tag:blogger.com,1999:blog-21308815.post6877601255724260587..comments2023-10-08T15:51:17.426+00:00Comments on Beyond Necessity: On the existence of root twoEdward Ockhamhttp://www.blogger.com/profile/07583379503310147119noreply@blogger.comBlogger45125tag:blogger.com,1999:blog-21308815.post-34147548342239300342012-04-21T06:33:06.757+00:002012-04-21T06:33:06.757+00:00>>And what is a finite number of steps, but ...>>And what is a finite number of steps, but a number of steps which can be completely counted?<br /><br />>>>> That is the essence of Cantor's original argument.<br />>>Which argument?<br /><br />I hope the <a href="http://ocham.blogspot.co.uk/2012/04/completing-infinite.html" rel="nofollow">latest post</a> may answer thisEdward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-68463883884404498772012-04-20T22:40:45.110+00:002012-04-20T22:40:45.110+00:00>> >> An infinite procedure is, by def...>> >> An infinite procedure is, by definition, a procedure which can never be completed<br /><br />>> No, an infinite process is one that can never be completed in a finite number of steps.<br /><br />And what is a finite number of steps, but a number of steps which can be completely counted?<br /><br />>> That is the essence of Cantor's original argument.<br /><br />Which argument?<br /><br />--<br /><br />>> To be free to stipulate means just that one <i>isn't</i> bound by some notion of correctness.<br /><br />I don't get it. Bound in what way?<br /><br />If his definition isn't correct (e.g. if he defines "=" such that 1.999...=3), it won't lead to useful results.<br /><br />>> He outlines how to construct a complete ordered field<br /><br />You forgot the scare quotes around "construct". Gowers even includes them in his own use of the word, presumably because he realizes he isn't outlining how to actually construct anything.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-49103982081131022092012-04-20T19:14:11.354+00:002012-04-20T19:14:11.354+00:00I have changed the period after which moderation i...I have changed the period after which moderation is required to 10 days. <br /><br />The number of comments on this post is now the maximum in the history of this blog. (The previous record was set by an Ayn Rand post, not surprisingly. So, philosophy of mathematics = Objectivism.Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-65931433925474388512012-04-20T18:31:33.200+00:002012-04-20T18:31:33.200+00:00>> I think if we have bought the idea of an ...>> I think if we have bought the idea of an infinite sequence, i.e. that such sequences exist, then we have bought pretty much everything that was for sale, no? <<<br /><br />I agree. If there is a stumbling block it's the infinite sequences.<br /><br />>> He is free to stipulate whatever he wants. But that doesn't mean he is correct. <<<br /><br />To be free to stipulate means just that one <i>isn't</i> bound by some notion of correctness.<br /><br />>> If anything he showed a somewhat more concrete realization of one type of real number - the square root of a rational number <<<br /><br />No, he does more. He outlines how to construct a complete ordered field---the article isn't a textbook---it's intended for mathematicians who can fill in the details---and shows how it contains an element that squares to 2. <br /><br />>> What would be the use of irrational numbers in a linear programming application... <<<br /><br />My rhetorical point was Should we deny ourselves, say, a 23 dimensional Euclidean space for theorising about the simplex algorithm in 23 variables because this 'space' is a spectacularly bad model for what we believe of physical space.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-14982110632018635112012-04-20T13:01:47.563+00:002012-04-20T13:01:47.563+00:00>> An infinite procedure is, by definition, ...>> An infinite procedure is, by definition, a procedure which can never be completed<br /><br />No, an infinite process is one that can never be completed in a <i>finite</i> number of steps. That is the essence of Cantor's original argument.Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-41859413191125679132012-04-20T12:08:08.632+00:002012-04-20T12:08:08.632+00:00"Anthony, having reread Gowers do you now see..."Anthony, having reread Gowers do you now see that he is completely free to stipulate that = means anything he likes with respect to his sequences?"<br /><br />He is free to stipulate whatever he wants. But that doesn't mean he is correct.<br /><br />"What he ends up with is a somewhat more concrete realisation of the real numbers."<br /><br />Not really. If anything he showed a somewhat more concrete realization of one type of real number - the square root of a rational number. If it were complete - and I argue it is not - then it would be a first step toward discussing real numbers in general. But only a first step, and not a particularly big one. There are only "countably many" square roots of rational numbers.