[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-detail-592766-en":59,"doc-seo-592766-105":81},{"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":5,"data":60},{"doc_id":61,"user_id":62,"nickname":63,"user_avatar":64,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":66,"doc_content":67,"file_id":68,"file_url":69,"file_type":70,"file_size":71,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":72,"language":73,"language_code":74,"site_id":75,"html_lang":74,"table_of_contents":76,"faqs":77,"seo_title":78,"seo_description":66,"update_tm":79,"read_time":80,"preview_page_count":4},592766,2336477982702,"Marglet","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs?","Examines the mismatch between formal mathematical logic techniques and the informal methods of mathematical practice when teaching and learning proofs at the college level. Investigates reasons for this discrepancy, its implications for proof-related activities, and summarizes college students’ views on proofs. Additionally, surveys the role of mathematical logic within Greek and Greek-Cypriot high-school curricula, concluding that formal logical methods are largely overestimated despite logic’s centrality.","Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs?  \nBy Michael Aristidou􀀍  \nAs it is already observed by mathematicians and educators, there is a discrepancy between the formal techniques of mathematical logic and the informal techniques of mathematics in regards to proof. We examine some of the reasons behind this discrepancy and to what degree it affects doing, teaching and learning mathematics in college. We also present some college students’opinions about proofs, and we briefly observe the situation in Greek and GreekCypriot high schools in which mathematical logic is part of the curriculum.  \nFinally, we argue that even though mathematical logic is central in mathematics, its formal methods are not really necessary in doing and teaching mathematical proofs and the role of those formalities has been, in general, overestimated by some educators.  \nKeywords: formal, logic, proof, student, teacher.  \nIntroduction  \nIn several colleges, some parts of mathematical logic (i.e. sets, propositional logic, and predicate logic) are usually taught in the early chapters of a discrete mathematics class, in order to prepare the students for the important chapter on proofs and proving techniques. Yet, most likely, students have already been exposed to proofs before the above-mentioned course in other mathematics courses or even in high school. Mathematical logic is to sharpen the logical and analytical skills of a student as these are necessary for the understanding and learning of mathematical proofs. Mathematical logic though is characterized by its symbolic presentation and formal rules. Mathematics, on the other hand, combines mathematical symbolism and natural language and its methods are rigorous yet less formal.  \nHistorically, logic is associated with Aristotle and his work the Organon in which he introduced terms like \"propositions\" and \"syllogisms\", the basics on categorical and hypothetical syllogism, and modal and inductive logic. It is also associated with the Stoics and their propositional logic, and their work on implication. Syllogistic logic and propositional logic led later to the development of predicate logic (or first order logic, i.e. the foundational logic for mathematics) by Frege and Hilbert in the 19th century. As Ferreiros said:  \n\"First-order logic emerged as an analysis of the most fundamental basis for the notion of mathematical proof. To put it otherwise, it emerged as the logic that is necessary  \n􀀍Associate Professor, American University of Kuwait, Kuwait.  \n[https://doi.org/10.30958/aje.7-1-5](https://doi.org/10.30958/aje.7-1-5) doi=10.30958/aje.7-1-5  \nand sufficient for codifying mathematical proofs, axiomatizing mathematical theories, and studying their metatheory.\" (Ferreiros, 2001, p. 479)  \nPredicate logic is also the foundation of modern mathematical logic. The latter is a subfield of mathematics that includes fields such as set theory, model theory and proof theory, and its primary interests are the foundations of mathematics and theoretical computer science.  \nThe interest in the foundations began in the 19th century with the development of axiomatic frameworks for geometry and arithmetic by Hilbert and Peano respectively. That led in the early 20th century to three main philosophies regarding the foundations of mathematics, namely Logicism, Formalism and Intuitionism, none of which adequately accounts for those foundations. Gödel pointed out the issues of consistency and completeness related to provability in general formal systems. Nevertheless, most mathematics can be formalized in terms of sets, and set theory serves nowadays as its foundation. In real mathematical practice though rarely one adheres to set-theoretical foundations to validate or refute mathematical questions. Each mathematical field has its own tools and methods and with general logical framework the predicate logic exploresits own questions, proves its own theorems, and establishes connections between ","cbCaicyICpJ5cwVQ","https://ap.wps.com/l/cbCaicyICpJ5cwVQ","pdf",1299672,23,"English","en",105,"# Introduction\n## What is Mathematical Proof?