c，1986：43）\u2079<\/p>\n

  • Original quote: \u2018So scheint das Ergebnis zu sein: Die Gedanken sind weder Dinge der Au\u00dfenwelt noch Vorstellungen. Ein drittes Reich muss anerkannt werden\u2019. \u539f\u6587\u4e3a：\u201cSo scheint das Ergebnis zu sein: Die Gedanken sind weder Dinge der Au\u00dfenwelt noch Vorstellungen. Ein drittes Reich muss anerkannt werden\u3002<\/li>\n

    A thought in Frege\u2019s sense is something, which is either true or false.\u00b9\u2070 What Frege called \u2018a third realm\u2019 \u2013 the term \u2018Drittes Reich\u2019 got a totally different meaning two decades later! \u2013 was later turned into \u2018world 3\u2019 by the Austrian-British philosopher Karl Popper (1980:157). At first glance, Popper\u2019s world 3 might look a bit Platonic (Hersh 1997: 220), but there are important differences. World 3 is a human creation, that is, it consists of the products of the human mind, whereas the Platonic ideal forms were regarded as eternal.\u00b9\u00b9 \u5728\u5f17\u96f7\u683c\u7684\u7406\u8bba\u4e2d，\u601d\u60f3\u662f\u5177\u6709\u771f\u5047\u503c\u7684\u4e8b\u7269\u00b9\u2070\u3002\u5f17\u96f7\u683c\u6240\u79f0\u7684\u201c\u7b2c\u4e09\u4e2a\u9886\u57df\u201d\u2014\u2014\u4e8c\u5341\u5e74\u540e，\u201c\u7b2c\u4e09\u5e1d\u56fd\u201d\u8fd9\u4e00\u8bcd\u6c47\u88ab\u8d4b\u4e88\u4e86\u5168\u7136\u4e0d\u540c\u7684\u542b\u4e49\u2014\u2014\u540e\u6765\u88ab\u5965\u88d4\u82f1\u56fd\u54f2\u5b66\u5bb6\u5361\u5c14\u00b7\u6ce2\u666e\u5c14\u53d1\u5c55\u4e3a\u201c\u4e16\u754c 3\u201d（\u6ce2\u666e\u5c14，1980：157）\u3002\u521d\u770b\u4e4b\u4e0b，\u6ce2\u666e\u5c14\u7684\u4e16\u754c 3 \u7406\u8bba\u4f3c\u4e4e\u5e26\u6709\u67cf\u62c9\u56fe\u4e3b\u4e49\u7684\u8272\u5f69（\u8d6b\u4ec0，1997：220），\u4f46\u4e8c\u8005\u5b58\u5728\u672c\u8d28\u533a\u522b：\u4e16\u754c 3 \u662f\u4eba\u7c7b\u7684\u521b\u9020\u7269，\u7531\u4eba\u7c7b\u601d\u7ef4\u7684\u4ea7\u7269\u6784\u6210；\u800c\u67cf\u62c9\u56fe\u6240\u63d0\u51fa\u7684\u7406\u5ff5\u5f62\u5f0f，\u5219\u88ab\u89c6\u4f5c\u6c38\u6052\u7684\u5b58\u5728\u00b9\u00b9\u3002<\/p>\n

  • \u2018nenne ich Gedanken etwas, bei dem \u00fcberhaupt Wahrheit in Frage kommen kann\u2019 (Frege 1986:33). \u5f17\u96f7\u683c\u5bf9\u601d\u60f3\u7684\u5b9a\u4e49\u4e3a：\u201c\u6211\u5c06\u90a3\u4e9b\u80fd\u591f\u63a2\u8ba8\u5176\u771f\u5047\u7684\u4e8b\u7269\u79f0\u4f5c\u601d\u60f3\u3002\u201d（\u5f17\u96f7\u683c，1986：33）\u3002 \u00a0<\/li>\n
  • The British mathematician Penrose also speaks about three worlds. But in his case, the world containing mathematics is closer to the Platonic realm (Penrose 1991, Livio 2009:2f.). \u82f1\u56fd\u6570\u5b66\u5bb6\u5f6d\u7f57\u65af\u4e5f\u63d0\u51fa\u4e86\u4e09\u4e2a\u4e16\u754c\u7684\u7406\u8bba，\u4f46\u5728\u4ed6\u7684\u7406\u8bba\u4e2d，\u5305\u542b\u6570\u5b66\u7684\u4e16\u754c\u66f4\u63a5\u8fd1\u67cf\u62c9\u56fe\u7684\u7406\u5ff5\u4e16\u754c（\u5f6d\u7f57\u65af，1991；\u5229\u7ef4\u5965，2009：2 \u53ca\u4e0b\u9875）\u3002<\/li>\n

    Please note that the conclusions in the next section do not presuppose the existence of world 3. I hope that my arguments will also convince readers who do not share my view on the ontology of mathematics. \u9700\u8bf4\u660e\u7684\u662f，\u4e0b\u4e00\u90e8\u5206\u7684\u7ed3\u8bba\u5e76\u4e0d\u4ee5\u4e16\u754c 3 \u7684\u5b58\u5728\u4e3a\u524d\u63d0，\u7b14\u8005\u5e0c\u671b\u76f8\u5173\u8bba\u8bc1\u4e5f\u80fd\u8ba9\u90a3\u4e9b\u4e0e\u7b14\u8005\u5728\u6570\u5b66\u672c\u4f53\u8bba\u95ee\u9898\u4e0a\u6301\u4e0d\u540c\u89c2\u70b9\u7684\u8bfb\u8005\u4fe1\u670d\u3002<\/p>\n

    How can one detect divine elements in mathematics?<\/h3>\n

    \u5982\u4f55\u5728\u6570\u5b66\u4e2d\u53d1\u73b0\u795e\u5723\u7684\u5143\u7d20？<\/h3>\n

    How can it be argued today that God speaks through the language of mathematics? Although a pragmatic Platonism is still alive among mathematicians, I will not build on it in the following section, because I think that mathematics is a combination of inventions and discoveries, see above.\u00b9\u00b2 \u5982\u4eca，\u6211\u4eec\u8be5\u5982\u4f55\u8bba\u8bc1\u4e0a\u5e1d\u901a\u8fc7\u6570\u5b66\u7684\u8bed\u8a00\u8a00\u8bf4？\u5c3d\u7ba1\u5b9e\u7528\u67cf\u62c9\u56fe\u4e3b\u4e49\u5728\u6570\u5b66\u5bb6\u4e2d\u4ecd\u6709\u5f71\u54cd，\u4f46\u7b14\u8005\u5728\u4e0b\u6587\u4e2d\u4e0d\u4f1a\u4ee5\u8fd9\u4e00\u7406\u8bba\u4e3a\u57fa\u7840，\u56e0\u7b14\u8005\u8ba4\u4e3a，\u6570\u5b66\u662f\u53d1\u660e\u4e0e\u53d1\u73b0\u7684\u7ed3\u5408，\u8fd9\u4e00\u70b9\u524d\u6587\u5df2\u4f5c\u9610\u91ca\u00b9\u00b2\u3002<\/p>\n

  • Note that in Kessler (2019:51), I referred to pragmatic Platonism as one of the four gateways to spirituality. This is because my approach there was purely phenomenological. I have not discussed to what extent these gateways could be justified theologically or philosophically. \u9700\u8bf4\u660e\u7684\u662f，\u7b14\u8005\u5728\u514b\u65af\u52d2（2019：51）\u4e2d，\u5c06\u5b9e\u7528\u67cf\u62c9\u56fe\u4e3b\u4e49\u89c6\u4f5c\u901a\u5f80\u7075\u6027\u7684\u56db\u5927\u9014\u5f84\u4e4b\u4e00，\u8fd9\u662f\u56e0\u4e3a\u8be5\u7814\u7a76\u91c7\u7528\u7684\u662f\u7eaf\u7cb9\u7684\u73b0\u8c61\u5b66\u7814\u7a76\u65b9\u6cd5，\u5e76\u672a\u4ece\u795e\u5b66\u6216\u54f2\u5b66\u89d2\u5ea6\u8bba\u8bc1\u8fd9\u4e9b\u9014\u5f84\u7684\u5408\u7406\u6027\u3002<\/li>\n

    I do argue that some features of mathematics hint at divine attributes. I am not saying that one can detect God within mathematics, but that one can at least find traces of the divine. Reflections by mathematicians on their work (like in Cassaza et al. 2015) and the empirical study Witz (2007) provide some empirical evidence for the fact that one might detect divine elements in mathematics. \u7b14\u8005\u8ba4\u4e3a，\u6570\u5b66\u7684\u67d0\u4e9b\u7279\u5f81\u6697\u793a\u7740\u795e\u5723\u7684\u5c5e\u6027\u3002\u7b14\u8005\u5e76\u975e\u63d0\u51fa\u4eba\u4eec\u80fd\u5728\u6570\u5b66\u4e2d\u76f4\u63a5\u611f\u77e5\u4e0a\u5e1d，\u800c\u662f\u8ba4\u4e3a，\u4eba\u4eec\u81f3\u5c11\u80fd\u5728\u6570\u5b66\u4e2d\u53d1\u73b0\u795e\u5723\u7684\u8e2a\u8ff9\u3002\u6570\u5b66\u5bb6\u5bf9\u5176\u7814\u7a76\u7684\u53cd\u601d（\u5982\u5361\u8428\u624e\u7b49\u4eba，2015）\u4ee5\u53ca\u7ef4\u8328\u5f00\u5c55\u7684\u5b9e\u8bc1\u7814\u7a76（2007），\u5747\u4e3a\u201c\u6570\u5b66\u4e2d\u5b58\u5728\u795e\u5723\u5143\u7d20\u201d\u8fd9\u4e00\u89c2\u70b9\u63d0\u4f9b\u4e86\u5b9e\u8bc1\u4f9d\u636e\u3002<\/p>\n

    Mathematics shows that there is indisputable truth<\/h4>\n

    \u6570\u5b66\u5f70\u663e\u4e86\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406\u7684\u5b58\u5728<\/h4>\n

    Let us start with a personal remark by the journalist Masha Gessen. In her work on the mathematician Grigori Perelman, she also shares her own experiences with math clubs in the former Soviet Union. \u2018To a Soviet child, the after-school math club was a miracle\u2019 (Gessen 2009:21). The ordinary school system followed the ideal of uniformity and taught the doctrines of the communist party. By contrast, the math clubs encouraged individuality and creativity. One could argue with others, and statements could be proved! Each student could receive merits by solving a math problem, regardless of his or her gender, ethnic background, political conviction or whatever. \u2018(I)t felt like love, truth, hope and justice all handed to me at once\u2019 (p. 22). Gessen mentions four transcendentals \u2013 love, truth, hope and justice \u2013 which we consider to be divine attributes, at least in the Jewish-Christian tradition. Thus, mathematics obviously can lead to the discovery that there are transcendentals. \u6211\u4eec\u5148\u6765\u770b\u8bb0\u8005\u739b\u838e\u00b7\u683c\u68ee\u7684\u4e00\u6bb5\u4e2a\u4eba\u611f\u609f\u3002\u683c\u68ee\u5728\u64b0\u5199\u6570\u5b66\u5bb6\u683c\u91cc\u6208\u91cc\u00b7\u4f69\u96f7\u5c14\u66fc\u7684\u4f20\u8bb0\u65f6，\u4e5f\u5206\u4eab\u4e86\u81ea\u5df1\u5728\u524d\u82cf\u8054\u6570\u5b66\u4ff1\u4e50\u90e8\u7684\u7ecf\u5386：\u201c\u5bf9\u4e00\u4e2a\u82cf\u8054\u5b69\u5b50\u800c\u8a00，\u8bfe\u5916\u6570\u5b66\u4ff1\u4e50\u90e8\u5b9b\u5982\u4e00\u4e2a\u5947\u8ff9\u3002\u201d（\u683c\u68ee，2009：21）\u3002\u5f7c\u65f6\u7684\u5e38\u89c4\u5b66\u6821\u6559\u80b2\u5949\u884c\u7edf\u4e00\u5316\u7684\u7406\u5ff5，\u704c\u8f93\u5171\u4ea7\u515a\u7684\u6559\u4e49；\u800c\u6570\u5b66\u4ff1\u4e50\u90e8\u5219\u9f13\u52b1\u4e2a\u6027\u4e0e\u521b\u65b0，\u4eba\u4eec\u53ef\u4ee5\u76f8\u4e92\u8fa9\u8bba，\u547d\u9898\u53ef\u4ee5\u88ab\u8bc1\u660e！\u6bcf\u4e2a\u5b66\u751f\u90fd\u80fd\u51ed\u501f\u89e3\u51fa\u6570\u5b66\u96be\u9898\u83b7\u5f97\u8ba4\u53ef，\u65e0\u8bba\u5176\u6027\u522b\u3001\u79cd\u65cf\u3001\u653f\u6cbb\u7acb\u573a\u5982\u4f55\u3002\u201c\u90a3\u4e00\u523b，\u6211\u4eff\u4f5b\u540c\u65f6\u62e5\u6709\u4e86\u7231\u3001\u771f\u7406\u3001\u5e0c\u671b\u4e0e\u6b63\u4e49\u3002\u201d（\u540c\u4e0a，\u7b2c 22 \u9875）\u3002\u683c\u68ee\u63d0\u53ca\u4e86\u7231\u3001\u771f\u7406\u3001\u5e0c\u671b\u3001\u6b63\u4e49\u8fd9\u56db\u79cd\u8d85\u9a8c\u7684\u4ef7\u503c，\u81f3\u5c11\u5728\u72b9\u592a\u6559\u4e0e\u57fa\u7763\u6559\u4f20\u7edf\u4e2d，\u8fd9\u4e9b\u4ef7\u503c\u88ab\u89c6\u4f5c\u795e\u5723\u7684\u5c5e\u6027\u3002\u7531\u6b64\u53ef\u89c1，\u6570\u5b66\u663e\u7136\u80fd\u5f15\u5bfc\u4eba\u4eec\u53d1\u73b0\u8d85\u9a8c\u4ef7\u503c\u7684\u5b58\u5728\u3002<\/p>\n

    Most importantly, in mathematics, one can learn that there is an indisputable truth. A mathematical formula is either true or false. Every mathematician would agree that Fermat\u2019s Last Theorem actually was true even before Andrew Wiles proved it in 1994 (Singh 1997). There is no need for Platonism to share this point of view. It is the belief in the objectivity of mathematical discourse, without committing to the belief that mathematical objects exist in a Platonic realm (Livio 2009:243). \u6700\u91cd\u8981\u7684\u662f，\u4eba\u4eec\u80fd\u4ece\u6570\u5b66\u4e2d\u8ba4\u8bc6\u5230，\u5b58\u5728\u7740\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406\u3002\u4e00\u4e2a\u6570\u5b66\u516c\u5f0f\u975e\u771f\u5373\u5047\u3002\u6240\u6709\u6570\u5b66\u5bb6\u90fd\u4f1a\u8ba4\u540c，\u5373\u4fbf\u5728\u5b89\u5fb7\u9c81\u00b7\u6000\u5c14\u65af 1994 \u5e74\u8bc1\u660e\u8d39\u9a6c\u5927\u5b9a\u7406\u4e4b\u524d，\u8fd9\u4e00\u5b9a\u7406\u672c\u8eab\u5c31\u662f\u6210\u7acb\u7684（\u8f9b\u683c，1997）\u3002\u8ba4\u540c\u8fd9\u4e00\u89c2\u70b9，\u5e76\u4e0d\u9700\u8981\u4ee5\u67cf\u62c9\u56fe\u4e3b\u4e49\u4e3a\u524d\u63d0，\u5b83\u4ec5\u4ec5\u662f\u5bf9\u6570\u5b66\u8bdd\u8bed\u5ba2\u89c2\u6027\u7684\u575a\u4fe1，\u800c\u975e\u8ba4\u5b9a\u6570\u5b66\u5ba2\u4f53\u5b58\u5728\u4e8e\u67cf\u62c9\u56fe\u7684\u7406\u5ff5\u4e16\u754c\u4e2d（\u5229\u7ef4\u5965，2009：243）\u3002<\/p>\n

    The detection of indisputable truth might lead people to look for the source of truth, which is linked to God, at least according to monotheistic religions (see, e.g. Job 28:23 in the Jewish-Christian Bible, Jn 14:6 in the New Testament). \u5bf9\u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406\u7684\u53d1\u73b0，\u53ef\u80fd\u4f1a\u5f15\u5bfc\u4eba\u4eec\u63a2\u5bfb\u771f\u7406\u7684\u672c\u6e90\u3002\u81f3\u5c11\u5728\u4e00\u795e\u6559\u7684\u4f20\u7edf\u4e2d，\u771f\u7406\u7684\u672c\u6e90\u4e0e\u4e0a\u5e1d\u76f8\u5173（\u4f8b\u5982\u53ef\u53c2\u89c1\u72b9\u592a\u6559\u4e0e\u57fa\u7763\u6559\u7ecf\u5178\u300a\u5723\u7ecf\u300b\u4e2d\u7684\u300a\u7ea6\u4f2f\u8bb0\u300b28 \u7ae0 23 \u8282\u3001\u300a\u7ea6\u7ff0\u798f\u97f3\u300b14 \u7ae0 6 \u8282）\u3002<\/p>\n

