{"id":8864,"date":"2023-02-11T18:48:06","date_gmt":"2023-02-11T11:48:06","guid":{"rendered":"https:\/\/rosetta.vn\/nguyenxuanxanh\/?p=8864"},"modified":"2024-05-01T22:45:33","modified_gmt":"2024-05-01T15:45:33","slug":"david-hilbert-nhan-thuc-tu-nhien-va-logic","status":"publish","type":"post","link":"https:\/\/rosetta.vn\/nguyenxuanxanh\/david-hilbert-nhan-thuc-tu-nhien-va-logic\/","title":{"rendered":"David Hilbert: <i>Nh\u1eadn th\u1ee9c t\u1ef1 nhi\u00ean v\u00e0 logic h\u1ecdc<\/i>"},"content":{"rendered":"<h1 style=\"text-align: center\"><span style=\"color: #008000\">NH\u1eacN TH\u1ee8C T\u1ef0 NHI\u00caN V\u00c0 LOGIC H\u1eccC<\/span><\/h1>\n<h2 style=\"text-align: center\"><span style=\"color: #000080\">NATURERKENNEN UND LOGIK<\/span><\/h2>\n<p style=\"text-align: center\"><span style=\"font-size: 14pt;color: #000080\"><strong>David Hilbert<\/strong><\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080\"><em>Di\u1ec5n t\u1eeb<\/em> t\u1ea1i K\u00f6nigsberg, 1930<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080\"><strong>Nguy\u1ec5n Xu\u00e2n Xanh<\/strong> <em><span style=\"color: #000080\">s\u01b0u t\u1ea7m v\u00e0 <\/span>chuy\u1ec3n ng\u1eef<\/em><\/span><\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-6321\" src=\"https:\/\/i0.wp.com\/rosetta.vn\/nguyenxuanxanh\/wp-content\/uploads\/sites\/6\/2021\/03\/picture-7.png?resize=74%2C48&#038;ssl=1\" alt=\"\" width=\"74\" height=\"48\" data-recalc-dims=\"1\" \/><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080\"><em>N\u1ebfu b\u1ea1n c\u00f3 th\u1ec3 nh\u00ecn v\u00e0o h\u1ea1t gi\u1ed1ng c\u1ee7a th\u1eddi gian,<\/em><\/span><br \/>\n<span style=\"color: #000080\"><em>V\u00e0 n\u00f3i h\u1ea1t n\u00e0o s\u1ebd ph\u00e1t tri\u1ec3n, v\u00e0 h\u1ea1t n\u00e0o s\u1ebd kh\u00f4ng,<\/em><\/span><br \/>\n<span style=\"color: #000080\"><em>Th\u00ec h\u00e3y n\u00f3i chuy\u1ec7n v\u1edbi t\u00f4i.<\/em><\/span><br \/>\n<span style=\"color: #000080\">William Shakespeare, <em>Macbeth\u00a0<\/em><\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080;font-size: 14pt\">Ngh\u1ec7 thu\u1eadt l\u00e0m to\u00e1n l\u00e0 \u1edf ch\u1ed7 t\u00ecm ra tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t ch\u1ee9a t\u1ea5t c\u1ea3 m\u1ea7m m\u1ed1ng c\u1ee7a t\u00ednh t\u1ed5ng qu\u00e1t.<\/span><\/p>\n<p style=\"text-align: center\"><strong><span style=\"color: #000080\">David Hilbert<\/span><\/strong><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080;font-size: 14pt\">Kh\u00f4ng c\u00f3 nh\u00e0 to\u00e1n h\u1ecdc n\u00e0o c\u00f3 t\u1ea7m v\u00f3c t\u01b0\u01a1ng \u0111\u01b0\u01a1ng \u0111\u00e3 v\u01b0\u01a1n l\u00ean t\u1eeb th\u1ebf h\u1ec7 c\u1ee7a ch\u00fang t\u00f4i\u2026 Hilbert \u0111\u1eb7c bi\u1ec7t kh\u00f4ng c\u00f3 \u0111\u1ecbnh ki\u1ebfn v\u1ec1 qu\u1ed1c gia v\u00e0 ch\u1ee7ng t\u1ed9c; trong t\u1ea5t c\u1ea3 c\u00e1c v\u1ea5n \u0111\u1ec1 li\u00ean quan \u0111\u1ebfn c\u00f4ng ch\u00fang, d\u00f9 l\u00e0 ch\u00ednh tr\u1ecb, x\u00e3 h\u1ed9i hay t\u00e2m linh, \u00f4ng \u1ea5y \u0111\u00e3 m\u00e3i m\u00e3i \u0111\u1ee9ng v\u1ec1 ph\u00eda t\u1ef1 do.<\/span><\/p>\n<p style=\"text-align: center\"><strong><span style=\"color: #000080\">Hermann Weyl<\/span><\/strong><\/p>\n<p><span style=\"font-size: 14pt;color: #000080\">T\u00f4i long tr\u1ecdng h\u1ecfi ng\u00e0i r\u1eb1ng li\u1ec7u b\u1eb1ng l\u1eddi th\u1ec1 \u0111\u00e3 \u0111\u1ecbnh, \u00f4ng c\u00f3 cam k\u1ebft h\u1ee9a v\u00e0 x\u00e1c nh\u1eadn m\u1ed9t c\u00e1ch t\u1eadn t\u00e2m nh\u1ea5t hay kh\u00f4ng, r\u1eb1ng ng\u00e0i s\u1ebd b\u1ea3o v\u1ec7 khoa h\u1ecdc ch\u00e2n ch\u00ednh m\u1ed9t c\u00e1ch can \u0111\u1ea3m, m\u1edf r\u1ed9ng v\u00e0 t\u00f4 \u0111i\u1ec3m n\u00f3, kh\u00f4ng ph\u1ea3i v\u00ec l\u1ee3i \u00edch ri\u00eang t\u01b0 hay \u0111\u1ec3 \u0111\u1ea1t \u0111\u01b0\u1ee3c \u00e1nh h\u00e0o quang h\u00e3o huy\u1ec1n, m\u00e0 \u0111\u1ec3 cho \u00e1nh s\u00e1ng ch\u00e2n l\u00fd c\u1ee7a Ch\u00faa t\u1ecfa s\u00e1ng v\u00e0 lan r\u1ed9ng.<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080\">(L\u1eddi th\u1ec1 \u0111\u01b0\u1ee3c khoa tr\u01b0\u1edfng khoa To\u00e1n \u0110\u1ea1i h\u1ecdc K\u00f6nigsberg chu\u1ea9n b\u1ecb \u0111\u1ec3 David Hilbert x\u00e1c nh\u1eadn tr\u01b0\u1edbc khi trao b\u1eb1ng Ti\u1ebfn s\u0129 Tri\u1ebft h\u1ecdc cho \u00f4ng)<\/span><\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/i0.wp.com\/miro.medium.com\/max\/800\/1*ahy12g2hFP21x7YolyQ_Ug.jpeg?resize=490%2C408&#038;ssl=1\" alt=\"On the Doctoral Thesis of David Hilbert | by Areeba Merriam ...\" width=\"490\" height=\"408\" data-recalc-dims=\"1\" \/><\/p>\n<p style=\"text-align: center\">David Hilbert (sinh ng\u00e0y <span class=\"LrzXr kno-fv wHYlTd z8gr9e\">23 th\u00e1ng 1, 1862, t\u1ea1i K\u00f6nigsberg, \u0110\u00f4ng Ph\u1ed5 &#8211; m\u1ea5t ng\u00e0y <\/span><span style=\"text-align: center\">14 th\u00e1ng 2, 1943, t\u1ea1i <\/span>G\u00f6ttingen, CHLB \u0110\u1ee9c)<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000080\"><strong><em>L\u1eddi n\u00f3i \u0111\u1ea7u<\/em><\/strong><\/span>. N\u0103m 1930, t\u1ee9c g\u1ea7n m\u1ed9t th\u1ebf k\u1ef7 tr\u01b0\u1edbc, t\u1ea1i <em>Hi\u1ec7p h\u1ed9i c\u00e1c nh\u00e0 khoa h\u1ecdc t\u1ef1 nhi\u00ean v\u00e0 b\u00e1c s\u0129 \u0110\u1ee9c\u00a0<\/em>(Gesellschaft der Deutschen Naturforscher und \u00c4rzte) di\u1ec5n ra \u1edf th\u00e0nh ph\u1ed1 K\u00f6nigsberg c\u1ee7a \u0110\u00f4ng Ph\u1ed5, nay l\u00e0 Kaliningrad thu\u1ed9c Nga, nh\u00e0 to\u00e1n h\u1ecdc David Hilbert, \u0111\u01b0\u1ee3c xem nh\u01b0 m\u1ed9t Euclid th\u1ee9 hai c\u1ee7a th\u1ebf k\u1ef7 20, \u0111\u1ecdc m\u1ed9t b\u00e1o c\u00e1o quan tr\u1ecdng c\u00f3 t\u00ean <em>Nh\u1eadn th\u1ee9c t\u1ef1 nhi\u00ean v\u00e0 Logic h\u1ecdc <\/em><a href=\"#_ftn1\" name=\"_ftnref1\">[1]<\/a> (Naturerkennen und Logik). N\u0103m \u0111\u00f3, Hilbert c\u0169ng ngh\u1ec9 h\u01b0u, v\u00e0 th\u00e0nh ph\u1ed1 K\u00f6nigsberg b\u1ea7u \u00f4ng l\u00e0m c\u00f4ng d\u00e2n danh d\u1ef1. \u00d4ng tr\u00ecnh b\u00e0y v\u1ec1 h\u1ec7 lu\u1eadn c\u1ee7a c\u00e1c kh\u00e1m ph\u00e1 l\u1edbn c\u1ee7a khoa h\u1ecdc di\u1ec5n ra t\u1eeb cu\u1ed1i th\u1ebf k\u1ef7 19 \u0111\u1ebfn nh\u1eefng th\u1eadp ni\u00ean \u0111\u1ea7u th\u1ebf k\u1ef7 20, c\u00f3 \u1ea3nh h\u01b0\u1edfng l\u1edbn l\u00ean tri\u1ebft h\u1ecdc t\u1ef1 nhi\u00ean (natural philosophy) v\u00e0 nh\u1eadn th\u1ee9c lu\u1eadn (epistemology) trong to\u00e1n h\u1ecdc v\u00e0 khoa h\u1ecdc t\u1ef1 nhi\u00ean. C\u00e1c ti\u1ebfn b\u1ed9 n\u00e0y xu\u1ea5t ph\u00e1t m\u1ea1nh m\u1ebd v\u00e0 ch\u1ee7 y\u1ebfu t\u1eeb nh\u1eefng kh\u00e1m ph\u00e1 trong l\u00e3nh v\u1ef1c v\u1eadt l\u00fd v\u00e0 thi\u00ean v\u0103n h\u1ecdc. Tri\u1ebft h\u1ecdc t\u1ef1 nhi\u00ean l\u00e0 m\u00f4n h\u1ecdc ng\u1ef1 tr\u1ecb l\u00e2u \u0111\u1eddi \u1edf ph\u01b0\u01a1ng T\u00e2y. Trong t\u00e1c ph\u1ea9m <em>Principia <\/em>c\u1ee7a Newton th\u1ebf k\u1ef7 17, n\u00f3 ch\u1ecbu m\u1ed9t kh\u00fac quanh \u0111\u1ea7u ti\u00ean l\u1edbn nh\u1ea5t trong l\u1ecbch s\u1eed r\u1ea5t t\u00edch c\u1ef1c, g\u1eafn li\u1ec1n v\u1edbi cu\u1ed9c c\u00e1ch m\u1ea1ng khoa h\u1ecdc v\u00e0 c\u00f3 \u1ea3nh h\u01b0\u1edfng l\u1edbn l\u00ean phong tr\u00e0o khai s\u00e1ng \u1edf ch\u00e2u \u00c2u. V\u00e0 250 n\u0103m sau, m\u1ed9t kh\u00fac quanh kh\u00e1c c\u00f3 l\u1ebd c\u00f2n tri\u1ec7t \u0111\u1ec3 h\u01a1n \u0111\u00e3 di\u1ec5n ra c\u0169ng t\u1ea1i ch\u00e2u \u00c2u. Hilbert mu\u1ed1n n\u00f3i v\u1ec1 kh\u00fac quanh \u0111\u00f3 m\u00e0 \u00f4ng l\u00e0 m\u1ed9t nh\u00e2n ch\u1ee9ng nh\u01b0 m\u1ed9t b\u00e0i h\u1ecdc nh\u1eadn th\u1ee9c lu\u1eadn cho c\u00f4ng ch\u00fang. B\u00e0i n\u00e0y kh\u00f4ng th\u1ec3 thi\u1ebfu cho <em>v\u0103n h\u00f3a khoa h\u1ecdc<\/em>.<\/p>\n<p>\u0110o\u1ea1n cu\u1ed1i c\u1ee7a b\u00e0i di\u1ec5n t\u1eeb (\u0111o\u1ea1n [6]) \u0111\u01b0\u1ee3c Hilbert l\u00e0m g\u1ecdn l\u1ea1i, v\u00e0 \u0111\u01b0\u1ee3c \u00f4ng \u0111\u1ecdc tr\u00ean radio trong b\u1ed1n ph\u00fat. \u0110\u00f3 l\u00e0 di\u1ec5n t\u1eeb r\u1ea5t \u1ea5n t\u01b0\u1ee3ng v\u00e0 n\u1ed5i ti\u1ebfng. C\u00f3 th\u1ec3 nghe gi\u1ecdng n\u00f3i c\u1ee7a \u00f4ng \u1edf \u0111\u00e2y:<\/p>\n<p><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"1140\" height=\"642\" src=\"https:\/\/www.youtube.com\/embed\/DJUQ2Q7Ivd0?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=vi&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation\"><\/iframe><\/span><\/p>\n<p>&nbsp;<\/p>\n<p>S\u1ef1 ph\u00e2n \u0111o\u1ea1n [1], [2], \u2026 trong b\u00e0i ph\u00e1t bi\u1ec3u c\u1ee7a Hilbert l\u00e0 do ng\u01b0\u1eddi d\u1ecbch th\u1ef1c hi\u1ec7n cho d\u1ec5 \u0111\u1ecdc h\u01a1n v\u1ec1 m\u1eb7t t\u00e2m l\u00fd. Trong \u0111o\u1ea1n [2] t\u00f4i c\u0169ng \u0111\u00e3 l\u01b0\u1ee3t b\u1edbt m\u1ed9t th\u00ed d\u1ee5 c\u1ee7a \u00f4ng. T\u00f4i tin kh\u00f4ng l\u00e0m \u1ea3nh h\u01b0\u1edfng \u0111\u1ebfn to\u00e0n b\u1ed9 \u00fd t\u01b0\u1edfng c\u1ee7a Hilbert, v\u1ed1n r\u1ea5t phong ph\u00fa.<\/p>\n<p>T\u00f4i r\u1ea5t mong nh\u1eadn \u0111\u01b0\u1ee3c c\u00e1c l\u1eddi b\u00ecnh c\u1ee7a c\u00e1c \u0111\u1ecdc gi\u1ea3 xa g\u1ea7n v\u1ec1 n\u1ed9i dung b\u00e0i vi\u1ebft c\u1ee7a Hilbert.<\/p>\n<p>&nbsp;<\/p>\n<h3 style=\"text-align: center\"><span style=\"color: #000080\"><strong><em>M\u1ed9t ch\u00fat ti\u1ec3u s\u1eed c\u1ee7a Hilbert<\/em><\/strong><\/span><\/h3>\n<p>Hilbert l\u00e0 &#8220;m\u1ed9t trong nh\u1eefng nh\u00e0 to\u00e1n h\u1ecdc v\u0129 \u0111\u1ea1i nh\u1ea5t c\u1ee7a m\u1ecdi th\u1eddi \u0111\u1ea1i&#8221; nh\u01b0 Heisenberg nh\u1eadn x\u00e9t. \u00d4ng sinh ra t\u1ea1i K\u00f6nigsberg, c\u0169ng l\u00e0 qu\u00ea h\u01b0\u01a1ng c\u1ee7a c\u00e1c nh\u00e0 tri\u1ebft h\u1ecdc Kant v\u00e0 Hamann, h\u1ecdc to\u00e1n t\u1ea1i \u0111\u1ea1i h\u1ecdc c\u1ee7a th\u00e0nh ph\u1ed1 v\u1edbi Ferdinand Lindemann, b\u1ea3o v\u1ec7 lu\u1eadn v\u0103n ti\u1ebfn s\u0129 n\u0103m 1885, gi\u1ea3ng d\u1ea1i v\u1edbi t\u01b0 c\u00e1ch privatdozent t\u1ea1i \u0111\u00f3 t\u1eeb 1889-92, v\u00e0 t\u1eeb 1893 l\u00e0 gi\u00e1o s\u01b0 th\u1ef1c th\u1ee5. N\u0103m 1895, th\u00f4ng qua s\u1ef1 \u1ee7ng h\u1ed9 c\u1ee7a Felix Klein, \u00f4ng nh\u1eadn l\u1eddi m\u1eddi l\u00e0m gi\u00e1o s\u01b0 t\u1ea1i \u0110\u1ea1i h\u1ecdc G\u00f6ttingen, v\u00e0 \u00f4ng t\u1eeb ch\u1ed1i nh\u1eefng l\u1eddi m\u1edbi c\u1ee7a nh\u1eefng n\u01a1i kh\u00e1c \u0111\u1ec3 \u1edf l\u1ea1i G\u00f6ttingen \u0111\u1ebfn cu\u1ed1i \u0111\u1eddi. V\u00e0 Hilbert \u0111\u00e3 t\u1ea1o s\u1ee9c h\u00fat m\u00e3nh li\u1ec7t, l\u00e0m cho tr\u01b0\u1eddng ph\u00e1i to\u00e1n c\u1ee7a Felix Klein cu\u1ed1i c\u00f9ng th\u0103ng hoa. Sinh vi\u00ean kh\u1eafp th\u1ebf gi\u1edbi \u0111\u1ebfn nghe c\u00e1c b\u00e0i gi\u1ea3ng truy\u1ec1n c\u1ea3m h\u1ee9ng v\u00e0 gi\u00e0u \u00fd t\u01b0\u1edfng c\u1ee7a \u00f4ng. Nh\u1eefng l\u00e3nh v\u1ef1c ch\u00ednh c\u1ee7a \u00f4ng l\u00e0: <em>Thuy\u1ebft bi\u1ebfn ph\u00e2n<\/em> (theory of variations), Thuy\u1ebft b\u1ea5t bi\u1ebfn (<em>Invariants<\/em>), Thuy\u1ebft s\u1ed1 (<em>Zahlentheorie<\/em>, 1893-99), Ti\u00ean \u0111\u1ec1 h\u00f3a h\u00ecnh h\u1ecdc (Axiomatik der Geometrie hay <em>Grundlagen der Geometrie,<\/em> 1899), Thuy\u1ebft c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh t\u00edch ph\u00e2n (<em>Integralgleichungen<\/em> 1901-12). N\u0103m 1897 Hilbert cho ra \u0111\u1eddi t\u00e1c ph\u1ea9m n\u1ed5i ti\u1ebfng <i>Zahlbericht<\/i> (&#8220;B\u00e1o c\u00e1o v\u1ec1 s\u1ed1&#8221;), nh\u1eb1m th\u1ed1ng nh\u1ea5t ng\u00e0nh s\u1ed1 h\u1ecdc \u0111\u1ea1i s\u1ed1\u00a0(<i>algebraic number theory<\/i>). N\u0103m 1915-16 quy\u1ec3n s\u00e1ch <em>Grundlagen der Physik<\/em> (C\u01a1 s\u1edf c\u1ee7a V\u1eadt l\u00fd) ra \u0111\u1eddi, sau \u0111\u00f3 <em>Grundlagen der Mathematik<\/em> (C\u01a1 s\u1edf c\u1ee7a To\u00e1n h\u1ecdc, 1923) v\u00e0 <em>Ph\u01b0\u01a1ng ph\u00e1p c\u1ee7a V\u1eadt l\u00fd to\u00e1n<\/em> (<em>Methoden der mathematischen Physik<\/em>, 1924, chung v\u1edbi Richard Courant, h\u1ecdc tr\u00f2 \u00f4ng).\u00a0 B\u00e0i ph\u00e1t bi\u1ec3u <em>Naturerkennen und Logik<\/em> m\u00e0 ch\u00fang ta s\u1ebd tr\u00ecnh b\u00e0y d\u01b0\u1edbi \u0111\u00e2y ph\u1ea3n \u1ea3nh l\u1eadp tr\u01b0\u1eddng tri\u1ebft h\u1ecdc c\u1ee7a \u00f4ng.