tag:blogger.com,1999:blog-20067416.post6692483677065937858..comments2023-12-21T06:35:36.624-05:00Comments on Recursivity: Misunderstood MathematicsUnknownnoreply@blogger.comBlogger81125tag:blogger.com,1999:blog-20067416.post-71284371397555515142018-05-07T18:54:22.614-04:002018-05-07T18:54:22.614-04:00By "the integers don't exist" I mean...By "the integers don't exist" I mean they don't exist as physical entities, but only as concepts in the brains of people thinking about them. <br /><br />I actually don't know what it means to say that real numbers exist, or don't exist, even though I use the language.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-44984159740091786142018-05-07T10:36:38.583-04:002018-05-07T10:36:38.583-04:00I don't use "completed totality" in ...I don't use "completed totality" in a strictly mathematical context. But in a philosophical discussion, especially about which axioms to accept or reject, I think the term can be useful. For example, the late Edward Nelson did not accept the consistency of first-order Peano arithmetic. Why not? In his article, "Taking Formalism Seriously," he starts off by arguing that "natural numbers do not exist." I think I know what he's getting at, but I don't think that this is the best way to express his point. After all, I see that in an earlier comment in this thread, you (Jeffrey Shallit) say that integers don't exist. But I suspect that you don't mean quite what Nelson meant, or at least that you're not drawing the same conclusions that Nelson did. I find it clearer to say that Nelson was denying the existence of the integers as a completed totality. That way, I know that Nelson is going to declare the meaninglessness (lack of semantic content) of any theorem whose proof makes essential use of the assumption that the set of all integers exists, but he will not (necessarily) react the same way to a theorem whose proof includes a statement like, "There exists a prime number between 100 and 200," even though the latter statement superficially appears to assume that "(some) integers exist." The term "completed totality" is useful when trying to categorize various views in the philosophy of mathematics.<br /><br />Tim ChowTimothy Chowhttps://www.blogger.com/profile/15157353087847193176noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-47750829974441506352018-05-07T06:13:50.329-04:002018-05-07T06:13:50.329-04:00So why, as a mathematician, use "completed to...So why, as a mathematician, use "completed totality" instead of set? It makes no sense to me.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-6122891549194688802018-05-06T21:58:41.673-04:002018-05-06T21:58:41.673-04:00The rough translation of "completed totality&...The rough translation of "completed totality" into modern language is simply "set."<br /><br />So for example, in classical philosophical language one might say, "the universe of all sets is not a completed totality" whereas today we would express more or less the same concept by saying, "The class of all sets is a proper class."<br /><br />In somewhat more detail: The intent of the classical distinction between "potential infinite" and "actual infinite" (or "completed totality") is to affirm the existence of infinitely many individual entities (such as integers) while denying the meaningfulness of speaking of (what in modern language we would call) the set of all such entities. The modern way of handling this distinction is to distinguish between working "inside the system" and "outside the system." So for example if I have a (standard, transitive) model M of ZFC that violates the power set axiom, and S ∈ M, then "outside the system" I can still form the set P of all subsets of S ∈ M, but the "completed totality" P might not be in M, even though all its members are. Similarly, outside the system, M is simply a set, but M ∉ M so the "set of all sets" doesn't exist as a "completed totality" inside the system.Timothy Chowhttps://www.blogger.com/profile/15157353087847193176noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-70950415799853271912018-05-06T20:06:35.643-04:002018-05-06T20:06:35.643-04:00What is a "completed totality"? I see t...What is a "completed totality"? I see that lots of people throw around this term as if it means something, but as far as I can see, it means nothing.<br /><br />I think you're wrong about Cantor crackpots, because I've talked to several of them. All of the ones I've talked to accept that there are infinite sets, and all of them make the same silly mistake: they don't understand that the set of rationals with terminating base-k expansions doesn't cover all the rationals. Sometimes they hide this mistake with layers of obfuscation, but it is at the heart of all the Cantor crackpots I've talked to.