    Mathematics shows that there is beauty<\/h4>\n

    \u6570\u5b66\u5f70\u663e\u4e86\u7f8e\u7684\u5b58\u5728<\/h4>\n

    In 1907, the British mathematician Bertrand Russell (1872\u20131970) wrote \u2018Mathematics, rightly viewed, possesses not only truth, but supreme beauty \u2013 a beauty cold and austere\u2019 (Russell 2014:49). His compatriot Hardy stated in 1940 that beauty is the first test for mathematics, because \u2018there is no permanent place in the world for ugly mathematics\u2019 (Hardy 2001:85). In 1952, the German mathematician Helmut Hasse (1898\u20131979) lectured on \u2018mathematics as science, art and power\u2019 (Hasse 1952). There he declared that beauty leads to mathematical truth (:19). He further stated that truth is a necessary condition for real mathematics but is not sufficient. There must also be beauty and harmony (:26). Sir Michael Atijah (1929\u20132019) was a strong advocate for beauty in mathematics.\u00b9\u00b3 He was convinced that mathematicians \u2018serve two masters, Beauty and Truth\u2019 (Atijah 2015:29; see also Atijah 1973). In the abovementioned survey, Aschbacher also asked her husband\u2019s colleagues why they were drawn to mathematics. A typical answer was \u2018drawn towards it by its beauty and elegance\u2019 (Aschbacher 2015:18). Heller (2018:229) regards mathematics to be \u2018the most beautiful product of human rationality\u2019. Zeki et al. (2014) report on a \u2018beauty competition\u2019 in which mathematicians rated formulae for beauty. The winner was the so-called Euler\u2019s identity (Kessler 2018:4\u20135). More information about the aesthetical dimension of mathematics can be found in the collective volume (Sinclair, Pimm & Higginson 2006) or in the special issue (Journal of Humanistic Mathematics 2016). Mathematical beauty can be found in the result itself, in the proof of the result, or as a guide to the right formula. 1907 \u5e74，\u82f1\u56fd\u6570\u5b66\u5bb6\u4f2f\u7279\u5170\u00b7\u7f57\u7d20（1872-1970）\u5199\u9053：\u201c\u6b63\u786e\u770b\u5f85\u7684\u6570\u5b66，\u4e0d\u4ec5\u62e5\u6709\u771f\u7406，\u8fd8\u62e5\u6709\u81f3\u9ad8\u65e0\u4e0a\u7684\u7f8e\u2014\u2014\u4e00\u79cd\u51b7\u5cfb\u800c\u5e84\u91cd\u7684\u7f8e\u3002\u201d（\u7f57\u7d20，2014：49）\u3002\u5176\u540c\u80de\u54c8\u4ee3\u5728 1940 \u5e74\u63d0\u51fa，\u7f8e\u662f\u8bc4\u5224\u6570\u5b66\u7684\u9996\u8981\u6807\u51c6，\u56e0\u4e3a\u201c\u4e11\u964b\u7684\u6570\u5b66\u5728\u4e16\u754c\u4e0a\u6ca1\u6709\u5bb9\u8eab\u4e4b\u5730\u201d（\u54c8\u4ee3，2001：85）\u30021952 \u5e74，\u5fb7\u56fd\u6570\u5b66\u5bb6\u8d6b\u5c14\u7a46\u7279\u00b7\u54c8\u585e（1898-1979）\u53d1\u8868\u4e86\u9898\u4e3a\u300a\u4f5c\u4e3a\u79d1\u5b66\u3001\u827a\u672f\u4e0e\u529b\u91cf\u7684\u6570\u5b66\u300b\u7684\u6f14\u8bb2（\u54c8\u585e，1952），\u4ed6\u5728\u6f14\u8bb2\u4e2d\u63d0\u51fa，\u7f8e\u662f\u901a\u5f80\u6570\u5b66\u771f\u7406\u7684\u6307\u5f15（\u540c\u4e0a，\u7b2c 19 \u9875），\u5e76\u8fdb\u4e00\u6b65\u6307\u51fa，\u771f\u7406\u662f\u771f\u6b63\u7684\u6570\u5b66\u7684\u5fc5\u8981\u6761\u4ef6，\u800c\u975e\u5145\u5206\u6761\u4ef6，\u771f\u6b63\u7684\u6570\u5b66\u8fd8\u5fc5\u987b\u517c\u5177\u7f8e\u4e0e\u548c\u8c10（\u540c\u4e0a，\u7b2c 26 \u9875）\u3002\u8fc8\u514b\u5c14\u00b7\u963f\u8482\u4e9a\u7235\u58eb（1929-2019）\u6781\u529b\u63a8\u5d07\u6570\u5b66\u4e2d\u7684\u7f8e\u00b9\u00b3，\u4ed6\u575a\u4fe1，\u6570\u5b66\u5bb6\u201c\u4f8d\u5949\u4e24\u4f4d\u4e3b\u4eba：\u7f8e\u4e0e\u771f\u7406\u201d（\u963f\u8482\u4e9a，2015：29；\u4ea6\u53ef\u53c2\u89c1\u963f\u8482\u4e9a，1973）\u3002\u5728\u4e0a\u8ff0\u963f\u5e0c\u5df4\u8d6b\u5f00\u5c55\u7684\u8c03\u67e5\u4e2d，\u5979\u8fd8\u8be2\u95ee\u4e86\u5176\u4e08\u592b\u7684\u540c\u4e8b\u88ab\u6570\u5b66\u5438\u5f15\u7684\u539f\u56e0，\u5178\u578b\u7684\u56de\u7b54\u662f：\u201c\u88ab\u6570\u5b66\u7684\u7f8e\u4e0e\u4f18\u96c5\u6240\u5438\u5f15\u3002\u201d（\u963f\u5e0c\u5df4\u8d6b，2015：18）\u3002\u6d77\u52d2\u5c06\u6570\u5b66\u89c6\u4f5c\u201c\u4eba\u7c7b\u7406\u6027\u6700\u7f8e\u4e3d\u7684\u4ea7\u7269\u201d（2018：229）\u3002\u6cfd\u57fa\u7b49\u4eba（2014）\u8bb0\u5f55\u4e86\u4e00\u573a\u7279\u6b8a\u7684\u201c\u7f8e\u4e4b\u7ade\u8d5b\u201d，\u6570\u5b66\u5bb6\u4eec\u4e3a\u6570\u5b66\u516c\u5f0f\u7684\u7f8e\u611f\u6253\u5206，\u6700\u7ec8\u6b27\u62c9\u6052\u7b49\u5f0f\u62d4\u5f97\u5934\u7b79（\u514b\u65af\u52d2，2018：4-5）\u3002\u5173\u4e8e\u6570\u5b66\u7684\u7f8e\u5b66\u7ef4\u5ea6，\u53ef\u53c2\u89c1\u8f9b\u514b\u83b1\u3001\u76ae\u59c6\u4e0e\u5e0c\u91d1\u68ee\u4e3b\u7f16\u7684\u8bba\u6587\u96c6（2006），\u6216\u300a\u4eba\u6587\u6570\u5b66\u671f\u520a\u300b\u7684\u76f8\u5173\u7279\u520a（2016）\u3002\u6570\u5b66\u4e4b\u7f8e\u65e2\u4f53\u73b0\u5728\u6570\u5b66\u7ed3\u8bba\u672c\u8eab，\u4e5f\u4f53\u73b0\u5728\u7ed3\u8bba\u7684\u8bc1\u660e\u8fc7\u7a0b\u4e2d，\u8fd8\u80fd\u4f5c\u4e3a\u6307\u5f15，\u5e2e\u52a9\u4eba\u4eec\u627e\u5230\u6b63\u786e\u7684\u516c\u5f0f\u3002<\/p>\n

  • Some of Atijah\u2019s talks are still available on YouTube. \u963f\u8482\u4e9a\u7684\u90e8\u5206\u6f14\u8bb2\u89c6\u9891\u4ecd\u53ef\u5728 YouTube \u5e73\u53f0\u89c2\u770b\u3002<\/li>\n

    When people experience beauty in mathematics, they might start wondering about the source of this beauty. The feeling might be comparable to arriving at the top of a high mountain, enjoying the beauty of the view. At such moments one is reminded of Psalm 19:1: \u2018The heavens declare the glory of God\u2019.\u00b9\u2074 Such a feeling might also come to one experiencing the beauty of maths. It is the \u2018shuddering before the beautiful\u2019 (Chandrasekhar 1987:541), an emotion similar to spiritual emotion called \u2018mysterium tremendum\u2019 by the German theologian Rudolf Otto (1936:12). This \u2018mysterium tremendum\u2019 contains elements of awefulness and overpoweringness (majestas) (:14, 20). \u5f53\u4eba\u4eec\u5728\u6570\u5b66\u4e2d\u611f\u53d7\u5230\u7f8e\u65f6，\u4fbf\u4f1a\u5f00\u59cb\u63a2\u5bfb\u8fd9\u4efd\u7f8e\u7684\u672c\u6e90\u3002\u8fd9\u79cd\u611f\u53d7，\u5982\u540c\u767b\u4e0a\u9ad8\u5c71\u4e4b\u5dc5，\u6b23\u8d4f\u773c\u524d\u7684\u58ee\u4e3d\u666f\u8272，\u5f7c\u65f6，\u4eba\u4eec\u4f1a\u60f3\u8d77\u300a\u8bd7\u7bc7\u300b19 \u7bc7 1 \u8282\u7684\u5185\u5bb9：\u201c\u8bf8\u5929\u8ff0\u8bf4\u4e0a\u5e1d\u7684\u8363\u8000\u3002\u201d\u00b9\u2074 \u4f53\u9a8c\u6570\u5b66\u4e4b\u7f8e\u65f6，\u4eba\u4eec\u4e5f\u4f1a\u4ea7\u751f\u8fd9\u6837\u7684\u611f\u53d7\u3002\u8fd9\u662f\u4e00\u79cd\u201c\u9762\u5bf9\u7f8e\u65f6\u7684\u9707\u64bc\u201d（\u94b1\u5fb7\u62c9\u585e\u5361，1987：541），\u4e0e\u5fb7\u56fd\u795e\u5b66\u5bb6\u9c81\u9053\u592b\u00b7\u5965\u6258\u63d0\u51fa\u7684\u201c\u795e\u79d8\u7684\u8d85\u9a8c\u6027\u201d\u8fd9\u4e00\u7075\u6027\u60c5\u611f\u76f8\u4f3c（\u5965\u6258，1936：12）\u3002\u8fd9\u79cd\u201c\u795e\u79d8\u7684\u8d85\u9a8c\u6027\u201d，\u8574\u542b\u7740\u4ee4\u4eba\u656c\u754f\u4e0e\u6298\u670d\u7684\u795e\u5723\u5a01\u4e25（\u5965\u6258，1936：14\u300120）\u3002<\/p>\n

  • Actually, in the Alps, there are several summit crosses quoting Psalm 19:1. \u4e8b\u5b9e\u4e0a，\u5728\u963f\u5c14\u5351\u65af\u5c71\u533a，\u591a\u5ea7\u5c71\u5cf0\u7684\u9876\u5cf0\u5341\u5b57\u67b6\u4e0a\u90fd\u523b\u6709\u300a\u8bd7\u7bc7\u300b19 \u7bc7 1 \u8282\u7684\u5185\u5bb9\u3002<\/li>\n

    Mathematics shows that there is beauty and glory, and the Bible links these attributes to God (e.g. Psalm 19:1, Isaiah 33:17, cf. Von Balthasar\u2019s seven volumes on the glory of God, Von Balthasar 1984). \u6570\u5b66\u5f70\u663e\u4e86\u7f8e\u4e0e\u8363\u8000\u7684\u5b58\u5728，\u800c\u300a\u5723\u7ecf\u300b\u5c06\u8fd9\u4e9b\u5c5e\u6027\u4e0e\u4e0a\u5e1d\u76f8\u8054\u7ed3（\u4f8b\u5982\u300a\u8bd7\u7bc7\u300b19 \u7bc7 1 \u8282\u3001\u300a\u4ee5\u8d5b\u4e9a\u4e66\u300b33 \u7bc7 17 \u8282，\u4ea6\u53ef\u53c2\u89c1\u51af\u00b7\u5df4\u5c14\u5854\u8428\u7684\u4e03\u5377\u672c\u8457\u4f5c\u300a\u4e0a\u5e1d\u7684\u8363\u8000\u300b，\u51af\u00b7\u5df4\u5c14\u5854\u8428，1984）\u3002<\/p>\n

    Mathematics shows the importance of community<\/h4>\n

    \u6570\u5b66\u5f70\u663e\u4e86\u5171\u540c\u4f53\u7684\u91cd\u8981\u6027<\/h4>\n

    Because human beings are social beings, there are many human activities demonstrating the importance of community. And each science has its own scientific community. But I think that the previous aspect of mathematical beauty (3.2) has an interesting side effect. \u4eba\u7c7b\u662f\u793e\u4f1a\u6027\u7684\u5b58\u5728，\u8bf8\u591a\u4eba\u7c7b\u6d3b\u52a8\u90fd\u80fd\u4f53\u73b0\u51fa\u5171\u540c\u4f53\u7684\u91cd\u8981\u6027，\u6bcf\u4e00\u95e8\u79d1\u5b66\u4e5f\u90fd\u6709\u5176\u4e13\u5c5e\u7684\u79d1\u5b66\u5171\u540c\u4f53\u3002\u4f46\u7b14\u8005\u8ba4\u4e3a，\u524d\u6587\u6240\u63a2\u8ba8\u7684\u6570\u5b66\u4e4b\u7f8e\u8fd9\u4e00\u7279\u8d28，\u8fd8\u5e26\u6765\u4e86\u4e00\u4e2a\u6709\u8da3\u7684\u884d\u751f\u6548\u5e94\u3002<\/p>\n

    Mathematics is a product of a community contributing together to create beauty. Usually, a piece of art or a symphony is the product of an individual. And in the case of a cathedral where many people have contributed at different times (632 years for the Cologne Cathedral!), they at least have to work at the same location. In mathematics, an invisible community consisting of many people in different places and at different times jointly contribute to the beauty of mathematics. In the case of Fermat\u2019s Last Theorem, it took 358 years (Singh 1997). \u6570\u5b66\u662f\u5171\u540c\u4f53\u643a\u624b\u5171\u521b\u7684\u7f8e\u7684\u7ed3\u6676\u3002\u4e00\u4ef6\u827a\u672f\u54c1\u3001\u4e00\u9996\u4ea4\u54cd\u66f2，\u901a\u5e38\u51fa\u81ea\u5355\u4e00\u521b\u4f5c\u8005\u4e4b\u624b；\u5373\u4fbf\u662f\u50cf\u79d1\u9686\u5927\u6559\u5802\u8fd9\u6837，\u7531\u4e0d\u540c\u65f6\u4ee3\u7684\u4eba\u4eec\u5386\u65f6 632 \u5e74\u5171\u540c\u5efa\u9020\u7684\u5efa\u7b51，\u521b\u4f5c\u8005\u4eec\u81f3\u5c11\u4e5f\u8eab\u5904\u540c\u4e00\u5730\u57df\u3002\u800c\u6570\u5b66\u7684\u7f8e，\u7531\u4e00\u4e2a\u8de8\u8d8a\u65f6\u7a7a\u7684\u65e0\u5f62\u5171\u540c\u4f53\u5171\u540c\u94f8\u5c31，\u8eab\u5904\u4e0d\u540c\u5730\u57df\u3001\u4e0d\u540c\u65f6\u4ee3\u7684\u6570\u5b66\u5bb6，\u90fd\u4e3a\u8fd9\u4efd\u7f8e\u8d21\u732e\u7740\u529b\u91cf\u3002\u4ec5\u8d39\u9a6c\u5927\u5b9a\u7406\u7684\u8bc1\u660e，\u4fbf\u51dd\u805a\u4e86 358 \u5e74\u7684\u96c6\u4f53\u667a\u6167（\u8f9b\u683c，1997）\u3002<\/p>\n

    In his personal account, Boas (2015:256) stresses the communal aspect of mathematics: \u2018After sitting at the feet of these gurus for a year, I was a lifelong convert to the religion of mathematics\u2019. \u535a\u963f\u65af\u5728\u5176\u4e2a\u4eba\u81ea\u8ff0\u4e2d，\u7740\u91cd\u5f3a\u8c03\u4e86\u6570\u5b66\u7684\u5171\u540c\u4f53\u5c5e\u6027：\u201c\u5728\u8fd9\u4e9b\u6570\u5b66\u5927\u5e08\u7684\u95e8\u4e0b\u5b66\u4e60\u4e00\u5e74\u540e，\u6211\u4fbf\u7ec8\u8eab\u7688\u4f9d\u4e86\u6570\u5b66\u8fd9\u4e00\u4fe1\u4ef0\u3002\u201d（\u535a\u963f\u65af，2015：256）\u3002<\/p>\n

