tag:blogger.com,1999:blog-5173950.post7444330350233792596..comments2014-04-18T15:46:06.416-04:00Comments on Grim's Hall: The Unity of Consciousness, Part IIGrimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comBlogger77125tag:blogger.com,1999:blog-5173950.post-64776596093743532462012-04-26T14:57:14.226-04:002012-04-26T14:57:14.226-04:00In fairness (as you know, having looked at him rec...In fairness (as you know, having looked at him recently) Kant is not an easy philosopher to understand. He's one of a few philosophers that there are competing schools about, too, where some think he meant to say X and others Y. <br /><br />For example, I belong to the school that interprets his moral theory less on the more-famous <i>Groundwork to the Metaphysics of Morals</i>, and more on the later <i>Metaphysics of Morals</i> -- but it's much less well known, though for the most part it's much more obviously sensible given the moral positions you'd expect from a European gentleman of cosmopolitan leanings in his time.<br /><br />However, I think it is fair to say that he is more popularly read (based on the <i>Groundwork</i>) as offering a program to radically rethink moral philosophy. I think he was just trying to make a secular argument for the ordinary moral beliefs of his day; but many like to take his <i>Groundwork</i> concepts about how to formalize morality and run with them into interesting and sometimes radical places.<br /><br />Rand may have had one of those teachers! But it's hard to see how she confuses his teachings with Hegel in any case. (Except that they're both German, and both take a huge amount of work to decipher! I assume she was against Hegel because Hegel was such a crucial figure for Marx.)Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-90202420414567653012012-04-26T09:59:59.550-04:002012-04-26T09:59:59.550-04:00P.S. - And I have noticed that any trained academi...P.S. - And I have noticed that any trained academic philosopher who reads Ayn Rand seems to agree that she did a terrible job describing the work of other writers, especially Kant. (And her epistemology relied heavily on psychological ideas, such as the <i>tabula rasa</i> that are simply not true; that spoils her <a href="http://www.scribd.com/doc/51894820/Ayn-Rand-The-Comprachicos" rel="nofollow">attack on public education</a>, more's the pity). <br /><br />It's a shame she took the path she did, because she really was a fantastic expositor of some very important ideas. Her essay, "The Anti-Industrial Revolution" - title essay in her book, <i>The New Left: The Anti-Industrial Revolution</i>, is an extremely good attack on "green" thought, and its appalling moral standing. (I read several books of her essays; I think that was the best one. On a good day, on the attack, she was truly unmatched.)Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-66546312156432682812012-04-26T09:28:55.513-04:002012-04-26T09:28:55.513-04:00Too right. May we live long enough to see that st...Too right. May we live long enough to see that start to happen!Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-64786994621506450172012-04-25T22:58:13.989-04:002012-04-25T22:58:13.989-04:00You know, one thing that Rand is particularly wron...You know, one thing that Rand is particularly wrong about in this piece is her assertion that the <i>Metaphysics</i> is "the basic book" of philosophy.<br /><br />In fact, students traditionally weren't allowed to read it until they had mastered the <i>Physics</i>, the logical texts (in the Middle Ages known as the 'Organon'), the works on the natural sciences, and a few other works like <i>De Anima</i>.<br /><br />Until the explosion of knowledge in the modern period, it was still possible for an individual to know everything that we knew as a species. That prevented talking-past-each-other discussions like this one, where backgrounds and languages are too different for a meaningful exchange of ideas; but at the cost of losing all that knowledge.<br /><br />It's a good argument for a better replacement for humanity -- or at least an upgrade for our minds and lifespans. We really can't do it anymore. Most of those I've studied with have at a pair of Ph.D.'s in order to keep up with some bare minimum: I know a philosopher of science who has Ph.D.'s in physics (focused on relativity) and metaphysics; another with degrees in math and philosophy of math; I have only a master's in history, to go with the doctorate in philosophy I'm after. Someday I hope to round it out as well.<br /><br />But even then, there's so much now that there isn't time to learn -- even if you don't take time off for wars, or families, or the joys of poker.