<\/p>\n<p>N\u0103m 1900, Hilbert, l\u00fac \u0111\u00f3 \u1edf tu\u1ed5i 38, \u0111\u00e3 t\u1ea1o ti\u1ebfng vang l\u1edbn khi \u0111\u01b0\u1ee3c c\u1eed \u0111\u1ecdc b\u00e0i <em>di\u1ec5n v\u0103n khai m\u1ea1c<\/em> cho H\u1ed9i ngh\u1ecb Qu\u1ed1c t\u1ebf c\u00e1c Nh\u00e0 to\u00e1n h\u1ecdc (ICM) l\u1ea7n th\u1ee9 hai \u1edf Paris Sorbonne v\u00e0 \u0111\u01b0a ra danh s\u00e1ch 23 b\u00e0i to\u00e1n (ch\u01b0a gi\u1ea3i) c\u1ee7a to\u00e1n h\u1ecdc, d\u01b0\u1edbi c\u00e1i t\u00ean \u201cC\u00e1c b\u00e0i to\u00e1n c\u1ee7a to\u00e1n h\u1ecdc\u201d (<em>Mathematische Probleme<\/em>), do ch\u00ednh \u00f4ng kh\u1ea3o s\u00e1t v\u00e0 \u0111\u1ec1 xu\u1ea5t, nh\u1eb1m \u0111\u1ecbnh h\u01b0\u1edbng ph\u00e1t tri\u1ec3n cho to\u00e1n h\u1ecdc trong nh\u1eefng th\u1eadp ni\u00ean sau \u0111\u00f3. Hilbert m\u1edf \u0111\u1ea7u b\u00e0i thuy\u1ebft trinh c\u1ee7a m\u00ecnh b\u1eb1ng tuy\u00ean b\u1ed1:<\/p>\n<p style=\"padding-left: 40px\">Ai trong ch\u00fang ta l\u1ea1i kh\u00f4ng sung s\u01b0\u1edbng khi v\u00e9n l\u00ean b\u1ee9c m\u00e0n che khu\u1ea5t t\u01b0\u01a1ng lai; \u0111\u1ec3 chi\u00eam ng\u01b0\u1ee1ng s\u1ef1 ph\u00e1t tri\u1ec3n s\u1eafp t\u1edbi c\u1ee7a khoa h\u1ecdc c\u1ee7a ch\u00fang ta v\u00e0 nh\u1eefng b\u00ed m\u1eadt c\u1ee7a s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a n\u00f3 trong nh\u1eefng th\u1ebf k\u1ef7 t\u1edbi? \u0110i\u1ec1u g\u00ec s\u1ebd l\u00e0 m\u1ee5c \u0111\u00edch cu\u1ed1i c\u00f9ng m\u00e0 tinh th\u1ea7n c\u1ee7a c\u00e1c th\u1ebf h\u1ec7 nh\u00e0 to\u00e1n h\u1ecdc t\u01b0\u01a1ng lai s\u1ebd h\u01b0\u1edbng t\u1edbi? Th\u1ebf k\u1ef7 m\u1edbi s\u1ebd ti\u1ebft l\u1ed9 nh\u1eefng ph\u01b0\u01a1ng ph\u00e1p n\u00e0o, nh\u1eefng s\u1ef1 ki\u1ec7n m\u1edbi n\u00e0o trong l\u0129nh v\u1ef1c r\u1ed9ng l\u1edbn v\u00e0 phong ph\u00fa c\u1ee7a t\u01b0 duy to\u00e1n h\u1ecdc?<\/p>\n<p>v\u00e0 \u00f4ng ti\u1ebfp t\u1ee5c:<\/p>\n<p style=\"padding-left: 40px\">L\u1ecbch s\u1eed d\u1ea1y ch\u00fang ta v\u1ec1 t\u00ednh li\u00ean t\u1ee5c c\u1ee7a s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a khoa h\u1ecdc. Ch\u00fang ta bi\u1ebft r\u1eb1ng m\u1ecdi th\u1eddi \u0111\u1ea1i \u0111\u1ec1u c\u00f3 nh\u1eefng v\u1ea5n \u0111\u1ec1 ri\u00eang c\u1ee7a n\u00f3, m\u00e0 th\u1eddi \u0111\u1ea1i s\u1eafp t\u1edbi s\u1ebd gi\u1ea3i quy\u1ebft ho\u1eb7c g\u1ea1t sang m\u1ed9t b\u00ean nh\u01b0 l\u00e0 v\u00f4 \u00edch v\u00e0 thay th\u1ebf ch\u00fang b\u1eb1ng nh\u1eefng v\u1ea5n \u0111\u1ec1 m\u1edbi. N\u1ebfu ch\u00fang ta mu\u1ed1n c\u00f3 \u0111\u01b0\u1ee3c \u00fd t\u01b0\u1edfng v\u1ec1 s\u1ef1 ph\u00e1t tri\u1ec3n c\u00f3 th\u1ec3 x\u1ea3y ra c\u1ee7a tri \u200b\u200b\u200b\u200bth\u1ee9c to\u00e1n h\u1ecdc trong t\u01b0\u01a1ng lai g\u1ea7n, ch\u00fang ta ph\u1ea3i \u0111\u1ec3 nh\u1eefng c\u00e2u h\u1ecfi m\u1edf l\u01b0\u1edbt qua t\u00e2m tr\u00ed m\u00ecnh v\u00e0 kh\u1ea3o s\u00e1t nh\u1eefng v\u1ea5n \u0111\u1ec1 m\u00e0 khoa h\u1ecdc hi\u1ec7n t\u1ea1i \u0111\u1eb7t ra v\u00e0 ch\u00fang ta mong \u0111\u1ee3i gi\u1ea3i ph\u00e1p c\u1ee7a ch\u00fang trong t\u01b0\u01a1ng lai. Ng\u00e0y nay, v\u00e0o th\u1eddi \u0111i\u1ec3m chuy\u1ec3n giao th\u1ebf k\u1ef7, \u0111\u1ed1i v\u1edbi t\u00f4i d\u01b0\u1eddng nh\u01b0 r\u1ea5t th\u00edch h\u1ee3p cho vi\u1ec7c xem x\u00e9t c\u00e1c v\u1ea5n \u0111\u1ec1 nh\u01b0 v\u1eady; v\u00ec nh\u1eefng kho\u1ea3ng th\u1eddi gian tuy\u1ec7t v\u1eddi kh\u00f4ng ch\u1ec9 m\u1eddi ch\u00fang ta nh\u00ecn l\u1ea1i qu\u00e1 kh\u1ee9 m\u00e0 c\u00f2n h\u01b0\u1edbng suy ngh\u0129 c\u1ee7a ch\u00fang ta \u0111\u1ebfn nh\u1eefng \u0111i\u1ec1u ch\u01b0a bi\u1ebft s\u1ebd x\u1ea3y ra.<\/p>\n<p>Sau khi tr\u00ecnh b\u00e0y c\u00e1c b\u00e0i to\u00e1n, Hilbert k\u1ebft th\u00fac b\u00e0i thuy\u1ebft tr\u00ecnh c\u1ee7a \u00f4ng khi n\u00f3i r\u1eb1ng ng\u01b0\u1eddi ta s\u1ee3 to\u00e1n h\u1ecdc b\u1ecb chia ra th\u00e0nh nhi\u1ec1u ng\u00e0nh \u0111\u1ed9c l\u1eadp. &#8220;T\u00f4i kh\u00f4ng tin v\u00e0 kh\u00f4ng mu\u1ed1n \u0111i\u1ec1u \u0111\u00f3.&#8221; \u00d4ng ti\u1ebfp:<\/p>\n<p style=\"padding-left: 40px\">T\u00ednh ch\u1ea5t th\u1ed1ng nh\u1ea5t c\u1ee7a to\u00e1n h\u1ecdc n\u1eb1m \u1edf b\u1ea3n ch\u1ea5t b\u00ean trong c\u1ee7a khoa h\u1ecdc n\u00e0y; b\u1edfi v\u00ec to\u00e1n h\u1ecdc l\u00e0 c\u01a1 s\u1edf c\u1ee7a m\u1ecdi nh\u1eadn th\u1ee9c khoa h\u1ecdc ch\u00ednh x\u00e1c. \u0110\u1ec3 n\u00f3 ho\u00e0n th\u00e0nh s\u1ee9 m\u1ec7nh cao c\u1ea3 n\u00e0y, mong nh\u1eefng b\u1eadc th\u1ea7y thi\u00ean t\u00e0i v\u00e0 v\u00f4 s\u1ed1 \u0111\u1ec7 t\u1eed c\u1ee7a h\u1ecd b\u1eebng b\u1eebng nhi\u1ec7t huy\u1ebft cao c\u1ea3 s\u1ebd xu\u1ea5t hi\u1ec7n trong th\u1ebf k\u1ef7 m\u1edbi!<\/p>\n<p>Hilbert nh\u1eafm \u0111\u1ebfn nh\u1eefng ph\u01b0\u01a1ng ph\u00e1p \u0111\u1ec3 &#8220;kh\u00e1m ph\u00e1 nh\u1eefng m\u1ed1i li\u00ean h\u1ec7 kh\u00f4ng ai ng\u1edd t\u1edbi gi\u1eefa c\u00e1c ng\u00e0nh tri th\u1ee9c \u0111\u1ebfn nay c\u00f2n \u0111\u1ee9ng ri\u00eang r\u1ebb&#8221; nh\u01b0 l\u1eddi \u00f4ng n\u00f3i. M\u1ed9t trong nh\u1eefng b\u00e0i to\u00e1n hi\u1ec7n nay v\u1eabn ch\u01b0a gi\u1ea3i \u0111\u01b0\u1ee3c l\u00e0 <em>Gi\u1ea3 thuy\u1ebft<\/em> Riemann.<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone wp-image-8928\" src=\"https:\/\/i0.wp.com\/rosetta.vn\/nguyenxuanxanh\/wp-content\/uploads\/sites\/6\/2023\/02\/Hilbert_-_Mathematische_Probleme.pdf_Adjusted.jpg?resize=379%2C445&#038;ssl=1\" alt=\"\" width=\"379\" height=\"445\" data-recalc-dims=\"1\" \/><\/p>\n<p style=\"text-align: center\">Trang \u0111\u1ea7u ti\u00ean b\u00e0i kh\u1ea3o s\u00e1t c\u00e1c b\u00e0i to\u00e1n c\u1ee7a Hilbert \u0111\u0103ng tr\u00ean <em>Tin t\u1ee9c c\u1ee7a Hi\u1ec7p h\u1ed9i Khoa h\u1ecdc \u1edf G\u00f6ttingen, l\u1edbp to\u00e1n-v\u1eadt l\u00fd<\/em><\/p>\n<p>M\u1ed9t th\u1eddi gian ng\u1eafn sau khi Hilbert m\u1ea5t, \u00c9lie Cartan (1869\u20131951),nh\u00e0 to\u00e1n h\u1ecdc Ph\u00e1p chuy\u00ean gia v\u1ec1 nh\u00f3m Lie, trong m\u1ed9t b\u1ee9c th\u01b0 g\u1eedi Constantin Caratheodory (1873\u20131950), nh\u1ea5n m\u1ea1nh t\u1ea7m quan tr\u1ecdng c\u1ee7a b\u00e0i ph\u00e1t bi\u1ec3u 1900 nh\u01b0 sau: &#8220;Ch\u00fang ta s\u1ebd kh\u00f4ng bao gi\u1edd nghe th\u1ea5y m\u1ed9t cu\u1ed9c n\u00f3i chuy\u1ec7n nh\u01b0 v\u1eady t\u1ea1i c\u00e1c \u0111\u1ea1i h\u1ed9i n\u1eefa.&#8221;<\/p>\n<p>S\u1ef1 quan t\u00e2m s\u00e2u s\u1eafc c\u1ee7a Hilbert v\u1edbi v\u1eadt l\u00fd to\u00e1n c\u0169ng g\u00f3p ph\u1ea7n t\u1ea1o n\u00ean danh ti\u1ebfng c\u1ee7a tr\u01b0\u1eddng \u0111\u1ea1i h\u1ecdc G\u00f6ttingen v\u1ec1 v\u1eadt l\u00fd. Ba ng\u01b0\u1eddi \u0111o\u1ea1t gi\u1ea3i Nobel V\u1eadt l\u00fd\u2014Max von Laue n\u0103m 1914, James Franck n\u0103m 1925 v\u00e0 Werner Heisenberg n\u0103m 1925. 1932\u2014d\u00e0nh ph\u1ea7n quan tr\u1ecdng trong s\u1ef1 nghi\u1ec7p c\u1ee7a h\u1ecd t\u1ea1i G\u00f6ttingen trong th\u1eddi gian Hilbert s\u1ed1ng \u1edf \u0111\u00f3. Nh\u00e0 to\u00e1n h\u1ecdc Hermann Minkowski (th\u1ea7y c\u1ee7a Einstein t\u1ea1i ETH Zurich) c\u0169ng t\u1eebng ho\u1ea1t \u0111\u1ed9ng \u1edf \u0111\u00e2y, th\u00fac \u0111\u1ea9y \u1ee9ng d\u1ee5ng m\u1edbi c\u1ee7a to\u00e1n h\u1ecdc v\u00e0o v\u1eadt l\u00fd cho \u0111\u1ebfn khi \u00f4ng qua \u0111\u1eddi v\u00e0o n\u0103m 1909, m\u1ed9t n\u0103m sau khi \u00f4ng \u0111\u00e3 h\u00ecnh h\u1ecdc h\u00f3a thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i h\u1eb9p c\u1ee7a Einstein th\u00e0nh kh\u00f4ng-th\u1eddi gian b\u1ed1n chi\u1ec1u, t\u1ea1o m\u1ed9t b\u01b0\u1edbc ti\u1ebfn to l\u1edbn trong vi\u1ec7c x\u00e1c \u0111\u1ecbnh khung h\u00ecnh h\u1ecdc cho thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i r\u1ed9ng sau \u0111\u00f3. Hilbert ti\u1ebfp t\u1ee5c th\u1ef1c hi\u1ec7n c\u00e1c b\u00e0i gi\u1ea3ng v\u00e0 seminar v\u1ec1 c\u00e1c \u0111\u1ec1 t\u00e0i v\u1eadt l\u00fd cho \u0111\u1ebfn 1930.<\/p>\n<p>M\u1eb7c d\u00f9 \u00f4ng r\u1ea5t th\u00edch th\u00fa \u0111\u01b0\u1ee3c m\u1edf r\u1ed9ng ch\u00e2n tr\u1eddi tri th\u1ee9c, ra s\u1ee9c th\u1eed v\u1edbi thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i r\u1ed9ng v\u00e0 c\u1ea3 thuy\u1ebft tr\u01b0\u1eddng th\u1ed1ng nh\u1ea5t, nh\u1eefng \u0111\u1ec1 t\u00e0i kh\u00f3 c\u1ee7a Einstein, nh\u01b0ng cu\u1ed1i c\u00f9ng thu ho\u1ea1ch c\u1ee7a \u00f4ng kh\u00f4ng th\u1ec3 so s\u00e1nh v\u1edbi th\u00e0nh qu\u1ea3 c\u1ee7a \u00f4ng trong l\u00e3nh v\u1ef1c to\u00e1n h\u1ecdc. \u0110\u1ec3 s\u00e1ng t\u1ea1o, nh\u00e0 khoa h\u1ecdc, v\u1eadt l\u00fd hay to\u00e1n h\u1ecdc, hay b\u1ea5t c\u1ee9 ng\u00e0nh h\u1ecdc n\u00e0o, c\u1ea7n c\u00f3 <em>tr\u1ef1c gi\u00e1c<\/em> t\u1ed1t. Poincar\u00e9 di\u1ec5n t\u1ea3 \u0111i\u1ec1u n\u00e0y r\u1ea5t s\u00fac t\u00edch: \u201c<em>Nh\u1edd logic ch\u00fang ta ch\u1ee9ng minh, trong khi nh\u1edd tr\u1ef1c gi\u00e1c ch\u00fang ta kh\u00e1m ph\u00e1<\/em>.\u201d (trong\u00a0<em>Mathematical definitions in education<\/em>, 1904), v\u00e0 \u201c<em>Do \u0111\u00f3, logic v\u1eabn l\u00e0 mi\u1ebfng \u0111\u1ea5t c\u1eb1n c\u1ed7i tr\u1eeb khi \u0111\u01b0\u1ee3c tr\u1ef1c gi\u00e1c l\u00e0m cho n\u00f3 m\u00e0u m\u1ee1<\/em>.\u201d (1908) Nh\u01b0ng kh\u00f4ng ph\u1ea3i Hilbert thi\u1ebfu tr\u1ef1c gi\u00e1c, m\u00e0 s\u1ef1 th\u1eadt l\u00e0 th\u1ef1c t\u1ebf c\u00f3 nhi\u1ec1u lo\u1ea1i tr\u1ef1c gi\u00e1c, v\u00e0 tr\u1ef1c gi\u00e1c to\u00e1n c\u00f3 th\u1ec3 kh\u00f4ng \u0111\u1ed3ng nh\u1ea5t v\u1edbi tr\u1ef1c gi\u00e1c v\u1eadt l\u00fd. Trong cu\u1ed9c &#8220;ch\u1ea1y \u0111ua&#8221; gi\u1eefa Hilbert v\u00e0 Einstein trong vi\u1ec7c thi\u1ebft l\u1eadp ph\u01b0\u01a1ng tr\u00ecnh tr\u01b0\u1eddng h\u1ea5p d\u1eabn n\u0103m 1915, Hilbert t\u1ecf ra r\u1ea5t m\u1ea1nh m\u1ebd v\u00e0 nhanh ch\u00f3ng v\u1ec1 m\u1eb7t to\u00e1n h\u1ecdc, gi\u1ed1ng nh\u01b0 tay \u0111ua c\u1eeb kh\u00f4i c\u1ee7a Formula 1. Nh\u01b0ng cu\u1ed1i c\u00f9ng kh\u00f4ng ph\u1ea3i \u00f4ng m\u00e0 Einstein \u0111\u00e3 thi\u1ebft l\u1eadp \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh tr\u01b0\u1eddng mong mu\u1ed1n. L\u00fd do \u0111\u01a1n gi\u1ea3n: Hilbert thi\u1ebfu tr\u1ef1c gi\u00e1c v\u1eadt l\u00fd c\u00e1i m\u00e0 Einstein c\u00f3. Xem b\u00e0i <a href=\"https:\/\/rosetta.vn\/nguyenxuanxanh\/vai-ngo-nhan-ve-albert-einstein\/\"><em>V\u00e0i ng\u1ed9 nh\u1eadn v\u1ec1 Albert Einstein<\/em>\u00a0<\/a>ph\u1ea7n I. Hilbert di\u1ec5n t\u1ea3 s\u1ef1 ki\u1ec7n n\u00e0y b\u1eb1ng c\u00e2u n\u00f3i d\u00ed d\u1ecfm: <em>&#8220;M\u1ed7i c\u1eadu h\u1ecdc sinh tr\u00ean \u0111\u01b0\u1eddng ph\u1ed1 G\u00f6ttingen c\u1ee7a ch\u00fang ta hi\u1ec3u v\u1ec1 h\u00ecnh h\u1ecdc b\u1ed1n chi\u1ec1u c\u00f2n nhi\u1ec1u h\u01a1n Einstein. M\u1eb7c d\u00f9 th\u1ebf Einstein l\u00e0 ng\u01b0\u1eddi \u0111\u00e3 l\u00e0m n\u00ean c\u00f4ng tr\u00ecnh ch\u1ee9 kh\u00f4ng ph\u1ea3i nh\u1eefng\u00a0 nh\u00e0 to\u00e1n h\u1ecdc.&#8221;\u00a0<\/em><\/p>\n<p>Theo nh\u1eadn x\u00e9t c\u1ee7a Murray Gell-Mann, cha \u0111\u1ebb c\u1ee7a h\u1ea1t quark c\u01a1 b\u1ea3n, c\u00e1ch l\u00e0m c\u1ee7a nh\u1eefng nh\u00e0 v\u1eadt l\u00fd l\u00e0, tr\u01b0\u1edbc nh\u1ea5t h\u1ecd kh\u00f4ng c\u1ea7n ph\u1ea3i h\u1ecdc qu\u00e1 nhi\u1ec1u to\u00e1n h\u1ecdc; ch\u1ec9 khi n\u00e0o g\u1eb7p kh\u00f3 kh\u0103n trong vi\u1ec7c gi\u1ea3i b\u00e0i to\u00e1n v\u1eadt l\u00fd c\u1ee7a m\u00ecnh, h\u1ecd m\u1edbi \u0111i t\u00ecm c\u00f4ng c\u1ee5 to\u00e1n h\u1ecdc th\u00edch h\u1ee3p cho n\u00f3. Ch\u1eb3ng h\u1ea1n nh\u01b0 Heisenberg v\u1edbi ma tr\u1eadn, hay Einstein v\u1edbi kh\u00f4ng gian Riemann, hay Gell-Mann v\u1edbi c\u00e1c nh\u00f3m Lie. \u0110\u1ed1i v\u1edbi Hilbert, nh\u1eefng ng\u01b0\u1eddi nh\u01b0 Einstein, Bohr, &#8220;m\u00f2 m\u1eabm&#8221; con \u0111\u01b0\u1eddng c\u1ee7a h\u1ecd trong &#8220;b\u00f3ng t\u1ed1i&#8221; \u0111\u1ec3 \u0111i \u0111\u1ebfn c\u00e1c concepts v\u1ec1 t\u01b0\u01a1ng \u0111\u1ed1i hay c\u1ea5u tr\u00fac nguy\u00ean t\u1eed, b\u1eb1ng nh\u1eefng lo\u1ea1i th\u00ed nghi\u1ec7m \u00fd t\u01b0\u1edfng v\u00e0 t\u01b0\u1edfng t\u01b0\u1ee3ng kh\u00e1c h\u01a1n c\u00e1ch c\u1ee7a nh\u00e0 to\u00e1n h\u1ecdc. C\u00f3 l\u1ebd Hilbert thi\u1ebfu nh\u1eefng tr\u1ea3i nghi\u1ec7m c\u1ee7a nh\u00e0 v\u1eadt l\u00fd. (Xem th\u00eam ch\u00fa th\u00edch [4] d\u01b0\u1edbi \u0111\u00e2y)<\/p>\n<p>N\u0103m 1910, khi gi\u1ea3i th\u01b0\u1edfng Bolyai th\u1ee9 hai, sau Poincar\u00e9 (1905), c\u1ee7a H\u00e0n l\u00e2m vi\u1ec7n khoa h\u1ecdc Hungary \u0111\u01b0\u1ee3c trao cho Hilbert, Poincar\u00e9 l\u00e0 ng\u01b0\u1eddi \u0111\u00e3 vi\u1ebft b\u00e0i khen ng\u1ee3i n\u1ed3ng nhi\u1ec7t. (Poincar\u00e9 sau n\u00e0y trong m\u1ed9t d\u1ecbp kh\u00e1c c\u0169ng vi\u1ebft m\u1ed9t b\u00e0i gi\u1edbi thi\u1ec7u n\u1ed3ng nhi\u1ec7t cho Einstein v\u00e0o m\u1ee5c \u0111\u00edch kh\u00e1c)<\/p>\n<p>Kh\u1ea3 n\u0103ng to\u00e1n h\u1ecdc c\u1ee7a \u00f4ng \u0111\u00e3 \u0111\u01b0\u1ee3c Otto Blumenthal, h\u1ecdc tr\u00f2 \u0111\u1ea7u ti\u00ean c\u1ee7a \u00f4ng t\u00f3m t\u1eaft nh\u01b0 th\u1ebf n\u00e0y:<\/p>\n<p style=\"padding-left: 40px\">Khi ph\u00e2n t\u00edch t\u00e0i n\u0103ng to\u00e1n h\u1ecdc, ng\u01b0\u1eddi ta ph\u1ea3i ph\u00e2n bi\u1ec7t gi\u1eefa kh\u1ea3 n\u0103ng t\u1ea1o ra c\u00e1c kh\u00e1i ni\u1ec7m m\u1edbi nh\u1eb1m sinh ra c\u00e1c lo\u1ea1i c\u1ea5u tr\u00fac t\u01b0 duy m\u1edbi v\u00e0 n\u0103ng khi\u1ebfu c\u1ea3m nh\u1eadn c\u00e1c m\u1ed1i li\u00ean h\u1ec7 s\u00e2u s\u1eafc h\u01a1n v\u00e0 s\u1ef1 th\u1ed1ng nh\u1ea5t \u1edf n\u1ec1n t\u1ea3ng. Trong tr\u01b0\u1eddng h\u1ee3p c\u1ee7a Hilbert, s\u1ef1 v\u0129 \u0111\u1ea1i c\u1ee7a \u00f4ng \u1ea5y n\u1eb1m \u1edf m\u1ed9t c\u00e1i nh\u00ecn th\u1ea5u su\u1ed1t v\u00f4 c\u00f9ng m\u1ea1nh m\u1ebd, xuy\u00ean th\u1ea5u v\u00e0o chi\u1ec1u s\u00e2u c\u1ee7a m\u1ed9t v\u1ea5n \u0111\u1ec1. T\u1ea5t c\u1ea3 c\u00e1c t\u00e1c ph\u1ea9m c\u1ee7a \u00f4ng \u0111\u1ec1u ch\u1ee9a \u0111\u1ef1ng nh\u1eefng v\u00ed d\u1ee5 t\u1eeb c\u00e1c l\u0129nh v\u1ef1c xa x\u00f4i m\u00e0 ch\u1ec9 \u00f4ng m\u1edbi c\u00f3 th\u1ec3 nh\u1eadn ra t\u00ednh li\u00ean k\u1ebft nhau v\u00e0 m\u1ed1i li\u00ean h\u1ec7 v\u1edbi v\u1ea5n \u0111\u1ec1 \u0111ang nghi\u00ean c\u1ee9u. T\u1eeb nh\u1eefng \u0111i\u1ec1u n\u00e0y, s\u1ef1 t\u1ed5ng h\u1ee3p, t\u00e1c ph\u1ea9m ngh\u1ec7 thu\u1eadt c\u1ee7a \u00f4ng \u1ea5y, cu\u1ed1i c\u00f9ng \u0111\u00e3 \u0111\u01b0\u1ee3c t\u1ea1o ra. Trong ch\u1eebng m\u1ef1c li\u00ean quan \u0111\u1ebfn vi\u1ec7c t\u1ea1o ra nh\u1eefng \u00fd t\u01b0\u1edfng m\u1edbi, t\u00f4i s\u1ebd \u0111\u1eb7t Minkowski cao h\u01a1n, v\u00e0 t\u1ea1o ra nh\u1eefng \u00fd t\u01b0\u1edfng v\u0129 \u0111\u1ea1i kinh \u0111i\u1ec3n, Gauss, Galois v\u00e0 Riemann. Nh\u01b0ng khi n\u00f3i \u0111\u1ebfn c\u00e1i nh\u00ecn xuy\u00ean su\u1ed1t, ch\u1ec9 m\u1ed9t v\u00e0i trong s\u1ed1 nh\u1eefng ng\u01b0\u1eddi v\u0129 \u0111\u1ea1i nh\u1ea5t l\u00e0 ngang h\u00e0ng v\u1edbi Hilbert.