    This communal contribution to beauty has interesting counterparts in Jewish-Christian theology. The Song of Songs in the Jewish Bible has also been interpreted as a metaphor for a marriage between God and His people. This metaphor is taken up in the New Testament calling the church \u2018Christ\u2019s bride\u2019 (Eph 5:31\u201332, 2 Cor 11:2). Note that this metaphor does not refer to a mystic union between God and an individual; the whole community of believers is involved. According to this Jewish-Christian metaphor, God is looking for community. \u8fd9\u79cd\u4e3a\u7f14\u9020\u7f8e\u800c\u51dd\u805a\u7684\u5171\u540c\u4f53\u7cbe\u795e，\u5728\u72b9\u592a\u6559-\u57fa\u7763\u6559\u795e\u5b66\u4e2d\u4e5f\u80fd\u627e\u5230\u6709\u8da3\u7684\u5bf9\u5e94\u3002\u72b9\u592a\u6559\u7ecf\u5178\u4e2d\u7684\u300a\u96c5\u6b4c\u300b，\u5e38\u88ab\u89e3\u8bfb\u4e3a\u4e0a\u5e1d\u4e0e\u5176\u5b50\u6c11\u4e4b\u95f4\u7ed3\u5408\u7684\u9690\u55bb；\u8fd9\u4e00\u9690\u55bb\u4e5f\u88ab\u300a\u65b0\u7ea6\u300b\u6cbf\u7528，\u5c06\u6559\u4f1a\u79f0\u4f5c\u201c\u57fa\u7763\u7684\u65b0\u5987\u201d（\u300a\u4ee5\u5f17\u6240\u4e66\u300b5 \u7ae0 31-32 \u8282\u3001\u300a\u54e5\u6797\u591a\u540e\u4e66\u300b11 \u7ae0 2 \u8282）\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f，\u8fd9\u4e00\u9690\u55bb\u6240\u6307\u7684\u5e76\u975e\u4e0a\u5e1d\u4e0e\u5355\u4e2a\u4fe1\u5f92\u4e4b\u95f4\u7684\u795e\u79d8\u7ed3\u5408，\u800c\u662f\u4e0a\u5e1d\u4e0e\u6574\u4e2a\u4fe1\u5f92\u5171\u540c\u4f53\u7684\u8054\u7ed3\u3002\u4ece\u8fd9\u4e00\u72b9\u592a\u6559-\u57fa\u7763\u6559\u7684\u9690\u55bb\u4e2d\u53ef\u4ee5\u770b\u51fa，\u4e0a\u5e1d\u4ea6\u5728\u5bfb\u6c42\u5171\u540c\u4f53\u7684\u8054\u7ed3\u3002<\/p>\n

    In addition, there is the Christian doctrine of the Trinity. According to this doctrine, there is already community within God; thus community could be considered a divine attribute, at least in Christian theology (see the wonderful icon The Trinity, painted in 1410 by Andrei Rublev). \u6b64\u5916，\u57fa\u7763\u6559\u7684\u4e09\u4f4d\u4e00\u4f53\u6559\u4e49\u4e5f\u8574\u542b\u7740\u5171\u540c\u4f53\u7684\u5185\u6db5\u3002\u8be5\u6559\u4e49\u63d0\u51fa，\u4e0a\u5e1d\u7684\u672c\u4f53\u4e4b\u4e2d\u4fbf\u5b58\u5728\u7740\u5171\u540c\u4f53，\u56e0\u6b64，\u81f3\u5c11\u5728\u57fa\u7763\u6559\u795e\u5b66\u4e2d，\u5171\u540c\u4f53\u53ef\u88ab\u89c6\u4f5c\u4e00\u79cd\u795e\u5723\u7684\u5c5e\u6027（\u53ef\u53c2\u89c1\u5b89\u5fb7\u70c8\u00b7\u9c81\u5e03\u5ed6\u592b\u4e8e 1410 \u5e74\u521b\u4f5c\u7684\u7ecf\u5178\u5723\u50cf\u753b\u300a\u4e09\u4f4d\u4e00\u4f53\u300b）\u3002<\/p>\n

    Mathematics shows that one can speak about infinity in reasonable terms<\/h4>\n

    \u6570\u5b66\u8868\u660e\u4eba\u4eec\u80fd\u5bf9\u65e0\u9650\u8fdb\u884c\u7406\u6027\u9610\u91ca<\/h4>\n

    There has been a long dispute about the understanding of infinity (Hilbert 1925). Is there an actual infinity? The German mathematician Georg Cantor (1845\u20131918) provided a new understanding of infinity by demonstrating that there are infinities of different sizes (Hilbert 1925:167). For example, the set of integers and the set of rational numbers are both infinite and countable, but the set of real numbers is infinite and uncountable, thus \u2018much larger\u2019 than the set of rational numbers. Cantor introduced transfinite numbers in order to distinguish infinite sets of different sizes. \u5b66\u754c\u5bf9\u65e0\u9650\u7684\u7406\u89e3，\u5386\u6765\u4e89\u8bae\u4e0d\u65ad（\u5e0c\u5c14\u4f2f\u7279，1925）\u3002\u5b9e\u65e0\u9650\u662f\u5426\u5b58\u5728？\u5fb7\u56fd\u6570\u5b66\u5bb6\u683c\u5965\u5c14\u683c\u00b7\u5eb7\u6258\u5c14（1845-1918）\u4e3a\u4eba\u4eec\u7406\u89e3\u65e0\u9650\u63d0\u4f9b\u4e86\u5168\u65b0\u7684\u89c6\u89d2，\u4ed6\u8bc1\u660e\u4e86\u65e0\u9650\u5b58\u5728\u4e0d\u540c\u7684\u5c42\u7ea7（\u5e0c\u5c14\u4f2f\u7279，1925：167）\u3002\u4f8b\u5982，\u6574\u6570\u96c6\u4e0e\u6709\u7406\u6570\u96c6\u90fd\u662f\u65e0\u9650\u7684\u3001\u53ef\u6570\u7684，\u800c\u5b9e\u6570\u96c6\u5219\u662f\u65e0\u9650\u7684\u3001\u4e0d\u53ef\u6570\u7684，\u5176\u201c\u89c4\u6a21\u201d\u8fdc\u5927\u4e8e\u6709\u7406\u6570\u96c6\u3002\u5eb7\u6258\u5c14\u8fd8\u63d0\u51fa\u4e86\u8d85\u7a77\u6570\u7684\u6982\u5ff5，\u7528\u4ee5\u533a\u5206\u4e0d\u540c\u5927\u5c0f\u7684\u65e0\u9650\u96c6\u5408\u3002<\/p>\n

    This discovery shows that it is reasonable to speak about infinity, thus demonstrating that speaking about an infinite God is not senseless per se. \u8fd9\u4e00\u53d1\u73b0\u8868\u660e，\u4eba\u4eec\u5bf9\u65e0\u9650\u7684\u63a2\u8ba8\u5177\u6709\u7406\u6027\u57fa\u7840，\u8fd9\u4e5f\u610f\u5473\u7740，\u5bf9\u65e0\u9650\u7684\u4e0a\u5e1d\u7684\u9610\u91ca，\u5176\u672c\u8eab\u5e76\u975e\u6beb\u65e0\u610f\u4e49\u3002<\/p>\n

    Studying different degrees of infinity might also show the mathematician that there is something, which is provably higher than him or herself. We only know a subsection of all infinite sets, and this is not only because our IQ might be too small, but because the structure of the system of thought has limitations.\u00b9\u2075 Thus the mathematician might come to the conclusion that there is somebody higher than him or herself (see Rm 1:20). \u5bf9\u4e0d\u540c\u5c42\u7ea7\u65e0\u9650\u7684\u7814\u7a76，\u4e5f\u4f1a\u8ba9\u6570\u5b66\u5bb6\u8ba4\u8bc6\u5230，\u5b58\u5728\u7740\u4e00\u79cd\u53ef\u88ab\u8bc1\u660e\u7684\u3001\u9ad8\u4e8e\u4eba\u7c7b\u81ea\u8eab\u7684\u5b58\u5728\u3002\u4eba\u7c7b\u5bf9\u65e0\u9650\u96c6\u5408\u7684\u8ba4\u77e5，\u4ec5\u4e3a\u51b0\u5c71\u4e00\u89d2，\u8fd9\u5e76\u975e\u4ec5\u4ec5\u56e0\u4e3a\u4eba\u7c7b\u7684\u667a\u529b\u6709\u9650，\u66f4\u6e90\u4e8e\u4eba\u7c7b\u601d\u7ef4\u4f53\u7cfb\u7684\u7ed3\u6784\u672c\u8eab\u5b58\u5728\u5c40\u9650\u6027\u00b9\u2075\u3002\u56e0\u6b64，\u6570\u5b66\u5bb6\u53ef\u80fd\u4f1a\u5f97\u51fa\u8fd9\u6837\u7684\u7ed3\u8bba：\u5b58\u5728\u7740\u9ad8\u4e8e\u4eba\u7c7b\u81ea\u8eab\u7684\u81f3\u9ad8\u5b58\u5728（\u53c2\u89c1\u300a\u7f57\u9a6c\u4e66\u300b1 \u7ae0 20 \u8282）\u3002<\/p>\n

  • I thank my former colleague Prof. Albrecht Beutelspacher, Giessen, for drawing my attention to this; see his e-mail 8 August 2020. \u5728\u6b64\u611f\u8c22\u7b14\u8005\u7684\u524d\u540c\u4e8b\u3001\u5409\u68ee\u5927\u5b66\u7684\u963f\u5c14\u5e03\u96f7\u5e0c\u7279\u00b7\u535a\u4f0a\u7279\u5c14\u65af\u5e15\u8d6b\u5c14\u6559\u6388，\u662f\u4ed6\u8ba9\u7b14\u8005\u5173\u6ce8\u5230\u8fd9\u4e00\u70b9，\u76f8\u5173\u89c2\u70b9\u53c2\u89c1\u5176 2020 \u5e74 8 \u6708 8 \u65e5\u7684\u90ae\u4ef6\u3002<\/li>\n

    Mathematics shows that reasoning about unseen and abstract objects can make sense<\/h4>\n

    \u6570\u5b66\u8868\u660e\u5bf9\u4e0d\u53ef\u89c1\u7684\u62bd\u8c61\u5ba2\u4f53\u7684\u63a8\u7406\u5177\u6709\u5408\u7406\u6027<\/h4>\n

    In her essay, Victoria Harrison argues that \u2018realism about mathematical objects can provide a model for thinking about realism within theology\u2019 (Harrison 2017:479). From a strictly empirical point of view, talking about God does not make much sense. Experiences might be interpreted as God speaking or acting, but our senses \u2018cannot provide direct knowledge of God\u2019 (p. 490). Theology shares this deficiency with mathematics because mathematical knowledge cannot be generated by our senses. Mathematical formulae are not part of the physical world. Still, mathematical language has proven to be meaningful and rational. This might serve as a hint that talking about theological objects like God might also be meaningful and rational. \u7ef4\u591a\u5229\u4e9a\u00b7\u54c8\u91cc\u68ee\u5728\u5176\u8bba\u6587\u4e2d\u63d0\u51fa，\u201c\u5bf9\u6570\u5b66\u5ba2\u4f53\u7684\u5b9e\u5728\u8bba\u89e3\u8bfb，\u53ef\u4e3a\u795e\u5b66\u4e2d\u7684\u5b9e\u5728\u8bba\u601d\u8003\u63d0\u4f9b\u8303\u672c\u201d（\u54c8\u91cc\u68ee，2017：479）\u3002\u4ece\u4e25\u683c\u7684\u7ecf\u9a8c\u4e3b\u4e49\u89c6\u89d2\u6765\u770b，\u5bf9\u4e0a\u5e1d\u7684\u63a2\u8ba8\u5e76\u65e0\u592a\u591a\u5b9e\u9645\u610f\u4e49\u3002\u4eba\u4eec\u6216\u8bb8\u4f1a\u5c06\u67d0\u4e9b\u4f53\u9a8c\u89e3\u8bfb\u4e3a\u4e0a\u5e1d\u7684\u8a00\u8bf4\u6216\u4f5c\u4e3a，\u4f46\u4eba\u7c7b\u7684\u611f\u5b98\u201c\u65e0\u6cd5\u76f4\u63a5\u8ba4\u77e5\u4e0a\u5e1d\u201d（\u540c\u4e0a，\u7b2c 490 \u9875）\u3002\u795e\u5b66\u7684\u8fd9\u4e00\u7279\u8d28\u4e0e\u6570\u5b66\u76f8\u901a，\u56e0\u4e3a\u6570\u5b66\u77e5\u8bc6\u540c\u6837\u65e0\u6cd5\u7531\u4eba\u7c7b\u7684\u611f\u5b98\u4ea7\u751f，\u6570\u5b66\u516c\u5f0f\u5e76\u975e\u7269\u8d28\u4e16\u754c\u7684\u7ec4\u6210\u90e8\u5206\u3002\u5373\u4fbf\u5982\u6b64，\u6570\u5b66\u8bed\u8a00\u5df2\u88ab\u8bc1\u660e\u662f\u5bcc\u6709\u610f\u4e49\u4e14\u5177\u6709\u7406\u6027\u7684，\u8fd9\u6216\u8bb8\u6697\u793a\u7740，\u5bf9\u4e0a\u5e1d\u8fd9\u7c7b\u795e\u5b66\u5ba2\u4f53\u7684\u63a2\u8ba8，\u540c\u6837\u53ef\u4ee5\u662f\u5bcc\u6709\u610f\u4e49\u4e14\u5177\u5907\u7406\u6027\u7684\u3002<\/p>\n

    The unreasonable effectiveness of mathematics<\/h4>\n

    \u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027<\/h4>\n

    The sections above dealt with mathematics only, looking for traces of the divine in the pure world of mathematics. This section will also look at the physical world and how mathematics is used to describe it. Physics Nobel laureate Eugene Wigner (1902\u20131995) once said: \u2018The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve\u2019 (quoted in Livio 2009:3). Wigner dubbed this \u2018the unreasonable effectiveness of mathematics\u2019 (4). Livio (2009:203) points out that this unreasonable effectiveness has two aspects. There is an active aspect when natural scientists develop a mathematical tool to describe a phenomenon they have observed. It is quite plausible that mathematical reasoning can lead to new knowledge about the physical world through this process. \u524d\u6587\u7684\u63a2\u8ba8\u4ec5\u56f4\u7ed5\u7eaf\u6570\u5b66\u5c55\u5f00，\u5728\u7eaf\u7cb9\u7684\u6570\u5b66\u4e16\u754c\u4e2d\u63a2\u5bfb\u795e\u5723\u7684\u8e2a\u8ff9\u3002\u672c\u8282\u5c06\u7ed3\u5408\u7269\u8d28\u4e16\u754c，\u63a2\u8ba8\u6570\u5b66\u5982\u4f55\u88ab\u7528\u4e8e\u63cf\u8ff0\u7269\u8d28\u4e16\u754c\u3002\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u5f97\u4e3b\u5c24\u91d1\u00b7\u7ef4\u683c\u7eb3（1902-1995）\u66fe\u8bf4：\u201c\u6570\u5b66\u8bed\u8a00\u7adf\u80fd\u5982\u6b64\u6070\u5207\u5730\u8868\u8ff0\u7269\u7406\u5b9a\u5f8b，\u8fd9\u4e00\u5947\u8ff9\u662f\u4e00\u4efd\u6211\u4eec\u65e2\u65e0\u6cd5\u7406\u89e3\u3001\u4e5f\u4e0d\u914d\u62e5\u6709\u7684\u73cd\u8d35\u793c\u7269\u3002\u201d（\u8f6c\u5f15\u81ea\u5229\u7ef4\u5965，2009：3）\u3002\u7ef4\u683c\u7eb3\u5c06\u8fd9\u4e00\u73b0\u8c61\u79f0\u4f5c\u201c\u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u201d（\u540c\u4e0a，\u7b2c 4 \u9875）\u3002\u5229\u7ef4\u5965（2009：203）\u6307\u51fa，\u8fd9\u79cd\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u4f53\u73b0\u5728\u4e24\u4e2a\u65b9\u9762\u3002\u5176\u4e00\u4e3a\u4e3b\u52a8\u5c42\u9762：\u81ea\u7136\u79d1\u5b66\u5bb6\u4e3a\u63cf\u8ff0\u6240\u89c2\u5bdf\u5230\u7684\u73b0\u8c61，\u7814\u53d1\u76f8\u5e94\u7684\u6570\u5b66\u5de5\u5177，\u901a\u8fc7\u8fd9\u4e00\u8fc7\u7a0b，\u6570\u5b66\u63a8\u7406\u80fd\u591f\u4e3a\u4eba\u7c7b\u5e26\u6765\u5173\u4e8e\u7269\u8d28\u4e16\u754c\u7684\u65b0\u8ba4\u77e5，\u8fd9\u4e00\u70b9\u662f\u5408\u4e4e\u60c5\u7406\u7684\u3002<\/p>\n

