{"id":178,"date":"2023-11-22T22:34:27","date_gmt":"2023-11-22T22:34:27","guid":{"rendered":"https:\/\/experimentalfrontiers.peachpuff-wolverine-566518.hostingersite.com\/?p=178"},"modified":"2023-11-24T16:25:12","modified_gmt":"2023-11-24T16:25:12","slug":"178","status":"publish","type":"post","link":"https:\/\/scienceblog.com\/experimentalfrontiers\/2023\/11\/22\/178\/","title":{"rendered":"What G\u00f6del Wrought"},"content":{"rendered":"<p data-pm-slice=\"1 1 []\"><em>The Enlightenment was a three-century European movement to tame the world under the reign of Reason. It began with Ren\u00e9 Descartes and Isaac Newton in the 17th century, expanding the domain of science; in the 18th century, Emmanuel Kant sought to rationalize faith itself, with his monograph, <\/em><a href=\"https:\/\/philpapers.org\/rec\/KANRWT#:~:text=A%20work%20of%20major%20importance,needs%20of%20an%20ethical%20life.\" target=\"_blank\" rel=\"noopener noreferrer nofollow\"><em>Religion within the Limits of Reason Alone<\/em><\/a><em>. In the 19th century, the Enlightenment really caught fire, and John Stuart Mill created (what he claimed as) an ethics of pure reason. It became fashionable to imagine that science would conquer all, and the world would be understood as a vast machine.\u00a0<\/em><\/p>\n<p><em>Some say that the Enlightenment faced a crisis with Nietzsche, and some say that the Enlightenment has yet to come to grips with quantum mechanics. I would argue that the Enlightenment ended in 1931, when a young Austrian mathematician named Kurt G\u00f6del wrote his epitaph on the tomb of All-Powerful Reason.<\/em><\/p>\n<hr \/>\n<h2>Whitehead and Russell<\/h2>\n<p>The story\u2019s prelude was in the 1910s, when a young Bertrand Russell teamed up with the foremost philosopher of his time, Alfred North Whitehead to write an encyclopedic, three-volume exposition of mathematical logic. The conquest of the world by Reason would begin with the conquest of mathematics. <strong><em>Every mathematical statement is either true or false. The way we can know this is that for every true statement there is a mathematical proof<\/em><\/strong>, derivable from a small number of postulates = self-evident statements about arithmetic. Examples of postulates are \u201cZero is a number\u201d; \u201cEvery number has a successor, which is also a number.\u201d Can\u2019t argue with that.<\/p>\n<p>Just as there are rules for arithmetic, there are rules for logic, and the latter can be used to rationalize the former. Whitehead and Russell set out to prove the bolded statement above \u2014 that every true statement could be proven. In practice, this is something that mathematicians had assumed since the time of Euclid. If a statement seemed true, it was reasonable to search for a proof without fear that the search for proof of a true statement would be doomed from the start. The very idea that Russell and Whitehead could question such a thing would have seemed like nitpicking to mathematicians just a few years earlier.<\/p>\n<p>But three volumes later, W &amp; R expressed frustration. Several times, they had felt themselves to be on the cusp of a proof, but there was an elusive gap that they could never quite close. They put the project aside in 1913, as Russell became an outspoken voice for peace during the Great War.<\/p>\n<blockquote><p>\u201cRussell&#8217;s activism against British participation in World War I led to fines, a loss of freedom of travel within Britain, and the non-renewal of his fellowship at Trinity College, Cambridge, and he was eventually sentenced to prison in 1918 on the tenuous grounds that he had interfered in British foreign policy.\u201d [<a href=\"https:\/\/en.wikipedia.org\/wiki\/Bertrand_Russell%27s_political_views\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">Wikipedia<\/a>]<\/p><\/blockquote>\n<h2>G\u00f6del\u2019s Bombshell<\/h2>\n<p>Before G\u00f6del, it was intuitively clear to everyone that any meaningful mathematical statement must be either true or false, \u201cthe law of the excluded middle\u201d. True statements can be proved, and false statements can be disproved.<\/p>\n<p>Not so, said G\u00f6del. There are an infinity of statements about which we may never know whether they are true or false.<\/p>\n<blockquote>\n<ul>\n<li>For example, \u201cThere is only one way to express any number as the product of prime numbers.\u201d This is a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Fundamental_theorem_of_arithmetic\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">classical truth<\/a>, proved in antiquity.<\/li>\n<li>For example, \u201cThere is an infinite number of Pythagorean Triples = sets of three integers {A, B, C} such that A<sup>2<\/sup> + B<sup>2<\/sup> = C<sup>2<\/sup> \u201d True and proven. It\u2019s true even if you don\u2019t count multiples of smaller Pythagorean Triples as distinct.