<\/p>\n<p>Felix Klein trong m\u1ed9t b\u00e0i di\u1ec5n v\u0103n n\u0103m 1893 t\u1ea1i Chicago \u0111\u00e3 chia c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc ra l\u00e0m ba nh\u00f3m: C\u00e1c nh\u00e0 logic h\u1ecdc (Logiker), c\u00e1c nh\u00e0 h\u00ecnh th\u1ee9c h\u1ecdc (Formalisten) v\u00e0 c\u00e1c nh\u00e0 tr\u1ef1c quan h\u1ecdc (Intuitiven). Weierstrass c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c x\u1ebfp v\u00e0o nh\u00f3m nh\u00e0 logic h\u1ecdc, Cayley nh\u00e0 h\u00ecnh th\u1ee9c h\u1ecdc, v\u00e0 Brouwer nh\u00e0 tr\u1ef1c quan h\u1ecdc. \u0110\u1ed1i v\u1edbi Hilbert, ng\u01b0\u1eddi ta n\u00f3i \u00f4ng l\u00e0 ng\u01b0\u1eddi \u0111\u1ee9ng t\u1ea1i giao l\u1ed9 c\u1ee7a ba nh\u00f3m.<\/p>\n<p>Hilbert kh\u00f4ng nh\u1eefng l\u00e0 m\u1ed9t h\u1ecdc gi\u1ea3 l\u1edbn, m\u00e0 c\u00f2n l\u00e0 m\u1ed9t ng\u01b0\u1eddi th\u1ea7y l\u1edbn m\u00e0 c\u00e1c sinh vi\u00ean v\u00e0 tr\u1ee3 l\u00fd c\u1ee7a \u00f4ng l\u00e0 nh\u1eefng nh\u00e2n ch\u1ee9ng. \u00d4ng \u0111\u00e3 chia s\u1ebb v\u1edbi h\u1ecd &#8220;ngh\u1ec7 thu\u1eadt th\u1ee7 c\u00f4ng&#8221; c\u1ee7a kh\u00e1m ph\u00e1 to\u00e1n h\u1ecdc, v\u00e0 \u0111\u1ec3 h\u1ecd tham gia v\u00e0o t\u00e1c ph\u1ea9m c\u1ee7a \u00f4ng. T\u00e1c ph\u1ea9m n\u1ed5i ti\u1ebfng A<em>nschauliche Geometrie<\/em> (H\u00ecnh h\u1ecdc tr\u1ef1c quan, t\u00ean b\u1ea3n ti\u1ebfng Anh: <i>Geometry and the Imagination<\/i>) c\u1ee7a \u00f4ng chung v\u1edbi nh\u00e0 to\u00e1n h\u1ecdc tr\u1ebb S. Cohn-Vossen l\u00e0 m\u1ed9t trong nh\u1eefng k\u1ebft qu\u1ea3 c\u1ee7a c\u00e1c ho\u1ea1t \u0111\u1ed9ng gi\u1ea3ng d\u1ea1y c\u1ee7a \u00f4ng. Trong 69 h\u1ecdc tr\u00f2 c\u1ee7a \u00f4ng, c\u00f3 nhi\u1ec1u ng\u01b0\u1eddi r\u1ea5t n\u1ed5i ti\u1ebfng, nh\u01b0\u00a0Otto Blumenthal, Felix Bernstein, Hermann Weyl, Richard Courant, Erich Hecke, Hugo Steinhaus, v\u00e0 Wilhelm Ackermann.<\/p>\n<p>Cu\u1ed1i \u0111\u1eddi, Hilbert \u0111\u00e3 ph\u1ea3i \u0111au kh\u1ed5 khi ch\u1ee9ng ki\u1ebfn Nazi ph\u00e1 h\u1ee7y mecca to\u00e1n h\u1ecdc c\u1ee7a G\u00f6ttingen th\u1ebf n\u00e0o m\u00e0 \u00f4ng \u0111\u00e3 g\u00f3p s\u1ee9c x\u00e2y d\u1ef1ng th\u1ebf n\u00e0o. Sau khi lo\u1ea1i b\u1ecf h\u1ebft c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc c\u00f3 d\u00f2ng m\u00e1u Jewish, hay c\u00f3 m\u1ed1i li\u00ean h\u1ec7 gia \u0111\u00ecnh v\u1edbi Jew, v\u1ecb B\u1ed9 tr\u01b0\u1edfng gi\u00e1o d\u1ee5c c\u1ee7a Nazi (Bernhard Rust) \u0111\u1ebfn h\u1ecfi Hilbert, &#8220;Sao, to\u00e1n h\u1ecdc \u1edf G\u00f6ttingen b\u00e2y gi\u1edd nh\u01b0 th\u1ebf n\u00e0o sau khi n\u00f3 \u0111\u01b0\u1ee3c gi\u1ea3i ph\u00f3ng kh\u1ecfi \u1ea3nh h\u01b0\u1edfng c\u1ee7a Do Th\u00e1i?&#8221;. Hilbert th\u1eb3ng th\u1eafn tr\u1ea3 l\u1eddi: \u201cTo\u00e1n h\u1ecdc \u1edf G\u00f6ttingen \u01b0? B\u00e2y gi\u1edd c\u00f2n \u0111\u00e2u n\u1eefa?\u201d Kh\u00f4ng ph\u1ea3i ch\u1ec9 c\u00f3 to\u00e1n h\u1ecdc, m\u00e0 c\u1ea3 n\u1ec1n khoa h\u1ecdc c\u1ee7a \u0110\u1ee9c c\u0169ng b\u1ecb \u0111\u00e1nh s\u1eadp, c\u00f9ng l\u00fac di\u1ec5n ra m\u1ed9t exodus c\u1ee7a c\u00e1c nh\u00e0 khoa h\u1ecdc t\u1eeb \u0110\u1ee9c v\u00e0 c\u00e1c qu\u1ed1c gia ch\u00e2u \u00c2u b\u1ecb Nazi \u0111e d\u1ecda sang Hoa K\u1ef3.<\/p>\n<p>Hilbert m\u1ea5t n\u0103m 1943 \u1edf tu\u1ed5i 81 t\u1ea1i G\u00f6ttingen. \u00d4ng may m\u1eafn kh\u00f4ng nh\u00ecn th\u1ea5y th\u00e0nh ph\u1ed1 qu\u00ea h\u01b0\u01a1ng th\u00e2n y\u00eau c\u1ee7a \u00f4ng K\u00f6nigsberg s\u1ebd kh\u00f4ng c\u00f2n thu\u1ed9c \u0110\u1ee9c n\u1eefa. Cu\u1ed9c kh\u1ee7ng ho\u1ea3ng ch\u00ednh tr\u1ecb cu\u1ed1i \u0111\u1eddi \u00f4ng tr\u00f9ng h\u1ee3p v\u1edbi cu\u1ed9c &#8220;kh\u1ee7ng ho\u1ea3ng&#8221; to\u00e1n h\u1ecdc, ng\u01b0\u1ee3c v\u1edbi ni\u1ec1m tin c\u1ee7a \u00f4ng, khi nh\u00e0 logic h\u1ecdc ng\u01b0\u1eddi \u00c1o Kurt G\u00f6del ch\u1ee9ng minh \u0111\u1ecbnh l\u00fd b\u1ea5t to\u00e0n, ngh\u0129a l\u00e0 c\u00f3 m\u1ed9t m\u1ec7nh \u0111\u1ec1 to\u00e1n h\u1ecdc m\u00e0 ng\u01b0\u1eddi ta s\u1ebd kh\u00f4ng quy\u1ebft \u0111\u1ecbnh \u0111\u01b0\u1ee3c n\u00f3 \u0111\u00fang hay sai. \u0110\u00f3 l\u00e0 m\u1ed9t cu\u1ed9c &#8220;t\u1ea5n c\u00f4ng&#8221; v\u00e0o c\u00e1c n\u1ec1n t\u1ea3ng logic c\u1ee7a to\u00e1n h\u1ecdc. D\u00f9 v\u1eady, \u00f4ng \u0111\u00e3 \u0111\u00f3ng g\u00f3p ph\u1ea7n l\u1edbn nh\u1ea5t c\u1ee7a s\u1ee9c l\u1ef1c \u00f4ng v\u00e0o to\u00e1n h\u1ecdc x\u00e2y d\u1ef1ng, nh\u1eefng g\u00ec c\u00f3 th\u1ec3 ch\u1ee9ng minh \u0111\u01b0\u1ee3c. Hilbert l\u00e0 con ng\u01b0\u1eddi \u0111\u1ea7y l\u1ea1c quan, b\u1eaft ngu\u1ed3n t\u1eeb m\u1ed9t n\u0103ng l\u01b0\u1ee3ng sung m\u00e3n c\u1ee7a \u00f4ng. Nh\u01b0ng n\u00f3 c\u0169ng ph\u1ea3n \u1ea3nh s\u1ef1 l\u1ea1c quan m\u1ea1nh m\u1ebd c\u1ee7a th\u1eddi \u0111\u1ea1i l\u00fac b\u1ea5y gi\u1edd, <em>zeitgeist<\/em>, tin v\u00e0o n\u0103ng l\u1ef1c kh\u00e1m ph\u00e1 v\u00e0 s\u00e1ng t\u1ea1o kh\u00f4ng bi\u00ean gi\u1edbi c\u1ee7a con ng\u01b0\u1eddi. \u00d4ng k\u1ebft th\u00fac b\u00e0i di\u1ec5n t\u1eeb n\u0103m 1930 b\u1eb1ng s\u00e1u ch\u1eef (\u0110\u1ee9c) n\u1ed5i ti\u1ebfng th\u1ec3 hi\u1ec7n nhi\u1ec7t huy\u1ebft c\u1ee7a \u00f4ng \u0111\u1ed1i v\u1edbi to\u00e1n h\u1ecdc v\u00e0 l\u00f2ng tin v\u00e0o n\u0103ng l\u1ef1c c\u1ee7a con ng\u01b0\u1eddi: <em>Wir m\u00fcssen wissen<\/em>\/ <em>Wir werden wissen\u00a0<\/em>(Ch\u00fang ta ph\u1ea3i bi\u1ebft\/ Ch\u00fang ta s\u1ebd bi\u1ebft), nh\u1eefng l\u1eddi sau \u0111\u00f3 \u0111\u01b0\u1ee3c kh\u1eafc l\u00ean bia m\u1ed9 \u00f4ng th\u1ec3 hi\u1ec7n t\u00e2m th\u1ebf c\u1ee7a \u00f4ng.<\/p>\n<p style=\"text-align: center\"><span style=\"color: #000080\"><strong>Nguy\u1ec5n Xu\u00e2n Xanh<\/strong><\/span>, 6\/2\/2023<\/p>\n<p style=\"text-align: center\">\u2b50\u2b50\u2b50<\/p>\n<p>&nbsp;<\/p>\n<h3 style=\"text-align: center\"><span style=\"font-size: 24pt;color: #808000\">DI\u1ec4N T\u1eea C\u1ee6A DAVID HILBERT<\/span><\/h3>\n<p style=\"text-align: center\"><span style=\"font-size: 14pt\"><strong><span style=\"color: #000080\">Naturerkennen und Logik<\/span><\/strong><\/span><\/p>\n<p style=\"text-align: center\">[1]<\/p>\n<p><span style=\"font-size: 24pt\"><strong>S<\/strong><\/span>\u1ef1 nh\u1eadn th\u1ee9c v\u1ec1 t\u1ef1 nhi\u00ean v\u00e0 cu\u1ed9c s\u1ed1ng l\u00e0 nhi\u1ec7m v\u1ee5 quan tr\u1ecdng nh\u1ea5t c\u1ee7a ch\u00fang ta. M\u1ecdi n\u1ed7 l\u1ef1c v\u00e0 \u00fd ch\u00ed c\u1ee7a con ng\u01b0\u1eddi \u0111\u1ec1u k\u1ebft th\u00fac \u1edf \u0111\u00f3, v\u00e0 ch\u00fang ta \u0111\u00e3 \u0111\u01b0\u1ee3c ban t\u1eb7ng cho th\u00e0nh c\u00f4ng ng\u00e0y c\u00e0ng nhi\u1ec1u h\u01a1n. Trong v\u00e0i th\u1eadp k\u1ef7 qua, ch\u00fang ta \u0111\u00e3 c\u00f3 \u0111\u01b0\u1ee3c nh\u1eadn \u200b\u200bth\u1ee9c phong ph\u00fa v\u00e0 s\u00e2u s\u1eafc h\u01a1n v\u1ec1 t\u1ef1 nhi\u00ean so v\u1edbi nhi\u1ec1u th\u1ebf k\u1ef7 tr\u01b0\u1edbc. H\u00f4m nay ch\u00fang ta mu\u1ed1n s\u1eed d\u1ee5ng t\u00ecnh hu\u1ed1ng thu\u1eadn l\u1ee3i n\u00e0y \u0111\u1ec3 x\u1eed l\u00fd m\u1ed9t v\u1ea5n \u0111\u1ec1 tri\u1ebft h\u1ecdc c\u0169 ph\u00f9 h\u1ee3p v\u1edbi ch\u1ee7 \u0111\u1ec1 c\u1ee7a ch\u00fang ta, \u0111\u00f3 l\u00e0 c\u00e2u h\u1ecfi c\u00f2n nhi\u1ec1u tranh c\u00e3i v\u1ec1 ph\u1ea7n \u0111\u00f3ng g\u00f3p m\u00e0 t\u01b0 duy m\u1ed9t m\u1eb7t v\u00e0 kinh nghi\u1ec7m m\u1eb7t kh\u00e1c d\u1ef1 ph\u1ea7n v\u00e0o nh\u1eadn \u200b\u200b\u200b\u200bth\u1ee9c c\u1ee7a ch\u00fang ta. C\u00e2u h\u1ecfi c\u0169 n\u00e0y l\u00e0 ch\u00ednh \u0111\u00e1ng, b\u1edfi v\u00ec tr\u1ea3 l\u1eddi n\u00f3 v\u1ec1 c\u01a1 b\u1ea3n c\u00f3 ngh\u0129a l\u00e0 x\u00e1c \u0111\u1ecbnh lo\u1ea1i nh\u1eadn \u200b\u200b\u200b\u200bth\u1ee9c khoa h\u1ecdc t\u1ef1 nhi\u00ean m\u00e0 ch\u00fang ta c\u00f3 n\u00f3i chung l\u00e0 lo\u1ea1i n\u00e0o, v\u00e0 theo ngh\u0129a n\u00e0o t\u1ea5t c\u1ea3 ki\u1ebfn \u200b\u200b\u200b\u200bth\u1ee9c m\u00e0 ch\u00fang ta thu th\u1eadp trong khoa h\u1ecdc t\u1ef1 nhi\u00ean l\u00e0 ch\u00e2n l\u00fd.<\/p>\n<p>Kh\u00f4ng t\u1ef1 ph\u1ee5 \u0111\u1ed1i v\u1edbi c\u00e1c nh\u00e0 tri\u1ebft h\u1ecdc v\u00e0 nh\u00e0 nghi\u00ean c\u1ee9u c\u1ed5 \u0111\u1ea1i, ng\u00e0y nay ch\u00fang ta c\u00f3 th\u1ec3 tin v\u00e0o m\u1ed9t c\u00e2u tr\u1ea3 l\u1eddi \u0111\u00fang \u0111\u1eafn cho c\u00e2u h\u1ecfi n\u00e0y m\u1ed9t c\u00e1ch ch\u1eafc ch\u1eafn h\u01a1n l\u00e0 h\u1ecd \u0111\u00e3 l\u00e0m, v\u00ec hai l\u00fd do: th\u1ee9 nh\u1ea5t l\u00e0 t\u1ed1c \u0111\u1ed9 ph\u00e1t tri\u1ec3n nhanh ch\u00f3ng \u0111\u00e3 \u0111\u01b0\u1ee3c \u0111\u1ec1 c\u1eadp \u1edf tr\u00ean m\u00e0 c\u00e1c ng\u00e0nh khoa h\u1ecdc c\u1ee7a ch\u00fang ta ng\u00e0y nay \u0111ang ph\u00e1t tri\u1ec3n.<\/p>\n<p>Nh\u1eefng kh\u00e1m ph\u00e1 quan tr\u1ecdng c\u1ee7a nh\u1eefng th\u1eddi k\u1ef3 tr\u01b0\u1edbc \u0111\u00f3, t\u1eeb Copernicus, Kepler, Galileo, Newton \u0111\u1ebfn Maxwell, tr\u1ea3i d\u00e0i trong g\u1ea7n b\u1ed1n th\u1ebf k\u1ef7. K\u1ef7 nguy\u00ean hi\u1ec7n \u0111\u1ea1i sau \u0111\u00f3 b\u1eaft \u0111\u1ea7u v\u1edbi vi\u1ec7c ph\u00e1t hi\u1ec7n ra s\u00f3ng Hertz. V\u00e0 sau \u0111\u00f3 nhanh ch\u00f3ng: R\u00f6ntgen ph\u00e1t hi\u1ec7n ra c\u00e1c tia quang tuy\u1ebfn, Curie hi\u1ec7n t\u01b0\u1ee3ng ph\u00f3ng x\u1ea1, Planck thi\u1ebft l\u1eadp thuy\u1ebft l\u01b0\u1ee3ng t\u1eed. V\u00e0 trong th\u1eddi gian g\u1ea7n \u0111\u00e2y nh\u1ea5t, c\u00e1c kh\u00e1m ph\u00e1 v\u1ec1 nh\u1eefng hi\u1ec7n t\u01b0\u1ee3ng m\u1edbi v\u00e0 nh\u1eefng m\u1ed1i li\u00ean h\u1ec7 \u0111\u00e1ng ng\u1ea1c nhi\u00ean \u0111ang \u1ed3 \u1ea1t k\u00e9o \u0111\u1ebfn khi\u1ebfn r\u1ea5t nhi\u1ec1u khu\u00f4n m\u1eb7t t\u1ecf ra b\u1ea5t an: Thuy\u1ebft ph\u00f3ng x\u1ea1 c\u1ee7a Rutherford, \u0111\u1ecbnh lu\u1eadt <em>hv<\/em> c\u1ee7a Einstein, c\u00e1ch gi\u1ea3i th\u00edch c\u1ee7a Bohr v\u1ec1 quang ph\u1ed5, c\u00e1ch \u0111\u00e1nh s\u1ed1 nguy\u00ean t\u1ed1 c\u1ee7a Moseley, thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i c\u1ee7a Einstein, s\u1ef1 ph\u00e2n h\u1ee7y nit\u01a1 c\u1ee7a Rutherford, c\u1ea5u tr\u00fac c\u00e1c nguy\u00ean t\u1ed1 c\u1ee7a Bohr, l\u00fd thuy\u1ebft \u0111\u1ed3ng v\u1ecb c\u1ee7a Aston.<\/p>\n<p>V\u00ec v\u1eady, ch\u1ec9 ri\u00eang trong l\u0129nh v\u1ef1c v\u1eadt l\u00fd, ch\u00fang ta \u0111\u00e3 c\u00f3 m\u1ed9t lo\u1ea1t kh\u00e1m ph\u00e1 kh\u00f4ng ng\u1eebng ngh\u1ec9, v\u00e0 c\u00e1c kh\u00e1m ph\u00e1 n\u00e0y quan tr\u1ecdng l\u00e0m sao! Kh\u00f4ng kh\u00e1m ph\u00e1 n\u00e0o trong s\u1ed1 \u0111\u00f3 thua k\u00e9m nh\u1eefng th\u00e0nh t\u1ef1u c\u1ee7a th\u1eddi tr\u01b0\u1edbc v\u1ec1 s\u1ef1 v\u0129 \u0111\u1ea1i, h\u01a1n n\u1eefa ch\u00fang b\u1ecb n\u00e9n l\u1ea1i g\u1ea7n nhau h\u01a1n v\u1ec1 th\u1eddi gian, nh\u01b0ng v\u1ec1 n\u1ed9i dung c\u0169ng \u0111a d\u1ea1ng nh\u01b0 th\u1eddi tr\u01b0\u1edbc. V\u00e0 trong \u0111\u00f3, l\u00fd thuy\u1ebft v\u00e0 th\u1ef1c h\u00e0nh, t\u01b0 duy v\u00e0 kinh nghi\u1ec7m kh\u00f4ng ng\u1eebng th\u1ec3 hi\u1ec7n s\u1ef1 g\u1eafn b\u00f3 m\u1eadt thi\u1ebft v\u1edbi nhau. Khi th\u00ec l\u00fd thuy\u1ebft \u0111i tr\u01b0\u1edbc, khi th\u00ec th\u1ef1c nghi\u1ec7m, lu\u00f4n lu\u00f4n x\u00e1c nh\u1eadn, b\u1ed5 sung v\u00e0 k\u00edch th\u00edch cho nhau. \u0110i\u1ec1u t\u01b0\u01a1ng t\u1ef1 c\u0169ng \u00e1p d\u1ee5ng cho h\u00f3a h\u1ecdc, thi\u00ean v\u0103n h\u1ecdc v\u00e0 c\u00e1c ng\u00e0nh sinh h\u1ecdc.