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-67215438333378128242018-05-06T17:34:44.864-04:002018-05-06T17:34:44.864-04:00Let me elaborate a bit on my previous comment that...Let me elaborate a bit on my previous comment that Cantor's argument contains some subtle, hidden assumptions, by offering two "perturbations" of the argument.<br /><br />First, let's observe that everything in the argument seems, at first glance, to be constructive and computable. So let's "prove" that Turing machines cannot compute every computable real number. (By the way, Goedel himself for a while believed that there could be no "canonical" definition of computability, largely on the basis of an argument similar to this one, so I'd be impressed if any ten-year-old could clearly identify and explain the flaw without coaching.)<br /><br />It is easy to write an algorithm that enumerates all Turing machines one by one. So now let us compute the binary expansion of a number x by diagonalizing: To get the nth digit of x, just enumerate Turing machines until you get the nth one, then use this Turing machine to compute n digits, and then reverse the nth digit to get the nth digit of x. Then x is not computed by any Turing machine, but we have just given an algorithm for computing it.<br /><br />The second perturbation is one I haven't seen published anywhere, so there is an implicit challenge here: Is there any way to develop axioms under which the following argument is actually correct?<br /><br />Suppose that we represent binary numbers (or subsets of integers) in the usual way as binary expansions, but that when we "list" the real numbers, the ordering that we use is not ω, the first infinite ordinal, but some other countable ordinal α. So α contains ω as an initial segment but also contains other elements, that appear later in the ordering. Then when we apply the Cantor diagonal construction, what we obtain is a number x that does not appear in the "ω part" of α, but there seems to be nothing to prevent x from showing up later in the list. This seems to block the contradiction. So, could it be possible that the natural numbers and the reals <i>as unordered sets</i> can be put into bijection with each other, but that the Cantor diagonal element tells us that we <i>cannot impose the ordering ω on the reals</i>?<br /><br />Somehow I suspect that this second perturbation cannot be turned into a meaningful proof of anything, since there are ways of formulating the diagonal argument without invoking a total ordering, but who knows? In any case my main point is that the Cantor argument, as usually presented, assumes without comment that the <i>same</i> total ordering is applied to the "rows" and the "columns." Seems obvious, but you can't be too careful when it comes to reasoning about infinite sets.<br /><br />Tim ChowTimothy Chowhttps://www.blogger.com/profile/15157353087847193176noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-12861934677654780152018-05-05T17:06:41.778-04:002018-05-05T17:06:41.778-04:00Since I'm a mathematician, I'm obviously o...Since I'm a mathematician, I'm obviously on Jeffrey Shallit's side for the most part, but at the same time, I don't think that Cantor skeptics are necessarily psychologically maladjusted in any simple sense. For example, let's consider the argument that professional mathematicians have studied the argument for a hundred years and all accept it. In any other field, this sort of argument does not carry a lot of weight, does it? Lots of wrong statements have been "universally" accepted for a hundred years, especially if we allow ourselves to restrict the "universe" to the subset of people who are "professionals." Refusing to accept the weight of opinion of a hundred years of "professionals" as being definitive is actually the sort of thing that I, as a mathematician, like to see in a student, and I certainly wouldn't regard it as indicative of any kind of psychiatric pathology.<br /><br />Mathematics is really a very peculiar field in that it concerns itself with <i>concepts that are sufficiently precise that questions about them can (sometimes) be definitively settled</i>. In almost any other arena, there's always some way to wiggle out of a seemingly crushing argument. You can challenge an assumption, or argue about a definition, or exploit some kind of vagueness somewhere. Never are you forced to concede that you're wrong and someone else is right. Even in sports or law, where there is a final court of appeal and you cannot win, you can still go away convinced that the judge or referee was wrong. Given that this is how all the rest of life works, it should not be surprising that people come to mathematics assuming that it works the same way. People have (almost) no experience with the idea that once you clarify your terms sufficiently, then the answer follows by inexorable logic, no matter what anyone says.