    But there is also a passive aspect, which is quite mysterious. In the history of science, it has happened several times that abstract mathematical theories were developed for pure mathematical reasons and later on were successfully applied to the natural sciences. Just to give three examples: Firstly, the complex number <\/p>\n

    i<\/p>\n

    i<\/p>\n

    <\/span><\/span>i<\/span><\/span><\/span><\/span><\/span> as a root of <\/p>\n

    (<\/p>\n

    \u2212<\/p>\n

    1<\/p>\n

    )<\/p>\n

    (-1)<\/p>\n

    <\/span><\/span>(<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> was invented in order to solve algebraic equations; nowadays it is heavily used in physics and electrical engineering. Secondly, the so-called Riemannian geometry was introduced by Riemann in a brilliant lecture in G\u00f6ttingen on June 10, 1854; half a century later, the physicist Albert Einstein used the Riemannian geometry to formulate his general theory of relativity. Thirdly, the \u2018story of knot theory demonstrates beautifully the unexpected power of mathematics\u2019 (Livio 2009: 217). The mathematical theory of knots was born in 1771; today it has significant applications in quantum field theory and in studying DNA. \u5176\u4e8c\u4e3a\u88ab\u52a8\u5c42\u9762，\u8fd9\u4e00\u5c42\u9762\u7684\u73b0\u8c61\u5219\u5145\u6ee1\u4e86\u795e\u79d8\u8272\u5f69\u3002\u5728\u79d1\u5b66\u53f2\u4e0a，\u591a\u6b21\u51fa\u73b0\u8fd9\u6837\u7684\u60c5\u51b5：\u6570\u5b66\u5bb6\u51fa\u4e8e\u7eaf\u6570\u5b66\u7814\u7a76\u7684\u76ee\u7684\u63d0\u51fa\u62bd\u8c61\u7684\u6570\u5b66\u7406\u8bba，\u800c\u540e\u8fd9\u4e9b\u7406\u8bba\u88ab\u6210\u529f\u5e94\u7528\u4e8e\u81ea\u7136\u79d1\u5b66\u9886\u57df\u3002\u4ec5\u4e3e\u4e09\u4f8b：\u7b2c\u4e00，\u4f5c\u4e3a<\/p>\n

    \u2212<\/p>\n

    1<\/p>\n

    -1<\/p>\n

    <\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u5e73\u65b9\u6839\u7684\u865a\u6570<\/p>\n

    i<\/p>\n

    i<\/p>\n

    <\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>，\u6700\u521d\u662f\u4e3a\u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u800c\u88ab\u63d0\u51fa，\u5982\u4eca\u5374\u5728\u7269\u7406\u5b66\u4e0e\u7535\u6c14\u5de5\u7a0b\u9886\u57df\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528；\u7b2c\u4e8c，\u9ece\u66fc\u4e8e 1854 \u5e74 6 \u6708 10 \u65e5\u5728\u54e5\u5ef7\u6839\u5927\u5b66\u7684\u4e00\u573a\u7ecf\u5178\u6f14\u8bb2\u4e2d\u63d0\u51fa\u4e86\u9ece\u66fc\u51e0\u4f55，\u534a\u4e2a\u4e16\u7eaa\u540e，\u7269\u7406\u5b66\u5bb6\u963f\u5c14\u4f2f\u7279\u00b7\u7231\u56e0\u65af\u5766\u8fd0\u7528\u9ece\u66fc\u51e0\u4f55\u6784\u5efa\u4e86\u5e7f\u4e49\u76f8\u5bf9\u8bba；\u7b2c\u4e09，\u201c\u7ebd\u7ed3\u8bba\u7684\u53d1\u5c55\u5386\u7a0b，\u5b8c\u7f8e\u5370\u8bc1\u4e86\u6570\u5b66\u8d85\u4e4e\u9884\u671f\u7684\u529b\u91cf\u201d（\u5229\u7ef4\u5965，2009：217），\u7ebd\u7ed3\u8bba\u4e8e 1771 \u5e74\u8bde\u751f，\u5982\u4eca\u5df2\u5728\u91cf\u5b50\u573a\u8bba\u4e0e DNA \u7814\u7a76\u9886\u57df\u53d1\u6325\u7740\u91cd\u8981\u4f5c\u7528\u3002<\/p>\n

    The issue of chance was long seen as a theological problem. How can the existence of chance be combined with a rational and omniscient God? In his book on the Philosophy of Chance, Polish cosmologist and theologian Michael Heller argue that chance is not irrational and \u2018does not destroy the mathematicality of the world\u2019 (Heller 2018:152). \u2018The world turns out to be mathematical \u2026 also in its own random and probabilistic behaviours\u2019 (p. 79). Heller (p. 143) also refers to the \u2018effective\u2019 characterisation of the physical universe by mathematical structure, even in the area of probabilities. \u5076\u7136\u6027\u95ee\u9898\u957f\u671f\u4ee5\u6765\u88ab\u89c6\u4f5c\u4e00\u4e2a\u795e\u5b66\u96be\u9898：\u5076\u7136\u6027\u7684\u5b58\u5728，\u5982\u4f55\u4e0e\u7406\u6027\u3001\u5168\u77e5\u7684\u4e0a\u5e1d\u76f8\u8c03\u548c？\u6ce2\u5170\u5b87\u5b99\u5b66\u5bb6\u3001\u795e\u5b66\u5bb6\u7c73\u54c8\u4f0a\u5c14\u00b7\u6d77\u52d2\u5728\u5176\u8457\u4f5c\u300a\u5076\u7136\u6027\u7684\u54f2\u5b66\u300b\u4e2d\u63d0\u51fa，\u5076\u7136\u6027\u5e76\u975e\u975e\u7406\u6027\u7684\u5b58\u5728，\u5b83\u201c\u4e0d\u4f1a\u7834\u574f\u4e16\u754c\u7684\u6570\u5b66\u672c\u8d28\u201d（\u6d77\u52d2，2018：152）\u3002\u201c\u5373\u4fbf\u5728\u968f\u673a\u4e0e\u6982\u7387\u6027\u7684\u884c\u4e3a\u4e2d，\u4e16\u754c\u7684\u672c\u8d28\u4f9d\u7136\u662f\u6570\u5b66\u7684\u201d（\u540c\u4e0a，\u7b2c 79 \u9875）\u3002\u6d77\u52d2（\u540c\u4e0a，\u7b2c 143 \u9875）\u8fd8\u6307\u51fa，\u5373\u4fbf\u5728\u6982\u7387\u9886\u57df，\u6570\u5b66\u7ed3\u6784\u4ecd\u80fd\u5bf9\u7269\u7406\u5b87\u5b99\u4f5c\u51fa\u201c\u6709\u6548\u7684\u201d\u63cf\u8ff0\u3002<\/p>\n

    Einstein raised the question: \u2018How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?\u2019 (quoted in Livio 2009:1). It is hard to give a rational explanation for this \u2018unreasonable effectiveness of mathematics\u2019.\u00b9\u2076 Wigner called it a \u2018gift\u2019. It could also be called a miracle, which brings us to theology again. \u7231\u56e0\u65af\u5766\u66fe\u63d0\u51fa\u8fd9\u6837\u7684\u7591\u95ee：\u201c\u6570\u5b66\u4f5c\u4e3a\u4eba\u7c7b\u601d\u7ef4\u7684\u4ea7\u7269，\u72ec\u7acb\u4e8e\u7ecf\u9a8c\u4e4b\u5916，\u4e3a\u4f55\u80fd\u5982\u6b64\u5b8c\u7f8e\u5730\u5951\u5408\u7269\u7406\u5b9e\u5728\u7684\u5ba2\u4f53？\u201d（\u8f6c\u5f15\u81ea\u5229\u7ef4\u5965，2009：1）\u3002\u4eba\u4eec\u5f88\u96be\u4e3a\u8fd9\u79cd\u201c\u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u201d\u7ed9\u51fa\u4e00\u4e2a\u7406\u6027\u7684\u89e3\u91ca\u00b9\u2076\u3002\u7ef4\u683c\u7eb3\u5c06\u5176\u79f0\u4f5c\u4e00\u4efd\u201c\u793c\u7269\u201d，\u800c\u8fd9\u4efd\u793c\u7269\u4e5f\u53ef\u88ab\u89c6\u4f5c\u4e00\u4e2a\u5947\u8ff9，\u8fd9\u4e00\u7ed3\u8bba\u53c8\u5c06\u6211\u4eec\u7684\u63a2\u8ba8\u5e26\u56de\u4e86\u795e\u5b66\u9886\u57df\u3002<\/p>\n

  • Livio (2009) lists some attempts to explain this effectiveness. \u5229\u7ef4\u5965（2009）\u5728\u5176\u8457\u4f5c\u4e2d\u5217\u4e3e\u4e86\u82e5\u5e72\u8bd5\u56fe\u89e3\u91ca\u8fd9\u4e00\u6709\u6548\u6027\u7684\u5c1d\u8bd5\u3002<\/li>\n

    Creation theology as understood in the Judeo-Christian tradition offers an explanatory model for the phenomenon of \u2018the unreasonable effectiveness\u2019. God created the world with wisdom (Pr 8:22\u201331) and He created human beings in His own image (Gn 1:26). Thus, human beings are enabled to discover the laws that God put into His creation. I am not saying that the study of mathematics will automatically lead people to believe in creation theology. But obviously, one can find traces in mathematics that might point to the Creator-God. \u72b9\u592a\u6559-\u57fa\u7763\u6559\u4f20\u7edf\u4e2d\u7684\u521b\u9020\u795e\u5b66，\u4e3a\u201c\u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u201d\u8fd9\u4e00\u73b0\u8c61\u63d0\u4f9b\u4e86\u4e00\u5957\u89e3\u91ca\u8303\u5f0f\u3002\u4e0a\u5e1d\u4ee5\u667a\u6167\u521b\u9020\u4e86\u4e16\u754c（\u300a\u7bb4\u8a00\u300b8 \u7ae0 22-31 \u8282），\u5e76\u6309\u7167\u81ea\u5df1\u7684\u5f62\u8c61\u9020\u4eba（\u300a\u521b\u4e16\u8bb0\u300b1 \u7ae0 26 \u8282）\u3002\u6b63\u56e0\u5982\u6b64，\u4eba\u7c7b\u624d\u62e5\u6709\u4e86\u53d1\u73b0\u4e0a\u5e1d\u8d4b\u4e88\u5176\u521b\u9020\u7269\u7684\u6cd5\u5219\u7684\u80fd\u529b\u3002\u7b14\u8005\u5e76\u975e\u63d0\u51fa，\u6570\u5b66\u7814\u7a76\u5fc5\u7136\u4f1a\u8ba9\u4eba\u4eec\u76f8\u4fe1\u521b\u9020\u795e\u5b66，\u800c\u662f\u8ba4\u4e3a，\u4eba\u4eec\u663e\u7136\u80fd\u5728\u6570\u5b66\u4e2d\u53d1\u73b0\u6307\u5411\u9020\u7269\u4e3b\u4e0a\u5e1d\u7684\u8e2a\u8ff9\u3002<\/p>\n

    Conclusion<\/h3>\n

    \u7ed3\u8bba<\/h3>\n

    The question of this article was \u2018Does God speak through the language of mathematics?\u2019 Being a mathematician and a Christian theologian, I am convinced that the answer should be \u2018yes\u2019. But I am also aware that the evidence for this answer is not as obvious as it was several centuries ago. The changes in mathematics made it necessary to rethink the evidence for this answer. We have learned that mathematical objects are created by human beings. Thus, it is not possible to argue that mathematical formulae exist in a divine mind. However, I think that there are hints in mathematics that seem to point towards attributes of God. \u672c\u6587\u7684\u6838\u5fc3\u95ee\u9898\u662f：\u201c\u4e0a\u5e1d\u662f\u5426\u901a\u8fc7\u6570\u5b66\u7684\u8bed\u8a00\u8a00\u8bf4？\u201d\u4f5c\u4e3a\u4e00\u540d\u6570\u5b66\u5bb6\u4e0e\u57fa\u7763\u6559\u795e\u5b66\u5bb6，\u7b14\u8005\u575a\u4fe1\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002\u4f46\u7b14\u8005\u4e5f\u8ba4\u8bc6\u5230，\u5982\u4eca\u652f\u6491\u8fd9\u4e00\u7b54\u6848\u7684\u8bc1\u636e，\u5df2\u4e0d\u518d\u50cf\u6570\u4e2a\u4e16\u7eaa\u524d\u90a3\u822c\u663e\u800c\u6613\u89c1\u3002\u6570\u5b66\u9886\u57df\u7684\u53d8\u9769，\u4f7f\u5f97\u6211\u4eec\u6709\u5fc5\u8981\u91cd\u65b0\u5ba1\u89c6\u8fd9\u4e9b\u8bc1\u636e\u3002\u6211\u4eec\u5df2\u7136\u8ba4\u8bc6\u5230，\u6570\u5b66\u5ba2\u4f53\u662f\u4eba\u7c7b\u7684\u521b\u9020\u7269，\u56e0\u6b64，\u65e0\u6cd5\u518d\u8bba\u8bc1\u6570\u5b66\u516c\u5f0f\u5b58\u5728\u4e8e\u795e\u5723\u7684\u601d\u7ef4\u4e4b\u4e2d\u3002\u4f46\u7b14\u8005\u8ba4\u4e3a，\u6570\u5b66\u4e2d\u4ecd\u5b58\u5728\u8bf8\u591a\u7ebf\u7d22，\u6307\u5411\u4e0a\u5e1d\u7684\u795e\u5723\u5c5e\u6027\u3002<\/p>\n

    I have mentioned six traces of the divine: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. I think that each of these arguments has a certain weight, but I do not think that they have the same weight. If one looks at the current discussion, arguments (1), (2) and (6) are probably the most convincing. \u7b14\u8005\u5728\u6587\u4e2d\u63d0\u51fa\u4e86\u795e\u5723\u7684\u516d\u79cd\u8e2a\u8ff9：(1) \u4e0d\u5bb9\u7f6e\u7591\u7684\u771f\u7406\u7684\u5b58\u5728；(2) \u7f8e\u7684\u5b58\u5728；(3) \u5171\u540c\u4f53\u7684\u91cd\u8981\u6027；(4) \u5bf9\u65e0\u9650\u8fdb\u884c\u7406\u6027\u9610\u91ca；(5) \u53d1\u73b0\u5bf9\u4e0d\u53ef\u89c1\u7684\u62bd\u8c61\u5ba2\u4f53\u8fdb\u884c\u8a00\u8bf4\u5177\u6709\u5408\u7406\u6027；(6) \u6570\u5b66\u8d85\u4e4e\u5e38\u7406\u7684\u6709\u6548\u6027\u3002\u7b14\u8005\u8ba4\u4e3a，\u8fd9\u516d\u5927\u8bba\u8bc1\u5747\u5177\u6709\u4e00\u5b9a\u7684\u8bf4\u670d\u529b，\u4f46\u5e76\u975e\u540c\u7b49\u91cd\u8981\u3002\u4ece\u5f53\u4e0b\u7684\u5b66\u754c\u63a2\u8ba8\u6765\u770b，\u7b2c (1)､(2)､(6) \u79cd\u8bba\u8bc1\u6216\u8bb8\u662f\u6700\u5177\u8bf4\u670d\u529b\u7684\u3002<\/p>\n

    Acknowledgements<\/h3>\n

    \u81f4\u8c22<\/h3>\n

    Competing interests \u5229\u76ca\u51b2\u7a81 The author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. \u4f5c\u8005\u58f0\u660e，\u5728\u64b0\u5199\u672c\u6587\u7684\u8fc7\u7a0b\u4e2d，\u4e0d\u5b58\u5728\u4efb\u4f55\u53ef\u80fd\u9020\u6210\u4e0d\u5f53\u5f71\u54cd\u7684\u8d22\u52a1\u6216\u4e2a\u4eba\u5173\u7cfb\u3002<\/p>\n

    Author\u2019s contributions \u4f5c\u8005\u8d21\u732e V.K. is the sole author of this article. \u672c\u6587\u7531 V\u00b7\u514b\u65af\u52d2\u72ec\u7acb\u64b0\u5199\u5b8c\u6210\u3002<\/p>\n

    Ethical considerations \u4f26\u7406\u8003\u91cf This article followed all ethical standards for research without direct contact with human or animal subjects. \u672c\u6587\u7684\u7814\u7a76\u672a\u76f4\u63a5\u6d89\u53ca\u4eba\u7c7b\u6216\u52a8\u7269\u53d7\u8bd5\u5bf9\u8c61，\u5b8c\u5168\u9075\u5faa\u76f8\u5173\u7814\u7a76\u4f26\u7406\u51c6\u5219\u3002<\/p>\n

    Funding information \u57fa\u91d1\u8d44\u52a9 This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. \u672c\u7814\u7a76\u672a\u83b7\u5f97\u4efb\u4f55\u516c\u5171\u3001\u5546\u4e1a\u6216\u975e\u8425\u5229\u6027\u8d44\u52a9\u673a\u6784\u7684\u4e13\u9879\u8d44\u52a9\u3002<\/p>\n