<\/p>\n<p>V\u00ec v\u1eady, \u0111\u1ed1i v\u1edbi c\u00e1c tri\u1ebft gia x\u01b0a h\u01a1n tr\u01b0\u1edbc \u0111\u00e2y, ch\u00fang ta c\u00f3 l\u1ee3i th\u1ebf l\u00e0 \u0111ang ch\u1ee9ng ki\u1ebfn \u200b\u200bm\u1ed9t s\u1ed1 l\u01b0\u1ee3ng l\u1edbn nh\u1eefng kh\u00e1m ph\u00e1 nh\u01b0 v\u1eady v\u00e0 quen bi\u1ebft \u0111\u01b0\u1ee3c nh\u1eefng quan \u0111i\u1ec3m m\u1edbi \u0111\u01b0\u1ee3c t\u1ea1o ra trong qu\u00e1 tr\u00ecnh h\u00ecnh th\u00e0nh c\u1ee7a ch\u00fang. Trong s\u1ed1 nh\u1eefng kh\u00e1m ph\u00e1 m\u1edbi, c\u00f3 nhi\u1ec1u c\u00e1i \u0111\u00e3 l\u00e0m thay \u0111\u1ed5i nh\u1eefng quan \u0111i\u1ec3m v\u00e0 t\u01b0 t\u01b0\u1edfng c\u0169 \u0111\u00e3 b\u00e1m r\u1ec5 s\u00e2u, ho\u1eb7c lo\u1ea1i b\u1ecf ch\u00fang ho\u00e0n to\u00e0n. V\u00ed d\u1ee5, ch\u1ec9 c\u1ea7n ngh\u0129 v\u1ec1 kh\u00e1i ni\u1ec7m m\u1edbi v\u1ec1 th\u1eddi gian trong thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i, ho\u1eb7c v\u1ec1 s\u1ef1 ph\u00e2n h\u1ee7y c\u1ee7a c\u00e1c nguy\u00ean t\u1ed1 h\u00f3a h\u1ecdc, v\u00e0 c\u00e1ch m\u00e0 qua \u0111\u00f3 nh\u1eefng \u0111\u1ecbnh ki\u1ebfn \u0111\u00e3 b\u1ecb lo\u1ea1i b\u1ecf, nh\u1eefng th\u1ee9 \u200b\u200bm\u00e0 tr\u01b0\u1edbc \u0111\u00e2y kh\u00f4ng ai d\u00e1m ngh\u0129 s\u1ebd thay \u0111\u1ed5i.<\/p>\n<p style=\"text-align: center\">[2]<\/p>\n<p>Nh\u01b0ng v\u1eabn c\u00f2n m\u1ed9t t\u00ecnh hu\u1ed1ng th\u1ee9 hai ng\u00e0y nay gi\u00fap ch\u00fang ta gi\u1ea3i quy\u1ebft v\u1ea5n \u0111\u1ec1 tri\u1ebft h\u1ecdc c\u0169 x\u01b0a \u0111\u00f3. Kh\u00f4ng ch\u1ec9 k\u1ef9 thu\u1eadt th\u00ed nghi\u1ec7m v\u00e0 ngh\u1ec7 thu\u1eadt \u0111\u1ec3 x\u00e2y d\u1ef1ng c\u00e1c t\u00f2a nh\u00e0 v\u1eadt l\u00fd-l\u00fd thuy\u1ebft \u0111\u00e3 \u0111\u1ea1t \u0111\u1ebfn m\u1ed9t tr\u00ecnh \u0111\u1ed9 ch\u01b0a t\u1eebng \u0111\u1ea1t t\u1edbi, m\u00e0 \u0111\u1ed1i t\u00e1c c\u1ee7a n\u00f3, c\u1ee5 th\u1ec3 l\u00e0 khoa h\u1ecdc logic, c\u0169ng \u0111\u00e3 \u0111\u1ea1t \u0111\u01b0\u1ee3c nh\u1eefng ti\u1ebfn b\u1ed9 \u0111\u00e1ng k\u1ec3. Ng\u00e0y nay, c\u00f3 m\u1ed9t ph\u01b0\u01a1ng ph\u00e1p chung \u0111\u1ec3 gi\u1ea3i quy\u1ebft l\u00fd thuy\u1ebft c\u00e1c c\u00e2u h\u1ecfi khoa h\u1ecdc t\u1ef1 nhi\u00ean; n\u00f3 gi\u00fap d\u1ec5 d\u00e0ng x\u00e1c \u0111\u1ecbnh v\u1ea5n \u0111\u1ec1 h\u01a1n trong m\u1ecdi tr\u01b0\u1eddng h\u1ee3p, v\u00e0 gi\u00fap chu\u1ea9n b\u1ecb \u0111\u01b0a ra gi\u1ea3i ph\u00e1p cho b\u00e0i to\u00e1n, c\u1ee5 th\u1ec3 \u0111\u00f3 l\u00e0 ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 (axiomatic method).<\/p>\n<p>\u00dd ngh\u0129a c\u1ee7a ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 (Axiomatik) \u0111\u01b0\u1ee3c nh\u1eafc \u0111\u1ebfn nhi\u1ec1u ng\u00e0y nay l\u00e0 g\u00ec? H\u00e3y xem, \u00fd t\u01b0\u1edfng c\u01a1 b\u1ea3n d\u1ef1a tr\u00ean th\u1ef1c t\u1ebf l\u00e0, ngay c\u1ea3 trong nhi\u1ec1u l\u0129nh v\u1ef1c tri \u200b\u200bth\u1ee9c s\u00e2u r\u1ed9ng, m\u1ed9t v\u00e0i m\u1ec7nh \u0111\u1ec1 &#8211; \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 ti\u00ean \u0111\u1ec1 &#8211; th\u01b0\u1eddng l\u00e0 \u0111\u1ee7 \u0111\u1ec3 x\u00e2y d\u1ef1ng to\u00e0n b\u1ed9 t\u00f2a nh\u00e0 c\u1ee7a l\u00fd thuy\u1ebft m\u1ed9t c\u00e1ch thu\u1ea7n t\u00fay logic. Nh\u01b0ng nh\u1eadn x\u00e9t n\u00e0y ch\u01b0a n\u00f3i h\u1ebft t\u1ea7m quan tr\u1ecdng c\u1ee7a n\u00f3. C\u00e1c v\u00ed d\u1ee5 c\u00f3 th\u1ec3 gi\u1ea3i th\u00edch t\u1ed1t nh\u1ea5t ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 cho ch\u00fang ta. V\u00ed d\u1ee5 l\u00e2u \u0111\u1eddi nh\u1ea5t v\u00e0 n\u1ed5i ti\u1ebfng nh\u1ea5t c\u1ee7a ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 l\u00e0 h\u00ecnh h\u1ecdc Euclid. Tuy nhi\u00ean, t\u00f4i mu\u1ed1n minh h\u1ecda ng\u1eafn g\u1ecdn ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 b\u1eb1ng m\u1ed9t v\u00ed d\u1ee5 r\u1ea5t r\u00f5 r\u00e0ng t\u1eeb sinh h\u1ecdc hi\u1ec7n \u0111\u1ea1i. [\u2026]<a href=\"#_ftn2\" name=\"_ftnref2\">[2]<\/a><\/p>\n<p>M\u1ed9t v\u00ed d\u1ee5 kh\u00e1c v\u1ec1 ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 trong m\u1ed9t l\u0129nh v\u1ef1c ho\u00e0n to\u00e0n kh\u00e1c l\u00e0 nh\u01b0 sau:<\/p>\n<p>Trong c\u00e1c ng\u00e0nh khoa h\u1ecdc l\u00fd thuy\u1ebft c\u1ee7a ch\u00fang ta, ch\u00fang ta \u0111\u00e3 quen v\u1edbi vi\u1ec7c \u00e1p d\u1ee5ng c\u00e1c qu\u00e1 tr\u00ecnh t\u01b0 duy h\u00ecnh th\u1ee9c v\u00e0 c\u00e1c ph\u01b0\u01a1ng ph\u00e1p tr\u1eebu t\u01b0\u1ee3ng. Ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1 thu\u1ed9c v\u1ec1 logic h\u1ecdc. Khi nghe \u0111\u1ebfn t\u1eeb logic, ng\u01b0\u1eddi ta ngh\u0129 \u0111\u1ebfn m\u1ed9t \u0111i\u1ec1u r\u1ea5t nh\u00e0m ch\u00e1n v\u00e0 kh\u00f3 kh\u0103n. Ng\u00e0y nay, khoa h\u1ecdc logic \u0111\u00e3 tr\u1edf n\u00ean d\u1ec5 hi\u1ec3u v\u00e0 r\u1ea5t th\u00fa v\u1ecb. Ch\u1eb3ng h\u1ea1n, ng\u01b0\u1eddi ta \u0111\u00e3 nh\u1eadn ra r\u1eb1ng c\u00e1c ph\u01b0\u01a1ng ph\u00e1p v\u00e0 kh\u00e1i ni\u1ec7m \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng trong cu\u1ed9c s\u1ed1ng h\u00e0ng ng\u00e0y \u0111\u00f2i h\u1ecfi m\u1ee9c \u0111\u1ed9 tr\u1eebu t\u01b0\u1ee3ng cao v\u00e0 ch\u1ec9 c\u00f3 th\u1ec3 hi\u1ec3u \u0111\u01b0\u1ee3c th\u00f4ng qua \u1ee9ng d\u1ee5ng kh\u00f4ng \u00fd th\u1ee9c c\u1ee7a c\u00e1c ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1. Ch\u1eb3ng h\u1ea1n, qu\u00e1 tr\u00ecnh t\u1ed5ng qu\u00e1t c\u1ee7a ph\u1ee7 \u0111\u1ecbnh (Negation) v\u00e0 \u0111\u1eb7c bi\u1ec7t kh\u00e1i ni\u1ec7m \u201cv\u00f4 c\u1ef1c\u201d (Unendlich). Li\u00ean quan \u0111\u1ebfn thu\u1eadt ng\u1eef &#8220;v\u00f4 c\u1ef1c&#8221;, ch\u00fang ta ph\u1ea3i th\u1ea5y r\u00f5 r\u1eb1ng &#8220;v\u00f4 c\u1ef1c&#8221; kh\u00f4ng c\u00f3 \u00fd ngh\u0129a tr\u1ef1c quan v\u00e0, n\u1ebfu kh\u00f4ng c\u00f3 nh\u1eefng nghi\u00ean c\u1ee9u th\u00eam, n\u00f3 s\u1ebd kh\u00f4ng c\u00f3 \u00fd ngh\u0129a g\u00ec c\u1ea3. B\u1edfi v\u00ec \u1edf kh\u1eafp m\u1ecdi n\u01a1i ch\u1ec9 c\u00f3 nh\u1eefng th\u1ee9 h\u1eefu h\u1ea1n. Kh\u00f4ng c\u00f3 t\u1ed1c \u0111\u1ed9 v\u00f4 h\u1ea1n v\u00e0 kh\u00f4ng c\u00f3 l\u1ef1c ho\u1eb7c hi\u1ec7u \u1ee9ng lan truy\u1ec1n nhanh v\u00f4 h\u1ea1n. Ngo\u00e0i ra, t\u00e1c \u0111\u1ed9ng (Wirkung) c\u00f3 b\u1ea3n ch\u1ea5t r\u1eddi r\u1ea1c v\u00e0 ch\u1ec9 t\u1ed3n t\u1ea1i \u1edf d\u1ea1ng l\u01b0\u1ee3ng t\u1eed. Kh\u00f4ng c\u00f3 g\u00ec li\u00ean t\u1ee5c c\u1ea3 \u0111\u1ec3 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c chia ra v\u00f4 t\u1eadn. Ngay c\u1ea3 \u00e1nh s\u00e1ng c\u0169ng c\u00f3 c\u1ea5u tr\u00fac nguy\u00ean t\u1eed, c\u0169ng nh\u01b0 \u0111\u1ed9 l\u1edbn c\u1ee7a t\u00e1c \u0111\u1ed9ng. T\u00f4i ch\u1eafc ch\u1eafn r\u1eb1ng ngay c\u1ea3 kh\u00f4ng gian c\u0169ng c\u00f3 ph\u1ea1m vi h\u1eefu h\u1ea1n, v\u00e0 m\u1ed9t ng\u00e0y n\u00e0o \u0111\u00f3 c\u00e1c nh\u00e0 thi\u00ean v\u0103n h\u1ecdc s\u1ebd c\u00f3 th\u1ec3 k\u1ec3 cho ch\u00fang ta bi\u1ebft kh\u00f4ng gian d\u00e0i, cao v\u00e0 r\u1ed9ng bao nhi\u00eau km. Ngay c\u1ea3 khi trong th\u1ef1c t\u1ebf th\u01b0\u1eddng c\u00f3 nh\u1eefng tr\u01b0\u1eddng h\u1ee3p s\u1ed1 l\u01b0\u1ee3ng r\u1ea5t l\u1edbn, v.d. kho\u1ea3ng c\u00e1ch c\u1ee7a c\u00e1c ng\u00f4i sao t\u00ednh b\u1eb1ng km, ho\u1eb7c s\u1ed1 l\u01b0\u1ee3ng c\u00e1c v\u00e1n c\u1edd c\u00f3 th\u1ec3 kh\u00e1c nhau v\u1ec1 c\u01a1 b\u1ea3n, th\u00ec t\u00ednh b\u1ea5t t\u1eadn v\u00e0 v\u00f4 t\u1eadn, nh\u1eefng kh\u00e1i ni\u1ec7m ph\u1ee7 \u0111\u1ecbnh c\u00e1c t\u00ecnh hu\u1ed1ng ph\u1ed5 bi\u1ebfn \u1edf kh\u1eafp m\u1ecdi n\u01a1i, l\u00e0 m\u1ed9t s\u1ef1 tr\u1eebu t\u01b0\u1ee3ng h\u00f3a kh\u1ee7ng khi\u1ebfp &#8211; ch\u1ec9 c\u00f3 th\u1ec3 th\u1ef1c hi\u1ec7n \u0111\u01b0\u1ee3c th\u00f4ng qua \u1ee9ng d\u1ee5ng c\u00f3 \u00fd th\u1ee9c ho\u1eb7c v\u00f4 th\u1ee9c c\u1ee7a ph\u01b0\u01a1ng ph\u00e1p ti\u00ean \u0111\u1ec1. Quan ni\u1ec7m v\u1ec1 c\u00e1i v\u00f4 h\u1ea1n n\u00e0y, c\u00e1i m\u00e0 t\u00f4i \u0111\u00e3 thi\u1ebft l\u1eadp th\u00f4ng qua c\u00e1c nghi\u00ean c\u1ee9u c\u1eb7n k\u1ebd, gi\u1ea3i quy\u1ebft m\u1ed9t lo\u1ea1t c\u00e2u h\u1ecfi c\u01a1 b\u1ea3n, \u0111\u1eb7c bi\u1ec7t qua \u0111\u00f3, nh\u1eefng ngh\u1ecbch l\u00fd c\u1ee7a Kant v\u1ec1 kh\u00f4ng gian v\u00e0 v\u1ec1 kh\u1ea3 n\u0103ng ph\u00e2n chia v\u00f4 h\u1ea1n \u0111\u00e3 tr\u1edf n\u00ean kh\u00f4ng c\u00f2n hi\u1ec7u l\u1ef1c, v\u00e0 nh\u1eefng kh\u00f3 kh\u0103n n\u1ea3y sinh theo \u0111\u00f3 c\u0169ng \u0111\u00e3 \u0111\u01b0\u1ee3c gi\u1ea3i quy\u1ebft.<\/p>\n<h3 style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-6321\" src=\"https:\/\/i0.wp.com\/rosetta.vn\/nguyenxuanxanh\/wp-content\/uploads\/sites\/6\/2021\/03\/picture-7.png?resize=74%2C48&#038;ssl=1\" alt=\"\" width=\"74\" height=\"48\" data-recalc-dims=\"1\" \/><\/h3>\n<h3 id=\"firstHeading\" class=\"firstHeading mw-first-heading\" style=\"text-align: center\"><span class=\"mw-page-title-main\" style=\"color: #000080\">Ngh\u1ecbch l\u00fd Hilbert c\u1ee7a Kh\u00e1ch s\u1ea1n L\u1edbn<\/span><\/h3>\n<p style=\"text-align: justify;padding-left: 40px\"><span style=\"color: #000080\">N\u00f3i v\u1ec1 v\u00f4 c\u1ef1c. H\u00e3y xem m\u1ed9t kh\u00e1ch s\u1ea1n l\u1edbn, v\u00e2ng, c\u00f3 v\u00f4 c\u1ef1c nh\u01b0ng \u0111\u1ebfm \u0111\u01b0\u1ee3c s\u1ed1 ph\u00f2ng. Gi\u1ea3 thi\u1ebft n\u00f3 \u0111ang \u0111\u1ea7y kh\u00e1ch. Ch\u1ee3t c\u00f3 m\u1ed9t v\u1ecb kh\u00e1ch m\u1edbi \u0111\u1ebfn xin thu\u00ea ph\u00f2ng. Ng\u01b0\u1eddi ch\u1ee7 n\u00f3i: Kh\u00f4ng c\u00f3 v\u1ea5n \u0111\u1ec1. \u00d4ng ta cho ng\u01b0\u1eddi \u0111ang \u1edf ph\u00f2ng 1 chuy\u1ec3n sang ph\u00f2ng 2, ng\u01b0\u1eddi \u1edf ph\u00f2ng 2 chuy\u1ec3n sang ph\u00f2ng 3, &#8230; ngh\u0129a l\u00e0 ng\u01b0\u1eddi \u1edf ph\u00f2ng s\u1ed1 n chuy\u1ec3n d\u00f9m qua ph\u00f2ng s\u1ed1 n+1. V\u1eady th\u00ec t\u1ea5t c\u1ea3 kh\u00e1ch \u0111\u1ec1u c\u00f3 ph\u00f2ng, kh\u00f4ng ai b\u1ecb b\u1ecf r\u01a1i c\u1ea3. Sau \u0111\u00f3 c\u00f3 m\u1ed9t ng\u01b0\u1eddi kh\u00e1c n\u1eefa \u0111\u1ebfn thu\u00ea ph\u00f2ng. \u00d4ng ch\u1ee7 n\u00f3i: c\u0169ng kh\u00f4ng c\u00f3 v\u1ea5n \u0111\u1ec1 chi. \u00d4ng cho d\u1eddi ph\u00f2ng y chang nh\u01b0 tr\u01b0\u1edbc, v\u00e0 m\u1ecdi vi\u1ec7c \u0111\u01b0\u1ee3c gi\u1ea3i quy\u1ebft. C\u1ee9 th\u1ebf, kh\u00e1ch s\u1ea1n tuy \u0111\u1ea7y nh\u01b0ng c\u00f3 th\u1ec3 nh\u1eadn th\u00eam m\u1ed9t s\u1ed1 h\u1eefu h\u1ea1n N kh\u00e1ch m\u1edbi m\u00e0 v\u1eabn ch\u1ee9a \u0111\u01b0\u1ee3c.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/f9\/David_Hilbert_1886.jpg\/190px-David_Hilbert_1886.jpg\" width=\"224\" height=\"311\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/6e\/David_Hilbert%2C_1907.jpg\/184px-David_Hilbert%2C_1907.jpg\" width=\"217\" height=\"310\" \/><\/p>\n<p style=\"text-align: center\">Hilbert n\u0103m 1886 v\u00e0 Hilbert n\u0103m 1907<\/p>\n<p style=\"text-align: center\">[3]<\/p>\n<p>N\u1ebfu b\u00e2y gi\u1edd chuy\u1ec3n sang ch\u00ednh v\u1ea5n \u0111\u1ec1 c\u1ee7a ch\u00fang ta, t\u1ef1 nhi\u00ean v\u00e0 t\u01b0 duy c\u00f3 li\u00ean h\u1ec7 v\u1edbi nhau nh\u01b0 th\u1ebf n\u00e0o, th\u00ec ch\u00fang ta h\u00e3y \u0111\u01b0a ra ba \u0111i\u1ec3m ch\u00ednh \u1edf \u0111\u00e2y. M\u1ed1i quan h\u1ec7 \u0111\u1ea7u ti\u00ean li\u00ean quan \u0111\u1ebfn v\u1ea5n \u0111\u1ec1 v\u1ec1 v\u00f4 c\u1ef1c v\u1eeba \u0111\u01b0\u1ee3c th\u1ea3o lu\u1eadn. Ch\u00fang ta \u0111\u00e3 th\u1ea5y: c\u00e1i v\u00f4 h\u1ea1n kh\u00f4ng \u0111\u01b0\u1ee3c th\u1ef1c hi\u1ec7n \u1edf b\u1ea5t c\u1ee9 n\u01a1i n\u00e0o; n\u00f3 kh\u00f4ng c\u00f3 trong t\u1ef1 nhi\u00ean v\u00e0 c\u0169ng kh\u00f4ng \u0111\u01b0\u1ee3c ph\u00e9p l\u00e0m c\u01a1 s\u1edf trong suy ngh\u0129 c\u1ee7a ch\u00fang ta m\u00e0 kh\u00f4ng c\u00f3 s\u1ef1 \u0111\u1ec1 ph\u00f2ng \u0111\u1eb7c bi\u1ec7t. \u1ede \u0111\u00e2y, t\u00f4i \u0111\u00e3 nh\u00ecn th\u1ea5y m\u1ed9t s\u1ef1 song song quan tr\u1ecdng gi\u1eefa t\u1ef1 nhi\u00ean v\u00e0 t\u01b0 duy, m\u1ed9t s\u1ef1 th\u1ed1ng nh\u1ea5t c\u01a1 b\u1ea3n gi\u1eefa kinh nghi\u1ec7m v\u00e0 l\u00fd thuy\u1ebft.<\/p>\n<p>Ch\u00fang ta nh\u1eadn th\u1ea5y m\u1ed9t s\u1ef1 song song kh\u00e1c (th\u1ee9 hai): suy ngh\u0129 c\u1ee7a ch\u00fang ta h\u01b0\u1edbng \u0111\u1ebfn s\u1ef1 th\u1ed1ng nh\u1ea5t v\u00e0 t\u00ecm c\u00e1ch x\u00e2y d\u1ef1ng s\u1ef1 th\u1ed1ng nh\u1ea5t; ch\u00fang ta quan s\u00e1t s\u1ef1 th\u1ed1ng nh\u1ea5t c\u1ee7a v\u1eadt li\u1ec7u (stuff) trong v\u1eadt ch\u1ea5t, v\u00e0 ch\u00fang ta nh\u1eadn th\u1ea5y s\u1ef1 th\u1ed1ng nh\u1ea5t c\u1ee7a c\u00e1c quy lu\u1eadt t\u1ef1 nhi\u00ean \u1edf m\u1ecdi n\u01a1i. Th\u1ef1c t\u1ebf, t\u1ef1 nhi\u00ean r\u1ea5t h\u1ed7 tr\u1ee3 nghi\u00ean c\u1ee9u c\u1ee7a ch\u00fang ta, nh\u01b0 th\u1ec3 n\u00f3 s\u1eb5n s\u00e0ng ti\u1ebft l\u1ed9 b\u00ed m\u1eadt c\u1ee7a m\u00ecnh. S\u1ef1 ph\u00e2n b\u1ed1 th\u01b0a th\u1edbt c\u1ee7a kh\u1ed1i l\u01b0\u1ee3ng trong kh\u00f4ng gian thi\u00ean th\u1ec3 cho ph\u00e9p kh\u00e1m ph\u00e1 v\u00e0 x\u00e1c nh\u1eadn ch\u00ednh x\u00e1c h\u01a1n \u0111\u1ecbnh lu\u1eadt Newton. B\u1ea5t ch\u1ea5p t\u1ed1c \u0111\u1ed9 \u00e1nh s\u00e1ng cao, Michelson \u0111\u00e3 c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh m\u1ed9t c\u00e1ch ch\u1eafc ch\u1eafn r\u1eb1ng \u0111\u1ecbnh lu\u1eadt c\u1ed9ng c\u1ee7a v\u1eadn t\u1ed1c l\u00e0 kh\u00f4ng \u0111\u00fang, b\u1edfi v\u00ec tr\u00e1i \u0111\u1ea5t c\u1ee7a ch\u00fang ta v\u1eabn quay quanh m\u1eb7t tr\u1eddi \u0111\u1ee7 nhanh \u0111\u1ec3 l\u00e0m \u0111i\u1ec1u n\u00e0y. Sao Th\u1ee7y \u0111ang l\u00e0m vui l\u00f2ng ch\u00fang ta b\u1eb1ng c\u00e1ch th\u1ef1c hi\u1ec7n chuy\u1ec3n \u0111\u1ed9ng \u0111i\u1ec3m c\u1eadn nh\u1eadt \u0111\u1ec3 ch\u00fang ta qua \u0111\u00f3 c\u00f3 th\u1ec3 ki\u1ec3m tra l\u00fd thuy\u1ebft c\u1ee7a Einstein. V\u00e0 tia s\u00e1ng c\u1ee7a ng\u00f4i sao c\u1ed1 \u0111\u1ecbnh \u0111i qua m\u1eb7t tr\u1eddi sao cho s\u1ef1 l\u1ec7ch c\u1ee7a n\u00f3 \u0111\u01b0\u1ee3c quan s\u00e1t th\u1ea5y.