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-85042619910149581342012-04-25T22:43:45.736-04:002012-04-25T22:43:45.736-04:00As for Rand, I can't say I've ever found h...As for Rand, I can't say I've ever found her interesting enough to read. I still don't; I got about two or three paragraphs into her piece, then started trying to find the part where she talks about Kant; discovered she's running Kantians and Hegelians together; well, in any case, Kant wasn't a bad guy at all. He was a hypochondriac, who set up a very rational system for himself to avoid admitting disorder into his life. He meant well, but his order is so orderly that even Kantians normally shy from it (especially when he turns, as he does now and then, to the subjects of capital punishment or breast-feeding).Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-82032568612053265042012-04-25T22:38:26.999-04:002012-04-25T22:38:26.999-04:00...and we leave aside the poor cranks, neither sci......and we leave aside the poor cranks, neither scientist nor philosopher, who are told that they can't square the circle or trisect the angle by the rules of Greek geometry, and waste lives and paper trying to prove they've done it. <br /><br />(In that case, thanks to analytic geometry and more besides, the proofs of impossibility are out there.)Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-57086787072354578212012-04-25T22:36:20.205-04:002012-04-25T22:36:20.205-04:00It is (and I think that some Muslims describe God ...It is (and I think that some Muslims describe God as the "ultiimate deceiver" as well). <br /><br /><i>At least you've probably learned that telling a philosopher that something is impossible is likely to produce an attempt to do the impossible thing...</i><br /><br />No doubt. Depending on just what it is, many scientists are similar. Sometimes, though, the question is just nonsensical. A quote from my freshman chemistry textbook -- <br /><br />"People sometimes ask me whether there might be elements other than those in the Periodic Table. I tell them that this is like wondering whether there could be another whole number between 6 and 7. Unfortunately, some people think that is a good question, too..."Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-18320115993844542972012-04-25T22:26:19.487-04:002012-04-25T22:26:19.487-04:00...not only can God not lie...
That's funny, ...<i>...not only can God not lie...</i><br /><br />That's funny, because I just attended a colloquium on Medieval Jewish and Islamic philosophy in which the translator (who could read Hebrew, Arabic and Greek) discussed Maimonides' praise for God's capacity for the ruse. I'm not sure that goes as far as a lie, but it's an interesting concept.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-49098643093366882512012-04-25T22:22:51.676-04:002012-04-25T22:22:51.676-04:00I've been waiting for two days for you to cite...I've been waiting for two days for you to cite that last xckd. :)<br /><br />It's true, in this case, that I don't have the background to do more than punch at you and see what you come up with. My background in symbolic logic is pretty good; in math general, it's largely limited to a fairly complete understanding of the odds of poker.<br /><br />We're in the place where we are because you haven't done the <i>Physics</i>, and I haven't done the math. You keep raising arguments that you haven't thought through to the same degree that Aristotle did; and I don't really understand the mathematical concepts or terms, as you rightly point out from time to time.<br /><br />I'm not sure we have enough common ground for me to explain to you why your arguments aren't going to shake out, or for you to explain to me why my attempts to explain them to you in mathematical terms aren't convincing. <br /><br />Not yet. But if we keep at it for another few years, who knows? Maybe we'll learn something. (At least you've probably learned that telling a philosopher that something is impossible is likely to produce an attempt to do the impossible thing; and I'm still not convinced we can't sort out a good way, once we make up our minds and sort out our terms.)Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-70786016545359146032012-04-25T22:04:50.718-04:002012-04-25T22:04:50.718-04:00There are also Catholic philosophers and theologia...<i>There are also Catholic philosophers and theologians who defend it to the point of saying that even God cannot do something that violates that particular law, or other laws of logic. This is often floated as a defense against atheistic arguments like, "Can God make a boulder so heavy he cannot lift it?" Either yes or no must be true; either claim is incompatible with Catholic claims about God. So the argument that God is ruled by the laws of logic is a way of avoiding that kind of anti-theistic argument.