Sociology and Pure Maths - Computer Science is more rigorous than Pure Maths: the case for certificates | N J Wildberger

When is a definition valid? Can we create a criterion that will allow us to check whether a given object fulfills the requirements of a given definition? One approach is the concept of a certificate, which plays a huge role in the theoretical development of Computer Science (CS), but only a minor role in Pure Mathematics. The idea of having definitions come with certificates of fulfillment is a natural and obvious one: so why don't we consistently require such in Pure Maths? The reason is depressing: it's because many of our definitions simply cannot support certificates: they are ultimately a dance of quantifiers that only appear to be rigorous. A great example of this is the modern analysis definition of "continuous function", such as the "cosine function". But in fact there are many others too, not only in modern analysis, but also in related disciplines. Our current acceptance of the ZFC "axiomatic system" which posits things that we would like to be true but that are far from obvious, and in fact even non-sensical, is a big part of our difficulties. In particular, the "Axiom of Choice" pretty well guarantees that many of the constructs and claims of modern Analysis are effectively fairy tales, outside the realm of concrete verification. Is it time to stop fooling ourselves, and our students, and adopt the more careful, logically clear approach of CS? ******* Research Gate page: https://www.researchgate.net/profile/Norman_Wildberger Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon: https://www.patreon.com/njwildberger Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriFv3G22iOUidUhkIGXuh Insights into Mathematics Playlists: https://www.youtube.com/playlist?list=PLIljB45xT85CL0-5wO4P-OwuH7SzBUyC0 (31 videos) https://www.youtube.com/playlist?list=PLIljB45xT85B0aMG-G9oqj-NPIuBMnq8z (18 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DHcleY93bd2WlywQYxHv7H (6 videos) https://www.youtube.com/playlist?list=PLIljB45xT85AZjh2GUdwsMuAWTEHXCLb- (4 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DeX148Q2T2elycnIYHTxdP (44 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A6D0eylgAM4Xc1b_dDTgHW (8 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DGxj1x_dyaSggbauAgrB6R (226 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A_VZD126sND3XECDSI-qGs (26 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DfokRNdx5eQ_qA43BwyM-_ (45 videos) https://www.youtube.com/playlist?list=PLIljB45xT85CvrEUnhS4dkTm9Lci5JTYS (43 videos) https://www.youtube.com/playlist?list=PLIljB45xT85D7wczwyUQdwDe2duZ7wPTf (40 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Ar_tuHmRDI2Xr5tWOTI1oW (55 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DPSrQpExd2L870ggxd2KEB (34 videos) https://www.youtube.com/playlist?list=PLIljB45xT85C3-5f6-qQS3YhgygZgnGYs (8 videos) https://www.youtube.com/playlist?list=PLIljB45xT85D7hfVLKyEJbQxOJ4yYm1-9 (9 videos) https://www.youtube.com/playlist?list=PLIljB45xT85BtiJIcDF2uuFc7_ndd1UL6 ( 46 videos) https://www.youtube.com/playlist?list=PLIljB45xT85B6MYBrWldlJ3w-Q1LaEdoy (40 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DIpzaWggqJDIK8m2u09dQL (7 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Ag6ragdmHaxUOKcQsIsrXZ (7 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A5w_vZ5BB1IQAxAMhz5o5S (19 videos) https://www.youtube.com/playlist?list=PLIljB45xT85BjhYV7QswANMF3gKPx9BqT (21 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Bi7PvrTkAaAswTwsfsR5HV (10 videos) https://www.youtube.com/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh (94 videos) Wild Egg Maths Playlists: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5m3XKK0OPc31l-GTqDiAjs (4 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz4CVqTYS1fwInPOZxBxDaQf (64 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7PpsRFHuGb3EhwZtEOdRjV (45 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz6HqaFPB4HTZOQ8jgDAVnuz (20 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz71VHhAIUN-7Y6-JVcwF-xA (8 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz75-t4UaYvHwPe5AHGA_EWe (9 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5W25vaXSreTMTwpPhWs1w2 (52 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz4DCenuIAr9R2kXLq4ywRAm (9 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7OynfQubQrYkxYurauO-HD (14 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7cyg5JdKBpzB4fTQT7f-Xl (8 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5Bcq1VN_R7zYDZH3u49FH8 (25 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5flY1TFd7cekRMZRvPGkzM (30 videos)