tag:blogger.com,1999:blog-3722233.post538283051254228068..comments2017-06-24T09:52:46.896-04:00Comments on Computational Complexity: Why did 1+1=2 take Russell and Whitehead 300 pages?Lance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.comBlogger32125tag:blogger.com,1999:blog-3722233.post-15771428848823615482017-04-05T07:53:42.109-04:002017-04-05T07:53:42.109-04:00I come from philosophy and (i) Quine wrote his dis...I come from philosophy and (i) Quine wrote his dissertation on PM, so it certainly was a big influence on the most influential American philosopher of the 20th century, and (ii) I believe pretty much all of the logical notation I teach comes from PM.HenningShttp://www.blogger.com/profile/09989327109380649680noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-70656577494395646152016-12-16T18:27:46.281-05:002016-12-16T18:27:46.281-05:001=(a pencil I hold in my hand)
There is no way to ...1=(a pencil I hold in my hand)<br />There is no way to duplicate its molecular structure down to infinity, therefore there is no 1+1 as I have defined 1<br />We cannot prove anything equals anything and we have to improve beyond infinity, which we dont even have proof of, that the endless 9's after a decimal cannot also be divided and how much.<br />Can we even know, since we don’t know infinity?<br />Proofs are all amazingly good for humanity in that they are all subjective, but must be tested to become useful. But cant any proof become questionable in time. And 1+1=2 is less factual than 1+1 doesnt = 2D Lhttp://www.blogger.com/profile/00328176738196347189noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-7080389164814612252016-07-26T17:08:58.397-04:002016-07-26T17:08:58.397-04:00However, Socrates soon came to the conclusion that...However, Socrates soon came to the conclusion that he was not right for this sort of inquiry: his speculations so confused him that he began to unlearn everything he had previously thought he knew. For instance, Socrates no longer knows even how to give an account of how one and one equals two. He finds it hard to believe that the reason for their becoming two is simply the fact that they were brought together. Nor can he believe that when one is divided in two, the reason for its becoming two is the division. In the first case, one becomes two through addition, in the second case, one becomes two through division: how can both addition and division be the reasons for one becoming two? Utterly confused, Socrates rejected these explanations, seeking a method of his own instead.....Chuckhttp://www.blogger.com/profile/05238615891442189866noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-6753023676581649532016-06-15T19:45:19.596-04:002016-06-15T19:45:19.596-04:00Ok...if 1 + 1 does not equal 2 then how want child...Ok...if 1 + 1 does not equal 2 then how want children do I have that I can claim tax benefits for...I am kind of hoping that one of you can give an answer of say..10? I could then safley quit work and live off the benefits...Mark McNultyhttp://www.blogger.com/profile/05789176067213866285noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-34581886290689817462016-01-11T15:58:27.231-05:002016-01-11T15:58:27.231-05:00from the introduction to Principia Mathematica, wr...from the introduction to Principia Mathematica, written by Russell: "the chief reason in favour of any theory theory on the principles of mathematics must always be inductive, i.e. it must lie in the fact that theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence, the early deductions, until they reach this point, give reasons rather for believing the premisses because true consequences follow from them, than for believing the consequences because they follow from the premisses."Moiseshttp://www.blogger.com/profile/17820950882286476615noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-10581323603178609682015-12-03T21:08:42.230-05:002015-12-03T21:08:42.230-05:00I have proof of Equation 1+1=2 shorter, with beaut...I have proof of Equation 1+1=2 shorter, with beauty and great, yes first I proof it in 64 pages, and my second proof is 15 pages and in my 3rd proof is 3 pages etc.gerry pajarillonoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-67479762960290835462015-11-29T17:19:20.276-05:002015-11-29T17:19:20.276-05:00Actually, even in modulo 2, 1+1=2. It just also ha...Actually, even in modulo 2, 1+1=2. It just also happens to equal 0, since 0 = 2 modulo 2. 1+1=2=0 mod 2Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-11878436773217586372015-10-06T03:04:09.987-04:002015-10-06T03:04:09.987-04:00The empty set is a subset of every set. That's...The empty set is a subset of every set. That's why it's always included in the power set. A subset is a set which only contains elements from another specified set. That says nothing about having to contain any of those elements.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-50684610095749696542015-08-11T12:53:06.073-04:002015-08-11T12:53:06.073-04:00Sorry, but 1+1=2 only in some cases. It is not tru...Sorry, but 1+1=2 only in some cases. It is not true when working in arithmetic modulo 2, for example. In that case, 1+1=0. :-)<br /><br />I see this as akin to Euler's quest to remove the fifth axiom from his geometry. The nett effect, eventually, was to discover spherical and hyperbolic geometries (as well as proving that the fifth axiom was indeed necessary to plane geometry).<br /><br />Insisting that "the math works" is to deny the source of some of the most profound insights ever made in mathematics.