</i><br /><br />Yep, I once heard William F. Buckley and Jerry Falwell agree that, not only can God not lie, he can't sin either. And since, to all appearances, God is an imaginary construct, they can give him whatever attributes they like. The difference between what they were doing, and what I am doign here, is that I do know something tangible about the mathematics we're discussing, have done some of it, used some of it, and worked through some of the proofs. (Sometimes long ago - which was why I was a little slow to remember just how the diagonal slash worked, for example.) So I am not simply repeating the dogmas of a Church, or demanding that an invisible, undetectable Deity conform to these, but looking at what men have actually done. And in doing so, encouraging you to stay out of <a href="http://xkcd.com/675/" rel="nofollow">this territory.</a> (Three xckd's in one thread! That cartoonist had truly added something to this world.) Were I wrong, who knows, you might be able to <i>show</i> something irrational or contradictory in the mathematics we're discussing. You just haven't done so yet.Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-9100861136392978222012-04-25T21:54:19.523-04:002012-04-25T21:54:19.523-04:00Many philosophers would have absolute fits at the ...<i>Many philosophers would have absolute fits at the idea that the law of noncontradiction can be violated (let alone is being violated, ever at all; and especially is violated by something as rational as mathematics). They'd want to join you in finding some sort of distinction that makes it all rational and orderly.</i><br /><br />But I don't need to "find some sort of distinction" - it already is rational. (I'm not sure what you mean by "orderly" in this context.) It is sometimes counterintuitive, but I have never yet seen advanced mathematics that is irrational. You've asserted once or twice that it is, and violates non-contradiction, but you haven't shown how...or rather, your arguments that it does break down once you look at the true definitions we're working with.<br /><br />Interestingly, this puts you in company with a philosopher who doesn't get much time from the academics - Ayn Rand. The <i>Introduction to Objectivist Epistemology</i> includes the transcript of a little seminar with her and some academics, where she declares that "mathematics after Bertrand Russell" is a fundamentally irrational enterprise. Which I'm afraid showed only that she did not understand modern mathematics, and some of his pop-quotes about "the science where we never know what we are talking about" had quite bamboozled her. (Interestingly, she liked to frame the battle of good and evil on the earth as a battle between Aristotle and Kant - Kant she blamed for the worst excesses of "we can't know anything" indecisiveness - see <a href="http://www.tracyfineart.com/usmc/philosophy_who_needs_it.htm" rel="nofollow">her West Point address</a> for an example.)Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-34380512889302518902012-04-25T21:43:39.288-04:002012-04-25T21:43:39.288-04:00But I don't actually need this many: I only wa...<i>But I don't actually need this many: I only want to hit the irrational points, so the rational ones ("r") will be left out. I therefore should only need R-r rounds to hit all the points I want. Since I ordered R rounds, I'll have extra!</i><br /><br />R and r, as you have defined them, are <i>sets</i>, not numbers. When you treat them as numbers, and say "I need only this many rounds," you are quite misusing the concept, and your statement doesn't make sense. <br /><br />Now, if your intent was to subtract one infinity from another, that doesn't make sense either - infinities are not numbers that can be manipulated the way 6 and pi can. <br /><br />If your intent is to take the real numbers as the unity of the rational and the irrational, well and good. You can certainly consider a subset of an infinite set and -- this is the key point -- infinite sets can have infinite subsets as well as finite subsets. Thus, "the set of all even numbers" is a countably infinite set; "the set of all numbers divisible by 100" is a subset of this set; but both are countably infinite. And in the sense that Cantor used the term, neither of them is "smaller" than the other, even though one is a subset of the other. <br /><br />In any caes, R and r and the other objects we're talking about are still sets, and they are not numbers. Discrete objects, like "rounds," can be paired with the positive integers and are therefore, at most, countably infinite. <br /><br /><i>Now my claim amounts to this: squaring infinity doesn't make any more difference than multiplying it by 1/2.