{"id":235331,"date":"2026-04-16T06:09:26","date_gmt":"2026-04-16T11:09:26","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2026\/04\/for-much-of-history-many-mathematicians"},"modified":"2026-04-16T06:09:26","modified_gmt":"2026-04-16T11:09:26","slug":"for-much-of-history-many-mathematicians","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2026\/04\/for-much-of-history-many-mathematicians","title":{"rendered":"For much of history, many mathematicians\u2014"},"content":{"rendered":"<p><\/p>\n<p><iframe style=\"display: block; margin: 0 auto; width: 100%; aspect-ratio: 4\/3; object-fit: contain;\" src=\"https:\/\/www.youtube.com\/embed\/rHtqGrtcB1w?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope;\n   picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>Following thinkers like Aristotle\u2014viewed infinity as a never-ending process rather than a completed object. In the late 19th century, Georg Cantor revolutionized this view by treating infinite sets as mathematical objects that could be compared and studied. His work showed that not all infinities are equal, and that there are infinitely many different sizes of infinity. While his ideas are foundational in modern mathematics, some philosophical schools, such as finitism and ultrafinitism, continue to question whether infinite objects meaningfully exist.<\/p>\n<p>I subscribe to The Economist for their science and tech coverage. As a TOE listener, get 35% off! No other podcast has this: <a href=\"https:\/\/economist.com\/TOE\">https:\/\/economist.com\/TOE<\/a><\/p>\n<p>FOLLOW:<br \/> Spotify: <a href=\"https:\/\/open.spotify.com\/show\/4gL14b9\">https:\/\/open.spotify.com\/show\/4gL14b9<\/a>\u2026 <\/p>\n<li>Substack: <a href=\"https:\/\/curtjaimungal.substack.com\/su\">https:\/\/curtjaimungal.substack.com\/su<\/a>\u2026 <\/li>\n<li>Twitter: <img alt=\"\" class=\"\"> \/ toewithcurt <\/li>\n<li>Discord Invite: <img alt=\"\" class=\"\"> \/ discord <\/li>\n<li>Crypto: <a href=\"https:\/\/commerce.coinbase.com\/checkou\">https:\/\/commerce.coinbase.com\/checkou<\/a>\u2026 <\/li>\n<li>PayPal: <a href=\"https:\/\/www.paypal.com\/donate?hosted_\">https:\/\/www.paypal.com\/donate?hosted_<\/a>\u2026 <\/li>\n<p> TIMESTAMPS: <\/p>\n<ul class=\"\" dir=\"ltr\">\n<li>00:00 \u2014 Potential vs. Actual Infinity <\/li>\n<li>03:12 \u2014 Cardinality and Aleph-Null <\/li>\n<li>06:12 \u2014 Diagonalization and Uncountability <\/li>\n<li>09:21 \u2014 ZFC and Logical Independence <\/li>\n<li>12:23 \u2014 Finitism and Ultrafinitism <\/li>\n<li>15:26 \u2014 Continuum Hypothesis Paradoxes <\/li>\n<li>16:00 \u2014 Foundational Mathematical Crisis <\/li>\n<\/ul>\n<p> LINKS MENTIONED: <\/p>\n<ul class=\"\" dir=\"ltr\">\n<li>The Most Abused Theorem in Math [TOE]: <img alt=\"\" class=\"\"> \u2022 The Most Abused Theorem in Math (G\u00f6del\u2019s I\u2026 <\/li>\n<li>Dror Bar Natan [TOE]: <img alt=\"\" class=\"\"> \u2022 Dror Bar Natan: Knot Theory &amp; Quantum Fiel\u2026 <\/li>\n<li>Hilbert\u2019s Problems: <a href=\"https:\/\/mathworld.wolfram.com\/Hilbert\">https:\/\/mathworld.wolfram.com\/Hilbert<\/a>\u2026 <\/li>\n<li>The Independence of the Continuum Hypothesis [paper]: <a href=\"https:\/\/www.pnas.org\/doi\/pdf\/10.1073\/.\">https:\/\/www.pnas.org\/doi\/pdf\/10.1073\/.<\/a>\u2026 <\/li>\n<li>Peano arithmetic: <a href=\"https:\/\/ncatlab.org\/nlab\/show\/Peano+a\">https:\/\/ncatlab.org\/nlab\/show\/Peano+a<\/a>\u2026 <\/li>\n<li>Cantor\u2019s Diagonal Argument: <a href=\"https:\/\/www.researchgate.net\/publicat\">https:\/\/www.researchgate.net\/publicat<\/a>\u2026 <\/li>\n<li> Hartog\u2019s Construction: paultaylor.eu\/trans\/HartogsF-wellord.pdf <\/li>\n<li>Cohen\u2019s Forcing Method: <a href=\"https:\/\/timothychow.net\/forcing.pdf\">https:\/\/timothychow.net\/forcing.pdf<\/a> <\/li>\n<li>Norman Wildberger [TOE]: <img alt=\"\" class=\"\"> \u2022 Norman Wildberger: The Problem with Infini\u2026 <\/li>\n<li>Woodin\u2019s lecture: <img alt=\"\" class=\"\"> \u2022 The Continuum Hypothesis and the search fo\u2026 <\/li>\n<\/ul>\n<p>In Search of Ultimate-L [paper]: <a href=\"https:\/\/www.jstor.org\/stable\/44164514\">https:\/\/www.jstor.org\/stable\/44164514<\/a> <\/p>\n<ul class=\"\" dir=\"ltr\">\n<li>Emily Riehl [TOE]: <img alt=\"\" class=\"\"> \u2022 Emily Riehl Makes Infinity Categories Elem\u2026 <\/li>\n<li>Sir Roger Penrose [TOE]: <img alt=\"\" class=\"\"> \u2022 Roger Penrose: Quantum Theory Is Wrong, No\u2026 <\/li>\n<li>Why Write? [article]: <a href=\"https:\/\/curtjaimungal.substack.com\/p\/.