    Data availability \u6570\u636e\u53ef\u7528\u6027 Data sharing is not applicable to this article as no new data were created or analysed in this study. \u672c\u6587\u672a\u751f\u6210\u6216\u5206\u6790\u65b0\u7684\u7814\u7a76\u6570\u636e，\u56e0\u6b64\u4e0d\u6d89\u53ca\u6570\u636e\u5171\u4eab\u76f8\u5173\u4e8b\u5b9c\u3002<\/p>\n

    Disclaimer \u514d\u8d23\u58f0\u660e The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. \u672c\u6587\u6240\u8868\u8fbe\u7684\u89c2\u70b9\u4e0e\u89c1\u89e3\u5747\u4e3a\u4f5c\u8005\u4e2a\u4eba\u89c2\u70b9，\u5e76\u4e0d\u4e00\u5b9a\u4ee3\u8868\u4f5c\u8005\u6240\u5c5e\u673a\u6784\u7684\u5b98\u65b9\u653f\u7b56\u6216\u7acb\u573a\u3002<\/p>\n

    References<\/h3>\n

    \u53c2\u8003\u6587\u732e<\/h3>\n

    Aschbacher, P., 2015, \u2018What are mathematicians really like? Observations of a spouse\u2019, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 16\u201328, The Mathematical Association of America, Washington, DC. \u963f\u5e0c\u5df4\u8d6b,P.,2015,\u300a\u6570\u5b66\u5bb6\u7a76\u7adf\u662f\u600e\u6837\u7684\u4eba?\u4e00\u4f4d\u914d\u5076\u7684\u89c2\u5bdf\u300b,\u8f7d\u4e8e\u5361\u8428\u624e,P.､\u514b\u5170\u8328,S.､\u9c81\u767b,R.(\u4e3b\u7f16),\u300a\u6211,\u6570\u5b66\u5bb6\u300b,\u7b2c 16-28 \u9875,\u7f8e\u56fd\u6570\u5b66\u534f\u4f1a,\u534e\u76db\u987f\u54e5\u4f26\u6bd4\u4e9a\u7279\u533a｡<\/p>\n

    Atiyah, M., 1973, \u2018Mathematics: Art and science\u2019, Bulletin of the American Mathematical Society 43(1), 87\u201388. https:\/\/doi.org\/10.1090\/S0273-0979-05-01095-5 \u963f\u8482\u4e9a,M.,1973,\u300a\u6570\u5b66:\u827a\u672f\u4e0e\u79d1\u5b66\u300b,\u300a\u7f8e\u56fd\u6570\u5b66\u4f1a\u901a\u62a5\u300b,43(1),\u7b2c 87-88 \u9875｡<\/p>\n

    Atiyah, M., 2015, \u2018Mathematics: Art and science\u2019, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 29\u201330, The Mathematical Association of America, Washington, DC. \u963f\u8482\u4e9a,M.,2015,\u300a\u6570\u5b66:\u827a\u672f\u4e0e\u79d1\u5b66\u300b,\u8f7d\u4e8e\u5361\u8428\u624e,P.､\u514b\u5170\u8328,S.､\u9c81\u767b,R.(\u4e3b\u7f16),\u300a\u6211,\u6570\u5b66\u5bb6\u300b,\u7b2c 29-30 \u9875,\u7f8e\u56fd\u6570\u5b66\u534f\u4f1a,\u534e\u76db\u987f\u54e5\u4f26\u6bd4\u4e9a\u7279\u533a｡<\/p>\n

    Boas, H.P., 2015, \u2018Why I became a Mathematician: A personal account\u2019, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 255\u2013256, The Mathematical Association of America, Washington, DC. \u535a\u963f\u65af,H\u00b7P.,2015,\u300a\u6211\u4e3a\u4f55\u6210\u4e3a\u6570\u5b66\u5bb6:\u4e2a\u4eba\u81ea\u8ff0\u300b,\u8f7d\u4e8e\u5361\u8428\u624e,P.､\u514b\u5170\u8328,S.､\u9c81\u767b,R.(\u4e3b\u7f16),\u300a\u6211,\u6570\u5b66\u5bb6\u300b,\u7b2c 255-256 \u9875,\u7f8e\u56fd\u6570\u5b66\u534f\u4f1a,\u534e\u76db\u987f\u54e5\u4f26\u6bd4\u4e9a\u7279\u533a｡<\/p>\n

    Bromand, J. & Kreis, G. (eds.), 2011, Gottesbeweise von Anselm bis G\u00f6del, Suhrkamp, Berlin. \u5e03\u7f57\u66fc\u5fb7,J.､\u514b\u96f7\u65af,G.(\u4e3b\u7f16),2011,\u300a\u4ece\u5b89\u745f\u4f26\u5230\u54e5\u5fb7\u5c14\u7684\u795e\u5b58\u5728\u8bc1\u660e\u300b,\u82cf\u5c14\u574e\u666e\u51fa\u7248\u793e,\u67cf\u6797｡<\/p>\n

    Cassaza, P., Krantz, S. & Ruden, R. (eds.), 2015, I, Mathematician, The Mathematical Association of America, Washington, DC. \u5361\u8428\u624e,P.､\u514b\u5170\u8328,S.､\u9c81\u767b,R.(\u4e3b\u7f16),2015,\u300a\u6211,\u6570\u5b66\u5bb6\u300b,\u7f8e\u56fd\u6570\u5b66\u534f\u4f1a,\u534e\u76db\u987f\u54e5\u4f26\u6bd4\u4e9a\u7279\u533a｡<\/p>\n

    Chandrasekhar, S., 1987, Truth and beauty: Aesthetics and motivations in science, Chicago University Press, Chicago, IL. \u94b1\u5fb7\u62c9\u585e\u5361,S.,1987,\u300a\u771f\u7406\u4e0e\u7f8e:\u79d1\u5b66\u4e2d\u7684\u7f8e\u5b66\u4e0e\u52a8\u673a\u300b,\u829d\u52a0\u54e5\u5927\u5b66\u51fa\u7248\u793e,\u4f0a\u5229\u8bfa\u4f0a\u5dde\u829d\u52a0\u54e5\u5e02｡<\/p>\n

    Davies, P., 1992, The mind of God: The scientific basis for a rational world, Simon & Schuster, New York, NY. \u6234\u7ef4\u65af,P.,1992,\u300a\u4e0a\u5e1d\u7684\u601d\u7ef4:\u7406\u6027\u4e16\u754c\u7684\u79d1\u5b66\u57fa\u7840\u300b,\u897f\u8499\u4e0e\u8212\u65af\u7279\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Descartes, R., 1976, Meditationen \u00fcber die Grundlagen der Philosophie, Felix Meiner, Hamburg. \u7b1b\u5361\u5c14,R.,1976,\u300a\u7b2c\u4e00\u54f2\u5b66\u6c89\u601d\u96c6\u300b,\u8d39\u5229\u514b\u65af\u00b7\u8fc8\u7eb3\u51fa\u7248\u793e,\u6c49\u5821｡<\/p>\n

    Frege, G., 1986, \u2018Der Gedanke\u2019, in G. Frege (ed.), Logische Untersuchungen, 3rd edn., pp. 30\u201353, Vandenhoeck & Ruprecht, G\u00f6ttingen. \u5f17\u96f7\u683c,G.,1986,\u300a\u601d\u60f3\u300b,\u8f7d\u4e8e\u5f17\u96f7\u683c,G.(\u4e3b\u7f16),\u300a\u903b\u8f91\u7814\u7a76\u300b(\u7b2c 3 \u7248),\u7b2c 30-53 \u9875,\u8303\u767b\u970d\u514b\u4e0e\u9c81\u666e\u96f7\u5e0c\u7279\u51fa\u7248\u793e,\u54e5\u5ef7\u6839｡<\/p>\n

    Gessen, M., 2009, Perfect rigor: A genius and the mathematical breakthrough, Houghton Mifflin Harcourt, New York, NY. \u683c\u68ee,M.,2009,\u300a\u5b8c\u7f8e\u7684\u4e25\u8c28:\u4e00\u4f4d\u5929\u624d\u4e0e\u6570\u5b66\u7684\u7a81\u7834\u300b,\u970d\u987f\u00b7\u7c73\u592b\u6797\u00b7\u54c8\u8003\u7279\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Hardy, G.H., 2001, A mathematician\u2019s apology, Cambridge University Press, Cambridge. \u54c8\u4ee3,G\u00b7H.,2001,\u300a\u4e00\u4e2a\u6570\u5b66\u5bb6\u7684\u8fa9\u767d\u300b,\u5251\u6865\u5927\u5b66\u51fa\u7248\u793e,\u5251\u6865｡<\/p>\n

    Harrison, V.S., 2017, \u2018Mathematical objects and the object of the theology\u2019, Religious Studies 53(4), 479\u2013496. https:\/\/doi.org\/10.1017\/S0034412516000238 \u54c8\u91cc\u68ee,V\u00b7S.,2017,\u300a\u6570\u5b66\u5ba2\u4f53\u4e0e\u795e\u5b66\u7684\u5ba2\u4f53\u300b,\u300a\u5b97\u6559\u7814\u7a76\u300b,53(4),\u7b2c 479-496 \u9875｡<\/p>\n

    Hasse, H., 1952, Mathematik als Wissenschaft, Kunst und Macht, Verlag f\u00fcr Angewandte Wissenschaften, Wiesbaden. \u54c8\u585e,H.,1952,\u300a\u4f5c\u4e3a\u79d1\u5b66､\u827a\u672f\u4e0e\u529b\u91cf\u7684\u6570\u5b66\u300b,\u5e94\u7528\u79d1\u5b66\u51fa\u7248\u793e,\u5a01\u65af\u5df4\u767b｡<\/p>\n

    Heller, M., 2018, Philosophy of chance, 2nd edn., Copernicus Center, Krak\u00f3w. \u6d77\u52d2,M.,2018,\u300a\u5076\u7136\u6027\u7684\u54f2\u5b66\u300b(\u7b2c 2 \u7248),\u54e5\u767d\u5c3c\u7814\u7a76\u4e2d\u5fc3,\u514b\u62c9\u79d1\u592b｡<\/p>\n

    Hersh, R., 1997, What is mathematics, really?, Jonathan Cape, London. \u8d6b\u4ec0,R.,1997,\u300a\u6570\u5b66\u7a76\u7adf\u662f\u4ec0\u4e48?\u300b,\u4e54\u7eb3\u68ee\u00b7\u51ef\u666e\u51fa\u7248\u793e,\u4f26\u6566｡<\/p>\n

    Hilbert, D., 1925, \u2018\u00dcber das Unendliche\u2019, Mathematische Annalen 95, 161\u2013190. https:\/\/doi.org\/10.1007\/BF01206605 \u5e0c\u5c14\u4f2f\u7279,D.,1925,\u300a\u8bba\u65e0\u9650\u300b,\u300a\u6570\u5b66\u5e74\u520a\u300b,95,\u7b2c 161-190 \u9875｡<\/p>\n

    Jonas, H., 1966, \u2018Is God a mathematician?\u2019 in H. Jonas (ed.), The phemomenon of life: Toward a philosophical biology, pp. 64\u201393, Harper & Row, New York, NY. \u7ea6\u7eb3\u65af,H.,1966,\u300a\u4e0a\u5e1d\u662f\u6570\u5b66\u5bb6\u5417?\u300b,\u8f7d\u4e8e\u7ea6\u7eb3\u65af,H.(\u4e3b\u7f16),\u300a\u751f\u547d\u73b0\u8c61:\u8d70\u5411\u54f2\u5b66\u751f\u7269\u5b66\u300b,\u7b2c 64-93 \u9875,\u54c8\u73c0\u4e0e\u7f57\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Journal of Humanistic Mathematics, 2016, \u2018Special issue on the nature and experience of mathematical beauty\u2019, Journal of Humanistic Mathematics 6(1), 3\u20137. https:\/\/doi.org\/10.5642\/jhummath.201601.03 \u300a\u4eba\u6587\u6570\u5b66\u671f\u520a\u300b,2016,\u300a\u6570\u5b66\u4e4b\u7f8e\u7684\u672c\u8d28\u4e0e\u4f53\u9a8c\u300b(\u7279\u520a),\u300a\u4eba\u6587\u6570\u5b66\u671f\u520a\u300b,6(1),\u7b2c 3-7 \u9875｡<\/p>\n

    Kanigel, R. 1991, The man who knew infinity, Abacus, London. \u5361\u5c3c\u683c\u5c14,R.,1991,\u300a\u77e5\u65e0\u6daf\u8005:\u62c9\u9a6c\u52aa\u91d1\u4f20\u300b,\u7b97\u76d8\u51fa\u7248\u793e,\u4f26\u6566｡<\/p>\n

    Kant, I., 1966, Kritik der reinen Vernunft, Reclam, Stuttgart. \u5eb7\u5fb7,I.,1966,\u300a\u7eaf\u7cb9\u7406\u6027\u6279\u5224\u300b,\u83b1\u514b\u62c9\u59c6\u51fa\u7248\u793e,\u65af\u56fe\u52a0\u7279｡<\/p>\n

    Kessler, V., 1994, \u2018Crossed product orders and non-commutative arithmetic\u2019, Journal of Number Theory 46(3), 255\u2013302. https:\/\/doi.org\/10.1006\/jnth.1994.1015 \u514b\u65af\u52d2,V.,1994,\u300a\u4ea4\u53c9\u79ef\u5e8f\u4e0e\u975e\u4ea4\u6362\u7b97\u672f\u300b,\u300a\u6570\u8bba\u671f\u520a\u300b,46(3),\u7b2c 255-302 \u9875｡<\/p>\n

    Kessler, V., 2018, \u2018God becomes beautiful \u2026 in mathematics\u2019, HTS Teologiese Studies \/ Theological Studies 74(1), 4886. https:\/\/doi.org\/10.4102\/hts.v74i1.4886 \u514b\u65af\u52d2,V.,2018,\u300a\u4e0a\u5e1d\u5728\u6570\u5b66\u4e2d\u5f70\u663e\u5176\u7f8e\u2026\u2026\u300b,\u300a\u795e\u5b66\u7814\u7a76\u300b,74(1),4886｡<\/p>\n

    Kessler, V., 2019, \u2018Spirituality in mathematics\u2019, Journal for the Study of Spirituality 9(1), 49\u201361. https:\/\/doi.org\/10.1080\/20440243.2019.1581384 \u514b\u65af\u52d2,V.,2019,\u300a\u6570\u5b66\u4e2d\u7684\u7075\u6027\u300b,\u300a\u7075\u6027\u7814\u7a76\u671f\u520a\u300b,9(1),\u7b2c 49-61 \u9875｡<\/p>\n

    Koetsier, T. & Bergmann, L. (eds.), 2005, Mathematics and the divine: A historical study, Elsevier, Amsterdam. \u79d1\u9f50\u5c14,T.､\u4f2f\u683c\u66fc,L.(\u4e3b\u7f16),2005,\u300a\u6570\u5b66\u4e0e\u795e\u5723:\u4e00\u9879\u5386\u53f2\u7814\u7a76\u300b,\u7231\u601d\u552f\u5c14\u51fa\u7248\u793e,\u963f\u59c6\u65af\u7279\u4e39｡<\/p>\n

    Livio, M., 2009, Is God a mathematician?, Simon & Schuster, New York, NY. \u5229\u7ef4\u5965,M.,2009,\u300a\u4e0a\u5e1d\u662f\u6570\u5b66\u5bb6\u5417?\u300b,\u897f\u8499\u4e0e\u8212\u65af\u7279\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Martin, G., 1960, Leibniz: Logik und Metaphysik, K\u00f6lner Universit\u00e4tsverlag, K\u00f6ln. \u9a6c\u4e01,G.,1960,\u300a\u83b1\u5e03\u5c3c\u8328:\u903b\u8f91\u4e0e\u5f62\u800c\u4e0a\u5b66\u300b,\u79d1\u9686\u5927\u5b66\u51fa\u7248\u793e,\u79d1\u9686｡<\/p>\n

    Mendelson, E., 1987, Introduction to mathematical logic, 3rd edn., Wadsworth & Brooks\/Cole, Monterey. \u95e8\u5fb7\u5c14\u677e,E.,1987,\u300a\u6570\u7406\u903b\u8f91\u5bfc\u8bba\u300b(\u7b2c 3 \u7248),\u6c83\u5179\u6c83\u601d\u4e0e\u5e03\u9c81\u514b\u65af\/\u79d1\u5c14\u51fa\u7248\u793e,\u8499\u7279\u96f7｡<\/p>\n

    Meyer, R.K., 1987, \u2018God exists!', No\u00fbs 21(3), 345\u2013361. https:\/\/doi.org\/10.2307\/2215186 \u8fc8\u8036,R\u00b7K.,1987,\u300a\u4e0a\u5e1d\u5b58\u5728!\u300b,\u300a\u8bfa\u52aa\u65af\u300b,21(3),\u7b2c 345-361 \u9875｡<\/p>\n

