tag:blogger.com,1999:blog-20067416.post4904550219896234256..comments2024-09-19T03:58:58.450-04:00Comments on Recursivity: Can Irrational Numbers be Represented in a Computer?Unknownnoreply@blogger.comBlogger22125tag:blogger.com,1999:blog-20067416.post-41920492227118790602020-10-04T11:41:14.650-04:002020-10-04T11:41:14.650-04:00Well only one example is possible, which is the nu...Well only one example is possible, which is the number specified by what I said:<br />0.100100001000000100....<br />It has a 1 in the first digit after the decimal point, in the 4th, in the 9th, etc. and 0's everywhere else.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-28781739463725911872020-10-02T11:30:32.020-04:002020-10-02T11:30:32.020-04:00can you explain what you mean by "consider th...can you explain what you mean by "consider the base-10 number defined by having a "1" in the i'th digit if i is a perfect square (like 1,4,9,16, etc.), and 0 otherwise", and give an example?Anonymoushttps://www.blogger.com/profile/17274277069952461507noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-5670424744476059462020-03-29T07:25:17.895-04:002020-03-29T07:25:17.895-04:00You're making two fundamental mistakes here:
...You're making two fundamental mistakes here:<br /><br />#1: confusing the decimal representation of a number with the number itself. These are two different things.<br /><br />#2: pi is an irrational number, and this fact has nothing to do with how many digits you calculate it to.<br /><br />What does "the computer cannot identify irrational numbers" mean? Here is one model where this is clearly false. Given an automaton M that on input n in base k, returns the n'th digit of a number x written in base b, call x(M) the resulting real number. Then there is a decision procedure that, on input M, tells you whether x(M) is irrational.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-37375110334808359662020-03-29T00:41:35.448-04:002020-03-29T00:41:35.448-04:00I'd love to see you try and get the computer t...I'd love to see you try and get the computer to plot the dirichlet function. You gave a pretty nice explanation, but even if you calculate pi upto a trillion digits, it is still rational. The computer cannot identify irrational numbers , like pi, because if it does, it technically should never stop computing it. For as soon as it does, pi becomes rationalFarhan Islamhttps://www.blogger.com/profile/04530373576683625004noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-85707903252763938802013-11-08T20:08:27.365-05:002013-11-08T20:08:27.365-05:00Isn't there this ``interval-based'' re...<i>Isn't there this ``interval-based'' representation that can represent any real numbers with arbitrary precision?</i><br /><br />Yes, but it is not the slightest bit relevant to what we are discussing.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-83496454414418141982011-05-15T19:51:55.942-04:002011-05-15T19:51:55.942-04:00Isn't there this ``interval-based'' re...Isn't there this ``interval-based'' representation that can represent any real numbers with arbitrary precision?<br /><br />-A Nanny MouseAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-13834976997314894812011-05-09T16:55:42.036-04:002011-05-09T16:55:42.036-04:00To: Takis Konstantopoulos:
The religious apologet...To: Takis Konstantopoulos:<br /><br />The religious apologetic is definitely<br />a Western phenomenon, though it could be picked up by Catholics, the idea seems to be practised largely in the Protestant world. <br /><br />In the US religion is a kind of product that has its salesmen. Churches come in a huge variety.<br /><br />As with any other type of consumerism, the salesman has to convince the potential customer that he/she has a need that will be met by the product, and that not having the product will make you somehow "deficient".<br /><br />This is different from places where there is one dominant church, particularly if that church is supported by the government, or in some way intermingled with the government. In this way, the Church is more of an institution and less of a product.John Stockwellhttps://www.blogger.com/profile/03496308585336775569noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-83840852523829886082011-05-08T05:40:38.240-04:002011-05-08T05:40:38.240-04:00John Stockwell:
Religious people still ask such n...John Stockwell:<br /><br />Religious people still ask such naive questions: Have you ever seen an ape-human? No? So evolution can't be true.<br /><br />I have a question. Even though I did grow up in a religious country (Greece, where religion is not separated from the state), I've never heard such things. It therefore appears that utter stupidies of the above type are arguments used by specific groups of religious fundamentalists.Takis Konstantopouloshttps://www.blogger.com/profile/14675216467783238403noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-41838046391184459952011-05-02T20:22:35.679-04:002011-05-02T20:22:35.679-04:00Monton is exhibiting apologist thinking. There ar...Monton is exhibiting apologist thinking. There are only two types of apologist argument. The first might be called an "argument from the credulity of the opponent", and the other is an "argument from ignorance". <br /><br />When I was a kid, religious apologists would ask "have you ever seen an atom", implying that we take atoms on faith, and should take other things we can't see on faith, as well.<br /><br />The second argument was "bumble bees cannot fly". This was an argument from the ignorance in that because the best models of flight at the time could <br />not account for the disparate wing loading/mass ratio of bumblebees.