<br /><br />In the case of Cantor's argument, the issue is compounded by the slipperiness of the concept of infinity. Eamon Knight asks an excellent question, "Could a different (but consistent) theory of infinities be constructed by choosing to privilege a different set of intuitions?" Most certainly this is possible. This is something that mathematicians tend to forget, since we're so used to the standard approach. I think that it is unlikely that there is any alternative theory of infinity that is as rich and intellectually fruitful as the standard mathematical one, but there are certainly "dull" theories of infinity that don't support Cantor's argument. For starters, one can simply refuse to accept the concept of an infinite set. Alternatively, one can accept that the set of integers exists as a completed totality, but deny that it is meaningful to speak of the power set of the set of integers. To accept Cantor's argument, we implicitly have to accept these assumptions, as well as other assumptions, such as that it is meaningful to talk about infinite lists and mappings and so forth. These assumptions about infinity are far from self-evident, and we also know (thanks to Goedel) that there is not any fully satisfactory sense in which we can <i>prove that these assumptions are self-consistent</i>.<br /><br />None of this is to exonerate your average Cantor skeptic, who is nowhere near being able to formulate as cogent an objection as, "I reject the power set axiom." But I think it does mean that there need not be anything more going on in their minds than something like, "This argument sounds fishy; it seems to depend on a lot of hidden premises; I can't put my finger on what exactly is wrong, but the conclusion can't be right." Throw in a bit of argumentativeness and hubris and voilà, you have your standard Cantor skeptic.<br /><br />Tim ChowTimothy Chowhttps://www.blogger.com/profile/15157353087847193176noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-45470120567623655762014-08-19T21:00:35.819-04:002014-08-19T21:00:35.819-04:00I believe it was Bertrand Russell who said : A stu...I believe it was Bertrand Russell who said : A stupid man's understanding of what a clever man says can never be accurate, because he unconsciously translates what he hears into something he can understand. <br /><br />I give you full marks for your patience Dr Shallit. Good luck with your book. I hope it is as amusing as this blog.Anonymoushttps://www.blogger.com/profile/09718770312625590169noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-91262127373223044482014-03-21T19:05:11.730-04:002014-03-21T19:05:11.730-04:00Going back to the part of the thread with Eamon Kn...Going back to the part of the thread with Eamon Knight (about 4 years ago now), I can think of a somewhat natural notion of cardinality that would set |2Z+1| = |2Z|, but would not be equivalent to the usual notion:<br /><br />|A| = |B| iff there is some indexing A_k and B_k of A and B such that {A_k – B_k} is bounded.<br /><br />Clearly (at least by my intuition) not as deep as the standard notion of cardinality, but I could imagine it might be good for a few cute theorems with a combinatorial flavor.Benjamin Schakhttps://www.blogger.com/profile/02091757155440747266noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-58874209496906413482013-11-09T17:16:15.768-05:002013-11-09T17:16:15.768-05:00Entertaining crackpottery, John. Of course real n...Entertaining crackpottery, John. Of course real numbers don't "exist" and neither do integers. Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-80326824950524301612013-11-09T08:01:44.437-05:002013-11-09T08:01:44.437-05:00An irrefutable proof that real numbers don't e...An irrefutable proof that real numbers don't exist:<br /><br />http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-49028344093204263262013-08-27T08:23:07.230-04:002013-08-27T08:23:07.230-04:00What if it's possible to enumerate N using onl...<i>What if it's possible to enumerate N using only a subset of N for the digits?</i><br /><br />I don't even know what this is supposed to mean. Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-69711283675829753372013-08-26T22:45:06.123-04:002013-08-26T22:45:06.123-04:00Hi, it's been a while.