<\/li>\n<li>For example, \u201cNo set of four numbers {A, B, C, N} exists such that A<sup>N<\/sup> + B<sup>N<\/sup> = C<sup>N<\/sup> except if N=2.\u201d This was called \u201c<a href=\"https:\/\/en.wikipedia.org\/wiki\/Wiles%27s_proof_of_Fermat%27s_Last_Theorem\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">Fermat\u2019s last theorem<\/a>\u201d and no one knew whether it was true or false until 1994. It\u2019s true.<\/li>\n<li>For example, \u201cFor every even number E, you can find two prime numbers that add up to E.\u201d This seems to be true, but no one knows for sure.<\/li>\n<\/ul>\n<\/blockquote>\n<p>How can simple integers defy our logical assault? How is it possible to know what can and cannot be known?<\/p>\n<p>I didn\u2019t understand what G\u00f6del\u2019s theorem was about until I read a book by <a href=\"https:\/\/www.amazon.com\/G%C3%B6dels-Proof-Ernest-Nagel\/dp\/0814758371\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">Nagel and Newman<\/a>. I think it doesn\u2019t take a book to understand G\u00f6del\u2019s proof, and if you have a little acquaintance with math and a nerdy disposition, it\u2019s fun.<\/p>\n<h2>G\u00f6del\u2019s Proof \u2014 a Cook\u2019s Tour<\/h2>\n<h3>1. Countable infinities<\/h3>\n<p>I\u2019ll start with something much simpler, but counter-intuitive in its own right. The simplest infinity, the \u201csmallest\u201d infinity, is called by mathematicians the \u201cnatural numbers\u201d, {0, 1, 2, 3, \u2026}. This is a \u201ccountable infinity\u201d.<\/p>\n<p>The next larger infinity you might imagine is the \u201crational numbers\u201d, which are numbers that can be expressed as a fraction like \u00bd or 7936\/55. Intuitively, there are a whole lot more rational numbers than natural numbers. But, no! The infinity of rational numbers is the same as the infinity of counting numbers. The proof is that you can \u201ccount\u201d them, meaning you can set them in an order such that you\u2019re sure you\u2019ll mention each one at least once. Here\u2019s how.<\/p>\n<p>0\/1, 1\/1, 1\/2, 2\/1, 1\/3, 2\/3, 3\/3, 3\/1, 3\/2, 3\/3, \u2026<\/p>\n<p>The trick is to make all the fractions you can just with 0 and 1; then with 0, 1, and 2; then with 0, 1, 2, and 3; then keep expanding outward, using one more integer each time and listing all the possibilities.<\/p>\n<p>But suppose you add irrational numbers \u2014 all the square roots and cube roots and numbers you might not even be able to write down that way, like the number that satisfies the equation 4x<sup>4<\/sup> + 7x<sup>3<\/sup> &#8211; 6x<sup>2<\/sup> + 19x = 5. This set is called the \u201calgebraic numbers\u201d, and again we would think there is a greater infinity of algebraic numbers than counting numbers, but again our intuition fails. The infinity of algebraic numbers is the same as the infinity of counting numbers. The proof is an extension of what we just did.<\/p>\n<p>Suppose you add in the numbers that don\u2019t solve any algebraic equation, numbers like \u03c0 and <em>e<\/em>. They are infinite, non-repeating decimals, but unlike \u221a2, they are not solutions to any equation involving integers. These are called \u201ctranscendental numbers,\u201d and indeed they cannot be \u201ccounted\u201d. There is no ordering that you can make that would include all transcendental numbers. The number of all non-repeating decimals is an \u201cuncountable infinity\u201d, the next larger infinity after the counting numbers.<\/p>\n<h3>2. Countable statements, countable proofs<\/h3>\n<p>Any statement you can make about arithmetic can be coded using a small set of symbols. In addition to the digits, you have +, -, * and \u00f7.\u00a0 With these you can make statements like \u201c23 + 45 = 68\u201d. Add a few symbols like ( and ). You can use \u2283 for \u201cimplies\u201d and \u2203\u2026\u220b for \u201cthere exists a number\u2026 such that\u201d and \u2200, meaning \u201ctrue for all numbers\u201d and ~ for \u201cnot\u201d. Now you can code all the statements in the above list, and every other statement about integers.<\/p>\n<div class=\"pullquote\">\n<table bgcolor=\"yellow\">\n<tbody>\n<tr>\n<td>You might have fun defining \u201cprime\u201d using the symbols I\u2019ve listed. If this kind of thing is fun for you..<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>G\u00f6del began by assigning a number to each of the symbols that he needed for a language encompassing all statements about arithmetic of the integers. This is easy. There are only a few such symbols. Let\u2019s call the list A.<\/p>\n<\/div>\n<p>The next step was to combine these symbols into statements about mathematics. Using the same trick we used for rational numbers, he made an (infinite) list of all possible mathematical statements. This is List B.<\/p>\n<p>Note: The list is infinite, but a \u201ccountable\u201d infinity. The point is that he could assign a unique integer to every possible logical\/arithmetic statement.