    Mukunda, N., 2015, \u2018Mathematics as the language of nature \u2013 A historical view\u2019, Proceedings of the Indian National Science Academy 81(2), 437\u2013356. https:\/\/doi.org\/10.16943\/ptinsa\/2015\/v81i2\/48092 \u7a46\u6606\u8fbe,N.,2015,\u300a\u4f5c\u4e3a\u81ea\u7136\u8bed\u8a00\u7684\u6570\u5b66:\u4e00\u79cd\u5386\u53f2\u89c6\u89d2\u300b,\u300a\u5370\u5ea6\u56fd\u5bb6\u79d1\u5b66\u9662\u9662\u520a\u300b,81(2),\u7b2c 347-356 \u9875｡<\/p>\n

    Otto, R., 1936, The idea of the holy: An inquiry into the non-rational factor in the idea of the divine and its relation to the rational, Oxford University Press, Oxford. \u5965\u6258,R.,1936,\u300a\u795e\u5723\u7684\u89c2\u5ff5:\u5bf9\u795e\u5723\u89c2\u5ff5\u4e2d\u975e\u7406\u6027\u56e0\u7d20\u53ca\u5176\u4e0e\u7406\u6027\u5173\u7cfb\u7684\u63a2\u7a76\u300b,\u725b\u6d25\u5927\u5b66\u51fa\u7248\u793e,\u725b\u6d25｡<\/p>\n

    Penrose, R., 1991, The emperor\u2019s new mind: Concerning computers, minds, and the laws of physics, Penguin Books, New York, NY. \u5f6d\u7f57\u65af,R.,1991,\u300a\u7687\u5e1d\u65b0\u8111:\u6709\u5173\u7535\u8111､\u4eba\u8111\u4e0e\u7269\u7406\u5b9a\u5f8b\u300b,\u4f01\u9e45\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Phillips, S., 2009, The mathematical connection between religion and sciences, Anthony Rowe, Chippenham. \u83f2\u5229\u666e\u65af,S.,2009,\u300a\u5b97\u6559\u4e0e\u79d1\u5b66\u4e4b\u95f4\u7684\u6570\u5b66\u8054\u7ed3\u300b,\u5b89\u4e1c\u5c3c\u00b7\u7f57\u51fa\u7248\u793e,\u5947\u5f6d\u7eb3\u59c6｡<\/p>\n

    Poincar\u00e9, H., 1905, Science and hypothesis, Walter Scott, London. \u5e9e\u52a0\u83b1,H.,1905,\u300a\u79d1\u5b66\u4e0e\u5047\u8bbe\u300b,\u6c83\u5c14\u7279\u00b7\u65af\u79d1\u7279\u51fa\u7248\u793e,\u4f26\u6566｡<\/p>\n

    Popper, K., 1980, Three worlds, The Tanner lecture on Human values, viewed 11 January 2021, from https:\/\/tannerlectures.utah.edu\/_resources\/documents\/a-to-z\/p\/popper80.pdf. \u6ce2\u666e\u5c14,K.,1980,\u300a\u4e09\u4e2a\u4e16\u754c\u300b,\u5766\u7eb3\u4eba\u7c7b\u4ef7\u503c\u8bb2\u5ea7,2021 \u5e74 1 \u6708 11 \u65e5\u67e5\u9605,\u81ea https:\/\/tannerlectures.utah.edu\/_resources\/documents\/a-to-z\/p\/popper80.pdf｡<\/p>\n

    Russell, B., 2014, Mysticism and logic and other essays, Martino Publishing, Mansfield Centre, Mansfield. \u7f57\u7d20,B.,2014,\u300a\u795e\u79d8\u4e3b\u4e49\u4e0e\u903b\u8f91\u53ca\u5176\u4ed6\u8bba\u6587\u300b,\u9a6c\u8482\u8bfa\u51fa\u7248\u793e,\u66fc\u65af\u83f2\u5c14\u5fb7\u4e2d\u5fc3｡<\/p>\n

    Shapiro, S., 2000, Thinking about mathematics: The philosophy of mathematics, Oxford University Press, Oxford. \u590f\u76ae\u7f57,S.,2000,\u300a\u601d\u8003\u6570\u5b66:\u6570\u5b66\u54f2\u5b66\u300b,\u725b\u6d25\u5927\u5b66\u51fa\u7248\u793e,\u725b\u6d25｡<\/p>\n

    Sinclair, N., Pimm, D. & Higginson, W. (eds.), 2006, Mathematics and the aesthetic: New approaches to an ancient affinity, Canadian Mathematical Society No. 25, Springer, New York, NY. \u8f9b\u514b\u83b1,N.､\u76ae\u59c6,D.､\u5e0c\u91d1\u68ee,W.(\u4e3b\u7f16),2006,\u300a\u6570\u5b66\u4e0e\u7f8e\u5b66:\u5bf9\u4e00\u79cd\u53e4\u8001\u8054\u7ed3\u7684\u65b0\u63a2\u8ba8\u300b,\u52a0\u62ff\u5927\u6570\u5b66\u4f1a\u4e1b\u4e66(\u7b2c 25 \u5377),\u65af\u666e\u6797\u683c\u51fa\u7248\u793e,\u7ebd\u7ea6\u5dde\u7ebd\u7ea6\u5e02｡<\/p>\n

    Singh, S., 1997, Fermat\u2019s last theorem: The story of a riddle that confounded the world\u2019s greatest minds for 358 years, Fourth Estate, London. \u8f9b\u683c,S.,1997,\u300a\u8d39\u9a6c\u5927\u5b9a\u7406:\u4e00\u4e2a\u56f0\u60d1\u4e86\u4e16\u95f4\u667a\u8005 358 \u5e74\u7684\u8c1c\u9898\u300b,\u7b2c\u56db\u7b49\u7ea7\u51fa\u7248\u793e,\u4f26\u6566｡<\/p>\n

    Smit, P.-B. & De Heide, N., 2021, \u2018God, the beautiful and mathematics: A response\u2019, HTS Teologiese Studies \/ Theological Studies 77(4), a6208. https:\/\/doi.org\/10.4102\/hts.v77i4.6208 \u65af\u5bc6\u7279,P\u00b7B.､\u5fb7\u00b7\u6d77\u5fb7,N.,2021,\u300a\u4e0a\u5e1d､\u7f8e\u4e0e\u6570\u5b66:\u4e00\u5219\u56de\u5e94\u300b,\u300a\u795e\u5b66\u7814\u7a76\u300b,77(4),a6208｡<\/p>\n

    Stewart, I., 2017, Significant figures: Lives and works of trailblazing mathematicians, Profile Books, London. \u65af\u56fe\u5c14\u7279,I.,2017,\u300a\u4f1f\u5927\u7684\u6570\u5b66\u5bb6:\u4ece\u829d\u8bfa\u5230\u5e9e\u52a0\u83b1\u300b,\u666e\u7f57\u83f2\u52d2\u51fa\u7248\u793e,\u4f26\u6566｡<\/p>\n

    Von Balthasar, H.U., 1984, The glory of the Lord: A theological aesthetics, Seven volumes, T&T Clark, Edinburgh. \u51af\u00b7\u5df4\u5c14\u5854\u8428,H\u00b7U.,1984,\u300a\u4e0a\u5e1d\u7684\u8363\u8000:\u795e\u5b66\u7f8e\u5b66\u300b(\u4e03\u5377\u672c),T&T \u514b\u62c9\u514b\u51fa\u7248\u793e,\u7231\u4e01\u5821｡<\/p>\n

    Witz, K.G., 2007, Spiritual aspirations connected with mathematics: The experience of American university students, Edwin Mellen Press, Lewiston. \u7ef4\u8328,K\u00b7G.,2007,\u300a\u4e0e\u6570\u5b66\u76f8\u5173\u7684\u7075\u6027\u8ffd\u6c42:\u7f8e\u56fd\u5927\u5b66\u751f\u7684\u4f53\u9a8c\u300b,\u57c3\u5fb7\u6e29\u00b7\u6885\u4f26\u51fa\u7248\u793e,\u5218\u6613\u65af\u987f｡<\/p>\n

    Zeki, S., Romaya, J.P., Benincasa, D.M. & Atiyah, M., 2014, \u2018The experience of mathematical beauty and its neural correlates\u2019, Frontiers in Human Neuroscience 8(68), 1\u201312. https:\/\/doi.org\/10.3389\/fnhum.2014.00068 \u6cfd\u57fa,S.､\u7f57\u739b\u96c5,J\u00b7P.､\u8d1d\u5b81\u5361\u8428,D\u00b7M.､\u963f\u8482\u4e9a,M.,2014,\u300a\u6570\u5b66\u4e4b\u7f8e\u7684\u4f53\u9a8c\u53ca\u5176\u795e\u7ecf\u5173\u8054\u300b,\u300a\u4eba\u7c7b\u795e\u7ecf\u79d1\u5b66\u524d\u6cbf\u300b,8(68),\u7b2c 1-12 \u9875｡<\/p>\n

    \n

    \u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898：\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417？<\/h2>\n

    \u4e00\u3001\u5f15\u8a00：\u6570\u5b66\u7684"\u65e0\u7406\u7531\u6709\u6548\u6027"<\/h3>\n

    \u6570\u5b66\u5728\u63cf\u8ff0\u81ea\u7136\u73b0\u8c61\u65f6\u5c55\u73b0\u51fa\u7684\u975e\u51e1\u6548\u529b，\u6784\u6210\u4e86\u79d1\u5b66\u54f2\u5b66\u4e2d\u6700\u6df1\u523b\u7684\u8c1c\u9898\u4e4b\u4e00\u3002\u82f1\u56fd\u7269\u7406\u5b66\u5bb6\u8a79\u59c6\u65af\u00b7\u91d1\u65af（James Jeans, 1877\u20131946）\u66fe\u8a00："\u5b87\u5b99\u770b\u4e0a\u53bb\u662f\u7531\u4e00\u4f4d\u7406\u8bba\u6570\u5b66\u5bb6\u8bbe\u8ba1\u7684\u3002"\u8fd9\u4e00\u89c2\u5bdf\u6307\u5411\u4e00\u4e2a\u6839\u672c\u6027\u95ee\u9898：\u6570\u5b66\u4e3a\u4f55\u80fd\u591f\u5982\u6b64\u7cbe\u786e\u5730\u523b\u753b\u7269\u7406\u5b9e\u5728？<\/p>\n

    \u963f\u5c14\u4f2f\u7279\u00b7\u7231\u56e0\u65af\u5766\u4e8e 1934 \u5e74\u63d0\u51fa\u7684\u7591\u95ee\u66f4\u4e3a\u5c16\u9510：<\/p>\n

    \u201c\u6570\u5b66，\u8fd9\u4e2a\u72ec\u7acb\u4e8e\u7ecf\u9a8c\u7684\u4eba\u7c7b\u601d\u7ef4\u7684\u4ea7\u7269，\u4e3a\u4f55\u80fd\u5982\u6b64\u5b8c\u7f8e\u5730\u7b26\u5408\u7269\u7406\u5b9e\u5728\u4e2d\u7684\u5bf9\u8c61？\u201d<\/p>\n

    \u6b64\u5373\u8457\u540d\u7684"\u6570\u5b66\u65e0\u7406\u7531\u7684\u6709\u6548\u6027"（unreasonable effectiveness of mathematics）\u95ee\u9898，\u7531\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u5f97\u4e3b\u5c24\u91d1\u00b7\u7ef4\u683c\u7eb3（Eugene Wigner, 1902\u20131995）\u4e8e 1960 \u5e74\u7cfb\u7edf\u9610\u8ff0\u3002\u4fdd\u7f57\u00b7\u72c4\u62c9\u514b（Paul Dirac）\u4ea6\u66fe\u8868\u8ff0\u7c7b\u4f3c\u6d1e\u89c1：<\/p>\n

    "\u4e0a\u5e1d\u7528\u7f8e\u4e3d\u7684\u6570\u5b66\u521b\u9020\u4e16\u754c\u3002<\/p>\n

    \u4e8c\u3001\u6570\u5b66\u6709\u6548\u6027\u7684\u53cc\u91cd\u9762\u5411<\/h3>\n

    2.1 \u4e3b\u52a8\u5c42\u9762：\u4ece\u73b0\u8c61\u5230\u89c4\u5f8b<\/h4>\n

    \u7269\u7406\u5b66\u5bb6\u4e3b\u52a8\u8fd0\u7528\u6570\u5b66\u5de5\u5177\u6784\u5efa\u7406\u8bba，\u8fd9\u4e00\u8fc7\u7a0b\u672c\u8eab\u5373\u4ee4\u4eba\u60ca\u5f02\u3002\u827e\u8428\u514b\u00b7\u725b\u987f\u4ece\u82f9\u679c\u843d\u5730\u3001\u6708\u7403\u8f68\u9053\u4e0e\u6f6e\u6c50\u73b0\u8c61\u4e2d\u62bd\u8c61\u51fa\u666e\u9002\u7684\u6570\u5b66\u5b9a\u5f8b；\u8a79\u59c6\u65af\u00b7\u514b\u62c9\u514b\u00b7\u9ea6\u514b\u65af\u97e6（James Clerk Maxwell, 1831\u20131879）\u4ee5\u56db\u4e2a\u7b80\u6d01\u7684\u65b9\u7a0b\u6982\u62ec\u4e86\u7535\u78c1\u5b66\u7684\u5168\u90e8\u5df2\u77e5\u73b0\u8c61；\u7231\u56e0\u65af\u5766\u7684\u5e7f\u4e49\u76f8\u5bf9\u8bba\u5219\u4ee5\u7cbe\u786e\u7684\u6570\u5b66\u7ed3\u6784\u63ed\u793a\u65f6\u7a7a\u7684\u51e0\u4f55\u672c\u8d28\u3002<\/p>\n

    2.2 \u88ab\u52a8\u5c42\u9762：\u7eaf\u7cb9\u6570\u5b66\u7684\u610f\u5916\u5e94\u7528<\/h4>\n

    \u66f4\u4e3a\u795e\u79d8\u7684\u662f，\u6570\u5b66\u5bb6\u51fa\u4e8e\u7eaf\u7cb9\u7406\u8bba\u5174\u8da3\u53d1\u5c55\u7684\u62bd\u8c61\u7ed3\u6784，\u5f80\u5f80\u5728\u6570\u5341\u5e74\u751a\u81f3\u6570\u767e\u5e74\u540e\u88ab\u53d1\u73b0\u6070\u597d\u63cf\u8ff0\u7269\u7406\u5b9e\u5728\u3002<\/p>\n

    \u5178\u578b\u6848\u4f8b：<\/p>\n\n\u6570\u5b66\u7406\u8bba\u521b\u7acb\u8005\u4e0e\u65f6\u95f4\u7269\u7406\u5e94\u7528<\/tr>\n\n\n\n\n\n\n
    \u5706\u9525\u66f2\u7ebf<\/td>\n\u95e8\u5948\u8d6b\u83ab\u65af，\u7ea6\u516c\u5143\u524d 350 \u5e74<\/td>\n\u5f00\u666e\u52d2\u4e0e\u725b\u987f\u7684\u884c\u661f\u8f68\u9053\u7406\u8bba<\/td>\n<\/tr>\n
    \u9ece\u66fc\u51e0\u4f55<\/td>\n\u6ce2\u6069\u54c8\u5fb7\u00b7\u9ece\u66fc（Bernhard Riemann, 1826\u20131866），1854 \u5e74<\/td>\n\u7231\u56e0\u65af\u5766\u5e7f\u4e49\u76f8\u5bf9\u8bba\u7684\u6570\u5b66\u6846\u67b6<\/td>\n<\/tr>\n
    \u7fa4\u8bba<\/td>\n\u57c3\u74e6\u91cc\u65af\u7279\u00b7\u4f3d\u7f57\u74e6（\u00c9variste Galois, 1811\u20131832）<\/td>\n\u7c92\u5b50\u7269\u7406\u5b66\u4e2d\u7684\u5bf9\u79f0\u6027\u5206\u6790<\/td>\n<\/tr>\n
    \u8d39\u6839\u9c8d\u59c6\u5e38\u6570<\/td>\n\u7c73\u5947\u00b7\u8d39\u6839\u9c8d\u59c6（Mitchell Feigenbaum），1975 \u5e74<\/td>\n\u6709\u5e8f-\u6df7\u6c8c\u8fc7\u6e21\u7684\u666e\u9002\u6027\u89c4\u5f8b<\/td>\n<\/tr>\n
    \u7ebd\u7ed3\u7406\u8bba<\/td>\n19 \u4e16\u7eaa\u539f\u5b50\u6a21\u578b\u7814\u7a76<\/td>\nDNA \u5206\u5b50\u7ed3\u6784\u3001\u5f26\u8bba<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    \u53cd\u8bbd\u6848\u4f8b：\u82f1\u56fd\u6570\u5b66\u5bb6\u6208\u5f17\u96f7\u00b7\u54c8\u7f57\u5fb7\u00b7\u54c8\u4ee3（Godfrey Harold Hardy, 1877\u20131947）\u66fe\u81ea\u8c6a\u5730\u5ba3\u79f0\u5176\u7eaf\u6570\u5b66\u7814\u7a76"\u4eca\u5929\u6ca1\u6709\u3001\u5c06\u6765\u4e5f\u4e0d\u4f1a\u7ed9\u4e16\u754c\u5e26\u6765\u4e1d\u6beb\u5f71\u54cd"\u3002\u7136\u800c，\u5176\u53d1\u5c55\u7684\u6570\u8bba\u65b9\u6cd5\u4e0d\u4ec5\u6784\u6210\u4e86\u7fa4\u4f53\u9057\u4f20\u5b66\u4e2d\u54c8\u4ee3-\u6e29\u4f2f\u683c\u5b9a\u5f8b（Hardy-Weinberg law）\u7684\u57fa\u7840，\u66f4\u5728 1973 \u5e74\u7531\u514b\u5229\u798f\u5fb7\u00b7\u67ef\u514b\u65af（Clifford Cocks）\u5e94\u7528\u4e8e\u516c\u94a5\u5bc6\u7801\u5b66\u2014\u2014\u76f4\u63a5\u5426\u5b9a\u4e86\u54c8\u4ee3\u5173\u4e8e"\u6570\u8bba\u4e0d\u4f1a\u7528\u4e8e\u6218\u4e89\u76ee\u7684"\u7684\u65ad\u8a00\u3002<\/p>\n