<br /><br />His argument of 1.999...=2, were it complete, didn't even go that far, as 1.999..., if is a number and it equals 2, is a natural number.<br /><br />"So, if you don't buy into the existence of real numbers, but do buy rationals and infinite sequences, forget worries over reals and stick to Gowersian sequences."<br /><br />It's the infinite sequences I have a problem with. An infinite procedure is, by definition, a procedure which can never be completed. A completed infinity is a contradiction in terms.<br /><br />This doesn't mean that the square root of two does not exist. But it does mean that any construction of the square root of two which relies on an infinite sequence is not valid.<br /><br />"Could [philosophy] also be the study of the possible?"<br /><br />Yes, philosophy involves studying the possible, as well as the actual. But for something to be possible, there must be some evidence for it. "Possible" does not mean merely imaginable. Someone who claims that "anything is possible" is using an improper definition of "possible".Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-43447982636661393822012-04-20T07:38:05.711+00:002012-04-20T07:38:05.711+00:00>>but do buy rationals and infinite sequence...>>but do buy rationals and infinite sequences<br /><br />I think if we have bought the idea of an infinite sequence, i.e. that such sequences exist, then we have bought pretty much everything that was for sale, no?Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-52994973816820483042012-04-19T19:36:47.297+00:002012-04-19T19:36:47.297+00:00Also for David: What would be the use of irration...Also for David: What would be the use of irrational numbers in a linear programming application used for optimizing economic allocation problems?<br /><br />What is being optimized, anyway? Presumably it is something which is somehow "tied to the physical world".<br /><br />If you're talking about the notion of cardinal utility functions, then I reject that theory altogether.<br /><br />If not, then where do the irrational numbers come into play, and what is their meaning with regard to thing which is being optimized?Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-4437591029662155992012-04-19T15:08:12.257+00:002012-04-19T15:08:12.257+00:00Anthony, having reread Gowers do you now see that ...Anthony, having reread Gowers do you now see that he is completely free to stipulate that <b>=</b> means anything he likes with respect to his sequences? It's important that you get this. Of course, he wants it to mean something that bears a strong resemblance to <i>=</i>, and lo, he finds that there is a way of doing it. What he ends up with is a somewhat more concrete realisation of the real numbers. So, if you don't buy into the existence of real numbers, but do buy rationals and infinite sequences, forget worries over reals and stick to Gowersian sequences. They do the same job. The whole exercise is a typical bit of mathematical construction work, not too far removed from the way (something that can do the job of) the integers can be built as equivalence classes of pairs of naturals. Can you see that it isn't 'defining something into existence'? It's making it out of other stuff. There's nothing axiomatic at all in Gowers's piece.<br /><br />>> And philosophy is a study of the actual, not a study of the imaginary. <<<br /><br />Could it also be the study of the possible?<br /><br />Regarding 'well-definedness', this term has at least two usages I can think of. The first is in the avoidance of empty names in proofs. 'Let P be the point where lines AB and CD intersect' can get you into a lot of trouble if you don't at first show that AB and CD <i>do</i> intersect. Those amusing 'proofs' of geometrical and other impossibilities often rely on some name not being well-defined. Second is when you want to define some operation on equivalence classes by operating on arbitrary representatives. You have to show that the result class is independent of which representatives you choose, else your operation is not 'well-defined'. 'Well-definedness' is important but not the whole story.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-34090063763482986392012-04-19T14:11:56.637+00:002012-04-19T14:11:56.637+00:00In any case, emotions exist, but emotions are not ...In any case, emotions exist, but emotions are not entities, they are phenomena. David was talking about the existence of entities (which he called "mathematical entities"), not phenomena. Though I suppose maybe by "mathematical entities" he is talking about phenomena, and not entities. We'll have to ask him.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-30746313201972301212012-04-19T14:00:32.802+00:002012-04-19T14:00:32.802+00:00Perhaps I misunderstood David. David, how would y...Perhaps I misunderstood David. David, how would you define "physical", and how would you define "tied to the physical"?Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-58911461173752828882012-04-19T13:54:30.869+00:002012-04-19T13:54:30.869+00:00>> It is a reasonable question to ask whethe...>> It is a reasonable question to ask whether there are non-physical things (for example, are emotions physical or not)?<br /><br />Surely emotions are "tied to the physical world", which I put in quotes because it was a phrase used by David. A better phrasing would probably be "perceived or abstracted from that which we perceive".Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-87491228610175601692012-04-19T13:35:44.351+00:002012-04-19T13:35:44.351+00:00>>Call it a "tie to the physical world&...>>Call it a "tie to the physical world". <br /><br />That's quite unnecessary for "existence". It is a reasonable question to ask whether there are non-physical things (for example, are emotions physical or not)? Regardless of the answer, if the question makes sense at all, then 'existence' per se has no tie to the physical.Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-16869940463527455442012-04-19T12:40:42.054+00:002012-04-19T12:40:42.054+00:00FWIW, I reread the Gowers presentation, and I did ...FWIW, I reread the Gowers presentation, and I did find it to be a good (*) presentation that infinite decimals are in fact well-defined. It seems that he (and perhaps also you?) is saying that "[mathematical] existence" is tantamount to well-definedness. On the other hand, I am saying that existence requires more than just well-definedness. Call it a "tie to the physical world". Call it the application of the principle of parsimony. These two are, I believe, aspects of the same thing, and part of the meaning of "existence".<br /><br />(*) I wouldn't call it a proof as it includes some missing details, but I found no reason to doubt that those details have been proven elsewhere.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-87101415582066916392012-04-19T12:01:06.511+00:002012-04-19T12:01:06.511+00:00It's unclear to me what "mathematical exi...It's unclear to me what "mathematical existence" means if it doesn't mean "existence". If mathematical axioms are arbitrary, and one just defines entities into existence, then there's no point in discussing whether or not anything "mathematically exists". You just define it to exist, and it exists.<br /><br />It's also unclear to me what "a philosophy of mathematics" is if it is not a branch of philosophy. And philosophy is a study of the actual, not a study of the imaginary.<br /><br />Finally, you say "Where Gowers is maybe a bit quick is in proving that this correspondence works." The problem that I was addressing was with his claim that he didn't have to prove anything.<br /><br />As for whether or not 1.999...=2, I accept that if 1.999... exists, then 1.999... equals 2. On the other hand, the fact that the existence of reals implies the existence of 1.999... which implies that 1.999...=2, is a reason to question the existence of reals.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-14617051123420157292012-04-18T16:06:25.561+00:002012-04-18T16:06:25.561+00:00>> That's exactly what I'm questioni...>> That's exactly what I'm questioning. <<<br /><br />We are getting into the deep waters of what's meant by 'mathematical existence'. I doubt that a philosophy of mathematics would want to tie itself as strongly to the physical as you seem to be suggesting.* Would physical space not being Euclidean and three dimensional vitiate in any way the theory of linear programming (optimising economic allocation problems) which works in Euclidean space of arbitrary dimension? <br /><br />*Though in the paper that Belette recommended Newstead tells us that Aristotle's Phil of math is very much 'abstraction from the natural world'. How far can abstraction go?<br /><br /><br />>> He says "there can be no argument, since I am giving a definition". <<<br /><br />And that's right. Let's use bold type for sequence talk and italic type for real number talk. Gowers is defining the relations <b><</b> and <b>=</b> between sequences in such a way that there is a mapping between sequences and reals that preserves the relations, that is<br /><br /><b>a < b</b> iff <i>a < b</i><br /><b>a = b</b> iff <i>a = b</i><br /><br />when sequence <b>a</b> maps to real <i>a</i>, etc. True statements about sequences and their slightly weird ordering relations correspond with true statements about reals and their usual ordering relations. That's what I mean by saying that the sequences 'stand proxy' for real numbers. The structure of the bold entities and their relations 'fits on to' (is isomorphic with) the structure of italic entities and their relations. Where Gowers is maybe a bit quick is in proving that this correspondence works. But it does.