<\/p>\n<p>Another note: Mathematicians use letters for variables, but there are only 26 of them, and G\u00f6del needed an infinite supply. So he just used x with limitless subscripts \u2014 x<sub>1<\/sub>, x<sub>2<\/sub>, x<sub>3<\/sub>, \u2026<\/p>\n<p>Yet Another note: The vast majority of statements in this list are nonsense. For example, \u201c\u2203 x \u220b x+3=5\u201d means \u201cThere exists a number such that when you add it to 3 the result is 5.\u201d But\u00a0 \u201c\u2203\u2283\u2200\u2200\u220047\u201d is just a meaningless string of symbols. These meaningless statements are a \u201cwaste\u201d or inefficiency of our counting system, but what the heck \u2014 we have an infinite number of numbers, and we can afford to waste most of them, so long as we are sure we have covered all the meaningful statements.<\/p>\n<p>Next step: Once you have a list of statements, you can construct a list of lists of statements. This is list C. Every possible proof is in our list. Every possible proof has a number assigned to it. Of course, most of the numbers don\u2019t correspond to legitimate proofs. In fact, most of them don\u2019t make any sense at all, but again, that doesn\u2019t matter. Just so long as we are sure that every proof is assigned to a number, we don\u2019t care that most of the numbers don\u2019t correspond to a legit proof.<\/p>\n<p>We\u2019re almost done with the groundwork. One more list: This one is easier for us than for G\u00f6del because we are familiar with computers and the concept of an algorithm is an everyday idea. A \u201cfunction\u201d is a set of procedures, taking an input number (or several input numbers) and doing something to them, adding or dividing, possibly doing something else with the result. We\u2019ll need to add symbols for \u201cif\u201d and \u201cthen\u201d, \u201cand\u201d and \u201cor\u201d. But then we can imagine listing every possible computer program that takes a list of integers as an input and yields an answer as an output. This is list D. D for \u201cdone with the groundwork\u201d.<\/p>\n<h2>What can you do with all these lists?<\/h2>\n<p>This is where the magic begins. R &amp; W were looking to prove that every true statement about integers has an associated proof. For G\u00f6del to sink their boat, all he had to do was to come up with a single true statement about numbers for which no proof exists. Of course, it\u2019s relatively easy to find statements about numbers that seem to be true because we can\u2019t think of any counter-examples, but no proof has yet been discovered. But maybe there <strong><em>is<\/em><\/strong> such a proof, and no one has yet been clever enough to find it. How would you show that <strong><em>no proof is possible<\/em><\/strong>? That seems daunting.<\/p>\n<table bgcolor=\"yellow\">\n<tbody>\n<tr>\n<td>Pick a number, any number. If it\u2019s even, divide by 2. If it\u2019s odd, multiply by 3 and add 1. Repeat.<\/p>\n<p>If you try this, you\u2019ll find that whatever number you start with, you will always come back to 1.<\/p>\n<p>No one has found a number for which you <strong><em>don\u2019t<\/em><\/strong> come back to 1 eventually. Why not? If you get to 1, then you cycle around 1, 4, 2, 1, 4, 2, 1, \u2026 forever. Certainly there must be other cycles in which you can get caught like this, going around endlessly without ever escaping back to a lower or higher number.<\/p>\n<p>No one has found such a cycle, and no one has explained the reason why.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-pm-slice=\"1 1 []\">G\u00f6del\u2019s strategy started with the numbering process we illustrated above. Every statement about numbers was assigned a number (list A). Every proof was assigned a number (list B). Every function was assigned a number (list C). (A functions is a sequence of operations and conditional directions that take some input numbers and yield an output, equivalent to an algorithm, or computer program.)<\/p>\n<p>(In this article, I\u2019ve given a general idea how to construct these three lists, but G\u00f6del did so very explicitly, detailing the exact rules, so that given any statement, he could tell you what number it was on the list, and given a number on the list, he could tell you what statement (if any) corresponded to that number. For example, the rule for telling whether a given proof number corresponds to a given statement number was something G\u00f6del could write out explicitly.)<\/p>\n<p>Every statement <strong><em>about<\/em><\/strong> arithmetic has an associated number <strong><em>within<\/em><\/strong> arithmetic. So every proof of a statement about arithmetic corresponds to a proof about numbers, and <strong><em>that proof itself has an associated number<\/em><\/strong>.<\/p>\n<p>The relationship between a statement and its proof is a relationship between two numbers \u2014 the number of the statement from list A and the number of the proof from list B. And all possible relationships between numbers are listed in list C, so somewhere in list C is the procedure for determining whether a proof is legit.