    \u4e09\u3001\u4e09\u4e2a\u4e16\u754c\u7684\u8c1c\u9898<\/h3>\n

    \u725b\u6d25\u5927\u5b66\u6570\u5b66\u7269\u7406\u5b66\u5bb6\u7f57\u6770\u00b7\u5f6d\u7f57\u65af（Roger Penrose）\u5728\u5176\u8457\u4f5c\u300a\u7687\u5e1d\u7684\u65b0\u8111\u300b\u4e0e\u300a\u901a\u5411\u5b9e\u5728\u4e4b\u8def\u300b\u4e2d\u63d0\u51fa\u4e86\u4e00\u4e2a\u5206\u6790\u6846\u67b6，\u5c06\u5b9e\u5728\u5212\u5206\u4e3a\u4e09\u4e2a distinct \u9886\u57df：<\/p>\n\n\u4e16\u754c\u5185\u6db5\u5b9e\u4f8b<\/tr>\n\n\n\n\n
    \u4e16\u754c\u4e00：\u610f\u8bc6\u611f\u77e5\u4e16\u754c<\/td>\n\u4e3b\u89c2\u7cbe\u795e\u4f53\u9a8c\u7684\u603b\u548c<\/td>\n\u60c5\u611f\u3001\u5ba1\u7f8e\u4f53\u9a8c\u3001\u610f\u8bc6\u72b6\u6001<\/td>\n<\/tr>\n
    \u4e16\u754c\u4e8c：\u7269\u7406\u5b9e\u5728\u4e16\u754c<\/td>\n\u5ba2\u89c2\u7269\u8d28\u5b58\u5728<\/td>\n\u661f\u7cfb\u3001\u539f\u5b50\u3001\u751f\u7269\u4f53<\/td>\n<\/tr>\n
    \u4e16\u754c\u4e09：\u6570\u5b66\u67cf\u62c9\u56fe\u4e16\u754c<\/td>\n\u62bd\u8c61\u6570\u5b66\u5f62\u5f0f\u7684\u9886\u57df<\/td>\n\u81ea\u7136\u6570\u3001\u51e0\u4f55\u5b9a\u7406\u3001\u7269\u7406\u5b9a\u5f8b\u7684\u6570\u5b66\u7ed3\u6784<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    \u8fd9\u4e09\u4e2a\u4e16\u754c\u4e4b\u95f4\u5b58\u5728\u795e\u79d8\u7684\u5173\u8054：<\/p>\n

  • \u7269\u7406\u4e16\u754c\u4f3c\u4e4e\u9075\u5faa\u5b58\u5728\u4e8e\u6570\u5b66\u4e16\u754c\u4e2d\u7684\u6cd5\u5219；<\/li>\n
  • \u611f\u77e5\u5fc3\u667a\u6e90\u4e8e\u7269\u7406\u4e16\u754c（\u5fc3-\u8111\u95ee\u9898）；<\/li>\n
  • \u5fc3\u667a\u80fd\u591f\u53d1\u73b0\u5e76\u8868\u8ff0\u6570\u5b66\u771f\u7406，\u4ece\u800c\u8fdb\u5165\u6570\u5b66\u4e16\u754c\u3002<\/li>\n

    \u5f6d\u7f57\u65af\u5bf9\u6b64\u7684\u7ed3\u8bba\u662f\u8c26\u900a\u7684：\u201c\u6beb\u65e0\u7591\u95ee，\u5e76\u4e0d\u771f\u6b63\u5b58\u5728\u4e09\u4e2a\u4e16\u754c，\u800c\u662f\u53ea\u6709\u4e00\u4e2a\u4e16\u754c\u3002\u5e76\u4e14\u76f4\u5230\u76ee\u524d\u4e3a\u6b62，\u5bf9\u4e8e\u8fd9\u4e2a\u771f\u5b9e\u4e16\u754c\u7684\u672c\u8d28，\u6211\u4eec\u5bf9\u5b83\u7684\u8ba4\u8bc6\u751a\u81f3\u8fde\u80a4\u6d45\u4e5f\u8c08\u4e0d\u4e0a\u3002\u201d<\/p>\n

    \u56db\u3001\u53d1\u73b0\u8fd8\u662f\u53d1\u660e？\u672c\u4f53\u8bba\u4e4b\u4e89<\/h3>\n

    \u6570\u5b66\u54f2\u5b66\u7684\u6839\u672c\u5206\u6b67\u5728\u4e8e：\u6570\u5b66\u5bf9\u8c61\u662f\u72ec\u7acb\u4e8e\u4eba\u7c7b\u5fc3\u667a\u7684\u5ba2\u89c2\u5b58\u5728（\u53d1\u73b0\u8bba），\u8fd8\u662f\u4eba\u7c7b\u601d\u7ef4\u7684\u6784\u9020（\u53d1\u660e\u8bba）？<\/p>\n

    4.1 \u67cf\u62c9\u56fe\u4e3b\u4e49：\u6570\u5b66\u4f5c\u4e3a\u53d1\u73b0<\/h4>\n

    \u6cd5\u56fd\u6570\u5b66\u5bb6\u963f\u5170\u00b7\u5b54\u6d85（Alain Connes, 1982 \u5e74\u83f2\u5c14\u5179\u5956\u30012001 \u5e74\u514b\u62c9\u798f\u5fb7\u5956\u5f97\u4e3b）\u660e\u786e\u4e3b\u5f20：<\/p>\n

    \u201c\u6839\u636e\u6211\u7684\u89c2\u5bdf，\u8d28\u6570\u7ec4\u6210\u7684\u4e16\u754c，\u8fdc\u6bd4\u6211\u4eec\u5468\u56f4\u7684\u7269\u8d28\u4e16\u754c\u7a33\u5b9a\u3002\u6570\u5b66\u5bb6\u7684\u5de5\u4f5c\u53ef\u4ee5\u4e0e\u63a2\u9669\u5bb6\u53d1\u73b0\u4e16\u754c\u76f8\u5ab2\u7f8e\u3002\u201d<\/p>\n

    \u5bf9\u8fd9\u4e00\u89c2\u70b9\u7684\u652f\u6301\u5305\u62ec：<\/p>\n

      \n
    • \u6570\u5b66\u771f\u7406\u7684\u5fc5\u7136\u6027：\n

      3<\/p>\n

      2<\/p>\n

      +<\/p>\n

      4<\/p>\n

      2<\/p>\n

      =<\/p>\n

      5<\/p>\n

      2<\/p>\n

      3^2 + 4^2 = 5^2<\/p>\n

      <\/span><\/span>3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>4<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>5<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0d\u4f9d\u8d56\u4e8e\u4eba\u7c7b\u8ba4\u77e5；<\/li>\n

    • \u6570\u5b66\u7ed3\u8bba\u7684\u6c38\u6052\u6027：\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u5b9a\u7406\u5386\u7ecf\u4e24\u5343\u4f59\u5e74\u4f9d\u7136\u6709\u6548，\u800c\u7269\u7406\u5b66\u7406\u8bba\u4e0d\u65ad\u88ab\u4fee\u6b63；<\/li>\n
    • \u6570\u5b66\u7684\u53ef\u9519\u6027（fallibilism）\u4ec5\u9650\u4e8e\u8bc1\u660e\u8fc7\u7a0b\u7684\u4e25\u8c28\u6027，\u800c\u975e\u7ed3\u8bba\u672c\u8eab\u7684\u771f\u4f2a\u3002<\/li>\n<\/ul>\n

      \u6570\u5b66\u79d1\u666e\u4f5c\u5bb6\u9a6c\u4e01\u00b7\u52a0\u5fb7\u7eb3（Martin Gardner）\u4ee5\u751f\u52a8\u6bd4\u55bb\u652f\u6301\u6b64\u8bf4：\u201c\u5982\u679c\u68ee\u6797\u4e2d\u6709 <\/p>\n

      2<\/p>\n

      2<\/p>\n

      <\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> \u53ea\u6050\u9f99\u4e0e\u53e6\u5916 <\/p>\n

      2<\/p>\n

      2<\/p>\n

      <\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> \u53ea\u6050\u9f99\u76f8\u9047，\u4e0d\u7ba1\u5468\u56f4\u662f\u5426\u6709\u4eba\u7c7b\u5728\u89c2\u5bdf，\u90a3\u513f\u90fd\u4f1a\u6709 <\/p>\n

      4<\/p>\n

      4<\/p>\n

      <\/span><\/span>4<\/span><\/span><\/span><\/span><\/span> \u53ea\u6050\u9f99\u3002\u201d<\/p>\n

      4.2 \u6784\u9020\u4e3b\u4e49：\u6570\u5b66\u4f5c\u4e3a\u53d1\u660e<\/h4>\n

      \u53cd\u5bf9\u610f\u89c1\u540c\u6837\u6709\u529b\u3002\u82f1\u56fd\u6570\u5b66\u5bb6\u8fc8\u514b\u5c14\u00b7\u963f\u8482\u4e9a\u7235\u58eb（Sir Michael Atiyah, 1966 \u5e74\u83f2\u5c14\u5179\u5956\u30012004 \u5e74\u963f\u8d1d\u5c14\u5956\u5f97\u4e3b）\u8d28\u7591：<\/p>\n

      \u201c\u5047\u5982\u5b87\u5b99\u662f\u4e00\u7ef4\u7a7a\u95f4，\u6216\u8005\u751a\u81f3\u662f\u79bb\u6563\u7684，\u5f88\u96be\u60f3\u8c61\u51e0\u4f55\u5b66\u5728\u8fd9\u4e2a\u4e00\u7ef4\u7a7a\u95f4\u4e2d\u662f\u5982\u4f55\u5b55\u80b2\u53d1\u5c55\u7684\u3002\u201d<\/p>\n

      \u963f\u8482\u4e9a\u4e3b\u5f20：\u201c\u901a\u8fc7\u7406\u60f3\u5316\u548c\u62bd\u8c61\u7269\u7406\u4e16\u754c\u4e2d\u7684\u90a3\u4e9b\u57fa\u672c\u8981\u7d20，\u4eba\u7c7b\u521b\u9020\u4e86\u6570\u5b66\u3002\u201d<\/p>\n

      \u8ba4\u77e5\u79d1\u5b66\u5bb6\u4e54\u6cbb\u00b7\u83b1\u8003\u592b（George Lakoff）\u4e0e\u62c9\u6590\u5c14\u00b7\u52aa\u6d85\u65af（Rafael N\u00fa\u00f1ez）\u5728\u300a\u6570\u5b66\u6765\u81ea\u54ea\u91cc\u300b（Where Mathematics Comes From）\u4e2d\u63d0\u51fa：\u6570\u5b66\u6e90\u4e8e"\u6211\u4eec\u7684\u8eab\u4f53\u3001\u5927\u8111，\u4ee5\u53ca\u6211\u4eec\u5728\u8fd9\u4e2a\u4e16\u754c\u4e2d\u6bcf\u5929\u7684\u7ecf\u5386"\u3002<\/p>\n

      4.3 \u5173\u952e\u95ee\u9898<\/h4>\n

      \u8fd9\u4e00\u5206\u6b67\u5f15\u51fa\u6df1\u5c42\u95ee\u9898：<\/p>\n

        \n
      • \u82e5\u6570\u5b66\u4e3a\u4eba\u7c7b\u53d1\u660e，\u5176\u666e\u904d\u6027\u5982\u4f55\u4fdd\u8bc1？\u5916\u661f\u6587\u660e\u662f\u5426\u4f1a\u53d1\u5c55\u51fa\u4e0d\u540c\u7684\u6570\u5b66？<\/li>\n
      • \u82e5\u6570\u5b66\u72ec\u7acb\u5b58\u5728，\u4eba\u7c7b\u5982\u4f55\u8de8\u8d8a\u65f6\u7a7a\u9650\u5236\u63a5\u89e6\u8fd9\u4e00\u6c38\u6052\u9886\u57df？<\/li>\n
      • \u6570\u5b66\u4e0e\u8bed\u8a00\u3001\u827a\u672f\u7684\u672c\u8d28\u533a\u522b\u4f55\u5728？<\/li>\n<\/ul>\n

        \u5361\u5c14\u00b7\u8428\u6839（Carl Sagan）\u8ba4\u4e3a\u8d28\u6570\u53ef\u4f5c\u4e3a\u8de8\u6587\u660e\u7684\u901a\u7528\u4fe1\u53f7；\u53f2\u8482\u82ac\u00b7\u6c83\u5c14\u592b\u62c9\u59c6（Stephen Wolfram）\u5219\u63d0\u51fa\u4eba\u7c7b\u6570\u5b66\u4ec5\u662f\u4f17\u591a\u53ef\u80fd\u5f62\u5f0f\u4e2d\u7684\u4e00\u79cd\u3002<\/p>\n

        \u4e94\u3001\u6570\u5b66\u4e0e\u5176\u4ed6\u77e5\u8bc6\u4f53\u7cfb\u7684\u5dee\u5f02<\/h3>\n

        \u5728\u81ea\u7136\u79d1\u5b66\u4e2d，\u65e7\u7406\u8bba\u5e38\u88ab\u65b0\u89c2\u6d4b\u4e0e\u65b0\u7406\u8bba\u53d6\u4ee3\u3002\u4f8b\u5982 19 \u4e16\u7eaa\u539f\u5b50\u6a21\u578b\u5728\u77ed\u65f6\u95f4\u5185\u88ab\u4fee\u6b63，\u725b\u987f\u529b\u5b66\u5728\u9ad8\u901f\u4e0e\u5fae\u89c2\u5c3a\u5ea6\u4e0b\u88ab\u76f8\u5bf9\u8bba\u4e0e\u91cf\u5b50\u529b\u5b66\u62d3\u5c55\u3002<\/p>\n

        \u800c\u6570\u5b66\u5448\u73b0\u51fa\u4e0d\u540c\u7684\u6f14\u8fdb\u6a21\u5f0f：<\/p>\n

          \n
        • \u5df2\u88ab\u4e25\u683c\u8bc1\u660e\u7684\u6570\u5b66\u7ed3\u8bba，\u901a\u5e38\u4e0d\u4f1a\u88ab\u540e\u7eed\u7814\u7a76\u5426\u5b9a；<\/li>\n
        • \u65b0\u6570\u5b66\u6210\u679c\u53ef\u4ee5\u517c\u5bb9\u3001\u62d3\u5c55\u65e7\u6210\u679c，\u800c\u975e\u7b80\u5355\u629b\u5f03；<\/li>\n
        • \u516c\u5143\u524d 250 \u5e74\u963f\u57fa\u7c73\u5fb7\u8bc1\u660e\u7684\u7403\u8868\u9762\u79ef\u516c\u5f0f，\u5728\u5f53\u4ee3\u4ecd\u88ab\u4f7f\u7528\u3002<\/li>\n<\/ul>\n

          \u4f0a\u6069\u00b7\u65af\u56fe\u5c14\u7279（Ian Stewart）\u6307\u51fa：<\/p>\n

          \u201c\u5728\u6570\u5b66\u9886\u57df\u91cc，\u8c2c\u8bef\u4e00\u8bcd\u8868\u793a\u5148\u524d\u4ee5\u4e3a\u662f\u6b63\u786e\u7684\u3001\u800c\u540e\u6765\u5374\u53d1\u73b0\u6709\u9519\u8bef\u5e76\u88ab\u7ea0\u6b63\u7684\u7ed3\u8bba\u3002\u201d<\/p>\n

          \u8fd9\u4e9b\u7ed3\u8bba\u4e4b\u6240\u4ee5\u88ab\u5224\u5b9a\u4e3a\u8c2c\u8bef，\u662f\u7531\u4e8e\u6570\u5b66\u5185\u90e8\u66f4\u4e25\u683c\u7684\u68c0\u9a8c，\u800c\u975e\u5916\u90e8\u5b66\u79d1\u7684\u51b2\u51fb\u3002<\/p>\n

