{"id":11651,"date":"2021-01-20T04:52:30","date_gmt":"2021-01-20T12:52:30","guid":{"rendered":"https:\/\/www.oeconomist.com\/blogs\/daniel\/?p=11651"},"modified":"2021-01-20T16:10:47","modified_gmt":"2021-01-21T00:10:47","slug":"tiny-spaces","status":"publish","type":"post","link":"https:\/\/www.oeconomist.com\/blogs\/daniel\/?p=11651","title":{"rendered":"Tiny Spaces"},"content":{"rendered":"

Famously, the Euclidean axiomata for space<\/span> seemed necessary<\/em> to many, so that various philosophers concluded or argued that some knowledge<\/span> or something playing a rôle like that of knowledge<\/span> derived from something other than experience<\/span>. Yet there were doubters of one of these axiomata — that parallel lines would never intersect — and eventually physicists concluded that the universe would be better described were this axiom regarded as incorrect. Once one axiom was abandoned, the presumption of necessity<\/em> of the others evaporated.<\/p>

I think that our concept of space<\/span> is built upon an experience of an object sometimes affecting another in ways that it sometimes does not, with the first being classified as near<\/span> when it does and not near<\/span> when it does; which<\/em> ways are associated in the concept of near<\/span>-ness are selected by experience. The concept of distance<\/span> — variability of near<\/span>-ness — develops from the variability of how one object affects another; and it is experience that selects which variabilities are associated with distance<\/span>. Our concept of space<\/span> is that of potential (realized or not) of near<\/span>-ness.<\/p>

The axiomata of Euclid were, implicitly, an attempted codification of observed properties of distance<\/span>; in the adoption of this codification or of another, one might revise which variabilities one associated with distance<\/span>. One might, in fact, hold onto those axiomata exactly by revising which variabilities are associated with distance. In saying that space is non-Euclidean, one ought to mean that the Euclidean axiomata are not the best suited to physics.<\/p>

Just as the axiomata of Euclid become ill-suited to physics when distances become very large, they may be ill-suited when distances become very small.<\/p>

Space might not even be divisible without limit. The mathematical construct of continuity<\/span> may not apply to the physical world. At least some physical quantities that were once imagined potentially to have measures corresponding to any<\/em> real number are now regarded as having measures corresponding only to integer multiples of quanta<\/span>; perhaps distance cannot be reduced below some minimum.<\/p>

And, at some sub-atomic level, any useable rules of distance<\/span> might be more complex. On a larger scale, non-Euclidean spaces are sometimes imagined to have worm-holes<\/span>, which is really to say that some spaces would have near<\/span>-ness by peculiar paths. Perhaps worm-holes<\/span> or some discontinuous analogue thereöf are pervasive<\/em> at a sub-atomic level, making space into something of a rat's nest.<\/p> ","protected":false},"excerpt":{"rendered":"Famously, the Euclidean axiomata for space seemed necessary to many, so that various philosophers concluded or argued that some knowledge or something playing a rôle like that of knowledge derived from something other than experience. Yet there were doubters of one of these axiomata — that parallel lines would never intersect — and eventually physicists […]","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"footnotes":""},"categories":[720,719,676,4],"tags":[1578,851,1577],"class_list":["post-11651","post","type-post","status-publish","format-standard","hentry","category-epistemology","category-metaphysics","category-physical-science","category-public","tag-euclid","tag-geometry","tag-space"],"_links":{"self":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts\/11651","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11651"}],"version-history":[{"count":8,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts\/11651\/revisions"}],"predecessor-version":[{"id":11659,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts\/11651\/revisions\/11659"}],"wp:attachment":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11651"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11651"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11651"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}