<\/p>\n<p>Nh\u01b0ng \u0111\u1eadp v\u00e0o m\u1eaft h\u01a1n n\u1eefa l\u00e0 m\u1ed9t hi\u1ec7n t\u01b0\u1ee3ng (th\u1ee9 ba), m\u00e0 ch\u00fang t\u00f4i theo m\u1ed9t ngh\u0129a kh\u00e1c v\u1edbi Leibniz, g\u1ecdi l\u00e0 s\u1ef1 <em>h\u00e0i h\u00f2a ti\u1ec1n \u0111\u1ecbnh<\/em> (pre-established harmony), th\u1ef1c t\u1ebf l\u00e0 m\u1ed9t s\u1ef1 hi\u1ec7n th\u00e2n v\u00e0 hi\u1ec7n th\u1ef1c h\u00f3a c\u00e1c \u00fd t\u01b0\u1edfng to\u00e1n h\u1ecdc. C\u00e1c v\u00ed d\u1ee5 c\u0169 h\u01a1n v\u1ec1 \u0111i\u1ec1u n\u00e0y l\u00e0 c\u00e1c ti\u1ebft di\u1ec7n h\u00ecnh n\u00f3n, \u0111\u00e3 \u0111\u01b0\u1ee3c nghi\u00ean c\u1ee9u t\u1eeb r\u1ea5t l\u00e2u tr\u01b0\u1edbc khi ng\u01b0\u1eddi ta ng\u1edd r\u1eb1ng c\u00e1c h\u00e0nh tinh c\u1ee7a ch\u00fang ta ho\u1eb7c th\u1eadm ch\u00ed c\u00e1c electron chuy\u1ec3n \u0111\u1ed9ng tr\u00ean nh\u1eefng qu\u1ef9 \u0111\u1ea1o nh\u01b0 v\u1eady. Nh\u01b0ng v\u00ed d\u1ee5 v\u0129 \u0111\u1ea1i nh\u1ea5t v\u00e0 tuy\u1ec7t v\u1eddi nh\u1ea5t v\u1ec1 s\u1ef1 h\u00e0i h\u00f2a ti\u1ec1n \u0111\u1ecbnh l\u00e0 thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i n\u1ed5i ti\u1ebfng c\u1ee7a Einstein. \u1ede \u0111\u00e2y, ch\u1ec9 b\u1eb1ng y\u00eau c\u1ea7u t\u1ed5ng qu\u00e1t v\u1ec1 t\u00ednh b\u1ea5t bi\u1ebfn k\u1ebft h\u1ee3p v\u1edbi nguy\u00ean l\u00fd v\u1ec1 t\u00ednh \u0111\u01a1n gi\u1ea3n l\u1edbn nh\u1ea5t m\u00e0 c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh vi ph\u00e2n cho th\u1ebf n\u0103ng h\u1ea5p d\u1eabn \u0111\u01b0\u1ee3c thi\u1ebft l\u1eadp r\u00f5 r\u00e0ng v\u1ec1 m\u1eb7t to\u00e1n h\u1ecdc; v\u00e0 s\u1ef1 thi\u1ebft l\u1eadp n\u00e0y s\u1ebd b\u1ea5t kh\u1ea3 thi n\u1ebfu kh\u00f4ng c\u00f3 nh\u1eefng nghi\u00ean c\u1ee9u to\u00e1n h\u1ecdc s\u00e2u xa v\u00e0 kh\u00f3 kh\u0103n c\u1ee7a Riemann \u0111\u00e3 t\u1ed3n t\u1ea1i t\u1eeb r\u1ea5t l\u00e2u tr\u01b0\u1edbc \u0111\u00f3. Trong th\u1eddi gian g\u1ea7n \u0111\u00e2y, ng\u00e0y c\u00e0ng c\u00f3 nhi\u1ec1u tr\u01b0\u1eddng h\u1ee3p trong \u0111\u00f3 c\u00e1c l\u00fd thuy\u1ebft to\u00e1n h\u1ecdc n\u1eb1m t\u1ea1i t\u00e2m \u0111i\u1ec3m c\u1ee7a m\u1ed1i quan t\u00e2m trong to\u00e1n h\u1ecdc \u0111\u1ed3ng th\u1eddi l\u1ea1i l\u00e0 nh\u1eefng l\u00fd thuy\u1ebft \u0111\u01b0\u1ee3c c\u1ea7n \u0111\u1ebfn trong v\u1eadt l\u00fd. T\u00f4i \u0111\u00e3 ph\u00e1t tri\u1ec3n l\u00fd thuy\u1ebft v\u1ec1 v\u00f4 s\u1ed1 bi\u1ebfn s\u1ed1 v\u00ec l\u1ee3i \u00edch to\u00e1n h\u1ecdc thu\u1ea7n t\u00fay, th\u1eadm ch\u00ed \u00e1p d\u1ee5ng thu\u1eadt ng\u1eef ph\u00e2n t\u00edch quang ph\u1ed5, m\u00e0 kh\u00f4ng ng\u1edd r\u1eb1ng nh\u1eefng \u0111i\u1ec1u n\u00e0y sau n\u00e0y s\u1ebd \u0111\u01b0\u1ee3c nh\u1eadn ra trong quang ph\u1ed5 th\u1ef1c c\u1ee7a v\u1eadt l\u00fd.<\/p>\n<p>Ch\u00fang ta ch\u1ec9 c\u00f3 th\u1ec3 hi\u1ec3u \u0111\u01b0\u1ee3c s\u1ef1 th\u1ed1ng nh\u1ea5t n\u00e0y gi\u1eefa t\u1ef1 nhi\u00ean v\u00e0 t\u01b0 duy, gi\u1eefa th\u1ef1c nghi\u1ec7m v\u00e0 l\u00fd thuy\u1ebft, n\u1ebfu ch\u00fang ta xem x\u00e9t y\u1ebfu t\u1ed1 h\u00ecnh th\u1ee9c v\u00e0 c\u01a1 ch\u1ebf li\u00ean quan tr\u00ean c\u1ea3 hai m\u1eb7t c\u1ee7a t\u1ef1 nhi\u00ean v\u00e0 gi\u00e1c t\u00ednh (Verstand, understanding) c\u1ee7a ch\u00fang ta. Qu\u00e1 tr\u00ecnh to\u00e1n h\u1ecdc c\u1ee7a s\u1ef1 lo\u1ea1i b\u1ecf d\u01b0\u1eddng nh\u01b0 cung c\u1ea5p c\u00e1c \u0111i\u1ec3m d\u1eebng v\u00e0 tr\u1ea1m ngh\u1ec9 n\u01a1i c\u00e1c v\u1eadt th\u1ec3 trong th\u1ebf gi\u1edbi th\u1ef1c c\u0169ng nh\u01b0 nh\u1eefng \u00fd t\u01b0\u1edfng trong th\u1ebf gi\u1edbi tinh th\u1ea7n tr\u00fa ng\u1ee5, v\u00e0 do \u0111\u00f3 ch\u00fang cho ph\u00e9p ch\u00fang ta xem x\u00e9t v\u00e0 so s\u00e1nh.<\/p>\n<p>Tuy nhi\u00ean, ngay c\u1ea3 s\u1ef1 h\u00e0i h\u00f2a ti\u1ec1n \u0111\u1ecbnh n\u00e0y v\u1eabn ch\u01b0a n\u00f3i h\u1ebft c\u00e1c m\u1ed1i quan h\u1ec7 gi\u1eefa t\u1ef1 nhi\u00ean v\u00e0 t\u01b0 duy, v\u00e0 ch\u01b0a l\u00e0m nh\u1eefng b\u00ed \u1ea9n s\u00e2u s\u1eafc nh\u1ea5t v\u1ec1 v\u1ea5n \u0111\u1ec1 c\u1ee7a ch\u00fang ta l\u1ed9 r\u00f5. \u0110\u1ec3 hi\u1ec3u \u0111\u01b0\u1ee3c \u0111i\u1ec1u n\u00e0y, ch\u00fang ta h\u00e3y xem x\u00e9t to\u00e0n b\u1ed9 ph\u1ee9c h\u1ee3p tri \u200b\u200b\u200b\u200bth\u1ee9c v\u1eadt l\u00fd-thi\u00ean v\u0103n. L\u00fac \u0111\u00f3, ch\u00fang ta nh\u1eadn th\u1ea5y trong khoa h\u1ecdc ng\u00e0y nay m\u1ed9t quan \u0111i\u1ec3m v\u01b0\u1ee3t xa c\u00e1c c\u00e1ch \u0111\u1eb7t v\u1ea5n \u0111\u1ec1 v\u00e0 m\u1ee5c ti\u00eau c\u0169 tr\u01b0\u1edbc \u0111\u00e2y c\u1ee7a khoa h\u1ecdc c\u1ee7a ch\u00fang ta: \u0111\u00f3 l\u00e0 th\u1ef1c t\u1ebf, r\u1eb1ng khoa h\u1ecdc ng\u00e0y nay, kh\u00f4ng ch\u1ec9 theo ngh\u0129a c\u01a1 h\u1ecdc c\u1ed5 \u0111i\u1ec3n t\u1eeb d\u1eef li\u1ec7u c\u1ee7a hi\u1ec7n t\u1ea1i, d\u1ea1y cho ch\u00fang ta x\u00e1c \u0111\u1ecbnh tr\u01b0\u1edbc c\u00e1c chuy\u1ec3n \u0111\u1ed9ng trong t\u01b0\u01a1ng lai v\u00e0 c\u00e1c hi\u1ec7n t\u01b0\u1ee3ng \u0111\u01b0\u1ee3c mong \u0111\u1ee3i, nh\u01b0ng n\u00f3 c\u0169ng c\u00f2n ch\u1ec9 ra r\u1eb1ng c\u00e1c tr\u1ea1ng th\u00e1i th\u1ef1c t\u1ebf hi\u1ec7n t\u1ea1i c\u1ee7a v\u1eadt ch\u1ea5t tr\u00ean tr\u00e1i \u0111\u1ea5t v\u00e0 trong v\u0169 tr\u1ee5 kh\u00f4ng ph\u1ea3i l\u00e0 ng\u1eabu nhi\u00ean hay t\u00f9y ti\u1ec7n, m\u00e0 tu\u00e2n th\u1ee7 c\u00e1c quy lu\u1eadt v\u1eadt l\u00fd.<\/p>\n<p>Nh\u1eefng b\u1eb1ng ch\u1ee9ng quan tr\u1ecdng nh\u1ea5t cho \u0111i\u1ec1u n\u00e0y l\u00e0 c\u00e1c m\u00f4 h\u00ecnh nguy\u00ean t\u1eed c\u1ee7a Bohr, c\u1ea5u tr\u00fac c\u1ee7a th\u1ebf gi\u1edbi c\u00e1c v\u00ec sao, v\u00e0 cu\u1ed1i c\u00f9ng l\u00e0 to\u00e0n b\u1ed9 l\u1ecbch s\u1eed ph\u00e1t tri\u1ec3n c\u1ee7a s\u1ef1 s\u1ed1ng h\u1eefu c\u01a1. Vi\u1ec7c theo \u0111u\u1ed5i c\u00e1c ph\u01b0\u01a1ng ph\u00e1p n\u00e0y d\u01b0\u1eddng nh\u01b0 th\u1ef1c s\u1ef1 ph\u1ea3i d\u1eabn \u0111\u1ebfn m\u1ed9t h\u1ec7 th\u1ed1ng c\u00e1c quy lu\u1eadt t\u1ef1 nhi\u00ean ph\u00f9 h\u1ee3p v\u1edbi th\u1ef1c t\u1ebf trong t\u1ed3ng th\u1ec3 c\u1ee7a ch\u00fang, v\u00e0 sau \u0111\u00f3 t\u1ea5t c\u1ea3 nh\u1eefng g\u00ec th\u1ef1c s\u1ef1 c\u1ea7n l\u00e0 t\u01b0 duy, ngh\u0129a l\u00e0 suy lu\u1eadn c\u00f3 t\u00ednh kh\u00e1i ni\u1ec7m \u0111\u1ec3 \u0111\u1ea1t \u0111\u01b0\u1ee3c t\u1ea5t c\u1ea3 c\u00e1c tri \u200b\u200b\u200b\u200bth\u1ee9c v\u1eadt l\u00fd; l\u00fac \u0111\u00f3 Hegel \u0111\u00e1ng l\u1ebd s\u1ebd \u0111\u00fang khi kh\u1eb3ng \u0111\u1ecbnh r\u1eb1ng t\u1ea5t c\u1ea3 c\u00e1c s\u1ef1 ki\u1ec7n t\u1ef1 nhi\u00ean \u0111\u1ec1u c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c suy ra t\u1eeb c\u00e1c kh\u00e1i ni\u1ec7m. Nh\u01b0ng k\u1ebft lu\u1eadn n\u00e0y l\u00e0 kh\u00f4ng ch\u00ednh x\u00e1c. B\u1edfi v\u00ec ngu\u1ed3n g\u1ed1c c\u1ee7a c\u00e1c \u0111\u1ecbnh lu\u1eadt th\u1ebf gi\u1edbi th\u00ec th\u1ebf n\u00e0o? Ch\u00fang ta suy ch\u00fang t\u1eeb \u0111\u00e2u? V\u00e0 ai d\u1ea1y cho ch\u00fang ta, r\u1eb1ng ch\u00fang ph\u00f9 h\u1ee3p v\u1edbi th\u1ef1c t\u1ea1i? C\u00e2u tr\u1ea3 l\u1eddi l\u00e0 ch\u1ec9 c\u00f3 kinh nghi\u1ec7m m\u1edbi gi\u00fap ch\u00fang ta c\u00f3 \u0111\u01b0\u1ee3c nh\u1eefng th\u1ee9 \u0111\u00f3. Ng\u01b0\u1ee3c v\u1edbi Hegel, ch\u00fang ta nh\u1eadn th\u1ee9c r\u1eb1ng c\u00e1c quy lu\u1eadt c\u1ee7a th\u1ebf gi\u1edbi kh\u00f4ng th\u1ec3 \u0111\u1ea1t \u0111\u01b0\u1ee3c b\u1eb1ng c\u00e1ch n\u00e0o kh\u00e1c ngo\u00e0i t\u1eeb kinh nghi\u1ec7m. C\u00f3 th\u1ec3 nhi\u1ec1u quan \u0111i\u1ec3m suy \u0111o\u00e1n \u0111a d\u1ea1ng c\u00f9ng t\u00e1c \u0111\u1ed9ng \u0111\u1ebfn vi\u1ec7c x\u00e2y d\u1ef1ng khung kh\u00e1i ni\u1ec7m v\u1eadt l\u00fd: Li\u1ec7u c\u00e1c \u0111\u1ecbnh lu\u1eadt \u0111\u00e3 \u0111\u01b0\u1ee3c thi\u1ebft l\u1eadp v\u00e0 khung logic c\u1ee7a c\u00e1c kh\u00e1i ni\u1ec7m \u0111\u01b0\u1ee3c x\u00e2y d\u1ef1ng t\u1eeb ch\u00fang c\u00f3 \u0111\u00fang hay kh\u00f4ng, \u0111i\u1ec1u \u0111\u00f3 ch\u1ec9 c\u00f3 kinh nghi\u1ec7m m\u1edbi c\u00f3 th\u1ec3 quy\u1ebft \u0111\u1ecbnh. \u0110\u00f4i khi m\u1ed9t \u00fd t\u01b0\u1edfng c\u00f3 ngu\u1ed3n g\u1ed1c ban \u0111\u1ea7u t\u1eeb t\u01b0 duy thu\u1ea7n t\u00fay, ch\u1eb3ng h\u1ea1n nh\u01b0 \u00fd t\u01b0\u1edfng v\u1ec1 thuy\u1ebft nguy\u00ean t\u1eed c\u1ee7a Democrit, trong khi s\u1ef1 t\u1ed3n t\u1ea1i c\u1ee7a nguy\u00ean t\u1eed ch\u1ec9 \u0111\u01b0\u1ee3c ch\u1ee9ng minh b\u1edfi v\u1eadt l\u00fd th\u1ef1c nghi\u1ec7m hai thi\u00ean ni\u00ean k\u1ef7 sau. \u0110\u00f4i khi kinh nghi\u1ec7m \u0111i tr\u01b0\u1edbc v\u00e0 \u00e1p \u0111\u1eb7t quan \u0111i\u1ec3m suy \u0111o\u00e1n l\u00ean tinh th\u1ea7n. V\u00ec v\u1eady, ch\u00fang ta c\u1ea3m \u01a1n s\u1ef1 th\u00fac \u0111\u1ea9y m\u1ea1nh m\u1ebd c\u1ee7a th\u00ed nghi\u1ec7m c\u1ee7a Michelson m\u00e0 v\u00ec th\u1ebf \u0111\u1ecbnh ki\u1ebfn \u200b\u200b\u200b\u200bb\u00e1m r\u1ec5 v\u1eefng ch\u1eafc v\u1ec1 th\u1eddi gian tuy\u1ec7t \u0111\u1ed1i c\u00f3 th\u1ec3 b\u1ecb x\u00f3a s\u1ea1ch v\u00e0 cu\u1ed1i c\u00f9ng \u00fd t\u01b0\u1edfng v\u1ec1 thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i r\u1ed9ng c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c Einstein t\u00ecm ra.<a href=\"#_ftn3\" name=\"_ftnref3\">[3]<\/a><\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/i0.wp.com\/cdn.britannica.com\/08\/3008-004-F87BC031\/Godfrey-Kneller-Isaac-Newton-portrait-1689.jpg?resize=256%2C314&#038;ssl=1\" alt=\"Isaac Newton | Biography, Facts, Discoveries, Laws, &amp; Inventions | Britannica\" width=\"256\" height=\"314\" data-recalc-dims=\"1\" \/>\u00a0<img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/f2\/Kant_gemaelde_3.jpg\/220px-Kant_gemaelde_3.jpg\" alt=\"Kant gemaelde 3.jpg\" width=\"237\" height=\"309\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/i0.wp.com\/cdn.britannica.com\/09\/75509-050-86D8CBBF\/Albert-Einstein.jpg?resize=210%2C312&#038;ssl=1\" alt=\"Albert Einstein | Biography, Education, Discoveries, &amp; Facts | Britannica\" width=\"210\" height=\"312\" data-recalc-dims=\"1\" \/><\/p>\n<p style=\"text-align: center\">Isaac Newton, Immanual Kant v\u00e0 Albert Einstein<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/i0.wp.com\/e.khoahoc.tv\/photos\/image\/2021\/04\/07\/Carl-Gauss-1.jpg?resize=248%2C279&#038;ssl=1\" alt=\"Gau\u00df c\u0169ng l\u00e0 m\u1ed9t ng\u01b0\u1eddi r\u1ea5t c\u00e1 t\u00ednh v\u00e0 th\u00edch t\u1ef1 m\u00ecnh quy\u1ebft \u0111\u1ecbnh h\u01b0\u1edbng \u0111i c\u1ee7a m\u00ecnh.\" width=\"248\" height=\"279\" data-recalc-dims=\"1\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/82\/Georg_Friedrich_Bernhard_Riemann.jpeg\/250px-Georg_Friedrich_Bernhard_Riemann.jpeg\" alt=\"Georg Friedrich Bernhard Riemann.jpeg\" width=\"256\" height=\"280\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c7\/Hermann_von_Helmholtz.jpg\/220px-Hermann_von_Helmholtz.jpg\" alt=\"Hermann von Helmholtz.jpg\" width=\"215\" height=\"279\" \/><\/p>\n<p style=\"text-align: center\">Carl Friedrich Gauss, Bernhard Riemann v\u00e0 Hermann von Helmholtz<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-6321\" src=\"https:\/\/i0.wp.com\/rosetta.vn\/nguyenxuanxanh\/wp-content\/uploads\/sites\/6\/2021\/03\/picture-7.png?resize=74%2C48&#038;ssl=1\" alt=\"\" width=\"74\" height=\"48\" data-recalc-dims=\"1\" \/><\/p>\n<p style=\"text-align: center\"><strong><span style=\"color: #000080\">M\u1ed1i li\u00ean h\u1ec7 m\u1eadt thi\u1ebft \u0111\u00e1ng ng\u1ea1c nhi\u00ean g<\/span><span style=\"color: #000080\">i\u1eefa v\u1eadt l\u00fd v\u00e0 to\u00e1n h\u1ecdc<\/span><\/strong><\/p>\n<p style=\"padding-left: 40px\"><span style=\"color: #000080\">Eugene Wigner, nh\u00e0 v\u1eadt l\u00fd gi\u1ea3i Nobel n\u0103m 1963, \u0111\u00e3 b\u00e0y t\u1ecf s\u1ef1 kinh ng\u1ea1c v\u1ec1 m\u1ed1i li\u00ean h\u1ec7 n\u00e0y trong <em>B\u00e0i gi\u1ea3ng Richard Courant<\/em> t\u1ea1i \u0110\u1ea1i h\u1ecdc New York ng\u00e0y 11, th\u00e1ng 5, 1959, cho r\u1eb1ng m\u1ed1i li\u00ean h\u1ec7 m\u1eadt thi\u1ebft n\u00e0y l\u00e0 unreasonable, kh\u00f4ng h\u1ee3p l\u00fd, r\u1eb1ng &#8220;t\u00ednh h\u1eefu \u00edch to l\u1edbn c\u1ee7a to\u00e1n h\u1ecdc trong khoa h\u1ecdc t\u1ef1 nhi\u00ean l\u00e0 m\u1ed9t c\u00e1i g\u00ec \u0111\u00f3\u00a0mang t\u00ednh b\u00ed \u1ea9n v\u00e0 kh\u00f4ng c\u00f3 l\u1eddi gi\u1ea3i th\u00edch h\u1ee3p l\u00fd cho n\u00f3&#8221;. Wigner vi\u1ebft:<\/span><\/p>\n<p style=\"padding-left: 80px\"><span style=\"color: #000080\">Ph\u00e9p l\u1ea1 v\u1ec1 s\u1ef1 ph\u00f9 h\u1ee3p (appropriateness) c\u1ee7a ng\u00f4n ng\u1eef to\u00e1n h\u1ecdc \u0111\u1ec3 x\u00e2y d\u1ef1ng c\u00e1c \u0111\u1ecbnh lu\u1eadt v\u1eadt l\u00fd l\u00e0 m\u1ed9t m\u00f3n qu\u00e0 tuy\u1ec7t v\u1eddi m\u00e0 ch\u00fang ta kh\u00f4ng hi\u1ec3u v\u00e0 c\u0169ng kh\u00f4ng x\u1ee9ng \u0111\u00e1ng. Ch\u00fang ta n\u00ean bi\u1ebft \u01a1n \u0111i\u1ec1u \u0111\u00f3 v\u00e0 hy v\u1ecdng r\u1eb1ng n\u00f3 s\u1ebd v\u1eabn c\u00f3 hi\u1ec7u l\u1ef1c trong nghi\u00ean c\u1ee9u t\u01b0\u01a1ng lai v\u00e0 r\u1eb1ng n\u00f3 s\u1ebd m\u1edf r\u1ed9ng, d\u00f9 t\u1ed1t hay x\u1ea5u, v\u1edbi ni\u1ec1m vui, m\u1eb7c d\u00f9 c\u0169ng c\u00f3 th\u1ec3 v\u1edbi s\u1ef1 kinh ng\u1ea1c cho ch\u00fang ta, \u0111\u1ebfn nh\u1eefng ng\u00e0nh h\u1ecdc r\u1ed9ng l\u1edbn kh\u00e1c.