</i><br /> <br />That's right. Neither one of these operations makes any sense at all, no more than dividing by zero does. Dividing two by zero doesn't give you a number twice as big as dividing one by zero. Both are undefined and neither makes any sense. "Taking the square root" of any infinity, that makes no sense either. <br /><br />(The way you work with infinities in applied mathematics is by using a variable, and taking the limit as that variable goes to infinity. Now if you were to take an formula like 2x/y, and take the limit as x and y both go to infinity, the limit is obviously 2, not because one "infinity" is bigger than another or because you're actually "dividing infinity by infinity," but because in the <i>process</i> of going to the limit, the expression on top stays twice the size of the expression on the bottom.)Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-29865666570005745892012-04-25T19:28:34.748-04:002012-04-25T19:28:34.748-04:00It also occurs to me that the irrationals are goin...It also occurs to me that the irrationals are going to be the same set of numbers between each integer, which offers interesting potential: maybe we can triple-pair them, so that you'd have one integer to denote which integer provided the field, and the second to denote the specific irrational being paired. Pi could be 3.0 -- that is, 3 to indicate it was in the space between 3 and 4; and 0 to indicate it was the first paired irrational. We'd need a table after that, since you can't go directly from Pi to 'the next smallest' or 'next largest,' but in theory once you map the set for one integer range, you've mapped it for all of them.<br /><br />Now, of course, this can't actually be done: but it leads me to think it might be a way of squaring the idea of the sizes of the two infinities being the same. In this way we can see that there's really only one irrational set; it just happens to recur between each integer.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-89969725135650822672012-04-25T07:39:59.122-04:002012-04-25T07:39:59.122-04:00By the way:
Many philosophers would have absolute...By the way:<br /><br />Many philosophers would have absolute fits at the idea that the law of noncontradiction can be violated (let alone <i>is</i> being violated, ever at all; and especially <i>is</i> violated by something as rational as mathematics). They'd want to join you in finding some sort of distinction that makes it all rational and orderly.<br /><br />There are also Catholic philosophers and theologians who defend it to the point of saying that even God cannot do something that violates that particular law, or other laws of logic. This is often floated as a defense against atheistic arguments like, "Can God make a boulder so heavy he cannot lift it?" Either yes or no must be true; either claim is incompatible with Catholic claims about God. So the argument that God is ruled by the laws of logic is a way of avoiding that kind of anti-theistic argument.<br /><br />Neoplatonic philosophy differs in that it believes that at the level of <i>nous</i>, noncontradiction breaks down. Actual infinites (as opposed to potential infinites) do not exist outside the realm of the intellect; therefore, it isn't important that noncontradiction should apply. It can be true both that there is a 1/2 ratio of all integers to even integers, and that this truth doesn't apply to the set. It can be true that we'll have exactly enough rounds, and more than we need. This is nonproblematic, because at the noetic level contradictories coexist in a single unextended whole.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-19383974651554684352012-04-25T06:28:58.855-04:002012-04-25T06:28:58.855-04:00Or rather, it does and it doesn't seem to be t...Or rather, it does and it doesn't seem to be true: in the same way that I expect for my supply of R rounds to be both <i>more than enough</i>, and <i>exactly the right amount</i>.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-82820049664566863532012-04-25T06:28:11.579-04:002012-04-25T06:28:11.579-04:00As for special relativity, I include the analogy o...As for special relativity, I include the analogy only as an analogy: I mean to say not that there is a relationship between c and infinity, but that we can recall the problems of c in relativity theory to remind us that we sometimes come up with unexpected limits. I think this is one area where very elegant mathematics are hitting a metaphysical limit; I believe I understand Cantor's argument, but it strikes me as fundamentally the same as the claim about evens and all integers. It's clear enough, looking at the number line (or the diagonal line); but it doesn't seem to be true.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-89053713848624182182012-04-25T06:25:54.800-04:002012-04-25T06:25:54.800-04:00One of the things that Aristotle often says in the...One of the things that Aristotle often says in the <i>Physics</i> is, "Let us make a fresh start."