<\/p>\n<p>So now we have a glimmer of how we might prove that \u201cno proof exists\u201d. It\u2019s equivalent to saying that we can find a number <strong>a<\/strong> (corresponding to a statement from list A) for which there is no corresponding number <strong>b<\/strong> (corresponding to a proof of that statement from list B).<\/p>\n<p>G\u00f6del could write a function (number <strong>c<\/strong>) that tells you (yes or no, 1 or 0) whether a list of statements (number\u00a0 <strong>b<\/strong>) is a legitimate proof of statement number <strong>a<\/strong>. Using this, he could write a sentence\u00a0 made of math symbols that says, for all numbers <strong>b<\/strong>, each and every one is not a proof of a given statement, number <strong>a<\/strong>. And that sentence is a statement, and so it has a statement number from list A.<\/p>\n<p>Now, suppose that statement number happened to be <strong>a<\/strong> itself. Then the content of statement <strong>a<\/strong> would be that no proof of <strong>a<\/strong> exists. If statement <strong>a<\/strong> is true, we\u2019re in just the kind of trouble that W &amp; R hoped to avoid \u2014 it would be a true statement about numbers for which there was no proof. But suppose statement <strong>a<\/strong> is false. Then we are in much more trouble. All of math is in trouble, in fact, because it would mean that there <strong><em>does exist<\/em><\/strong> a proof of <strong>a<\/strong> even though <strong>a<\/strong> is false.\u00a0A single inconsistency is fatal to any system of logic.<\/p>\n<h2>What is the number that corresponds to the statement, \u201cthere is no proof of this statement\u201d?<\/h2>\n<p>The crux of G\u00f6del\u2019s idea was to create a mapping from statements about arithmetic to logical statements. Every logical statement had a counterpart in a statement about arithmetic (though not <em>vice versa<\/em>) so that he could come up with a statement about numbers that we know to be true because the corresponding logical statement is true, even though the statement can never be proved within the system of logic he started with.<\/p>\n<p>I described how he went about this, but in the above description, I have swindled you in the crucial last step in this proof. We have a statement that has the symbol <strong>a<\/strong> in it, a generic number, and we want to substitute for the symbol <strong>a <\/strong>a specific number, a position within the list of all statements corresponding to the statement itself. We can calculate the number corresponding to the statement we have (with the symbol <strong>a<\/strong>), and put that number into the statement where the symbol <strong>a<\/strong> was. But as soon as we substitute that number for the symbol, we have changed the statement, so it has a different number. The number <strong>a<\/strong> corresponds to the statement that has the <strong>variable<\/strong> in it, but to make the statement really apply to itself, it has to have <strong><em>its own list number<\/em><\/strong> instead of the variable.<\/p>\n<p>G\u00f6del was able to close the gap in this proof because he knew the numbering system, and so he knew how changing from one symbol to another symbol would affect the number. He calculated this explicitly, and then was able to calculate the number of the statement that had its own proper statement number embedded. It\u2019s analogous to what we do when we have an equation with x on both the left and right of the equation, but with a little algebraic manipulation we can \u201csolve for x\u201d.<\/p>\n<h2>So, the system of arithmetic contains just one true statement that can\u2019t be proved?<\/h2>\n<p>We might protest: Fine. This is just one statement. Why not add it to the list of postulates, and now the system no longer contains statements that are \u201ctrue but not provable from the postulates\u201d. G\u00f6del is ahead of us. Already in his first published paper, he shows that if we do this, then we can recreate his same proof, starting over with the expanded set of postulates, and come up with a different G\u00f6del statement that is true but not provable. And so forth, if we added this new statement to our postulates. So we can\u2019t have a complete body of provable theorems unless we\u2019re willing to start with an infinite set of postulates.<\/p>\n<p>This infinity of unprovable statements with expanded postulates are certainly not provable with the original set of postulates. So this logic shows there is not one but an infinity of true, unprovable statements about the integers.<\/p>\n<div class=\"pullquote\">\n<p>In their later years at the Institute for Advanced Study in Princeton, G\u00f6del and Einstein<br \/>\nwere frequent companions, walking on the campus and exchanging ethereal thoughts.