[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-431223-105":59,"doc-detail-431223-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","paradoxes-and-common-sense-introduction","PARADOXES and COMMON SENSE - INTRODUCTION","","Paradoxes and Commonsense explains what constitutes a “true paradox” as a chain of flawless reasoning that reaches a result known to be impossible, contrasting it with puzzles and games after logical resolution. It traces the word’s Greek origin and compares a paradox to a miracle as a violation of the laws of mind. The introduction discusses why a single genuine mathematical paradox would collapse mathematical science and surveys upcoming paradoxes in logic and 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is a “true paradox” according to the introduction?","Question",{"text":112,"@type":113},"A true paradox is a chain of flawless reasoning that ends in a result known to be impossible and not true.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does a paradox differ from a miracle and from a puzzle?",{"text":117,"@type":113},"A miracle violates the laws of nature, while a paradox violates the laws of mind and thought; once explained, a paradox stops being a true paradox and becomes a puzzle, game, or logical exercise.",{"name":119,"@type":110,"acceptedAnswer":120},"Why would a single mathematical paradox undermine mathematical science?",{"text":121,"@type":113},"The introduction argues that the existence of one true mathematical paradox would allow every mathematical statement to be proved logically, making proofs spurious and unraveling the logical fabric.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},431223,1791464739,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":14,"category_name":15,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},962085564807,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","PARADOXES  \nandCOMMONSENSE ayAUBREY J.KEMPNER  \nD.VAN NOSTRAND COMPANY,INC.PRINCETON,NEW JERSEYTORONTONEW YORK  \n10000  \nPARADOXESAND COMMON SENSE  \nby  \nAUBREY J.KEMPNER  \nProfessor of Mathematics,EmeritusUniversity of Colorado  \nD.VAN NOSTRAND COMPANY,INC.PRINCETON,NEW JERSEY  \nLONDON  \nTORONTO  \nCONTENTS  \n1  \nINTRODUCTION  \nSOME PARADOXES IN LOGIC2The Greek Lawyer and His Pupil2Richard’s Antinomy2MATHEMATICAL PARADOXES42=1 and Consequences4All Triangles Are Isosceles4Moebius Surfaces with One Side6Achilles and the Tortoise6The Bouncing Ball71-1+1-1+1+一...=1/28What Becomes of a+b=b+a?8As Many Even Numbers as There Are Whole Numbers11The Two Diagonals of a Square Equal the Four Sides,andπ=212A Segment Is Twice as Long as Itself13Distance Between Two Points Inside a Square Inch Greater thanDistance from Earth to Sun14Pythagoras and His Paradox17Non-Euclidean Geometry,and Conclusion19  \n,  \nCoPYRIGHT ◎1959 BY D.VAN NOSTRAND COMPANY,INc.  \nPublished simultaneously in Canada by D.VAN NoSTRAND Co.(Canada),ITD.  \nLibrary of Congress Catalog Card No.59-15867  \n·  \n# INTRODUCTION\n\nA true paradox represents a chain of flawless reasoning which leadsto a result known to be impossible and not true.  \nIn a serious way,the paradoxes have occupied leaders of thought fromthe time of the Golden Age of Greek science,more than twenty centuriesago.The very word is of Greek origin(para=contrary to;doxos=opinion;plural,paradoxa).In German,a paradox is sometimes calleda\"Schein-Wahrheit,\"a phantom truth,a very apt characterization.  \nA true paradox is an object of awe and wonder.It ranks in impor-tance with the miracle.A miracle is a violation of the laws of nature,a paradox is a violation of the laws of mind,of thought.The conse-quences of a miracle and of a paradox are equally devastating.Themiracle destroys our confidence in the stability of the world around us,the paradox destroys our confidence in our inner world,the world ofthe mind.  \nWhether there exists in science any true paradox,we do not know.It is to be hoped there is none,for it has been shown that in the vastdomain of mathematics the existence of a single true paradox will utterlydestroy mathematical science and therefore all science that is built onthe same laws of logic.The pattern of our logical thinking is woven soclosely that an incurable flaw at any point unravels the whole fabric.The manner in which this appears is that,from the existence of a singlemathematical paradox,it is shown that every mathematical statement,whether true or false,can be proved logically.All mathematical proofsthus become spurious.  \nOnce a paradox has received its logical explanation,it ceases to be atrue paradox and becomes a puzzle,a game,or an exercise in logic.  \nThe 18th century French philosopher,Rousseau,is reported to havesaid,“A paradox is a truth that has come into the world a centurytoo early.”  \nWe shall consider a few paradoxes originating in logic and a numberof mathematical paradoxes.All of them have been resolved,broughtinto line with our laws of thought,sometimes by displaying possibilitiesthat had been overlooked in the analysis of the situation (these are thesimple ones)and sometimes through gradual changes in our fundamentalconcepts—the change often stretching over centuries.  \nWe have tried to avoid shallow examples which are often nothing butparlor-game puzzles with hidden clues.The one coming closest to this  \n.  \nis perhaps“All Triangles Are Isosceles.”For allits apparent obvious-ness,the one called“2=1”has deeper foundations.  \n# SOME PARADOXES IN LOGIC\n\nThe Greek Lawyer and His Pupil  \nThe key to this paradox lies in the fact that two different criteria areused interchangeably in the interpretation of a statement.  \nA famous lawyer and a student he has undertaken to coach in the lawconclude the following contract:If the student wins his first case,hemust pay a stipulated fee.If he loses his first case,he is to pay nothing.  \nBecause the student has no case in court for some tim","cbCaibqbx7lV4AJA","https://ap.wps.com/l/cbCaibqbx7lV4AJA","pdf",2015012,28,"English","# INTRODUCTION\n## Some Paradoxes in Logic\n### The Greek Lawyer and His Pupil\n### Richard's Antinomy","[{\"question\":\"What is a “true paradox” according to the introduction?\",\"answer\":\"A true paradox is a chain of flawless reasoning that ends in a result known to be impossible and not true.\"},{\"question\":\"How does a paradox differ from a miracle and from a puzzle?\",\"answer\":\"A miracle violates the laws of nature, while a paradox violates the laws of mind and thought; once explained, a paradox stops being a true paradox and becomes a puzzle, game, or logical exercise.\"},{\"question\":\"Why would a single mathematical paradox undermine mathematical science?\",\"answer\":\"The introduction argues that the existence of one true mathematical paradox would allow every mathematical statement to be proved logically, making proofs spurious and unraveling the logical fabric.\"}]","PARADOXES and COMMON SENSE - INTRODUCTION | PDF",1790654855,43]