<br /><br />There is a reason why it is impossible to divide a line both at a given point and the point next to that first point. The reason is that 'the point next to it' is a phrase that does not refer to any extant object. The reason is not that you cannot divide at an infinite number of points, or indeed at any other point that does exist; it's just that the concept "next to it" doesn't fit.<br /><br />Thus, let's say I wish to assign a round to every irrational 'point' on the line that composes the set of real numbers (we shall call this set "R"). I place an order for R rounds.<br /><br />But I don't actually need this many: I only want to hit the irrational points, so the rational ones ("r") will be left out. I therefore should only need R-r rounds to hit all the points I want. Since I ordered R rounds, I'll have extra!<br /><br />Or possibly not; since the set of irrationals is an uncountable infinity just like R, the fact that I'm subtracting a countable infinity may leave me with just enough. So, I will have either more than I need or exactly as many as I need. <br /><br />But we can further specify the relation. The difference between the countable infinity r and the uncountable infinity of the irrationals is also able to be specified: between each member of r and the next member of r, there is an infinite number of irrationals. Thus, the irrational set is equal to r multiplied by infinity. <br /><br />Now my claim amounts to this: squaring infinity doesn't make any more difference than multiplying it by 1/2. The set "r" is infinite by nature; thus, if you take the square root of r and R, you'll end up not with two sets of different 'sizes,' but with something like "1=1".Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-2098011779011354472012-04-25T01:51:49.573-04:002012-04-25T01:51:49.573-04:00A countable infinity is supposed to be a different...<i>A countable infinity is supposed to be a different size than an uncountable one because it is a subset...</i><br /><br />No, sir, that's your idea, not Cantor's. A countable infinity is supposed to be a different "size" than an uncountable one -- and note that Cantor has expanded the concept of "size" to account for these different kinds of infinities -- because the one can be matched, by rule, to the set of integers and the other can't. The set of positive even integers is a subset of the positive integers, but they are both countable infinities and are the <i>same</i> size -- as a matter of definition. <br /><br />Your "shooting" analogy is an intuitive one, the kind of thing that's very useful for understanding problems on the human scale. It doesn't work so well for infinities. As I mentioned above, comparing infinities is a matter of comparing processes - do you have a set of numbers to which the positive integers can "catch up" by a process you can define? If yes, it's countable; if not, no. The set you've defined is uncountable, because if you pair it with the natural numbers, by any means whatsoever, it never catches up - an X can always be defined that your process won't reach. <br /><br />This is not the case wiht the rational numbers; the positive integers can "catch up," and so it is a countable set. That's the best metaphor I can think of - but going with metaphors and intuitive analogies gets you only so far in mathematics or modern physics, and then you are <a href="http://xkcd.com/895/" rel="nofollow">here</a>. <br /><br /><i>Our diagonal proof is elegant because it shows you a proof that there will be elements not included, etc.</i><br /><br />Right...<br /><br /><i>That's the same kind of argument about the even set and all sets. It's not, in fact, 'intuitive' in the sense I think you are using the word, but rather a logical argument that shows that the proof follows from the definitions.</i><br /><br />Exactly. <br /><br /><i>Thus, your arguments are completely rational -- but the physics seems to show that they aren't true.</i><br /><br />Now you're mixing up a couple of different things -- special relativity, which is what you're talking about, has nothing to do with the difference between countable and uncountable infinities - not as far as I know. <br /><br />Off the top of my head, in fact, I didn't know <i>any</i> physics that deals with that distinction - I did a quick Google and found an abstract or two I did not understand; so I think there may be some, but these simple problems in special relativity you're describing aren't among them. <br /><br /><i>It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not rational given the system, but it is true.