<\/span><\/p>\n<p style=\"padding-left: 120px;text-align: right\"><span style=\"color: #000080\">Eugene Wigner<em>, The unreasonable effectiveness of mathematics in the natural sciences <\/em><\/span><\/p>\n<p style=\"padding-left: 40px;text-align: justify\"><span style=\"color: #000080\">Tuy nhi\u00ean c\u0169ng c\u1ea7n b\u1ed5 sung th\u00eam nh\u1eadn \u0111\u1ecbnh c\u1ee7a Einstein. Trong b\u00e0i b\u00e1o c\u00e1o tr\u01b0\u1edbc H\u00e0n L\u00e2m vi\u1ec7n Ph\u1ed5 c\u00f3 t\u00ean <em>H\u00ecnh h\u1ecdc v\u00e0 Kinh nghi\u1ec7m<\/em> (Geometrie und Erfahrung, 1921), \u00f4ng n\u00f3i, nh\u01b0 m\u1ed9t s\u1ef1 \u0111\u00fac k\u1ebft t\u1eeb nh\u1eefng nghi\u00ean c\u1ee9u c\u00f3 t\u00ednh c\u00e1ch m\u1ea1ng c\u1ee7a \u00f4ng trong thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i: <em>Bao l\u00e2u c\u00e1c \u0111\u1ecbnh l\u00fd to\u00e1n h\u1ecdc n\u00f3i v\u1ec1 th\u1ef1c t\u1ea1i, th\u00ec ch\u00fang kh\u00f4ng ch\u1eafc ch\u1eafn, v\u00e0 bao l\u00e2u ch\u00fang ch\u1eafc ch\u1eafn, th\u00ec ch\u00fang kh\u00f4ng n\u00f3i v\u1ec1 th\u1ef1c t\u1ea1i<\/em>. \u0110\u00f3 l\u00e0 c\u00e2u tr\u1ea3 l\u1eddi ng\u1eafn g\u1ecdn c\u1ee7a Einstein cho \u0111i\u1ec1u b\u1ea5t \u1ed5n sau \u0111\u00e2y: &#8220;L\u00e0m sao to\u00e1n h\u1ecdc, v\u1ed1n l\u00e0 m\u1ed9t s\u1ea3n ph\u1ea9m c\u1ee7a t\u01b0 duy con ng\u01b0\u1eddi v\u00e0 \u0111\u1ed9c l\u1eadp v\u1edbi m\u1ecdi kinh nghi\u1ec7m, l\u1ea1i c\u00f3 th\u1ec3 h\u00f9 h\u1ee3p v\u1edbi c\u00e1c \u0111\u1ed1i t\u01b0\u1ee3ng c\u1ee7a th\u1ef1c t\u1ea1i m\u1ed9t c\u00e1ch tuy\u1ec7t h\u1ea3o nh\u01b0 th\u1ebf. C\u00f3 th\u1eadt l\u00fd tr\u00ed con ng\u01b0\u1eddi c\u00f3 th\u1ec3 l\u00fd gi\u1ea3i h\u1ebft c\u00e1c t\u00ednh ch\u1ea5t c\u1ee7a s\u1ef1 v\u1eadt hi\u1ec7n th\u1ef1c ch\u1ec9 b\u1eb1ng t\u01b0 duy thu\u1ea7n t\u00fay th\u00f4i m\u00e0 kh\u00f4ng c\u1ea7n \u0111\u1ebfn kinh nghi\u1ec7m hay sao?&#8221; (Xem b\u00e0i <em>H\u00ecnh h\u1ecdc v\u00e0 Kinh nghi\u1ec7m<\/em> trong s\u00e1ch <em>Thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i h\u1eb9p v\u00e0 r\u1ed9ng<\/em> c\u1ee7a Einstein, tr. 189 tr\u1edf \u0111i). T\u00ednh ch\u1ea5t ti\u00ean nghi\u1ec7m, <em>a priory<\/em>, c\u1ee7a m\u1ed9t s\u1ed1 quan \u0111i\u1ec3m tri\u1ebft h\u1ecdc, \u0111\u00e3 tr\u1edf n\u00ean h\u1eadu nghi\u1ec7m, <em>a posteriori<\/em>, m\u1edbi th\u00edch h\u1ee3p v\u1edbi th\u1ef1c t\u1ea1i, nh\u01b0 kh\u00f4ng gian ch\u00fang ta s\u1ed1ng kh\u00f4ng ph\u1ea3i l\u00e0 Euclid m\u00e0 phi-Euclid, v\u00e0 th\u1eddi gian c\u0169ng kh\u00f4ng ph\u1ea3i l\u00e0 tuy\u1ec7t \u0111\u1ed1i, m\u00e0 t\u01b0\u01a1ng \u0111\u1ed1i.\u00a0<\/span><\/p>\n<p style=\"text-align: center\">H\u1ebft ph\u1ea7n th\u00eam<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[4]<\/p>\n<p>Tuy nhi\u00ean, ai mu\u1ed1n ph\u1ee7 nh\u1eadn r\u1eb1ng c\u00e1c quy lu\u1eadt c\u1ee7a th\u1ebf gi\u1edbi xu\u1ea5t ph\u00e1t t\u1eeb kinh nghi\u1ec7m \u0111\u1ec1u ph\u1ea3i kh\u1eb3ng \u0111\u1ecbnh r\u1eb1ng c\u00f3 m\u1ed9t ngu\u1ed3n tri th\u1ee9c th\u1ee9 ba b\u00ean c\u1ea1nh suy lu\u1eadn v\u00e0 kinh nghi\u1ec7m.<a href=\"#_ftn4\" name=\"_ftnref4\">[4]<\/a><\/p>\n<p>Tr\u00ean th\u1ef1c t\u1ebf, c\u00e1c tri\u1ebft gia &#8211; v\u00e0 Kant l\u00e0 \u0111\u1ea1i di\u1ec7n kinh \u0111i\u1ec3n c\u1ee7a quan \u0111i\u1ec3m n\u00e0y &#8211; \u0111\u00e3 kh\u1eb3ng \u0111\u1ecbnh r\u1eb1ng, ngo\u00e0i logic v\u00e0 kinh nghi\u1ec7m, ch\u00fang ta v\u1eabn c\u00f3 m\u1ed9t lo\u1ea1i tri th\u1ee9c ti\u00ean nghi\u1ec7m (a priori)<a href=\"#_ftn5\" name=\"_ftnref5\">[5]<\/a> n\u00e0o \u0111\u00f3 v\u1ec1 th\u1ef1c t\u1ea1i. Th\u1eadt s\u1ef1, t\u00f4i th\u1eeba nh\u1eadn, m\u1ed9t s\u1ed1 nh\u1eadn th\u1ee9c ti\u00ean nghi\u1ec7m nh\u1ea5t \u0111\u1ecbnh l\u00e0 c\u1ea7n thi\u1ebft cho vi\u1ec7c x\u00e2y d\u1ef1ng c\u00e1c khung l\u00fd thuy\u1ebft v\u00e0 r\u1eb1ng s\u1ef1 ph\u00e1t tri\u1ec3n tri \u200b\u200bth\u1ee9c c\u1ee7a ch\u00fang ta lu\u00f4n d\u1ef1a tr\u00ean nh\u1eefng hi\u1ec3u bi\u1ebft nh\u01b0 v\u1eady. T\u00f4i tin r\u1eb1ng ki\u1ebfn \u200b\u200bth\u1ee9c to\u00e1n h\u1ecdc cu\u1ed1i c\u00f9ng c\u0169ng d\u1ef1a tr\u00ean m\u1ed9t lo\u1ea1i nh\u1eadn th\u1ee9c tr\u1ef1c quan nh\u01b0 th\u1ebf. V\u00e0 r\u1eb1ng, th\u1eadm ch\u00ed \u0111\u1ec3 x\u00e2y d\u1ef1ng l\u00fd thuy\u1ebft s\u1ed1 ch\u00fang ta a priori c\u0169ng c\u1ea7n m\u1ed9t quan ni\u1ec7m tr\u1ef1c quan nh\u1ea5t \u0111\u1ecbnh. Do \u0111\u00f3, \u00fd t\u01b0\u1edfng c\u01a1 b\u1ea3n chung nh\u1ea5t c\u1ee7a nh\u1eadn th\u1ee9c lu\u1eadn c\u1ee7a Kant v\u1eabn gi\u1eef \u0111\u01b0\u1ee3c t\u1ea7m quan tr\u1ecdng c\u1ee7a n\u00f3: c\u1ee5 th\u1ec3, v\u1ea5n \u0111\u1ec1 tri\u1ebft h\u1ecdc l\u00e0 l\u00e0m sao x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c nh\u1eadn th\u1ee9c tr\u1ef1c quan \u0111\u00f3 m\u1ed9t c\u00e1ch ti\u00ean nghi\u1ec7m, v\u00e0 qua \u0111\u00f3 nghi\u00ean c\u1ee9u \u0111i\u1ec1u ki\u1ec7n v\u1ec1 kh\u1ea3 n\u0103ng c\u1ee7a m\u1ed7i nh\u1eadn th\u1ee9c c\u00f3 t\u00ednh kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ed3ng th\u1eddi c\u1ee7a m\u1ed7i kinh nghi\u1ec7m. T\u00f4i tin r\u1eb1ng \u0111i\u1ec1u n\u00e0y v\u1ec1 c\u01a1 b\u1ea3n \u0111\u00e3 x\u1ea3y ra trong c\u00e1c nghi\u00ean c\u1ee9u c\u1ee7a t\u00f4i v\u1ec1 c\u00e1c nguy\u00ean l\u00fd to\u00e1n h\u1ecdc. T\u00ednh ti\u00ean nghi\u1ec7m (a priori) kh\u00f4ng g\u00ec nhi\u1ec1u h\u01a1n hay \u00edt h\u01a1n l\u00e0 m\u1ed9t th\u00e1i \u0111\u1ed9 c\u0103n b\u1ea3n ho\u1eb7c s\u1ef1 di\u1ec5n \u0111\u1ea1t m\u1ed9t s\u1ed1 \u0111i\u1ec1u ki\u1ec7n ti\u00ean quy\u1ebft kh\u00f4ng th\u1ec3 thi\u1ebfu cho t\u01b0 duy v\u00e0 kinh nghi\u1ec7m. Nh\u01b0ng ranh gi\u1edbi gi\u1eefa m\u1ed9t b\u00ean l\u00e0 nh\u1eefng g\u00ec ch\u00fang ta s\u1edf h\u1eefu m\u1ed9t c\u00e1ch ti\u00ean nghi\u1ec7m v\u00e0 m\u1ed9t b\u00ean l\u00e0 nh\u1eefng g\u00ec c\u1ea7n thi\u1ebft cho kinh nghi\u1ec7m ph\u1ea3i \u0111\u01b0\u1ee3c v\u1ea1ch ra l\u00e0 kh\u00e1c v\u1edbi Kant; Kant \u0111\u00e3 \u0111\u00e1nh gi\u00e1 vai tr\u00f2 v\u00e0 ph\u1ea1m vi c\u1ee7a ti\u00ean nghi\u1ec7m qu\u00e1 cao.<\/p>\n<p>V\u00e0o th\u1eddi c\u1ee7a Kant, ng\u01b0\u1eddi ta c\u00f3 th\u1ec3 ngh\u0129 r\u1eb1ng c\u00e1c kh\u00e1i ni\u1ec7m v\u1ec1 kh\u00f4ng gian v\u00e0 th\u1eddi gian m\u00e0 ng\u01b0\u1eddi ta \u0111\u00e3 s\u1edf h\u1eefu c\u0169ng c\u00f3 th\u1ec3 \u00e1p d\u1ee5ng ph\u1ed5 bi\u1ebfn v\u00e0 tr\u1ef1c ti\u1ebfp v\u00e0o th\u1ef1c t\u1ea1i, ch\u1eb3ng h\u1ea1n nh\u1eefng \u00fd t\u01b0\u1edfng c\u1ee7a ch\u00fang ta v\u1ec1 s\u1ed1 l\u01b0\u1ee3ng, th\u1ee9 t\u1ef1 v\u00e0 k\u00edch th\u01b0\u1edbc, m\u00e0 ch\u00fang ta th\u01b0\u1eddng xuy\u00ean s\u1eed d\u1ee5ng trong c\u00e1c l\u00fd thuy\u1ebft to\u00e1n h\u1ecdc v\u00e0 v\u1eadt l\u00fd theo c\u00e1ch quen thu\u1ed9c \u0111\u1ed1i v\u1edbi ch\u00fang ta. V\u1eady th\u00ec, l\u00fd thuy\u1ebft v\u1ec1 kh\u00f4ng gian v\u00e0 th\u1eddi gian, \u0111\u1eb7c bi\u1ec7t l\u00e0 h\u00ecnh h\u1ecdc, th\u1ef1c s\u1ef1 s\u1ebd l\u00e0 m\u1ed9t c\u00e1i g\u00ec \u0111\u00f3 nh\u01b0 s\u1ed1 h\u1ecdc, \u0111i tr\u01b0\u1edbc m\u1ecdi nh\u1eadn \u200b\u200b\u200b\u200bth\u1ee9c v\u1ec1 t\u1ef1 nhi\u00ean. Nh\u01b0ng quan \u0111i\u1ec3m n\u00e0y c\u1ee7a Kant \u0111\u00e3 b\u1ecb b\u00e1c b\u1ecf tr\u01b0\u1edbc khi s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a v\u1eadt l\u00fd b\u1eaft bu\u1ed9c n\u00f3, \u0111\u1eb7c bi\u1ec7t b\u1edfi Riemann v\u00e0 Helmholtz\u2013 m\u1ed9t c\u00e1ch ch\u00ednh \u0111\u00e1ng; b\u1edfi v\u00ec h\u00ecnh h\u1ecdc kh\u00f4ng g\u00ec kh\u00e1c h\u01a1n l\u00e0 m\u1ed9t ph\u1ea7n c\u1ee7a to\u00e0n b\u1ed9 khung kh\u00e1i ni\u1ec7m v\u1eadt l\u00fd m\u00f4 t\u1ea3 c\u00e1c m\u1ed1i quan h\u1ec7 v\u1ecb tr\u00ed c\u00f3 th\u1ec3 c\u00f3 c\u1ee7a c\u00e1c v\u1eadt r\u1eafn v\u1edbi nhau trong th\u1ebf gi\u1edbi c\u1ee7a c\u00e1c v\u1eadt th\u1ec3 th\u1ef1c. R\u1eb1ng c\u00f3 nh\u1eefng v\u1eadt r\u1eafn c\u00f3 th\u1ec3 di chuy\u1ec3n \u0111\u01b0\u1ee3c v\u00e0 m\u1ed1i quan h\u1ec7 v\u1ecb tr\u00ed c\u1ee7a ch\u00fang l\u00e0 g\u00ec, nh\u1eefng \u0111i\u1ec1u \u0111\u00f3 ch\u1ec9 l\u00e0 v\u1ea5n \u0111\u1ec1 kinh nghi\u1ec7m. \u0110\u1ecbnh l\u00fd t\u1ed5ng c\u00e1c g\u00f3c trong m\u1ed9t tam gi\u00e1c l\u00e0 hai g\u00f3c vu\u00f4ng v\u00e0 ti\u00ean \u0111\u1ec1 song song, nh\u1eefng th\u1ee9 \u0111\u00f3 ch\u1ec9 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ki\u1ec3m ch\u1ee9ng ho\u1eb7c b\u00e1c b\u1ecf b\u1eb1ng th\u1ef1c nghi\u1ec7m, nh\u01b0 Gauss \u0111\u00e3 nh\u1eadn \u0111\u1ecbnh. N\u1ebfu ch\u1eb3ng h\u1ea1n, t\u1ea5t c\u1ea3 c\u00e1c s\u1ef1 ki\u1ec7n \u0111\u01b0\u1ee3c bi\u1ec3u th\u1ecb b\u1edfi c\u00e1c \u0111\u1ecbnh l\u00fd t\u01b0\u01a1ng \u0111\u1eb3ng (congruent <a href=\"#_ftn6\" name=\"_ftnref6\">[6]<\/a>) ch\u1ee9ng minh l\u00e0 ph\u00f9 h\u1ee3p v\u1edbi kinh nghi\u1ec7m, nh\u01b0ng m\u1eb7t kh\u00e1c, n\u1ebfu t\u1ed5ng c\u00e1c g\u00f3c trong m\u1ed9t tam gi\u00e1c \u0111\u01b0\u1ee3c t\u1ea1o b\u1edfi c\u00e1c thanh c\u1ee9ng nh\u1ecf h\u01a1n m\u1ed9t g\u00f3c vu\u00f4ng, th\u00ec kh\u00f4ng ai s\u1ebd ngh\u0129 r\u1eb1ng ti\u00ean \u0111\u1ec1 song song trong kh\u00f4ng gian c\u1ee7a v\u1eadt th\u1ec3 th\u1ef1c l\u00e0 c\u00f3 hi\u1ec7u l\u1ef1c.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[5]<\/p>\n<p>C\u1ea7n h\u1ebft s\u1ee9c th\u1eadn tr\u1ecdng khi \u0111\u01b0a m\u1ed9t nh\u1eadn th\u1ee9c n\u00e0o v\u00e0o kho ti\u00ean nghi\u1ec7m; nhi\u1ec1u nh\u1eadn th\u1ee9c tr\u01b0\u1edbc \u0111\u00e2y t\u1eebng \u0111\u01b0\u1ee3c coi l\u00e0 ti\u00ean nghi\u1ec7m gi\u1edd \u0111\u00e2y \u0111\u01b0\u1ee3c nh\u1eadn di\u1ec7n l\u00e0 kh\u00f4ng ch\u00ednh x\u00e1c n\u1eefa. V\u00ed d\u1ee5 \u1ea5n t\u01b0\u1ee3ng n\u1ed5i b\u1eadt nh\u1ea5t v\u1ec1 \u0111i\u1ec1u n\u00e0y l\u00e0 \u00fd t\u01b0\u1edfng v\u1ec1 hi\u1ec7n t\u1ea1i tuy\u1ec7t \u0111\u1ed1i. Kh\u00f4ng c\u00f3 th\u1ee9 g\u1ecdi l\u00e0 hi\u1ec7n t\u1ea1i tuy\u1ec7t \u0111\u1ed1i, cho d\u00f9 ch\u00fang ta \u0111\u00e3 quen ch\u1ea5p nh\u1eadn n\u00f3 t\u1eeb th\u1eddi th\u01a1 \u1ea5u \u0111\u1ebfn d\u01b0\u1eddng n\u00e0o, b\u1edfi v\u00ec trong cu\u1ed9c s\u1ed1ng h\u00e0ng ng\u00e0y ch\u00fang ta ch\u1ec9 xoay quanh nh\u1eefng kho\u1ea3ng c\u00e1ch ng\u1eafn v\u00e0 chuy\u1ec3n \u0111\u1ed9ng ch\u1eadm. N\u1ebfu \u0111i\u1ec1u n\u00e0y di\u1ec5n ra kh\u00e1c \u0111i, kh\u00f4ng ai c\u00f3 th\u1ec3 ngh\u0129 \u0111\u1ebfn vi\u1ec7c \u0111\u01b0a ra th\u1eddi gian tuy\u1ec7t \u0111\u1ed1i. Ngay c\u1ea3 nh\u1eefng nh\u00e0 t\u01b0 t\u01b0\u1edfng s\u00e2u s\u1eafc nh\u01b0 Newton v\u00e0 Kant c\u0169ng ch\u01b0a bao gi\u1edd ngh\u0129 \u0111\u1ebfn vi\u1ec7c nghi ng\u1edd t\u00ednh tuy\u1ec7t \u0111\u1ed1i c\u1ee7a th\u1eddi gian. Newton th\u1eadn tr\u1ecdng th\u1eadm ch\u00ed c\u00f2n di\u1ec5n t\u1ea3 y\u00eau c\u1ea7u n\u00e0y (c\u1ee7a th\u1eddi gian tuy\u1ec7t \u0111\u1ed1i) m\u1ed9t c\u00e1ch r\u00f5 r\u00e0ng nh\u1ea5t c\u00f3 th\u1ec3: th\u1eddi gian th\u1ef1c tuy\u1ec7t \u0111\u1ed1i t\u1ef1 n\u00f3 v\u00e0 theo b\u1ea3n ch\u1ea5t c\u1ee7a n\u00f3 tr\u00f4i ch\u1ea3y \u0111\u1ec1u v\u00e0 kh\u00f4ng li\u00ean quan \u0111\u1ebfn b\u1ea5t k\u1ef3 \u0111\u1ed1i t\u01b0\u1ee3ng n\u00e0o. V\u1edbi m\u00f4 t\u1ea3 \u0111\u00f3, Newton \u0111\u00e3 th\u00e0nh th\u1eadt ng\u0103n ch\u1eb7n b\u1ea5t k\u1ef3 s\u1ef1 r\u00fat lui hay th\u1ecfa hi\u1ec7p n\u00e0o, v\u00e0 Kant, nh\u00e0 tri\u1ebft h\u1ecdc ph\u00ea ph\u00e1n, \u0111\u00e3 ch\u1ee9ng t\u1ecf m\u00ecnh l\u00e0 ng\u01b0\u1eddi kh\u00f4ng ph\u00ea ph\u00e1n \u1edf \u0111\u00e2y b\u1eb1ng c\u00e1ch s\u1eb5n s\u00e0ng ch\u1ea5p nh\u1eadn Newton m\u00e0 kh\u00f4ng c\u1ea7n suy ngh\u0129. Ch\u1ec9 c\u00f3 Einstein cu\u1ed1i c\u00f9ng m\u1edbi gi\u1ea3i ph\u00f3ng ch\u00fang ta kh\u1ecfi \u0111\u1ecbnh ki\u1ebfn n\u00e0y &#8211; \u0111\u00f3 s\u1ebd lu\u00f4n l\u00e0 m\u1ed9t trong nh\u1eefng chi\u1ebfn c\u00f4ng phi th\u01b0\u1eddng nh\u1ea5t c\u1ee7a tinh th\u1ea7n con ng\u01b0\u1eddi &#8211; v\u00e0 l\u00fd thuy\u1ebft ti\u00ean nghi\u1ec7m \u0111\u01b0\u1ee3c \u00e1p d\u1ee5ng s\u00e2u r\u1ed9ng \u0111\u1ebfn nay kh\u00f4ng th\u1ec3 b\u1ecb d\u1eabn ch\u1ee9ng l\u00e0 phi l\u00fd m\u1ed9t c\u00e1ch thuy\u1ebft ph\u1ee5c h\u01a1n b\u1edfi s\u1ef1 ti\u1ebfn b\u1ed9 n\u00e0y c\u1ee7a khoa h\u1ecdc v\u1eadt l\u00fd. Gi\u1ea3 \u0111\u1ecbnh v\u1ec1 th\u1eddi gian tuy\u1ec7t \u0111\u1ed1i, m\u1ed9t trong nh\u1eefng h\u1ec7 qu\u1ea3 kh\u00e1c, l\u00e0 k\u00e9o theo h\u1ec7 qu\u1ea3 cho \u0111\u1ecbnh l\u00fd v\u1ec1 ph\u00e9p c\u1ed9ng v\u1eadn t\u1ed1c khi hai v\u1eadn t\u1ed1c \u0111\u01b0\u1ee3c k\u1ebft h\u1ee3p l\u1ea1i \u2014 b\u1ea3n th\u00e2n c\u0169ng l\u00e0 m\u1ed9t \u0111\u1ecbnh l\u00fd d\u01b0\u1eddng nh\u01b0 kh\u00f3 c\u00f3 th\u1ec3 v\u01b0\u1ee3t qua v\u1ec1 m\u1eb7t hi\u1ec5n nhi\u00ean v\u00e0 t\u00ednh d\u1ec5 hi\u1ec3u ph\u1ed5 bi\u1ebfn c\u1ee7a n\u00f3 \u2014 nh\u01b0ng c\u00e1c th\u00ed nghi\u1ec7m \u0111a d\u1ea1ng nh\u1ea5t trong l\u0129nh v\u1ef1c quang h\u1ecdc, thi\u00ean v\u0103n h\u1ecdc v\u00e0 l\u00fd thuy\u1ebft \u0111i\u1ec7n l\u1ea1i cho th\u1ea5y m\u1ed9t c\u00e1ch thuy\u1ebft ph\u1ee5c r\u1eb1ng \u0111\u1ecbnh l\u00fd v\u1ec1 ph\u00e9p c\u1ed9ng v\u1eadn t\u1ed1c n\u00e0y l\u00e0 kh\u00f4ng ch\u00ednh x\u00e1c; th\u1ef1c t\u1ebf c\u00f3 m\u1ed9t \u0111\u1ecbnh lu\u1eadt kh\u00e1c ph\u1ee9c t\u1ea1p h\u01a1n cho s\u1ef1 k\u1ebft h\u1ee3p c\u1ee7a hai v\u1eadn t\u1ed1c.