{"id":1124,"date":"2018-06-13T04:28:58","date_gmt":"2018-06-13T04:28:58","guid":{"rendered":"https:\/\/kellyfish.me\/?p=1124"},"modified":"2018-11-04T21:18:18","modified_gmt":"2018-11-04T21:18:18","slug":"the-parallel-universe-of-01","status":"publish","type":"post","link":"https:\/\/kellyfish.me\/index.php\/2018\/06\/13\/the-parallel-universe-of-01\/","title":{"rendered":"The Parallel Universe of 0=1"},"content":{"rendered":"

What happens if mathematicians prove that 0=1? How would this affect our lives and understanding of the world? Would it prove the existence of parallel universes where GPS signals are routed differently?<\/span><\/p>\n

I learned proof by contradiction in geometry class. The technique – proving a proposition was true by showing the opposite could never occur – seemed unconventional. It greatly intrigued me and has been my favorite method.<\/span><\/p>\n

With a contradiction like 0=1, a wrong assumption must be used in the proof. In geometry class, I often wondered: What if, rather than tossing away this incorrect assumption, we got rid of Euclidean axioms, statements that were assumed to be correct? Few people challenged Euclidean axioms because they were deemed \u201cobvious and true\u201d, but I liked questioning the conventional system and considering the effects in altering it.<\/span><\/p>\n

If non-Euclidean geometry is used, like in Einstein’s relativity, it would be an unfamiliar universe to live in. Manufacturers might find themselves using wasteful packaging designs. Cars might be in aerodynamically inefficient shapes. The sum of angles of a triangle wouldn\u2019t be 180 degrees. My origami could be sloppily folded.<\/span><\/p>\n

Mathematical foundations are based on Axioms of Set Theory. In 1908, Zermelo proposed a set of axioms for set theory. This was later extended by Fraenkel. Together with the Axiom of Choice, it has been called the ZFC Theory. ZFC is currently the best recognized set of axioms forming the foundation of mathematics. From these axioms, logical rules are applied to derive more results. Like branches grown from tree roots, theorems are derived from axioms. If 0=1 can be deduced without false assumptions, this means the axioms are inconsistent, rendering the system useless. It would be like pulling the roots out of the ground, disrupting the entire tree.<\/span><\/p>\n

While my hands wrote the steps of proofs by contradiction in class, my mind wondered what steps were needed if 0=1 was indeed proven. Mathematicians would need to produce an alternative set of axioms that doesn’t reproduce the 0=1 result. If major theorems still follow, then life will remain the same. However, if important results are invalidated, then engineers will have to repair or even abandon countless machines and systems, affecting many parts of our lives.<\/span><\/p>\n

For example, if Euclidean axioms fail, it would change our understanding of space. If mathematical induction fails, then results in number theory and computer science may become invalid. That would have wide consequences. Laws like Newton’s \u201cF=ma\u201d aren’t axioms but formulas validated through experiments. From these laws, engineers utilize mathematical techniques to derive results to build bridges, computers, and nuclear reactors. However, if previously true theorems aren’t valid anymore, then derived results may be invalid. We can’t assure bridges are safe, or everything will work.<\/span><\/p>\n

So can\u2019t mathematicians do us a favor and prove that it is impossible to prove 0=1 in the axiomatic system? Well, long time ago, they did try, but without success. In 1931, Godel published the twin theorems of incompleteness. A consequence of Godel’s result is that it is not possible to prove the axiomatic system is consistent, unless the system is inconsistent. This defeats the purpose to seek finality. Disappointingly, there won’t be proof to show my original origami designs are optimal, or theorems learned in my geometry class are correct.<\/span><\/p>\n

I find this intellectually and philosophically unsatisfactory. We base our mathematical, scientific and engineering achievements on a system which we don’t know and can never prove is consistent. We live in one of the parallel universes but don’t and won’t know which one.<\/span><\/p>\n

Godel pioneered the meta-mathematics, the use of mathematical methods to study mathematics itself. However, he placed a boundary on the power of the axiomatic system, limiting its ability to answer questions about itself, or its \u201cself-discovery\u201d.<\/span><\/p>\n

After completing my geometry course, I started pondering about the human brain. Similar to an axiomatic system searching for validation from within itself, we as humans are engaging in our self-discovery using our neural expressions and experiences to probe into our neural networks. If biological incompleteness does exist, then it could constrain ultimate understanding of our neural functions.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

What happens if mathematicians prove that 0=1? How would this affect our lives and understanding of the world? Would it prove the existence of parallel universes where GPS signals are routed differently? I learned proof by contradiction in geometry class. The technique – proving a proposition was true by showing the opposite could never occur […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[28],"tags":[],"class_list":["post-1124","post","type-post","status-publish","format-standard","hentry","category-articles"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/comments?post=1124"}],"version-history":[{"count":4,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1124\/revisions"}],"predecessor-version":[{"id":1327,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1124\/revisions\/1327"}],"wp:attachment":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/media?parent=1124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/categories?post=1124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/tags?post=1124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}