What if it's possi...Hi, it's been a while.<br /><br />What if it's possible to enumerate N using only a subset of N for the digits? (This assumes using binary or higher base).<br /><br />If so, then all Cantor has proven is that the same is true with R. You only need a subset of R (aka N) digits to enumerate all of R.<br /><br />But since a subset of N can map one to one with N outside of the digit/row relationship, then it means it's also possible that a subset of R (aka N) can map to R.<br /><br />It doesn't mean that there is necessarily a bijection between N and R. But it would show a flaw in the proof, no?<br /><br />Would you agree with this? On the condition the first question above is true?<br /><br />The reason people have problem with Cantor's proof is that they believe the first question I posed above is true.<br /><br />Do you have a proof that it is false? Note that we're talking about the digits needed to represent N. Not if there is a bijection outside of this relationship. Also, we are talking about binary or higher base.<br />Vorlathnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-84270119274559450872013-02-26T18:45:33.062-05:002013-02-26T18:45:33.062-05:00Hey Mr. Reading Comprehension Guru, I never said t...<i>Hey Mr. Reading Comprehension Guru, I never said the book I edited was a book about the history of science. I said it included a chapter on the history of dating the Earth.</i><br /><br />Ok, now I <i>know</i> you are just trolling, since I was simply responding to <i>your</i> comment that "I like your assumption that a science publisher is automatically better than a religious publisher when it comes to the history of nineteenth century science". <br /><br />Go bother someone who gives a damn what you think.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-23864781957070609972013-02-26T16:18:28.202-05:002013-02-26T16:18:28.202-05:00"And yes, if someone chooses to publish about..."And yes, if someone chooses to publish about the history of science with a primarily religious publisher, that says volumes about their motivation. "<br />Hey Mr. Reading Comprehension Guru, I never said the book I edited was a book about the history of science. I said it included a chapter on the history of dating the Earth. And I didn't even say it was with a religious publisher.<br /><br />The term troll is subjective.<br /><br />I'll accept being called a coward for wishing to remain anonymous as long as you call all of your friends who also blog anonymously cowards. That link I provided above suggests you have many such friends.Winston Smithnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-68509671804327757262013-02-26T14:12:51.727-05:002013-02-26T14:12:51.727-05:00Hmm ... a single quote from more 44 years ago doe...Hmm ... a single quote from more 44 years ago does not make a very convincing argument. I believe astronomers are among the most open minded and diverse of scientists. You can't exactly travel to a gamma-ray burst to figure out what is really going on, so there are many competing theories.<br /><br />I recently heard a podcast about astronomer George Chapline, who doesn't believe in black holes. Instead, he replaces them by "dark-energy stars". I'm not sure if his theory has gained any acceptance.Georgehttps://www.blogger.com/profile/10140920751826036814noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-30402600829251750022013-02-26T09:34:41.093-05:002013-02-26T09:34:41.093-05:00I like your assumption that a science publisher is...<i>I like your assumption that a science publisher is automatically better than a religious publisher when it comes to the history of nineteenth century science.</i><br /><br />I said nothing about scientific publishers. Tell me, are you actually this incompetent at reading comprehension, or are you just trolling?<br /><br />And yes, if someone chooses to publish about the history of science with a primarily religious publisher, that says volumes about their motivation.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-55348936629683220402013-02-26T09:20:43.734-05:002013-02-26T09:20:43.734-05:00For Fernie's quote, go look up Fernie J D (196...For Fernie's quote, go look up Fernie J D (1969) "The period-luminosity relation: a historical review." Publ. Astron. Society Pacific, vol. 81 p707-731. I'm sure you will continue to believe that I took his quote out of context.<br /><br />I like your assumption that a science publisher is automatically better than a religious publisher when it comes to the history of nineteenth century science.Winston Smithnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-31490144346688768642013-02-26T09:16:22.728-05:002013-02-26T09:16:22.728-05:00I edited a book which included a chapter on the to...