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-17219639589417869732012-04-18T13:40:55.908+00:002012-04-18T13:40:55.908+00:00"So, in this respect at least, 1.999... and 2..."So, in this respect at least, 1.999... and 2.0... behave just like our pre-Gowers notion of the real number 2."<br /><br />You seem to be presenting an argument for 1.999...=2. Had Gowers done that, I wouldn't have "basically stopped paying serious attention" (I may or may not have been able to follow the argument, but I would have given it a try).<br /><br />But Gowers doesn't present an argument. Nor does he say that the proof is merely too big to fit within the margins, so he is omitting it for space considerations. He says "there can be no argument, since I am giving a definition".Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-11455695347229045762012-04-18T12:49:18.174+00:002012-04-18T12:49:18.174+00:00Besides, how do you know it's exactly root 2? ...Besides, how do you know it's exactly root 2? How do you know it's not root 2 rounded to the nearest planck length, or sub planck length?<br /><br />You could say that we inductively assume it's true, because we don't have any reason not to...but we do have reasons to doubt that standard euclidean geometry and physics work on the scale of the extremely small.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-91787070378545852932012-04-18T12:36:22.047+00:002012-04-18T12:36:22.047+00:00>> The [ratio of the] distance between the t...>> The [ratio of the] distance between the tips is then [a multiple of] root 2.<br /><br />Only in euclidean geometry, which the real world most emphatically is not.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-35940315721187797122012-04-18T12:34:48.984+00:002012-04-18T12:34:48.984+00:00>> "For a mathematical entity to exist,...>> "For a mathematical entity to exist, must it be 'instantiated', as it were, in the physical? "<br /><br />I'd remove the word "mathematical" from that sentence.<br /><br />>> "That would leave a lot of reals with 'nowhere to be'. Almost all of them, in fact."<br /><br />That's exactly what I'm questioning. I don't know enough about math to answer the question, but it seems to me that once we go beyond the "countably infinite" into the "uncountably infinite", we don't even have <i>potential</i> existence, let alone <i>actual</i> existence.<br /><br />Interestingly, from what I know of math and physics, it seems that imaginary numbers have more of a claim to existence than real numbers. But, again, I could just be missing something that the notion of the reals are actually useful for.Anthonyhttps://www.blogger.com/profile/15847046461397802596noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-914959382005100712012-04-17T15:49:16.242+00:002012-04-17T15:49:16.242+00:00>>every real is a multiple of root 2
Oh yes...>>every real is a multiple of root 2<br /><br />Oh yes silly me. You know what I meant. The ratio.Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-48968307553389555422012-04-17T14:30:37.848+00:002012-04-17T14:30:37.848+00:00Ed, every real is a multiple of root 2. We might ...Ed, every real is a multiple of root 2. We might say the <i>ratio</i> of the distance between the tips and the common length of the hands is root 2, but that's a ratio, not a length.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-52918104231468575022012-04-17T14:07:22.187+00:002012-04-17T14:07:22.187+00:00>>Can the length of anything in reality ever...>>Can the length of anything in reality ever actually be the square root of two?<br /><br />Well yes. The distance between the ends of the hands of a clock whose hands are the same length, at precisely 9 o'clock. You object that the hands cannot be exactly the same length. I reply, take any parts of the hands which are equidistant from the centre. <br /><br />The distance between the tips is then a multiple of root 2.Edward Ockhamhttps://www.blogger.com/profile/07583379503310147119noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-67329369362951010782012-04-17T13:06:24.148+00:002012-04-17T13:06:24.148+00:00>> for any sequence ε you like <<
oop...>> for any sequence ε you like <<<br /><br />oops. any sequence ε other than 0.0..., the latter being [0, 0, 0, 0,...] of course.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.comtag:blogger.com,1999:blog-21308815.post-67738969857108542892012-04-17T12:59:32.558+00:002012-04-17T12:59:32.558+00:00>> Okay, but at the atomic level, these theo...>> Okay, but at the atomic level, these theoretical underpinnings seem to be wrong. <<<br /><br />Yes, I have some sympathy with this. What on earth do we mean when we say that the quantum state of some system is represented by a ray in some Hilbert space over the complex numbers? How can the latter mathematical gobbledegook relate to the actual world? One answer is that it offers a calculus for predicting what happens in the real world that seems to work very well indeed.David Brightlyhttps://www.blogger.com/profile/06757969974801621186noreply@blogger.com