<a href=\"#_ftn7\" name=\"_ftnref7\">[7]<\/a><\/p>\n<p>Ch\u00fang ta c\u00f3 th\u1ec3 n\u00f3i: trong th\u1eddi gian g\u1ea7n \u0111\u00e2y, quan \u0111i\u1ec3m v\u1ec1 t\u00ednh ch\u1ea5t th\u1ef1c nghi\u1ec7m c\u1ee7a h\u00ecnh h\u1ecdc do Gauss v\u00e0 Helmholtz \u0111\u1ec1 x\u01b0\u1edbng \u0111\u00e3 tr\u1edf th\u00e0nh m\u1ed9t th\u00e0nh qu\u1ea3 ch\u1eafc ch\u1eafn c\u1ee7a khoa h\u1ecdc. Ng\u00e0y nay n\u00f3 ph\u1ea3i ph\u1ee5c v\u1ee5 nh\u01b0 m\u1ed9t \u0111i\u1ec3m quy chi\u1ebfu v\u1eefng ch\u1eafc cho m\u1ecdi suy \u0111o\u00e1n tri\u1ebft h\u1ecdc li\u00ean quan \u0111\u1ebfn kh\u00f4ng gian v\u00e0 th\u1eddi gian. V\u00ec l\u00fd thuy\u1ebft h\u1ea5p d\u1eabn c\u1ee7a Einstein l\u00e0m cho \u0111i\u1ec1u n\u00e0y r\u00f5 r\u00e0ng: h\u00ecnh h\u1ecdc kh\u00f4ng g\u00ec kh\u00e1c h\u01a1n l\u00e0 m\u1ed9t nh\u00e1nh c\u1ee7a v\u1eadt l\u00fd h\u1ecdc; c\u00e1c ch\u00e2n l\u00fd h\u00ecnh h\u1ecdc v\u1ec1 m\u1ecdi ph\u01b0\u01a1ng di\u1ec7n c\u01a1 b\u1ea3n kh\u00f4ng kh\u00e1c bi\u1ec7t g\u00ec v\u1edbi c\u00e1c ch\u00e2n l\u00fd v\u1eadt l\u00fd. V\u00ed d\u1ee5: \u0111\u1ecbnh l\u00fd Pythagore v\u00e0 \u0111\u1ecbnh lu\u1eadt h\u1ea5p d\u1eabn c\u1ee7a Newton c\u00f3 li\u00ean quan v\u1edbi nhau v\u1ec1 b\u1ea3n ch\u1ea5t, trong ch\u1eebng m\u1ef1c ch\u00fang b\u1ecb chi ph\u1ed1i b\u1edfi c\u00f9ng m\u1ed9t kh\u00e1i ni\u1ec7m v\u1eadt l\u00fd c\u01a1 b\u1ea3n, \u0111\u00f3 l\u00e0 th\u1ebf (potential). Nh\u01b0ng \u0111\u1ed1i v\u1edbi m\u1ecdi ng\u01b0\u1eddi s\u00e0nh s\u1ecfi v\u1ec1 thuy\u1ebft h\u1ea5p d\u1eabn c\u1ee7a Einstein c\u00f2n nhi\u1ec1u h\u01a1n n\u1eefa: hai \u0111\u1ecbnh lu\u1eadt n\u00e0y, r\u1ea5t kh\u00e1c nhau v\u00e0 cho \u0111\u1ebfn nay r\u00f5 r\u00e0ng l\u00e0 r\u1ea5t c\u00e1ch xa nhau, m\u1ed9t c\u00e1i l\u00e0 m\u1ed9t \u0111\u1ecbnh l\u00fd v\u1ec1 h\u00ecnh h\u1ecdc c\u0103n b\u1ea3n \u0111\u01b0\u1ee3c bi\u1ebft t\u1eeb th\u1eddi C\u1ed5 \u0111\u1ea1i v\u00e0 \u0111\u01b0\u1ee3c d\u1ea1y kh\u1eafp n\u01a1i trong tr\u01b0\u1eddng h\u1ecdc, c\u00e1i kia m\u1ed9t \u0111\u1ecbnh lu\u1eadt chi ph\u1ed1i t\u00e1c \u0111\u1ed9ng c\u1ee7a c\u00e1c kh\u1ed1i l\u01b0\u1ee3ng l\u00ean nhau, ch\u00fang kh\u00f4ng ch\u1ec9 c\u00f3 c\u00f9ng \u0111\u1eb7c t\u00ednh, m\u00e0 ch\u1ec9 l\u00e0 c\u00e1c ph\u1ea7n c\u1ee7a m\u1ed9t v\u00e0 c\u00f9ng m\u1ed9t \u0111\u1ecbnh lu\u1eadt t\u1ed5ng qu\u00e1t.<\/p>\n<p>S\u1ef1 gi\u1ed1ng nhau c\u01a1 b\u1ea3n c\u1ee7a c\u00e1c th\u1ef1c t\u1ebf h\u00ecnh h\u1ecdc v\u00e0 v\u1eadt l\u00fd kh\u00f3 c\u00f3 th\u1ec3 l\u1ed9 di\u1ec7n r\u00f5 r\u00e0ng h\u01a1n. T\u1ea5t nhi\u00ean, trong c\u1ea5u tr\u00fac logic th\u00f4ng th\u01b0\u1eddng v\u00e0 trong nh\u1eefng tr\u1ea3i nghi\u1ec7m th\u00f4ng th\u01b0\u1eddng h\u00e0ng ng\u00e0y quen thu\u1ed9c t\u1eeb th\u1eddi th\u01a1 \u1ea5u c\u1ee7a ch\u00fang ta, c\u00e1c m\u1ec7nh \u0111\u1ec1 h\u00ecnh h\u1ecdc v\u00e0 \u0111\u1ed9ng h\u1ecdc \u0111i tr\u01b0\u1edbc c\u00e1c m\u1ec7nh \u0111\u1ec1 \u0111\u1ed9ng l\u1ef1c h\u1ecdc, v\u00e0 ho\u00e0n c\u1ea3nh n\u00e0y gi\u1ea3i th\u00edch vi\u1ec7c ng\u01b0\u1eddi ta qu\u00ean r\u1eb1ng t\u1ea5t c\u1ea3 l\u00e0 kinh nghi\u1ec7m. V\u00ec v\u1eady, ch\u00fang ta th\u1ea5y: L\u00fd thuy\u1ebft ti\u00ean nghi\u1ec7m c\u1ee7a Kant v\u1eabn ch\u1ee9a \u0111\u1ef1ng nh\u1eefng r\u1ec9 s\u00e9t do con ng\u01b0\u1eddi t\u1ea1o ra, v\u00e0 n\u00f3 c\u1ea7n ph\u1ea3i \u0111\u01b0\u1ee3c gi\u1ea3i ph\u00f3ng kh\u1ecfi ch\u00fang, v\u00e0 sau qu\u00e1 tr\u00ecnh lo\u1ea1i b\u1ecf, ch\u1ec9 c\u00f2n l\u1ea1i c\u00e1i quan \u0111i\u1ec3m ti\u00ean nghi\u1ec7m ph\u1ee5c v\u1ee5 l\u00e0m n\u1ec1n t\u1ea3ng cho nh\u1eadn th\u1ee9c to\u00e1n h\u1ecdc thu\u1ea7n t\u00fay: v\u1ec1 c\u01a1 b\u1ea3n, \u0111\u00f3 l\u00e0 quan \u0111i\u1ec3m sau c\u00f9ng \u0111\u01b0\u1ee3c t\u00f4i \u0111\u1eb7c tr\u01b0ng trong nhi\u1ec1u c\u00f4ng tr\u00ecnh nghi\u00ean c\u1ee9u. <a href=\"#_ftn8\" name=\"_ftnref8\">[8]<\/a><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[6]<\/p>\n<p>C\u00f4ng cu\u0323 trung gian gi\u01b0\u0303a ly\u0301 thuy\u00ea\u0301t va\u0300 th\u01b0\u0323c ha\u0300nh, gi\u01b0\u0303a t\u01b0 duy va\u0300 quan sa\u0301t la\u0300 toa\u0301n ho\u0323c. No\u0301 x\u00e2y d\u01b0\u0323ng c\u00e2\u0300u n\u00f4\u0301i va\u0300 la\u0300m cho chi\u00ea\u0301c c\u00e2\u0300u n\u00e0y nga\u0300y ca\u0300ng v\u1eefng ch\u1eafc h\u01a1n. Cho n\u00ean, to\u00e0n b\u1ed9 n\u00ea\u0300n v\u0103n ho\u0301a hi\u00ea\u0323n ta\u0323i c\u1ee7a chu\u0301ng ta, trong ch\u01b0\u0300ng m\u01b0\u0323c d\u1ef1a tr\u00ean nh\u00e2\u0323n th\u01b0\u0301c v\u00ea\u0300 t\u01b0\u0323 nhi\u00ean va\u0300 s\u01b0\u0323 l\u00e0m cho no\u0301 c\u00f3 \u00edch, ti\u0300m th\u00e2\u0301y n\u00ea\u0300n ta\u0309ng cu\u0309a m\u00ecnh trong toa\u0301n ho\u0323c. Galilei \u0111\u00e3 t\u1eebng n\u00f3i: Ng\u01b0\u01a1\u0300i ta chi\u0309 co\u0301 th\u00ea\u0309 hi\u00ea\u0309u \u0111\u01b0\u01a1\u0323c t\u01b0\u0323 nhi\u00ean khi h\u1ecdc \u0111\u01b0\u1ee3c ng\u00f4n ng\u01b0\u0303 cu\u0309a n\u00f3 va\u0300 quen thu\u00f4\u0323c v\u1edbi ca\u0301c ky\u0301 hi\u00ea\u0323u ma\u0300 qua \u0111o\u0301 t\u01b0\u0323 nhi\u00ean no\u0301i v\u01a1\u0301i chu\u0301ng ta. Nh\u01b0ng ng\u00f4n ng\u01b0\u0303 n\u00e0y la\u0300 toa\u0301n ho\u0323c, va\u0300 ca\u0301c ky\u0301 hi\u00ea\u0323u chi\u0301nh la\u0300 nh\u01b0\u0303ng hi\u0300nh th\u1ec3 toa\u0301n ho\u0323c. Kant t\u01b0\u0300ng qua\u0309 quy\u00ea\u0301t: \u201cT\u00f4i cho r\u0103\u0300ng trong m\u00f4\u0303i m\u00f4n khoa ho\u0323c t\u01b0\u0323 nhi\u00ean \u0111\u1eb7c bi\u1ec7t, ch\u00fang ta chi\u0309 co\u0301 th\u00ea\u0309 ti\u0300m th\u00e2\u0301y khoa ho\u0323c \u0111i\u0301ch th\u01b0\u0323c trong ch\u01b0\u0300ng m\u01b0\u0323c n\u00f3 ch\u1ee9a n\u00f4\u0323i ha\u0300m toa\u0301n ho\u0323c trong \u0111o\u0301.\u201d S\u01b0\u0323 th\u00e2\u0323t: Chu\u0301ng ta chi\u0309 la\u0300m chu\u0309 \u0111\u01b0\u01a1\u0323c m\u00f4\u0323t ly\u0301 thuy\u00ea\u0301t khoa ho\u0323c khi chu\u0301ng ta t\u00e1ch \u0111\u01b0\u01a1\u0323c ca\u0301i nh\u00e2n toa\u0301n ho\u0323c c\u1ee7a n\u00f3 ra va\u0300 b\u00f3c tr\u1ea7n n\u00f3 hoa\u0300n to\u00e0n kh\u1ecfi v\u1ecf. Kh\u00f4ng co\u0301 toa\u0301n ho\u0323c, kh\u00f4ng th\u00ea\u0309 co\u0301 nga\u0300nh thi\u00ean v\u0103n h\u1ecdc v\u00e0 v\u00e2\u0323t ly\u0301 h\u1ecdc ng\u00e0y nay; nh\u01b0\u0303ng nga\u0300nh khoa ho\u0323c na\u0300y, trong nh\u1eefng ph\u1ea7n l\u00fd thuy\u1ebft c\u1ee7a ch\u00fang, \u0111\u1ec1u hi\u00ea\u0323n ra \u1edf d\u1ea1ng toa\u0301n ho\u0323c. Nh\u01b0\u0303ng \u01b0\u0301ng du\u0323ng na\u0300y, cu\u0303ng nh\u01b0 v\u00f4 s\u00f4\u0301 nh\u01b0\u0303ng \u01b0\u0301ng du\u0323ng kha\u0301c, la\u0300 nh\u01b0\u0303ng g\u00ec \u0111a\u0303 giu\u0301p cho toa\u0301n ho\u0323c la\u0300m n\u00ean t\u00ean tu\u00f4\u0309i m\u00e0 n\u00f3 c\u00f3 \u0111\u01b0\u1ee3c v\u1edbi \u0111\u00f4ng \u0111\u1ea3o c\u00f4ng chu\u0301ng.<\/p>\n<p>M\u1eb7c d\u00f9 v\u1eady, t\u1ea5t c\u1ea3 ca\u0301c nha\u0300 toa\u0301n ho\u0323c \u0111a\u0303 t\u01b0\u0300 ch\u00f4\u0301i l\u00e2\u0301y \u01b0\u0301ng du\u0323ng la\u0300m th\u01b0\u01a1\u0301c \u0111o gi\u00e1 tr\u1ecb cho toa\u0301n ho\u0323c. \u00d4ng ho\u00e0ng c\u1ee7a c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc, Gauss, ng\u01b0\u1eddi ch\u1eafc ch\u1eafn \u0111\u1ed3ng th\u1eddi l\u00e0 m\u1ed9t nh\u00e0 to\u00e1n h\u1ecdc \u1ee9ng d\u1ee5ng xu\u1ea5t s\u1eafc, ng\u01b0\u1eddi \u0111\u00e3 t\u1ea1o m\u1edbi to\u00e0n c\u1ea3 c\u00e1c ng\u00e0nh khoa h\u1ecdc, ch\u1eb3ng h\u1ea1n nh\u01b0 l\u00fd thuy\u1ebft sai s\u1ed1, tr\u1eafc \u0111\u1ecba, \u0111\u1ec3 l\u00e0m cho to\u00e1n h\u1ecdc \u0111\u00f3ng vai tr\u00f2 ch\u1ee7 \u0111\u1ea1o trong \u0111\u00f3, ng\u01b0\u1eddi m\u00e0 khi c\u00e1c nh\u00e0 thi\u00ean v\u0103n kh\u00e1m ph\u00e1 m\u1edbi h\u00e0nh tinh Ceres &#8211; m\u1ed9t h\u00e0nh tinh \u0111\u1eb7c bi\u1ec7t quan tr\u1ecdng v\u00e0 th\u00fa v\u1ecb &#8211; nh\u01b0ng \u0111\u1ec3 cho n\u00f3 th\u1ea5t l\u1ea1c v\u00e0 kh\u00f4ng th\u1ec3 t\u00ecm l\u1ea1i \u0111\u01b0\u1ee3c, \u0111\u00e3 s\u00e1ng t\u1ea1o ra m\u1ed9t l\u00fd thuy\u1ebft to\u00e1n h\u1ecdc m\u1edbi tr\u00ean c\u01a1 s\u1edf \u0111\u00f3 \u00f4ng ti\u00ean \u0111o\u00e1n ch\u00ednh x\u00e1c v\u1ecb tr\u00ed c\u1ee7a Ceres, ng\u01b0\u1eddi \u0111\u00e3 ph\u00e1t minh ra m\u00e1y \u0111i\u1ec7n b\u00e1o v\u00e0 nhi\u1ec1u th\u1ee9 thi\u1ebft th\u1ef1c kh\u00e1c, c\u00f3 c\u00f9ng quan \u0111i\u1ec3m. L\u00fd thuy\u1ebft s\u1ed1 thu\u1ea7n t\u00fay l\u00e0 l\u0129nh v\u1ef1c to\u00e1n h\u1ecdc ch\u01b0a t\u1eebng \u0111\u01b0\u1ee3c \u00e1p d\u1ee5ng tr\u01b0\u1edbc \u0111\u00e2y. Nh\u01b0ng ch\u00ednh l\u00fd thuy\u1ebft s\u1ed1 m\u00e0 Gauss g\u1ecdi l\u00e0 n\u1eef ho\u00e0ng c\u1ee7a to\u00e1n h\u1ecdc v\u00e0 \u0111\u01b0\u1ee3c \u00f4ng c\u0169ng nh\u01b0 h\u1ea7u h\u1ebft c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc v\u0129 \u0111\u1ea1i t\u00f4n vinh. Gauss n\u00f3i v\u1ec1 s\u1ee9c h\u1ea5p d\u1eabn k\u1ef3 di\u1ec7u \u0111\u00e3 l\u00e0m cho l\u00fd thuy\u1ebft s\u1ed1 tr\u1edf th\u00e0nh m\u00f4n khoa h\u1ecdc y\u00eau th\u00edch c\u1ee7a c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc \u0111\u1ea7u ti\u00ean, kh\u00f4ng k\u1ec3 \u0111\u1ebfn s\u1ef1 phong ph\u00fa b\u1ea5t t\u1eadn c\u1ee7a n\u00f3, m\u00e0 \u1edf \u0111\u00f3 n\u00f3 v\u01b0\u1ee3t xa t\u1ea5t c\u1ea3 c\u00e1c ph\u1ea7n kh\u00e1c c\u1ee7a to\u00e1n h\u1ecdc. Gauss m\u00f4 t\u1ea3 khi c\u00f2n tr\u1ebb \u0111\u00e3 b\u1ecb m\u00ea ho\u1eb7c th\u1ebf n\u00e0o b\u1edfi s\u1ef1 h\u1ea5p d\u1eabn c\u1ee7a c\u00e1c cu\u1ed9c nghi\u00ean c\u1ee9u l\u00fd thuy\u1ebft s\u1ed1 \u0111\u1ebfn m\u1ee9c \u00f4ng kh\u00f4ng th\u1ec3 c\u01b0\u1ee1ng l\u1ea1i ch\u00fang \u0111\u01b0\u1ee3c n\u1eefa. \u00d4ng ca ng\u1ee3i Fermat, Euler, Lagrange v\u00e0 Legendre l\u00e0 nh\u1eefng ng\u01b0\u1eddi n\u1ed5i ti\u1ebfng v\u00f4 song v\u00ec \u0111\u00e3 m\u1edf ra c\u00e1nh c\u1eeda \u0111\u1ebfn th\u00e1nh \u0111\u1ecba c\u1ee7a khoa h\u1ecdc th\u1ea7n th\u00e1nh n\u00e0y, v\u00e0 cho th\u1ea5y n\u00f3 ch\u1ee9a \u0111\u1ea7y nh\u1eefng kho b\u00e1u nh\u01b0 th\u1ebf n\u00e0o. V\u00e0 c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc tr\u01b0\u1edbc Gauss v\u00e0 nh\u1eefng ng\u01b0\u1eddi sau Gauss, ch\u1eb3ng h\u1ea1n nh\u01b0 Lejeune Dirichlet, Kummer, Hermite, Kronecker v\u00e0 Minkowski, c\u0169ng ph\u00e1t bi\u1ec3u m\u1ed9t c\u00e1ch nhi\u1ec7t t\u00ecnh t\u01b0\u01a1ng t\u1ef1. Kronecker so sa\u0301nh ca\u0301c nha\u0300 ly\u0301 thuy\u00ea\u0301t s\u00f4\u0301 v\u01a1\u0301i nh\u01b0\u0303ng ng\u01b0\u01a1\u0300i \u0103n ha\u0323t sen, m\u1ed9t khi \u0111\u00e3 n\u1ebfm th\u1eed m\u00f3n na\u0300y, h\u1ecd se\u0303 kh\u00f4ng bao gi\u01a1\u0300 d\u1eebng l\u1ea1i \u0111\u01b0\u01a1\u0323c.