<i>I edited a book which included a chapter on the topic. But sorry, I'm not saying which one.</i><br /><br />Coward. But I bet it either doesn't exist or it's with a primarily religious publisher.<br /><br />Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-783598323419724682013-02-26T09:08:11.399-05:002013-02-26T09:08:11.399-05:00I'm also very interested in the history of ide...I'm also very interested in the history of ideas and their acceptance (and herd mentality). I edited a book which included a chapter on the topic. But sorry, I'm not saying which one.<br />PS: What you're calling cavil, I call significant.<br />"Yes, I see there's a lot you don't understand." -- Hmmm, I can definitely see why you're fan of Callan Bentley.Winston Smithnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-13325396742770864412013-02-26T06:56:15.872-05:002013-02-26T06:56:15.872-05:00If a historian of math is the same as a historian ...<i>If a historian of math is the same as a historian of science, then you win. If not...<br /></i><br /><br />I fully expected cavilling about the point that mathematics is not science. True enough. The point remains, I'm very interested in the history of ideas and their acceptance, and I've even gotten a peer-reviewed paper published on that topic. Now, how about your expertise in the subject?Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-32185099552107981352013-02-26T04:29:26.184-05:002013-02-26T04:29:26.184-05:00I don't understand your point about my not men...<i>I don't understand your point about my not mentioning Kelvin. </i><br /><br />Yes, I see there's a lot you don't understand. I was responding to <i>your claim</i> that <i>I love how you "wonder" if I'm a young-earth crackpot because I believe there was a ton of groupthink going on in Kelvin's time</i>. How could that possibly have been my justification, when up to then you hadn't mentioned Kelvin? Try to pay attention.<br /><br />You don't seem to understand my other response, either. People make guesses about other people's behavior & motivations based on lots of social cues; these need not be based on "logic" and hence calling "ad hominem" or justifying your decision to hide behind a pseudonym misses the point entirely. You asked why I thought you were a creationist, and I gave my reasons for making that guess. <br /><br />Finally, as for "quote-mining". nobody serious (even historians of science) thinks you can answer questions about complex behaviors like "groupthink" by producing a single unsourced quote which is probably about something else entirely (but since you didn't cite the paper, we can't check so easily). That's typical creationist behavior.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-34375088967950713272013-02-25T22:37:12.720-05:002013-02-25T22:37:12.720-05:00I don't understand your point about my not men...I don't understand your point about my not mentioning Kelvin. An understanding of the history of the herd mentality in this issue can't be done without his name being front and center. a) This is a silly assumption, based on http://www.noforbiddenquestions.com/2011/04/should-i-blog-anonymously/ b) A quote that rubs you the wrong way is by definition "quote-mined". c) ad hominem.Winston Smithnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-15160208051704021612013-02-25T12:21:37.414-05:002013-02-25T12:21:37.414-05:00because I believe there was a ton of
groupthink g...<i>because I believe there was a ton of <br />groupthink going on in Kelvin's time</i><br /><br />Wrong. Up to now you hadn't mentioned Kelvin. I suspected you were a young-earth creationist because (a) you adopt a pseudonym to conceal your real name (b) you like to quote-mine to make points and (c) you provided no references and are incoherent in your points. Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-44055831855565636692013-02-25T12:05:31.132-05:002013-02-25T12:05:31.132-05:00Haven't read Dalrymple's, but I've rea...Haven't read Dalrymple's, but I've read Burchfield's book on Kelvin.<br /><br />I love how you "wonder" if I'm a young-earth crackpot because I believe there was a ton of <br />groupthink going on in Kelvin's time, and even today, according to my understanding of Fernie. <br />In your "wonder", there's an implied, though I'm sure unintended, admitting of deficiency of challenging the consensus going on in the science world. An insider charging "Groupthink"? Unthinkable! He must be a creationist! Sad.<br /><br />If a historian of math is the same as a historian of science, then you win. If not...<br />Winston Smithnoreply@blogger.com