<br /><br />Take another example: a polynomial of degree n has "up to n roots" when all we know are real numbers, but "the math works and has been right for many years". Explore it, and we discover complex numbers.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-56477119420920028332015-03-21T10:32:42.567-04:002015-03-21T10:32:42.567-04:00Yes, and that's why Logic and Set Theory are n...Yes, and that's why Logic and Set Theory are not exactly suited to express 1+1=2<br />My approach is: you don't need + because it postpones the operation of addition, which is immediate.<br />Naturally write 11 and there you have it. Of course if you want to you can put a decimal system on top of the natural numbers 1.. But that is another `story`.<br />A story that has `2` = 11 in the beginning.<br />Enjoy my old blog, with work in progress: http://iteror.blogspot.nl/<br />And my new blog too: https://exwaan.wordpress.com/<br />YouAsamkhyeya dâsahttp://www.blogger.com/profile/10538920755118262226noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-68630244783468718612014-12-12T17:03:09.988-05:002014-12-12T17:03:09.988-05:00PM is flawed. There is an error early on (p7 or p3...PM is flawed. There is an error early on (p7 or p3 perhaps, I can't quite recall). They assume that 'All sets are a sub set of some other set'. However they completely forgot that there is an exception: The Empty Set. Thus the whole of PM fails to establish that Mathematics is reducible to Logic. (it isn't, as the existence of Proof by Induction should already indicate).Sleeypsloonoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-7918285687197080722014-11-09T08:17:06.837-05:002014-11-09T08:17:06.837-05:00Actually this is not the real demonstration. It on...Actually this is not the real demonstration. It only occurs in volume two. In that number he just proves that if we take two different classes with only one element, we can form their sum, which is a class with two elements, that's all. Pay attention, he says "when arithmetical addition has been defined". I suggest you be careful about what people say of Principia: as all topics of knowledge, mathematics is divided into schools of thought, being Zermelo's the most used in modern times; so mathematicians tend to dislike Principia. The axiom of reducibility that causes the problem was fixed by Quine, which produced the New Foundations. If you want a modern account of Principia, just read Quine or Rosser. Just put in your mind the following: none of the schools is so successful as they think, nor unsuccessful as the adversaries think. Most of mathematics comes to opinion, and Russell's didn't please a lot of scholars. Read the introduction to the second edition of https://archive.org/details/principlesofmath005807mbp.Ricardo Aloysiohttp://www.blogger.com/profile/07033655473576006337noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-55216008338377267392014-07-26T20:11:09.032-04:002014-07-26T20:11:09.032-04:00Another opinion from someone who never read the bo...Another opinion from someone who never read the book: I'd think the first 300 pages develop a number of ideas unrelated to proving 1+1=2 and that they could have rearranged the presentation to prove it faster if that were the goal. Or maybe they even put it off as long as possible for dramatic effect, and it was after 300 pages that they could no longer avoid proving 1+1=2.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-66647718056688761162014-05-30T16:24:57.341-04:002014-05-30T16:24:57.341-04:00Have you heard of Russell's Paradox? If you ha...Have you heard of Russell's Paradox? If you have/had then you shouldn't/wouldn't think that what they did was of little importance.<br />Russell was a logician, if logic was broken where does that leave him?adamtract9http://www.blogger.com/profile/12638144738331351438noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-48906641851246385342014-05-09T22:58:31.898-04:002014-05-09T22:58:31.898-04:00They had to define what the the symbols "1&qu...They had to define what the the symbols "1", "+", "2", and "=" were. The symbols had not been as formally defined until the book was written. It took until page 300 to define the symbols and to prove that they could put the symbols 1+1= together before proving that 1+1=2. They went to the basics of basic in this proof. including defining what a "1" is. <br /><br />We, in school or at home, learn that 1 is a quantity equivalent to one object but that object is just a representation of a 1 not the definition of it. <br /><br />I hope that this makes some sense. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-35678313220044178422014-01-16T22:55:30.918-05:002014-01-16T22:55:30.918-05:00Why bother to prove 1+1=2 when we all know that th...Why bother to prove 1+1=2 when we all know that the maths works and has been right for many years. I feel Russel and Whiteheads exercise was a waste of brain time that could have been put to more constructive uses.Hugo Hallamhttp://www.blogger.com/profile/17537681278329703277noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-54368883392933715942012-01-03T10:12:22.753-05:002012-01-03T10:12:22.753-05:00Hmm. I wonder if Russell and Whitehead toyed with ...Hmm. I wonder if Russell and Whitehead toyed with Reductio Ad Absurdum. I mean, if 1+2 does not equal 2, would that not create some contradictions? <br /><br />How many hands do you have? Fourteen seems like the wrong answer.<br /><br />It must have been a drag for R&W to publish this enormous epic of the intellect, only to have Gödel almost immediately say, “Nope!”Neosimian Sapienshttp://neosimian-sapiens.