<\/p>\n<\/div>\n<h2>G\u00f6del\u2019s incompleteness theorem<\/h2>\n<p>G\u00f6del\u2019s theorem is sometimes called the \u201cincompleteness theorem\u201d because it says our postulates are incomplete or, if you like, our knowledge of the integers must always be incomplete. It\u2019s never called the \u201cinconsistency theorem\u201d because no mathematician wants to doubt that he can trust proofs. No one thinks it possible that the G\u00f6del statement is actually false, but provable.<\/p>\n<p><strong>Impact \u2014 <\/strong>Sure, the transcendental poets and the romantic composers and the pointillist painters were already chipping away at the central claims of the Enlightenment in the 19th century. And the stunning appearance of quantum mechanics in 1926 confronted the objectivists with the reality that physics itself could never separate the observer from the observed.<\/p>\n<p>But it was left to G\u00f6del to show that mathematics \u2014 the core of the scientific worldview \u2014 was itself incomplete, and must forever remain so. The completion of G\u00f6del\u2019s revolution will be to integrate reason and the scientific approach with other human faculties such as intuition, faith, and clairvoyance. We are not yet there.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Enlightenment was a three-century European movement to tame the world under the reign of Reason. It began with Ren\u00e9 Descartes and Isaac Newton in the 17th century, expanding the domain of science; in the 18th century, Emmanuel Kant sought to rationalize faith itself, with his monograph, Religion within the Limits of Reason Alone. In &#8230; <a title=\"What G\u00f6del Wrought\" class=\"read-more\" href=\"https:\/\/scienceblog.com\/experimentalfrontiers\/2023\/11\/22\/178\/\" aria-label=\"Read more about What G\u00f6del Wrought\">Read more<\/a><\/p>\n","protected":false},"author":65,"featured_media":179,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-178","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-50"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.4 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>What G\u00f6del Wrought - Experimental Frontiers, with Josh Mitteldorf<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/scienceblog.com\/experimentalfrontiers\/2023\/11\/22\/178\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What G\u00f6del Wrought\" \/>\n<meta property=\"og:description\" content=\"The Enlightenment was a three-century European movement to tame the world under the reign of Reason. It began with Ren\u00e9 Descartes and Isaac Newton in the 17th century, expanding the domain of science; in the 18th century, Emmanuel Kant sought to rationalize faith itself, with his monograph, Religion within the Limits of Reason Alone. In ... 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The surprising fact that our bodies are genetically programmed to age and to die offers an enormous opportunity for medical intervention. It may be that therapies to slow the progress of aging need not repair or regenerate anything, but only need to interfere with an existing program of self-destruction. Mitteldorf has taught a weekly yoga class for thirty years. He is an advocate for vigorous self care, including exercise, meditation and caloric restriction. After earning a PhD in astrophysicist, Mitteldorf moved to evolutionary biology as a primary field in 1996. He has taught at Harvard, Berkeley, Bryn Mawr, LaSalle and Temple University. He is presently affiliated with MIT as a visiting scholar. In private life, Mitteldorf is an advocate for election integrity as well as public health. He is an avid amateur musician, playing piano in chamber groups, French horn in community orchestras. His two daughters are among the first children adopted from China in the mid-1980s. Much to the surprise of evolutionary biologists, genetic experiments indicate that aging has been selected as an adaptation for its own sake. This poses a conundrum: the impact of aging on individual fitness is wholly negative, so aging must be regarded as a kind of evolutionary altruism. Unlike other forms of evolutionary altruism, aging offers benefits to the community that are weak, and not well focussed on near kin of the altruist. This makes the mechanism challenging to understand and to model. more at http:\\\/\\\/mathforum.org\\\/~josh\",\"sameAs\":[\"http:\\\/\\\/AgingAdvice.org\"],\"url\":\"https:\\\/\\\/scienceblog.com\\\/experimentalfrontiers\\\/author\\\/joshmitteldorf\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"What G\u00f6del Wrought - Experimental Frontiers, with Josh Mitteldorf","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/scienceblog.com\/experimentalfrontiers\/2023\/11\/22\/178\/","og_locale":"en_US","og_type":"article","og_title":"What G\u00f6del Wrought","og_description":"The Enlightenment was a three-century European movement to tame the world under the reign of Reason. 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