\n## Example","[{\"question\":\"Why is there a discrepancy between formal mathematical logic and informal proof techniques?\",\"answer\":\"The paper highlights differences between formal techniques used in mathematical logic and the informal techniques common in actual mathematical reasoning. It then analyzes reasons for this gap and how it influences proof practice, teaching, and learning.\"},{\"question\":\"What role does mathematical logic play in preparing students for proof in college?\",\"answer\":\"The paper notes that some logic topics, such as sets and propositional/predicate logic, are often taught early in discrete mathematics to prepare students for proofs. It also argues that students may have already encountered proofs earlier in other courses or even in high school.\"},{\"question\":\"Is formal mathematical logic necessary for teaching and doing mathematical proofs?\",\"answer\":\"The conclusion argues that even though mathematical logic is central to mathematics, its formal methods are not really necessary for doing and teaching mathematical proofs. It also claims the importance of those formalities has been generally overestimated by some educators.\"}]","Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs? | PDF",1792075481,58,{"code":4,"msg":82,"data":83},"ok",{"site_id":75,"language":74,"slug":84,"title":65,"keywords":85,"description":66,"schema_data":86,"social_meta":140,"head_meta":142,"extra_data":144,"updated_unix":145},"is-mathematical-logic-really-necessary-in-teaching-mathematical-proofs","",{"@graph":87,"@context":139},[88,102,122],{"@type":89,"itemListElement":90},"BreadcrumbList",[91,95,97,100],{"item":92,"name":93,"@type":94,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":96,"name":9,"@type":94,"position":14},"https://docshare.wps.com/document/",{"item":98,"name":40,"@type":94,"position":99},"https://docshare.wps.com/document/research-report/",3,{"item":101,"name":65,"@type":94,"position":19},"https://docshare.wps.com/document/is-mathematical-logic-really-necessary-in-teaching-mathematical-proofs/592766/",{"url":101,"name":65,"@type":103,"image":104,"author":109,"headline":65,"publisher":111,"fileFormat":114,"inLanguage":74,"description":66,"dateModified":115,"datePublished":116,"encodingFormat":114,"isAccessibleForFree":117,"interactionStatistic":118},"DigitalDocument",{"url":105,"@type":106,"width":107,"height":108},"https://docshare.wps.com/thumbnails/is-mathematical-logic-really-necessary-in-teaching-mathematical-proofs/592766.png","ImageObject",300,407,{"name":63,"@type":110},"Person",{"url":92,"name":112,"@type":113},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":119,"interactionType":120,"userInteractionCount":24},"InteractionCounter",{"@type":121},"ViewAction",{"@type":123,"mainEntity":124},"FAQPage",[125,131,135],{"name":126,"@type":127,"acceptedAnswer":128},"Why is there a discrepancy between formal mathematical logic and informal proof techniques?","Question",{"text":129,"@type":130},"The paper highlights differences between formal techniques used in mathematical logic and the informal techniques common in actual mathematical reasoning. It then analyzes reasons for this gap and how it influences proof practice, teaching, and learning.","Answer",{"name":132,"@type":127,"acceptedAnswer":133},"What role does mathematical logic play in preparing students for proof in college?",{"text":134,"@type":130},"The paper notes that some logic topics, such as sets and propositional/predicate logic, are often taught early in discrete mathematics to prepare students for proofs. It also argues that students may have already encountered proofs earlier in other courses or even in high school.",{"name":136,"@type":127,"acceptedAnswer":137},"Is formal mathematical logic necessary for teaching and doing mathematical proofs?",{"text":138,"@type":130},"The conclusion argues that even though mathematical logic is central to mathematics, its formal methods are not really necessary for doing and teaching mathematical proofs. It also claims the importance of those formalities has been generally overestimated by some educators.","https://schema.org",{"og:url":101,"og:type":141,"og:title":65,"og:site_name":112,"og:description":66},"article",{"robots":143,"canonical":101},"index,follow",{"doc_id":61,"site_id":75},1792231047]