          \u516d\u3001\u5386\u53f2\u89c6\u89d2：\u4ece\u6bd5\u8fbe\u54e5\u62c9\u65af\u5230\u73b0\u4ee3<\/h3>\n

          \u6570\u5b66\u4e0e\u5b9e\u5728\u7684\u5173\u7cfb\u95ee\u9898\u5e76\u975e\u73b0\u4ee3\u72ec\u6709\u3002\u6258\u9a6c\u65af\u00b7\u970d\u5e03\u65af（Thomas Hobbes, 1588\u20131679）\u5728\u300a\u5229\u7ef4\u5766\u300b\u4e2d\u5c06\u51e0\u4f55\u5b66\u89c6\u4e3a\u7406\u6027\u8bba\u8bc1\u7684\u5178\u8303：<\/p>\n

          \u201c\u5728\u51e0\u4f55\u5b66（\u8fd9\u662f\u8fc4\u4eca\u4e3a\u6b62\u552f\u4e00\u4ee4\u4e0a\u5e1d\u6ee1\u610f\u5e76\u6069\u8d50\u7ed9\u4eba\u7c7b\u7684\u5b66\u95ee）\u4e2d，\u4eba\u4eec\u9996\u5148\u786e\u5b9a\u540d\u79f0\u7684\u542b\u4e49（\u4ed6\u4eec\u628a\u786e\u5b9a\u542b\u4e49\u79f0\u4e3a\u2019\u5b9a\u4e49\u2019），\u5e76\u628a\u5b83\u4eec\u4f5c\u4e3a\u8ba4\u77e5\u7684\u8d77\u70b9\u3002\u201d<\/p>\n

          \u8fd9\u4e00\u4f20\u7edf\u53ef\u8ffd\u6eaf\u81f3\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u7684"\u4e07\u7269\u7686\u6570"\u4e0e\u67cf\u62c9\u56fe\u7684\u7406\u578b\u8bba\u3002\u7136\u800c，\u5386\u53f2\u4ea6\u663e\u793a\u6570\u5b66\u89c2\u7684\u6f14\u53d8：\u4ece\u53e4\u5e0c\u814a\u7684\u786e\u5b9a\u6027\u8ffd\u6c42，\u5230 19 \u4e16\u7eaa\u975e\u6b27\u51e0\u4f55\u5e26\u6765\u7684\u5371\u673a，\u518d\u5230 20 \u4e16\u7eaa\u516c\u7406\u5316\u65b9\u6cd5\u4e0e\u5f62\u5f0f\u4e3b\u4e49\u7684\u53d1\u5c55\u3002<\/p>\n

          \u4e03\u3001\u7ed3\u8bba：\u672a\u89e3\u4e4b\u8c1c\u7684\u6301\u7eed\u610f\u4e49<\/h3>\n

          "\u4e0a\u5e1d\u662f\u6570\u5b66\u5bb6\u5417？"\u8fd9\u4e00\u95ee\u9898\u2014\u2014\u65e0\u8bba\u662f\u795e\u5b66\u8fd8\u662f\u9690\u55bb\u610f\u4e49\u4e0a\u7684\u2014\u2014\u6307\u5411\u4eba\u7c7b\u77e5\u8bc6\u7684\u57fa\u7840\u7ed3\u6784\u3002\u6570\u5b66\u65e0\u7406\u7531\u7684\u6709\u6548\u6027\u8868\u660e，\u5728\u4eba\u7c7b\u7684\u62bd\u8c61\u601d\u7ef4\u4e0e\u81ea\u7136\u7684\u6df1\u5c42\u79e9\u5e8f\u4e4b\u95f4\u5b58\u5728\u67d0\u79cd\u6df1\u523b\u5173\u8054，\u5176\u672c\u8d28\u81f3\u4eca\u672a\u88ab\u5145\u5206\u7406\u89e3\u3002<\/p>\n

          \u6b63\u5982\u7ef4\u683c\u7eb3\u6240\u8a00，\u6570\u5b66\u4e0e\u7269\u7406\u7684\u5951\u5408\u662f"\u4e0a\u5929\u8d50\u4e88\u6211\u4eec\u7684\u7edd\u5999\u793c\u7269"\u2014\u2014\u6211\u4eec\u65e2\u672a\u771f\u6b63\u7406\u89e3\u8fd9\u4efd\u793c\u7269，\u4ea6\u53d7\u4e4b\u6709\u6127\u3002\u8fd9\u4e00\u8c1c\u9898\u5c06\u7ee7\u7eed\u6fc0\u53d1\u6570\u5b66\u5bb6\u3001\u7269\u7406\u5b66\u5bb6\u4e0e\u54f2\u5b66\u5bb6\u7684\u601d\u8003，\u63d0\u9192\u6211\u4eec：\u5728\u77e5\u8bc6\u7684\u8fb9\u754c，\u60ca\u5947\u672c\u8eab\u5373\u662f\u7406\u89e3\u7684\u8d77\u70b9\u3002<\/p>\n

          \u53c2\u8003\u6587\u732e<\/p>\n

          \u672c\u6587\u5185\u5bb9\u57fa\u4e8e\u9a6c\u91cc\u5965\u00b7\u5229\u7ef4\u5965（Mario Livio）\u8457\u4f5c\u300a\u6700\u540e\u7684\u6570\u5b66\u95ee\u9898\u300b（Is God a Mathematician?）\u7684\u601d\u60f3\u8109\u7edc\u6574\u7406\u3002<\/p>\n

          \u5229\u7ef4\u5965\u4e3a\u54c8\u52c3\u592a\u7a7a\u671b\u8fdc\u955c\u79d1\u5b66\u7814\u7a76\u6240\u5929\u4f53\u7269\u7406\u5b66\u5bb6，\u6570\u5b66\u4e0e\u79d1\u5b66\u53f2\u7814\u7a76\u8005，\u66fe\u83b7\u76ae\u4e9a\u8bfa\u5956\u4e0e\u56fd\u9645\u6bd5\u8fbe\u54e5\u62c9\u65af\u6570\u5b66\u7545\u9500\u4e66\u5956\u3002<\/p>\n

          \u5ef6\u4f38\u9605\u8bfb：\u56fe\u7075\u6570\u5b66\u53f2\u7cfb\u5217\u7b80\u4ecb<\/h3>\n
        • \n

          \u300a\u6700\u540e\u7684\u6570\u5b66\u95ee\u9898：\u4e0a\u5e1d\u662f\u6570\u5b66\u5bb6\u5417？\u300b \u4ee5\u6570\u5b66\u6f14\u5316\u4e3a\u4e3b\u7ebf，\u4ece\u54f2\u5b66\u3001\u5386\u53f2\u3001\u6587\u5316\u89c6\u89d2\u8ba8\u8bba\u6570\u5b66\u7684\u5b58\u5728\u5f62\u5f0f，\u68b3\u7406\u6570\u5b66\u4e0e\u7269\u7406\u4e16\u754c\u3001\u4eba\u7c7b\u601d\u7ef4\u7684\u5173\u7cfb，\u5448\u73b0\u8d2f\u7a7f\u601d\u60f3\u53f2\u7684\u95ee\u9898：\u6570\u5b66\u662f\u53d1\u660e\u8fd8\u662f\u53d1\u73b0\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u6570\u5b66\u90a3\u4e9b\u4e8b：\u4f1f\u5927\u7684\u95ee\u9898\u4e0e\u975e\u51e1\u7684\u4eba\u300b \u4ee5 A\u2013Z \u77ed\u6587\u7ed3\u6784，\u4ecb\u7ecd\u6570\u5b66\u4e2d\u91cd\u8981\u5b9a\u7406\u3001\u95ee\u9898\u4e0e\u4eba\u7269，\u83b7\u7f8e\u56fd\u51fa\u7248\u5546\u534f\u4f1a\u6570\u5b66\u4f73\u4f5c\u5956\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u4ee3\u6570\u7684\u5386\u53f2：\u4eba\u7c7b\u5bf9\u672a\u77e5\u91cf\u7684\u4e0d\u820d\u8ffd\u8e2a\u300b \u5448\u73b0\u4ee3\u6570\u5b66\u4ece\u672a\u77e5\u91cf\u3001\u65b9\u7a0b\u7406\u8bba\u5230\u62bd\u8c61\u4ee3\u6570\u7684\u53d1\u5c55\u8109\u7edc，\u5c55\u793a\u4ee3\u6570\u601d\u7ef4\u5bf9\u6570\u5b66\u4e0e\u8ba4\u77e5\u65b9\u5f0f\u7684\u5f71\u54cd\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u4e0d\u53ef\u80fd\u7684\u51e0\u4f55\u6311\u6218：\u6570\u5b66\u6c42\u7d22\u4e24\u5343\u5e74\u300b \u56f4\u7ed5\u5316\u5706\u4e3a\u65b9\u3001\u4e09\u7b49\u5206\u89d2\u3001\u500d\u7acb\u65b9\u3001\u4f5c\u5706\u5185\u63a5\u6b63\u591a\u8fb9\u5f62\u56db\u5927\u53e4\u5178\u95ee\u9898，\u5c55\u793a\u5176\u5bf9\u6570\u5b66\u53d1\u5c55\u7684\u6301\u7eed\u63a8\u52a8\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u60a0\u626c\u7684\u7d20\u6570：\u4e8c\u5343\u5e74\u6570\u5b66\u7edd\u5531\u9ece\u66fc\u5047\u8bbe\u300b \u4ee5\u9ece\u66fc\u5047\u8bbe\u4e3a\u7ebf\u7d22，\u56de\u987e\u7d20\u6570\u7814\u7a76\u7684\u5386\u53f2，\u53ca\u5176\u5728\u7535\u5b50\u5546\u52a1\u3001\u91cf\u5b50\u529b\u5b66\u3001\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u7684\u5f71\u54cd\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u6625\u591c\u5341\u8bdd：\u6570\u5b66\u4e0e\u60c5\u7eea\u300b \u4ece\u60c5\u7eea\u4e0e\u5fc3\u667a\u53d1\u5c55\u7684\u89c6\u89d2，\u8ba8\u8bba\u8ba4\u77e5\u3001\u6559\u80b2\u4e0e\u521b\u9020\u6027\u7684\u5173\u7cfb，\u5c06\u6570\u5b66\u7406\u89e3\u4e0e\u60c5\u611f\u3001\u4eba\u683c\u53d1\u5c55\u76f8\u8054\u7cfb\u3002<\/p>\n<\/li>\n

        • \n

          \u300a\u6211\u53ea\u4f1a\u7b97\u672f：\u5c0f\u5e73\u90a6\u5f66\u81ea\u4f20\u300b \u83f2\u5c14\u5179\u5956\u3001\u6c83\u5c14\u592b\u5956\u5f97\u4e3b\u5c0f\u5e73\u90a6\u5f66\u7684\u81ea\u4f20，\u56de\u987e\u5176\u5b66\u4e60\u4e0e\u7814\u7a76\u5386\u7a0b，\u53cd\u601d\u6570\u5b66\u601d\u7ef4\u4e0e\u6570\u5b66\u6559\u80b2\u3002<\/p>\n<\/li>\n

          \n

          reference<\/h2>\n
            \n
          • God is a Mathematician – Science and Nonduality (SAND) https:\/\/scienceandnonduality.com\/article\/god-is-a-mathematician\/<\/li>\n
          • Mathematics declaring the glory of God https:\/\/scielo.org.za\/pdf\/vee\/v43n1\/12.pdf<\/li>\n
          • \u56de\u7b54\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898：\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417？ https:\/\/mp.weixin.qq.com\/s\/MGdJW4P5EN7nkbPE6FHvvA<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

            \u6ce8：\u672c\u6587\u4e3a \u201c\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\u201d \u76f8\u5173\u5408\u8f91\u3002 \u82f1\u6587\u5f15\u6587，\u673a\u7ffb\u672a\u6821\u3002 \u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002 God is a Mathematician
            \n\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6
            \nArticle by Sunil Mehrotra \u4f5c\u8005 \u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9
            \nMathematics is abstract, symbolic, structured and precise. It is true everywhere and always and mat<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[9014],"topic":[],"class_list":["post-80365","post","type-post","status-publish","format-standard","hentry","category-server","tag-9014"],"yoast_head":"\n\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/80365.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u6ce8：\u672c\u6587\u4e3a \u201c\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\u201d \u76f8\u5173\u5408\u8f91\u3002 \u82f1\u6587\u5f15\u6587，\u673a\u7ffb\u672a\u6821\u3002 \u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002 God is a Mathematician \u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6 Article by Sunil Mehrotra \u4f5c\u8005 \u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9 Mathematics is abstract, symbolic, structured and precise. It is true everywhere and always and mat\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.wsisp.com\/helps\/80365.html\" \/>\n<meta property=\"og:site_name\" content=\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"article:published_time\" content=\"2026-03-04T14:41:09+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"46 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/80365.html\",\"url\":\"https:\/\/www.wsisp.com\/helps\/80365.html\",\"name\":\"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\",\"isPartOf\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/#website\"},\"datePublished\":\"2026-03-04T14:41:09+00:00\",\"dateModified\":\"2026-03-04T14:41:09+00:00\",\"author\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41\"},\"breadcrumb\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/80365.html#breadcrumb\"},\"inLanguage\":\"zh-Hans\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.wsisp.com\/helps\/80365.html\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/80365.html#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"\u9996\u9875\",\"item\":\"https:\/\/www.wsisp.com\/helps\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#website\",\"url\":\"https:\/\/www.wsisp.com\/helps\/\",\"name\":\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\",\"description\":\"\u9999\u6e2f\u670d\u52a1\u5668_\u9999\u6e2f\u4e91\u670d\u52a1\u5668\u8d44\u8baf_\u670d\u52a1\u5668\u5e2e\u52a9\u6587\u6863_\u670d\u52a1\u5668\u6559\u7a0b\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.wsisp.com\/helps\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"zh-Hans\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41\",\"name\":\"admin\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-Hans\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery\",\"contentUrl\":\"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery\",\"caption\":\"admin\"},\"sameAs\":[\"http:\/\/wp.wsisp.com\"],\"url\":\"https:\/\/www.wsisp.com\/helps\/author\/admin\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.wsisp.com\/helps\/80365.html","og_locale":"zh_CN","og_type":"article","og_title":"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","og_description":"\u6ce8：\u672c\u6587\u4e3a \u201c\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\u201d \u76f8\u5173\u5408\u8f91\u3002 \u82f1\u6587\u5f15\u6587，\u673a\u7ffb\u672a\u6821\u3002 \u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002 God is a Mathematician \u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6 Article by Sunil Mehrotra \u4f5c\u8005 \u82cf\u5c3c\u5c14\u30fb\u6885\u8d6b\u7f57\u7279\u62c9 Mathematics is abstract, symbolic, structured and precise. It is true everywhere and always and mat","og_url":"https:\/\/www.wsisp.com\/helps\/80365.html","og_site_name":"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","article_published_time":"2026-03-04T14:41:09+00:00","author":"admin","twitter_card":"summary_large_image","twitter_misc":{"\u4f5c\u8005":"admin","\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4":"46 \u5206"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/www.wsisp.com\/helps\/80365.html","url":"https:\/\/www.wsisp.com\/helps\/80365.html","name":"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","isPartOf":{"@id":"https:\/\/www.wsisp.com\/helps\/#website"},"datePublished":"2026-03-04T14:41:09+00:00","dateModified":"2026-03-04T14:41:09+00:00","author":{"@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41"},"breadcrumb":{"@id":"https:\/\/www.wsisp.com\/helps\/80365.html#breadcrumb"},"inLanguage":"zh-Hans","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.wsisp.com\/helps\/80365.html"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.wsisp.com\/helps\/80365.html#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"\u9996\u9875","item":"https:\/\/www.wsisp.com\/helps"},{"@type":"ListItem","position":2,"name":"\u6570\u5b66\u7684\u7ec8\u6781\u95ee\u9898\uff1a\u4e0a\u5e1d\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u5417\uff1f"}]},{"@type":"WebSite","@id":"https:\/\/www.wsisp.com\/helps\/#website","url":"https:\/\/www.wsisp.com\/helps\/","name":"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","description":"\u9999\u6e2f\u670d\u52a1\u5668_\u9999\u6e2f\u4e91\u670d\u52a1\u5668\u8d44\u8baf_\u670d\u52a1\u5668\u5e2e\u52a9\u6587\u6863_\u670d\u52a1\u5668\u6559\u7a0b","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.wsisp.com\/helps\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"zh-Hans"},{"@type":"Person","@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41","name":"admin","image":{"@type":"ImageObject","inLanguage":"zh-Hans","@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/image\/","url":"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery","contentUrl":"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery","caption":"admin"},"sameAs":["http:\/\/wp.wsisp.com"],"url":"https:\/\/www.wsisp.com\/helps\/author\/admin"}]}},"_links":{"self":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts\/80365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/comments?post=80365"}],"version-history":[{"count":0,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts\/80365\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/media?parent=80365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/categories?post=80365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/tags?post=80365"},{"taxonomy":"topic","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/topic?post=80365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}