</i><br /><br />Incorrect. In this case, it's not even a matter of developing a new kind of mathematics, but of changing some ideas about the physics, based on the experimental data. (Special relativity is derived using ordinary algebra and calculus; it's general relativity that requires the non-Euclidian geometry. Two spaceships approaching each other at uniform speeds require only special relativity to describe their motion.) The physical assumption that underlies it is - if spaceship A is firing a light beam at spaceship B, the light beam is travelling at speed c away from spaceship A. It is also travelling at speed c towards spaceship B; its velocity, relative to both these moving objects, is exactly the same. Which is a deeply strange thing, but if you change to this assumption, you get results that fit the world very well indeed (one result is that nothing will go faster than light relative to anything else; which is the result you are talking about).<br /><br />There is nothing <i>irrational</i> anywhere in the process, just as there is no "flagrant violation of the law of non-contradiction" in Cantor's way of comparing infinities, a separate topic.Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-60691649903110552742012-04-25T01:41:33.129-04:002012-04-25T01:41:33.129-04:00"Thus, your arguments are completely rational...<i>"Thus, your arguments are completely rational -- but the physics seems to show that they aren't true. Infinities do seem to work more like c: they serve as a ceiling, beyond which what ought to be true isn't true. It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not rational given the system, but it is true."</i><br /><br />But didn't we just determine that there are two different types of infinities, therefore, why would we assume that all infinities should follow the same rule? Also, you say it's not rational, based on our interaction with the universe, but we can't interact with things moving at hyper velocities for instance, and apparently physicists are making the calculations work if they cancel infinity- so apparently we do have some kind of rational handle on infinity. Now, it may turn out it's wrong later, like Newtonian physics, but for a while, we thought that was bulletproof. In the meantime, I'm still not persuaded we've negated the Kantian model (though I don't really believe it). So long as we can develop rationale for these ideas under which what we predict and what we observe are consistent, and we seem to be doing that, Kant could be right. I don't think it hinges on executability- only on the rationale and the evidence matching up, right?douglashttp://www.blogger.com/profile/17261739259295914188noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-15116453565341945722012-04-25T01:22:31.520-04:002012-04-25T01:22:31.520-04:00That last thing you said isn't so at all. To ...That last thing you said isn't so at all. To answer you best, let's go back to this -- <br /><br /><i>Since we're now 55 posts into this discussion, where do you think we stand on the question of whether we can attain mathematical truths by contemplation alone?</i><br /><br />Now, I thought it was <i>physical</i> truths we were talking about, derived through mathematics, but this is how it seems to me -- <br /><br />#1, mathematics, the kind we can apply to the physical world, <i>starts</i> with observable facts in the real world. (Objects add, multiply, and divide the way numbers do.) Observable reality gives us postulates we can use to derive more mathematical facts. The positive integers, the axioms of Euclid - these came about in this way. And the calculations we make using these postulates - correspond to the real world.<br /><br />#2 - Sometimes, however, our ideas about the world that lead to mathematics, whether based on intuition or observation, lead to results that <i>do not fit the world</i>. In that case, we have to change our assumptions in order to get math that works out physically. <br /><br />So, if we reject the idea that an infinite series can sum to a finite sum -- we get Zeno's paradoxes, which do not fit the world we observe. If we accept the idea, however counterintuitive it may be, and develop a mathematics accordingly, we get results that are astoundingly accurate. I gave you the example of the Central Limit Theorem before (which does require calculus and much more). Another famous example is non-Euclidian geometry - geometry that begins by changing the fifth axiom of Euclid; and as you probably know, fits elegeantly with general relativity, and is actually useful for relativistic calculations. <br /><br />So mathematical proofs are indeed an exercise in logic - but if the math is going to be fruitful, and the "truths" are going to correspond to something in the outside world, the assumptions from which we begin have got to be based in that world. And when the results don't fit the world, sometimes those assumptions have got to be modified. The reward is that mathematics takes us places where "raw" intuition never would -- in ways that fit the evidence.Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-42008073872521055122012-04-24T12:13:54.693-04:002012-04-24T12:13:54.693-04:00I read the proof, and your explanation of how you ...I read the proof, and your explanation of how you apply it, but I still disagree. We'll get to X, because we're firing an infinite number of rounds until we've hit every point between 2 and 3. It isn't important that we get it on the first stroke, because of course there will be an <i>infinite</i> number of X-type points that we miss to start with. The point is that the rule continues to fill in the gaps forever -- so it's OK if the gaps are infinitely big.<br /><br />The other point is that we calculate the place of the irrational not by its name, but by reference to the continuum -- thus, starting with pi -- so that we have a reliable way of bracketing. We're going to be shooting infinitely close to our first coordinate (and every other coordinate), so we need a coordinate that is precise and reliable.<br /><br />Actually, I was thinking a bit more about this, and it occurs to me that there's a significant difference between a line with points, and a number line with numbers. All the points are of the same type, but some (although vanishingly few, as it turns out) of the numbers will be rational. <br /><br />So we'll need to modify the rule to ensure that any pairing that finds a rational number is re-issued, because it won't do to pair a rational number.<br /><br />Now, pause and consider that the intuitive problem may be on your side. A countable infinity is supposed to be a different size than an uncountable one because it is a subset: the set of real numbers is uncountable, but the set of rational numbers is both countable and a subset. Our diagonal proof is elegant because it shows you a proof that there will be elements not included, etc.<br /><br />That's the same kind of argument about the even set and all sets. It's not, in fact, 'intuitive' in the sense I think you are using the word, but rather a logical argument that shows that the proof follows from the definitions. We've given definitions of sets and subsets and inclusion; and we've given a rule for creating an infinite set S0, and then S(m,n) pairings. In each case, the argument follows from the definition.<br /><br />Thus, your arguments are completely rational -- but the physics seems to show that they aren't true. Infinities do seem to work more like c: they serve as a ceiling, beyond which what ought to be true isn't true. It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not <i>rational</i> given the system, but it is true.<br /><br />So maybe you'll like it better if we say the world isn't rational, rather than that math isn't; but if math is going to map to the world, it's going to have to know when to bend or change its rules. What follows from reason ceases to apply.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-43354122033774945802012-04-24T08:03:30.150-04:002012-04-24T08:03:30.150-04:00What I'm actually trying to conclude is that e...<i>What I'm actually trying to conclude is that every infinity is the same size -- whether it is countable or uncountable is of no matter to me.</i><br /><br />The whole <i>point</i> of "countable" versus "uncountable" infinities is that this is not so - that is why Cantor is remembered at all. To do this, he had to come up with a concept for "size" that applies to infinities - which is a matter of pairing them off with mathematical sets, not as a final result, but as a process. <br /><br />After a good night's sleep, I woke up remembering just how to explain the diagonal slash in terms of your example - an interval bounded by pi. <br /><br />Imagine your numbers set out in grand array, paired off with the positive integers, like this:<br /><br />1 -- 3.1415926535....<br />2 -- 3.1415950112....<br />3 -- 3.1415926536....<br /><br />...and so on, so that every irrational number you're creating is paired off with an integer, by means of "bracketing fire" or whatever. <br /><br />Now, if I can create an irrational number that is <i>not</i> in your array, no matter how big your array gets, then you are not able to pair off all the irrationals in that interval with the positive integers. <br /><br />And I can do that. Call my new number X. The first six digits are the same as pi - 3.14159... The seventh digit is going to be a three (I added 1 to the 2 that occurs there in pi) <br /><br />For the eighth digit, take the eighth digit of the number you have paired with integer 2. Change that number to something else. That is the eighth digit of X. <br /><br />For the ninth digit, take the ninth digit of the number you have paired with integer 3. Change that digit to something else. That is the ninth digit of X. Keep going.