<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcRW4kkC-aJF5-OqjG1t5HJ7d_w7AKuaa-4ri4lSSiXhgtmlWzd1N2n5aWdSiuPHXmqlGnc&amp;usqp=CAU\" alt=\"Portrait Of The Astronomer Galileo Galilei Painting by Unknown Artist - Fine Art America\" width=\"254\" height=\"253\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcQAGE2qx-E7dcEGujAbgx0bmYSbbnXi946pCw&amp;usqp=CAU\" alt=\"Henri Poincar\u00e9, French mathematician, theoretical physicist Stock Photo | Adobe Stock\" width=\"189\" height=\"253\" \/> <img decoding=\"async\" loading=\"lazy\" class=\"\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/90\/Carl_Jacobi.jpg\/220px-Carl_Jacobi.jpg\" alt=\"Carl Jacobi.jpg\" width=\"209\" height=\"240\" \/><\/p>\n<p style=\"text-align: center\">Galileo Galilei, Henri Poincar\u00e9 v\u00e0 Carl Gustav Jacob Jacobi<\/p>\n<p>Poincar\u00e9, nh\u00e0 to\u00e1n h\u1ecdc l\u1ed7i l\u1ea1c nh\u1ea5t trong th\u1ebf h\u1ec7 c\u1ee7a \u00f4ng, ng\u01b0\u1eddi v\u1ec1 c\u01a1 b\u1ea3n \u0111\u1ed3ng th\u1eddi l\u00e0 nh\u00e0 v\u1eadt l\u00fd v\u00e0 nh\u00e0 thi\u00ean v\u0103n h\u1ecdc, c\u0169ng c\u00f3 c\u00f9ng quan \u0111i\u1ec3m. Poincar\u00e9 \u0111\u00e3 t\u1eebng quay sang ch\u1ed1ng l\u1ea1i Tolstoi v\u1edbi s\u1ef1 s\u1eafc b\u00e9n \u0111\u1eb7c bi\u1ec7t, ng\u01b0\u1eddi \u0111\u00e3 tuy\u00ean b\u1ed1 r\u1eb1ng \u0111\u00f2i h\u1ecfi &#8220;khoa h\u1ecdc v\u00ec l\u1ee3i \u00edch c\u1ee7a khoa h\u1ecdc&#8221; l\u00e0 \u0111i\u00ean r\u1ed3. &#8220;Li\u1ec7u ch\u00fang ta,&#8221; Tolstoi n\u00f3i, &#8220;h\u01b0\u1edbng d\u1eabn b\u1ea3n th\u00e2n trong vi\u1ec7c l\u1ef1a ch\u1ecdn ngh\u1ec1 nghi\u1ec7p c\u1ee7a m\u00ecnh theo t\u00ednh kh\u00ed c\u1ee7a \u00f3c t\u00f2 m\u00f2 c\u1ee7a ch\u00fang ta hay kh\u00f4ng? Quy\u1ebft \u0111\u1ecbnh d\u1ef1a tr\u00ean s\u1ef1 h\u1eefu d\u1ee5ng, t\u1ee9c l\u00e0, theo nhu c\u1ea7u th\u1ef1c t\u1ebf v\u00e0 \u0111\u1ea1o \u0111\u1ee9c c\u1ee7a ch\u00fang ta kh\u00f4ng t\u1ed1t h\u01a1n hay sao?\u201d K\u1ef3 l\u1ea1, ch\u00ednh Tolstoi l\u1ea1i l\u00e0 ng\u01b0\u1eddi m\u00e0 c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc ch\u00fang ta ph\u1ea3i b\u00e1c b\u1ecf nh\u01b0 m\u1ed9t ng\u01b0\u1eddi theo ch\u1ee7 ngh\u0129a hi\u1ec7n th\u1ef1c n\u00f4ng c\u1ea1n v\u00e0 theo ch\u1ee7 ngh\u0129a v\u1ecb l\u1ee3i h\u1eb9p h\u00f2i. Poincar\u00e9 l\u1eadp lu\u1eadn ch\u1ed1ng l\u1ea1i Tolstoi r\u1eb1ng, n\u1ebfu ng\u01b0\u1eddi ta ti\u1ebfn h\u00e0nh theo c\u00f4ng th\u1ee9c c\u1ee7a Tolstoi, th\u00ec khoa h\u1ecdc s\u1ebd kh\u00f4ng bao gi\u1edd ra \u0111\u1eddi. B\u1ea1n ch\u1ec9 c\u1ea7n m\u1edf m\u1eaft ra, Poincar\u00e9 k\u1ebft lu\u1eadn, \u0111\u1ec3 th\u1ea5y ch\u1eb3ng h\u1ea1n nh\u1eefng th\u00e0nh t\u1ef1u c\u1ee7a ng\u00e0nh c\u00f4ng nghi\u1ec7p s\u1ebd kh\u00f4ng bao gi\u1edd \u0111\u01b0\u1ee3c nh\u00ecn th\u1ea5y \u00e1nh s\u00e1ng n\u1ebfu nh\u1eefng ng\u01b0\u1eddi th\u1ef1c h\u00e0nh n\u00e0y t\u1ed3n t\u1ea1i m\u1ed9t m\u00ecnh, v\u00e0 n\u1ebfu nh\u1eefng th\u00e0nh t\u1ef1u n\u00e0y kh\u00f4ng \u0111\u01b0\u1ee3c th\u00fac \u0111\u1ea9y b\u1edfi nh\u1eefng k\u1ebb ngu xu\u1ea9n b\u1ea5t v\u1ee5 l\u1ee3i, nh\u1eefng ng\u01b0\u1eddi kh\u00f4ng bao gi\u1edd ngh\u0129 \u0111\u1ebfn vi\u1ec7c khai th\u00e1c h\u1eefu d\u1ee5ng. T\u1ea5t c\u1ea3 ch\u00fang ta \u0111\u1ec1u c\u00f3 c\u00f9ng quan \u0111i\u1ec3m.<\/p>\n<p>Nh\u00e0 to\u00e1n h\u1ecdc v\u0129 \u0111\u1ea1i c\u1ee7a ch\u00fang ta Jacobi \u01a1\u0309 K\u00f6nigsberg c\u0169ng ngh\u0129 v\u1eady, Jacobi, ng\u01b0\u1eddi m\u00e0 t\u00ean tu\u1ed5i \u0111\u1ee9ng c\u1ea1nh Gauss v\u00e0 v\u1eabn \u0111\u01b0\u1ee3c m\u1ecdi sinh vi\u00ean trong c\u00e1c m\u00f4n h\u1ecdc c\u1ee7a ch\u00fang ta g\u1ecdi v\u1edbi s\u1ef1 t\u00f4n k\u00ednh. Khi nh\u00e0 to\u00e1n h\u1ecdc n\u1ed5i ti\u1ebfng Fourier m\u1ed9t l\u1ea7n n\u00f3i r\u1eb1ng m\u1ee5c \u0111\u00edch ch\u00ednh c\u1ee7a to\u00e1n h\u1ecdc l\u00e0 \u0111\u1ec3 gi\u1ea3i th\u00edch c\u00e1c hi\u1ec7n t\u01b0\u1ee3ng t\u1ef1 nhi\u00ean, th\u00ec ch\u00ednh Jacobi \u0111\u00e3 qu\u1edf tr\u00e1ch \u00f4ng v\u1edbi t\u1ea5t c\u1ea3 t\u00ednh kh\u00ed s\u00f4i n\u1ed5i c\u1ee7a \u00f4ng. M\u1ed9t tri\u1ebft gia, nh\u01b0 ch\u00ednh Fourier, Jacobi k\u00eau l\u00ean, l\u1ebd ra ph\u1ea3i bi\u1ebft r\u1eb1ng s\u1ef1 t\u00f4n vinh tinh th\u1ea7n con ng\u01b0\u1eddi l\u00e0 m\u1ee5c \u0111\u00edch duy nh\u1ea5t c\u1ee7a m\u1ecdi ng\u00e0nh khoa h\u1ecdc, v\u00e0 r\u1eb1ng t\u1eeb quan \u0111i\u1ec3m n\u00e0y, m\u1ed9t v\u1ea5n \u0111\u1ec1 c\u1ee7a l\u00fd thuy\u1ebft s\u1ed1 thu\u1ea7n t\u00fay c\u0169ng \u0111\u00e1ng gi\u00e1 nh\u01b0 m\u1ed9t v\u1ea5n \u0111\u1ec1 ph\u1ee5c v\u1ee5 cho c\u00e1c \u1ee9ng d\u1ee5ng.<\/p>\n<p>B\u1ea5t c\u1ee9 ai c\u1ea3m nh\u1eadn \u0111\u01b0\u1ee3c ch\u00e2n l\u00fd c\u1ee7a l\u1ed1i suy ngh\u0129 v\u00e0 th\u1ebf gi\u1edbi quan r\u1ed9ng m\u1edf to\u00e1t ra t\u1eeb nh\u1eefng l\u1eddi n\u00e0y c\u1ee7a Jacobi s\u1ebd kh\u00f4ng r\u01a1i v\u00e0o ch\u1ee7 ngh\u0129a ho\u00e0i nghi tho\u00e1i lui v\u00e0 ph\u00f9 phi\u1ebfm; anh ta s\u1ebd kh\u00f4ng tin v\u00e0o nh\u1eefng k\u1ebb v\u01a1\u0301i v\u1ebb m\u1eb7t tri\u1ebft h\u1ecdc va\u0300 gio\u0323ng \u0111i\u1ec7u cao si\u00eau, ti\u00ean \u0111oa\u0301n v\u00ea\u0300 s\u01b0\u0323 suy ta\u0300n cu\u0309a v\u0103n ho\u0301a v\u00e0 r\u01a1i v\u00e0o ch\u1ee7 ngh\u0129a \u201cchu\u0301ng ta se\u0303 kh\u00f4ng bi\u00ea\u0301t\u201d (ignorabimus). \u0110\u00f4\u0301i v\u01a1\u0301i nh\u00e0 to\u00e1n h\u1ecdc kh\u00f4ng co\u0301 ignorabimus, va\u0300 theo t\u00f4i, cu\u0303ng kh\u00f4ng h\u1ec1 co\u0301 \u0111i\u00ea\u0300u \u0111o\u0301 \u0111\u1ed1i v\u1edbi khoa ho\u0323c. Nh\u00e0 tri\u1ebft h\u1ecdc Comte \u0111\u00e3 t\u1eebng n\u00f3i\u2014v\u1edbi \u00fd \u0111\u1ecbnh n\u00eau ra m\u1ed9t v\u1ea5n \u0111\u1ec1 ch\u1eafc ch\u1eafn kh\u00f4ng th\u1ec3 gi\u1ea3i quy\u1ebft \u0111\u01b0\u1ee3c\u2014r\u1eb1ng khoa h\u1ecdc s\u1ebd kh\u00f4ng bao gi\u1edd th\u00e0nh c\u00f4ng trong vi\u1ec7c kh\u00e1m ph\u00e1 b\u00ed m\u1eadt v\u1ec1 s\u1ef1 c\u1ea5u th\u00e0nh h\u00f3a h\u1ecdc c\u1ee7a c\u00e1c thi\u00ean th\u1ec3. V\u00e0i n\u0103m sau, v\u1ea5n \u0111\u1ec1 n\u00e0y \u0111\u00e3 \u0111\u01b0\u1ee3c gi\u1ea3i quy\u1ebft b\u1eb1ng ph\u00e2n t\u00edch quang ph\u1ed5 c\u1ee7a Kirchhoff v\u00e0 Bunsen, v\u00e0 ng\u00e0y nay c\u00f3 th\u1ec3 n\u00f3i r\u1eb1ng ch\u00fang ta s\u1eed d\u1ee5ng c\u00e1c ng\u00f4i sao xa x\u00f4i nh\u1ea5t l\u00e0m nh\u1eefng ph\u00f2ng th\u00ed nghi\u1ec7m v\u1eadt l\u00fd v\u00e0 h\u00f3a h\u1ecdc quan tr\u1ecdng nh\u1ea5t, \u0111i\u1ec1u m\u00e0 ch\u00fang ta ho\u00e0n to\u00e0n kh\u00f4ng t\u00ecm th\u1ea5y tr\u00ean tr\u00e1i \u0111\u1ea5t. Theo t\u00f4i, l\u00fd do th\u1ef1c s\u1ef1 khi\u1ebfn cho Comte kh\u00f4ng th\u00e0nh c\u00f4ng trong vi\u1ec7c t\u00ecm ra m\u1ed9t b\u00e0i to\u00e1n kh\u00f4ng th\u1ec3 gi\u1ea3i \u0111\u01b0\u1ee3c l\u00e0 kh\u00f4ng c\u00f3 c\u00e1i g\u1ecdi l\u00e0 b\u00e0i to\u00e1n kh\u00f4ng gi\u1ea3i \u0111\u01b0\u1ee3c. Thay v\u00ec \u201cch\u1ee7 ngh\u0129a ch\u00fang ta kh\u00f4ng bi\u1ebft\u201d \u0111i\u00ean kh\u00f9ng, gi\u1ea3i ph\u00e1p c\u1ee7a ch\u00fang ta ng\u01b0\u1ee3c l\u1ea1i l\u00e0:<\/p>\n<p style=\"text-align: center\"><em>Ch\u00fang ta ph\u1ea3i bi\u1ebft,<\/em><\/p>\n<p style=\"text-align: center\"><em>Ch\u00fang ta s\u1ebd bi\u1ebft<\/em><\/p>\n<p style=\"text-align: center\">(Wir m\u00fcssen wissen<\/p>\n<p style=\"text-align: center\">Wir werden wissen)<\/p>\n<p style=\"text-align: center\"><strong>David Hilbert<\/strong><\/p>\n<p><span style=\"color: #000080\"><em>Xem th\u00eam<\/em><\/span>:<\/p>\n<p class=\"search-title\" style=\"text-align: center\"><span style=\"font-size: 12pt\"><a title=\"David Hilbert: Di\u1ec5n t\u1eeb b\u1ed1n ph\u00fat b\u1ea5t h\u1ee7\u00a0tr\u00ean \u0111\u00e0i ph\u00e1t thanh n\u0103m 1930\u00a0\" href=\"https:\/\/rosetta.vn\/nguyenxuanxanh\/david-hilbert\/\" rel=\"bookmark\">DAVID HILBERT:\u00a0<i>DI\u1ec4N T\u1eea B\u1ed0N PH\u00daT B\u1ea4T H\u1ee6\u00a0TR\u00caN \u0110\u00c0I PH\u00c1T THANH N\u0102M 1930<\/i>\u00a0<\/a><\/span><\/p>\n<p><span style=\"color: #000000\">M\u1ed9t quy\u1ec3n s\u00e1ch r\u1ea5t hay n\u00f3i v\u1ec1 Hilbert v\u00e0 th\u1eddi \u0111\u1ea1i c\u1ee7a \u00f4ng:\u00a0<\/span><\/p>\n<p style=\"padding-left: 40px\"><span style=\"color: #000080\">-Constance Reid, <em>Hilbert<\/em>. Springer-Verlag, 1970.<\/span><\/p>\n<p><em>Tham kh\u1ea3o<\/em>:<\/p>\n<p><a href=\"#_ftnref1\" name=\"_ftn1\">[1]<\/a> Logic h\u1ecdc (Logik, hay logics) \u0111\u01b0\u1ee3c hi\u1ec3u l\u00e0 khoa h\u1ecdc v\u1ec1 t\u01b0 duy, v\u1ec1 c\u1ea5u tr\u00fac, h\u00ecnh th\u00e1i v\u00e0 quy lu\u1eadt c\u1ee7a t\u01b0 duy; h\u1ecdc thuy\u1ebft v\u1ec1 t\u01b0 duy logic, v\u1ec1 suy lu\u1eadn tr\u00ean c\u01a1 s\u1edf c\u00e1c m\u1ec7nh \u0111\u1ec1 \u0111\u00e3 cho. Ti\u1ebfng Anh \u0111\u01b0\u1ee3c vi\u1ebft: <em>Logic and the knowledge of nature<\/em>. Logic c\u0169ng c\u00f3 ngh\u0129a l\u00e0 logic h\u1ecdc. Ban \u0111\u1ea7u t\u00f4i c\u0169ng vi\u1ebft &#8220;logic&#8221; th\u00f4i v\u00e0 c\u00f3 ch\u1ee7 gi\u1ea3i, cho g\u1ecdn. M\u00f4n logic \u0111\u00e3 \u0111\u01b0\u1ee3c d\u1ea1y r\u1ed9ng r\u00e3i t\u1ea1i c\u00e1c \u0111\u1ea1i h\u1ecdc Trung c\u1ed5 nh\u01b0 m\u1ed9t s\u1ef1 b\u1eaft bu\u1ed9c \u0111\u1ec3 r\u00e8n luy\u1ec7n n\u0103ng l\u1ef1c t\u01b0 duy con ng\u01b0\u1eddi.<\/p>\n<p><a href=\"#_ftnref2\" name=\"_ftn2\">[2]<\/a> T\u00f4i l\u01b0\u1ee3t b\u1edbt m\u1ed9t th\u00ed d\u1ee5 li\u00ean quan \u0111\u1ebfn sinh h\u1ecdc \u0111\u1ec3 t\u1eadp trung h\u01a1n.<\/p>\n<p><a href=\"#_ftnref3\" name=\"_ftn3\">[3]<\/a> Th\u00ed nghi\u1ec7m Michelson g\u00f3p ph\u1ea7n v\u00e0o vi\u1ec7c x\u00e2y d\u1ef1ng ti\u00ean \u0111\u1ec1 cho Thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i h\u1eb9p nhi\u1ec1u h\u01a1n l\u00e0 thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i r\u1ed9ng.<\/p>\n<p><a href=\"#_ftnref4\" name=\"_ftn4\">[4]<\/a> C\u00e1c \u0111\u1ecbnh lu\u1eadt v\u1eadt l\u00fd \u0111\u1ec1u ph\u1ea3i xu\u1ea5t ph\u00e1t t\u1eeb kinh nghi\u1ec7m, r\u1ed3i tr\u1edf v\u1ec1 kinh nghi\u1ec7m nh\u01b0 Einstein n\u00f3i: &#8220;T\u1ea5t c\u1ea3 tri th\u1ee9c v\u1ec1 th\u1ef1c t\u1ea1i xu\u1ea5t ph\u00e1t t\u1eeb kinh nghi\u1ec7m, v\u00e0 cu\u1ed1i c\u00f9ng tr\u1edf v\u1ec1 n\u00f3&#8221;. \u0110i\u1ec1u \u0111\u00f3 \u0111\u00fang, nh\u01b0ng b\u1eb1ng con \u0111\u01b0\u1eddng n\u00e0o? Einstein quan ni\u1ec7m r\u1eb1ng ph\u1ea3i th\u00f4ng qua con \u0111\u01b0\u1eddng \u201ctr\u01b0\u1edbc t\u00e1c t\u1ef1 do\u201d (<em>freie Sch\u00f6pfung<\/em>) c\u1ee7a tinh th\u1ea7n con ng\u01b0\u1eddi, <em>kh\u00f4ng th\u1ec3 \u0111\u01b0\u1ee3c r\u00fat ra t\u1eeb nh\u1eefng tr\u1ea3i nghi\u1ec7m gi\u00e1c quan m\u1ed9t c\u00e1ch quy n\u1ea1p<\/em>. \u0110\u00f3 l\u00e0 ph\u1ea7n s\u00e1ng t\u1ea1o c\u1ee7a m\u1ed7i nh\u00e0 khoa h\u1ecdc. Kh\u00f4ng c\u00f3 con \u0111\u01b0\u1eddng suy lu\u1eadn logic thu\u1ea7n t\u00fay n\u00e0o d\u1eabn tr\u1ef1c ti\u1ebfp t\u1eeb kinh nghi\u1ec7m t\u1edbi c\u00e1c \u0111\u1ecbnh lu\u1eadt. C\u0169ng kh\u00f4ng c\u00f3 con \u0111\u01b0\u1eddng b\u1eb1ng thuy\u1ebft ti\u00ean nghi\u1ec7m, a priori, c\u1ee7a Kant. \u0110\u00e2y l\u00e0 m\u1ed9t \u0111\u1ec1 t\u00e0i quan tr\u1ecdng, t\u00f4i hy v\u1ecdng s\u1ebd c\u00f3 d\u1ecbp tr\u1edf l\u1ea1i sau tr\u00ecnh b\u00e0y nhi\u1ec1u h\u01a1n.<\/p>\n<p><a href=\"#_ftnref5\" name=\"_ftn5\">[5]<\/a> Immanual Kant tr\u00ecnh b\u00e0y \u00fd t\u01b0\u1edfng nh\u1eadn th\u1ee9c ti\u00ean nghi\u1ec7m (a priori) trong quy\u1ec3n s\u00e1ch <em>Ph\u00ea ph\u00e1n L\u00fd t\u00ednh thu\u1ea7n t\u00fay<\/em>. Xem b\u1ea3n d\u1ecbch c\u1ee7a h\u1ecdc gi\u1ea3 B\u00f9i V\u0103n Nam S\u01a1n.<\/p>\n<p><a href=\"#_ftnref6\" name=\"_ftn6\">[6]<\/a> C\u00f3 c\u00f9ng d\u1ea1ng v\u00e0 k\u00edch c\u1ee1<\/p>\n<p><a href=\"#_ftnref7\" name=\"_ftn7\">[7]<\/a> Xem s\u00e1ch <em>Thuy\u1ebft t\u01b0\u01a1ng \u0111\u1ed1i h\u1eb9p v\u00e0 r\u1ed9ng<\/em> c\u1ee7a Einstein, ch\u01b0\u01a1ng 6 v\u00e0 13, do Nguy\u1ec5n Xu\u00e2n Xanh chuy\u1ec3n ng\u1eef. Nxb T\u1ed5ng h\u1ee3p TP HCM. Einstein gi\u1ea3i th\u00edch r\u00f5 h\u01a1n.<\/p>\n<p><a href=\"#_ftnref8\" name=\"_ftn8\">[8]<\/a> 1 Vgl. \u00dcber das Unendliche, Mathem. Ann. 95; Die Grundlagen der Mathematik, Abh. a. d. mathem. Sem. d. Hamburgischen Universit\u00e4t 6. (ch\u00fa th\u00edch c\u1ee7a Hilbert)<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NH\u1eacN TH\u1ee8C T\u1ef0 NHI\u00caN V\u00c0 LOGIC H\u1eccC NATURERKENNEN UND LOGIK David Hilbert Di\u1ec5n t\u1eeb t\u1ea1i K\u00f6nigsberg, 1930 Nguy\u1ec5n Xu\u00e2n Xanh s\u01b0u t\u1ea7m v\u00e0 chuy\u1ec3n ng\u1eef N\u1ebfu b\u1ea1n c\u00f3 th\u1ec3 nh\u00ecn v\u00e0o h\u1ea1t gi\u1ed1ng c\u1ee7a th\u1eddi gian, V\u00e0 n\u00f3i h\u1ea1t n\u00e0o s\u1ebd ph\u00e1t tri\u1ec3n, v\u00e0 h\u1ea1t n\u00e0o s\u1ebd kh\u00f4ng, Th\u00ec h\u00e3y n\u00f3i chuy\u1ec7n v\u1edbi t\u00f4i. William<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/posts\/8864"}],"collection":[{"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/comments?post=8864"}],"version-history":[{"count":112,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/posts\/8864\/revisions"}],"predecessor-version":[{"id":11488,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/posts\/8864\/revisions\/11488"}],"wp:attachment":[{"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/media?parent=8864"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/categories?post=8864"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rosetta.vn\/nguyenxuanxanh\/wp-json\/wp\/v2\/tags?post=8864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}