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-26963905666026880662011-07-30T02:02:02.154-04:002011-07-30T02:02:02.154-04:00Norm Megill has a complete, hyperlinked formal pro...Norm Megill has a <a href="http://us.metamath.org/mpegif/2p2e4.htm" rel="nofollow">complete, hyperlinked formal proof of 2+2=4 from the axioms of predicate calculus and ZFC.</a> Fully-expanded, it consists of about 26,000 elementary steps.<br /><br />Norm <a href="http://us.metamath.org/mpegif/mmset.html#trivia" rel="nofollow">writes</a>, "One of the reasons that the proof of 2 + 2 = 4 is so long is that 2 and 4 are complex numbers—i.e. we are really proving (2+0i) + (2+0i) = (4+0i)—and these have a complicated construction but provide the most flexibility [...] In terms of textbook pages, the construction formalizes perhaps 70 pages from Takeuti and Zaring's detailed set theory book (and its first-order logic prerequisite) to obtain ordinal arithmetic, plus essentially all of Landau's 136-page Foundations of Analysis."Josh Jordanhttp://www.jordancurve.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-88496809691045026582011-07-28T09:53:49.086-04:002011-07-28T09:53:49.086-04:00I think Russell says in his autobiography that Göd...I think Russell says in his autobiography that Gödel might have been the only person ever to read the whole book.<br /><br />Also, I understand, as Micki mentions, that they define "1" as the class of all unit sets (perhaps all unit classes), where + is some kind of disjoint union, which would allow 1+1 to be equal to 2. Still, I am not sure, but I think that's roughly the idea.Janomahttp://www.blogger.com/profile/08125807104571129259noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-24751930784865665632011-07-27T19:30:31.487-04:002011-07-27T19:30:31.487-04:00Sam, by "1" I think they mean the class ...Sam, by "1" I think they mean the class of all unit classes, which if I recall rightly is so big it's not even a set.Mickihttp://www.blogger.com/profile/11592612022853202507noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-57145666292208502832011-07-27T19:25:54.187-04:002011-07-27T19:25:54.187-04:00Scott, your answer makes so much more sense if I r...Scott, your answer makes so much more sense if I read R&W as Russell and Whitehead instead of Read&Write. 8) Didn't Whitehead insist on top billing though?Mickihttp://www.blogger.com/profile/11592612022853202507noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-82863713592385036852011-07-26T14:03:35.567-04:002011-07-26T14:03:35.567-04:00I suspect the thing that confuses people about the...I suspect the thing that confuses people about the claim that R&W took "300 pages to prove 1+1=2", is that they interpret it to mean that R&W *discovered* 300 pages of nontrivial mathematical machinery underlying our naive intuition that 1+1=2 (in the same way, maybe, that sphere != donut takes several pages to prove despite its intuitive obviousness). But then they wonder what such machinery could possibly look like. And they're right to wonder! Their intuition is correct; there REALLY IS no mathematical demonstration that could make 1+1=2 more secure than it already is. If you don't already know what the positive integers are and how to manipulate them, then you certainly can't do first-order logic.<br /><br />So I think the right analogy to understand R&W would be a book on computer organization that spent hundreds of pages painstakingly setting out a specific computer architecture, and that only by page 300 (or Volume II, or whatever) was ready to explain what a simple program calculating 1+1=2 would do when compiled. That might STILL be a little weird, but there's nothing inconceivable about it.Scott Aaronsonhttp://www.scottaaronson.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-72454325532166165052011-07-25T21:08:54.360-04:002011-07-25T21:08:54.360-04:00In modern times, depending on what you mean by &qu...In modern times, depending on what you mean by "1", "2", and "+", 1+1=2 is either true by definition, or very trivial.<br /><br />For example, in ring theory, 2 is *defined* to be 1+1 (where 1 is the multiplicative identity and + is one of the ring operations). In this context, 1+1=2 is true by definition.<br /><br />In Peano Arithmetic, 1 might mean S(0) (the successor of 0) and 2 might mean S(S(0)), and proving that S(S(0))=S(0)+S(0) requires a small amount of work: maybe half a page, or maybe 4 lines, depending whether you're allowed to assume commutativity of + or have to prove that too, respectively. (For the 4-line proof where commutativity is assumed, see <a href="http://www.xamuel.com/proving-1-plus-1-equals-2/" rel="nofollow">here</a>)Sam Alexanderhttp://www.xamuel.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-11894952635275721842011-07-25T19:56:23.376-04:002011-07-25T19:56:23.376-04:00Would anyone agree with the amazon review statemen...Would anyone agree with the amazon review statement that any mathematician would call this the most influential book of the 20th century? It did have a big impact on logic, but certain papers like Godel's had a bigger impact, I think. It's a bit tricky because most of the most important works were papers, not books, but was PM more important than Knuth's Art of Computer Programming or Bourbaki? What is on your list?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-8702763053597985752011-07-25T18:02:59.285-04:002011-07-25T18:02:59.285-04:00I posted too soon before the DNS fully propagated....I posted too soon before the DNS fully propagated.Lancehttp://www.blogger.com/profile/06752030912874378610noreply@blogger.com