<br /><br />Of course, neither of these notions work as apologetics now, because the nonlinear aspects of<br />insect flight were studied decades ago, and atoms are easily imaged via scanning-tunneling microscopes.John Stockwellhttps://www.blogger.com/profile/03496308585336775569noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-32518843787325657732011-04-30T13:17:50.047-04:002011-04-30T13:17:50.047-04:00I think the bigger problem is that if you want to ...I think the bigger problem is that if you want to speak sensibly you have to talk about representing classes of numbers, not representing individual numbers!Sniffnoynoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-36829854134131169102011-04-29T23:03:53.871-04:002011-04-29T23:03:53.871-04:00"Storing each individual bit isn't the on..."Storing each individual bit isn't the only way to represent an algebraic number."<br /><br />I know that. But my take on the context is that Monton believes that it is.Johnhttps://www.blogger.com/profile/10876775111703252840noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-58165990331174930922011-04-29T15:52:00.577-04:002011-04-29T15:52:00.577-04:00After all, the exponent in universal gravitation i...After all, the exponent in universal gravitation is 2, but nobody says that the fact that it is such a simple integer is evidence we are living in a simulation.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-3622547069603276142011-04-29T15:50:25.117-04:002011-04-29T15:50:25.117-04:00Or, just as well, it could be evidence that the fu...Or, just as well, it could be evidence that the fundamental constants are themselves the result of some simple natural algorithmic process. I don't think either one is decidable.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-5950256461146177922011-04-29T15:45:11.154-04:002011-04-29T15:45:11.154-04:00I'm inclined to somewhat agree with his point ...I'm inclined to somewhat agree with his point in that context. If it did turn out that there were a handful of basic constants in the universe and they were all rational numbers with a power of 2 in the denominator, this would be evidence that we exist in a very naive simulation. But this would be *weak* evidence.<br /><br />Incidentally, does he propose a method for how we are supposed to even determine if a fundamental constant is irrational? That seems impossible given measurement constraints.Joshuahttps://www.blogger.com/profile/00637936588223855248noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-26487654724558112022011-04-29T15:24:49.815-04:002011-04-29T15:24:49.815-04:00The context was a discussion of whether we are liv...The context was a discussion of whether we are living in a computer simulation; he argues that if we discover the fundamental constants have decimal representations that end at 16 digits, this would be evidence in favor of this, and if the fundamental constants are irrational, this would be evidence against this. <br /><br />I think this is an extremely naive view of what is possible or not possible with computers.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-32264564334724252482011-04-29T14:31:27.080-04:002011-04-29T14:31:27.080-04:00So why was Monton making these claims? My guess wo...So why was Monton making these claims? My guess would have been along the lines of what Keith guessed. That seems to be a not uncommon line of argument among philosophers who want to argue that there's something intrinsically distinct about what humans do from what machines do. I have trouble coming up with other reasons an ID book would talk about this otherwise.Joshuahttps://www.blogger.com/profile/00637936588223855248noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-16575885449413780322011-04-29T10:39:55.197-04:002011-04-29T10:39:55.197-04:00John:
Actually, the base-2 representation of sqrt...John:<br /><br />Actually, the base-2 representation of sqrt(2) <i>can</i> be accurately represented - which was one of my points. Storing each individual bit isn't the only way to represent an algebraic number.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-63456165425188695572011-04-29T10:38:11.896-04:002011-04-29T10:38:11.896-04:00KeithB:
You'd lose that bet. The passage had...KeithB:<br /><br />You'd lose that bet. The passage had nothing to do with human or machine understanding.<br /><br />I don't even know what you are trying to say when you say "the underlying *hardware* can't understand the concept of sqrt(2)".Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-83962335528080051732011-04-29T10:17:50.141-04:002011-04-29T10:17:50.141-04:00John, I'll bet he was claiming that a human ca...John, I'll bet he was claiming that a human can understand the abstract concept of sqrt(2), while a computer can only understand a binary expansion to some number of decimal places.<br /><br />And Jeffery:<br />If the claim that the underlying *hardware* can't understand the concept of sqrt(2) he has a point. (Unless FPU's have a new capability I am not aware of.) However, since my neuron's don't have a concept of sqrt(2), I don't see his point.KeithBnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-3347529880440090982011-04-28T22:07:46.415-04:002011-04-28T22:07:46.415-04:00"In a computer (as we understand computers, a..."In a computer (as we understand computers, at least), irrational numbers can't be fully accurately represented."<br /><br />If this means the base-10 (or base-2) representation (I'm pretty sure it probably does), then it can't be accurately represented on paper, in our brains or anywhere else. So what was his point?Johnhttps://www.blogger.com/profile/10876775111703252840noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-63931707397461936792011-04-28T16:51:13.823-04:002011-04-28T16:51:13.823-04:00Of course!Of course!Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-50133820064864578652011-04-28T16:41:19.375-04:002011-04-28T16:41:19.375-04:00Instead of "all but a set of measure zero&quo...Instead of "all but a set of measure zero" can't we use the stronger claim of "all but a countable set"?Joshuahttps://www.blogger.com/profile/00637936588223855248noreply@blogger.com