{"id":2276,"date":"2015-04-22T03:08:51","date_gmt":"2015-04-22T07:08:51","guid":{"rendered":"https:\/\/scottaaronson.blog\/?p=2276"},"modified":"2016-12-10T04:43:10","modified_gmt":"2016-12-10T09:43:10","slug":"is-there-something-mysterious-about-math","status":"publish","type":"post","link":"https:\/\/scottaaronson.blog\/?p=2276","title":{"rendered":"“Is There Something Mysterious About Math?”"},"content":{"rendered":"

When it rains, it pours: after not blogging for a month, I now have a second thing to blog about in as many days. \u00a0Aeon<\/em>, an online magazine, asked me to write a short essay responding to the question above, so I did. \u00a0My essay is here<\/a>. \u00a0Spoiler alert: my thesis is that yes, there’s something “mysterious” about math,\u00a0but\u00a0the main mystery\u00a0is why there isn’t even\u00a0more<\/em> mystery than there is. \u00a0Also—shameless attempt to get you to click—the essay discusses\u00a0the “discrete math is just a disorganized mess of random statements”\u00a0view of Lubo\u0161 Motl, who’s useful for putting flesh on what might\u00a0otherwise be a strawman position. \u00a0Comments welcome (when aren’t they?). \u00a0You should\u00a0also read\u00a0other interesting responses to\u00a0the same question<\/a> by Penelope Maddy, James Franklin, and Neil Levy. \u00a0Thanks very much to Ed Lake at Aeon for commissioning these pieces.<\/p>\n

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Update (4\/22):<\/strong><\/span> On rereading my piece, I felt bad that it didn’t make a clear enough distinction between two separate questions:<\/p>\n

    \n
  1. Are there humanly-comprehensible explanations<\/em> for why the\u00a0mathematical statements that we care about are true or false—thereby rendering their truth or falsity “non-mysterious” to us?<\/li>\n
  2. Are there formal\u00a0proofs or disproofs<\/em> of the statements?<\/li>\n<\/ol>\n

    Interestingly, neither of the above implies the other. \u00a0Thus, to take an example from the essay, no one has any idea\u00a0how to prove that the digits 0 through 9 occur with equal frequency in the decimal expansion of \u03c0, and yet it’s utterly non-mysterious (at a “physics level of rigor”) why that particular statement should be true. \u00a0Conversely, there are many\u00a0examples of statements for which we do<\/em> have proofs, but which experts in the relevant fields still see\u00a0as “mysterious,” because the proofs aren’t illuminating or explanatory enough. \u00a0Any proofs that\u00a0require\u00a0gigantic manipulations of formulas, “magically” terminating in the desired outcome, probably fall into that class, as do proofs that require computer enumeration of cases (like that of the Four-Color Theorem).<\/p>\n

    But it’s not just that proof and explanation are incomparable; sometimes they might even\u00a0be\u00a0at odds. \u00a0In this MathOverflow post<\/a>, Timothy Gowers\u00a0relates an interesting\u00a0speculation\u00a0of Don Zagier, that statements like the equidistribution of the digits of \u03c0 might be unprovable from the usual axioms of set theory, precisely because<\/em> they’re so “obviously” true—and for that very reason, there need not be anything deeper underlying their truth. \u00a0As Gowers points out, we shouldn’t go overboard with this speculation, because there are plenty of\u00a0other examples of mathematical statements (the Green-Tao theorem, Vinogradov’s theorem, etc.) that also<\/em> seem like they might\u00a0be true “just because”—true only because their falsehood would require a statistical miracle—but\u00a0for which mathematicians nevertheless managed to give fully\u00a0rigorous proofs, in effect\u00a0formalizing<\/em>\u00a0the intuition that it would take a miracle to make them false.<\/p>\n

    Zagier’s speculation is related to another objection one could raise against my essay: while I said that the “G\u00f6delian gremlin” has remained surprisingly dormant in the 85 years since its discovery (and that this is a fascinating fact crying out for explanation), who’s to say that it’s not lurking in some of the very open problems that I mentioned, like \u03c0’s equidistribution, the Riemann Hypothesis, the Goldbach Conjecture, or P\u2260NP? \u00a0Conceivably, not only are all those conjectures unprovable from the usual axioms of set theory, but their unprovability is itself<\/em> unprovable, and so on,\u00a0so that we could never even have the satisfaction of knowing why we’ll never know.<\/p>\n

    My response to these objections is\u00a0basically just to appeal yet\u00a0again to\u00a0the empirical record. \u00a0First, while proof and explanation need not\u00a0go together and sometimes don’t, by and large they do<\/em>\u00a0go together: over thousands over years, mathematicians learned to seek formal proofs largely\u00a0because<\/em> they discovered\u00a0that without them, their understanding constantly went awry. \u00a0Also, while no one can rule out<\/em> that P vs. NP, the Riemann Hypothesis, etc., might be independent of set theory, there’s very little\u00a0in the history of math—including in the recent history, which saw spectacular proofs of (e.g.) Fermat’s Last Theorem and the Poincar\u00e9 Conjecture—that lends concrete\u00a0support to such fatalism.<\/p>\n

    So in summary, I’d say that history does<\/em>\u00a0present us with “two mysteries\u00a0of the mathematical supercontinent”—namely, why do so many\u00a0of the mathematical statements that humans\u00a0care about turn out to be\u00a0tightly linked in webs of explanation, and also<\/em> in webs of proof, rather than occupying separate islands?—and that these two mysteries are very closely related, if not quite the same.<\/p>\n","protected":false},"excerpt":{"rendered":"

    When it rains, it pours: after not blogging for a month, I now have a second thing to blog about in as many days. \u00a0Aeon, an online magazine, asked me to write a short essay responding to the question above, so I did. \u00a0My essay is here. \u00a0Spoiler alert: my thesis is that yes, there’s […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false},"categories":[12],"tags":[],"class_list":["post-2276","post","type-post","status-publish","format-standard","hentry","category-metaphysical-spouting"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts\/2276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2276"}],"version-history":[{"count":6,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts\/2276\/revisions"}],"predecessor-version":[{"id":2292,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts\/2276\/revisions\/2292"}],"wp:attachment":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}