\">https:\/\/curtjaimungal.substack.com\/p\/.<\/a>\u2026 <\/li>\n<\/ul>\n<p> ASSETS USED: <\/p>\n<ul class=\"\" dir=\"ltr\">\n<li>Infinity display: <img alt=\"\" class=\"\"> \u2022 1h Infinity Symbol Animation for Backgroun\u2026 <\/li>\n<li>Number counter: <img alt=\"\" class=\"\"> \u2022 Count up number 1\u20134000 overlay for edits <\/li>\n<\/ul>\n<p> Guests do not pay to appear. #science.<br \/> Substack: <a href=\"https:\/\/curtjaimungal.substack.com\/su\">https:\/\/curtjaimungal.substack.com\/su<\/a>\u2026<br \/> Twitter: \/ toewithcurt.<br \/> Discord Invite: \/ discord.<br \/> Crypto: <a href=\"https:\/\/commerce.coinbase.com\/checkou\">https:\/\/commerce.coinbase.com\/checkou<\/a>\u2026<br \/> PayPal: <a href=\"https:\/\/www.paypal.com\/donate?hosted_\">https:\/\/www.paypal.com\/donate?hosted_<\/a>\u2026<\/p>\n<p>TIMESTAMPS:<br \/> 00:00 \u2014 Potential vs. Actual Infinity.<br \/> 03:12 \u2014 Cardinality and Aleph-Null.<br \/> 06:12 \u2014 Diagonalization and Uncountability.<br \/> 09:21 \u2014 ZFC and Logical Independence.<br \/> 12:23 \u2014 Finitism and Ultrafinitism.<br \/> 15:26 \u2014 Continuum Hypothesis Paradoxes.<br \/> 16:00 \u2014 Foundational Mathematical Crisis.<\/p>\n<p>LINKS MENTIONED:<br \/> The Most Abused Theorem in Math [TOE]: \u2022 The Most Abused Theorem in Math (G\u00f6del\u2019s I\u2026<br \/> Dror Bar Natan [TOE]: \u2022 Dror Bar Natan: Knot Theory &amp; Quantum Fiel\u2026<br \/> Hilbert\u2019s Problems: <a href=\"https:\/\/mathworld.wolfram.com\/Hilbert\">https:\/\/mathworld.wolfram.com\/Hilbert<\/a>\u2026<br \/> The Independence of the Continuum Hypothesis [paper]: <a href=\"https:\/\/www.pnas.org\/doi\/pdf\/10.1073\/.\">https:\/\/www.pnas.org\/doi\/pdf\/10.1073\/.<\/a>\u2026<br \/> Peano arithmetic: <a href=\"https:\/\/ncatlab.org\/nlab\/show\/Peano+a\">https:\/\/ncatlab.org\/nlab\/show\/Peano+a<\/a>\u2026<br \/> Cantor\u2019s Diagonal Argument: <a href=\"https:\/\/www.researchgate.net\/publicat\">https:\/\/www.researchgate.net\/publicat<\/a>\u2026<br \/> Hartog\u2019s Construction: paultaylor.eu\/trans\/HartogsF-wellord.pdf.<br \/> Cohen\u2019s Forcing Method: <a href=\"https:\/\/timothychow.net\/forcing.pdf\">https:\/\/timothychow.net\/forcing.pdf<\/a>.<br \/> Norman Wildberger [TOE]: \u2022 Norman Wildberger: The Problem with Infini\u2026<br \/> Woodin\u2019s lecture: \u2022 The Continuum Hypothesis and the search fo\u2026<br \/> In Search of Ultimate-L [paper]: <a href=\"https:\/\/www.jstor.org\/stable\/44164514\">https:\/\/www.jstor.org\/stable\/44164514<\/a><br \/> Emily Riehl [TOE]: \u2022 Emily Riehl Makes Infinity Categories Elem\u2026<br \/> Sir Roger Penrose [TOE]: \u2022 Roger Penrose: Quantum Theory Is Wrong, No\u2026<br \/> Why Write? [article]: <a href=\"https:\/\/curtjaimungal.substack.com\/p\/.\">https:\/\/curtjaimungal.substack.com\/p\/.<\/a>\u2026<\/p>\n<p>ASSETS USED:<\/p>\n<div class=\"more-link-wrapper\"> <a class=\"more-link\" href=\"https:\/\/lifeboat.com\/blog\/2026\/04\/for-much-of-history-many-mathematicians\">Continue reading \u201cFor much of history, many mathematicians\u2014\u201d | &gt;<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Following thinkers like Aristotle\u2014viewed infinity as a never-ending process rather than a completed object. In the late 19th century, Georg Cantor revolutionized this view by treating infinite sets as mathematical objects that could be compared and studied. His work showed that not all infinities are equal, and that there are infinitely many different sizes of [\u2026]<\/p>\n","protected":false},"author":661,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[32,2229,1617],"tags":[],"class_list":["post-235331","post","type-post","status-publish","format-standard","hentry","category-education","category-mathematics","category-quantum-physics"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/235331","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/661"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=235331"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/235331\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=235331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=235331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=235331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}