<br /><br />Now, X is going to lie inside your interval, be a nonrepeating decimal, an irrational number. Yet it is different from every number you've got paired with an integer in the interval. Thus, they can't all be paired, not only not as a "final result" (which is by definition impossible), but not even as a <i>process</i> that takes you out to infinity. This works no matter how tightly you draw the interval. <br /><br />The rational numbers, as I showed you above, <i>can</i> be paired with the integers...not as a final result, but as a process that takes you out to infinity. <br /><br />I'm told Cantor actually had a different proof to start with, and developed this later. But this is the one that makes it into the pop-math books, because it's much easier for the readers to follow, and is beautiful in its elegance. <br /><br />But the real point is this - you were trying to say that mathematical results in this area are <i>irrational</i> -- and I am trying to show you that they are not irrational, even though they are counterintuitive, and understandably take an effort to grasp. <br /><br />The rest will have to wait 'til lunch break or nightfall.Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-83967628626980885142012-04-23T23:07:45.383-04:002012-04-23T23:07:45.383-04:00But let's back up. Here's a bit from the ...But let's back up. Here's a bit from the original post.<br /><br /><i>There is another problem, though, which is that we can also obtain knowledge through contemplation alone: for example, we can come to knowledge of mathematical truths simply by thinking. </i><br /><br />The Cantor proof you cited a moment ago is an example of this. The main thrust of our discussion has been against it -- in favor of needing 'to go back to the world' to learn about things like c behaving differently. <br /><br />Since we're now 55 posts into this discussion, where do you think we stand on the question of whether we can attain mathematical truths by contemplation alone?Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-19262575160561652682012-04-23T22:29:02.984-04:002012-04-23T22:29:02.984-04:00What I'm actually trying to conclude is that e...What I'm actually trying to conclude is that every infinity is the same size -- whether it is countable or uncountable is of no matter to me. That is, I'm concluding the very thing that modern physics appears to support. As cited above, "<i>Real objects cannot have infinite charge or mass or whatever. But when scientists in the 1950s started calculating those quantities with their latest and fanciest theories, infinities kept sprouting up and ruining things. Rather than abandon the theories, though, a few persistent scientists realized that they could do away with the infinities through mathematical prestidigitation. (Basically, they started calculating with and canceling out infinity like a regular old number, normally a big no-no.)</i>"<br /><br />The problem you're having in convincing me, I think, is that you haven't actually worked through the <i>Physics</i>. :)<br /><br />You keep trying to tell me that Aristotle would agree with you if only he'd had the benefit of your education, but perhaps there remain a few things to learn from the old master. All of these thought experiments you're citing are there -- the problem re: Plank length is a Book VIII problem, although obviously it's phrased in different terms. It's also a problem about the atomists, quite familiar to Aristotelian and neoplatonic thought. Your contention from a recent discussion that space and time can't exist separately, and so that there is no 'before' to talk about is a Book VII issue, in which Aristotle comes to the same conclusion that you want to come to as well.<br /><br />The problem about irrational numbers being infinite is a problem about the nature of a continuum, which Aristotle discusses at very great length in the <i>Physics</i>. <br /><br />There should be no conceptual problem in thinking of infinities as a kind of constant like the speed of light. Consider the famous schoolhouse problem of two cars driving towards each other, versus two starships roaring at each other at substantial percentages of c: in the one case you add the velocities together to get the speed with which they are closing, and in the other you need a whole different approach. <br /><br />That seems to be what physicists are doing by canceling infinities, and it seems to work. If the model says that can't be done, we just haven't figured out the right way of doing it; and you should join me in trying.<br /><br />Now that seems to me what Aristotle would say: it's empirical, in just the way that you (and he) like.Grimhttp://www.blogger.com/profile/07543082562999855432noreply@blogger.comtag:blogger.com,1999:blog-5173950.post-56809270780148485392012-04-23T22:01:48.400-04:002012-04-23T22:01:48.400-04:00I'm pretty sure it's not my intuition that...<i>I'm pretty sure it's not my intuition that's at work here. The definition of 'even number' is any number that can be divided by two.</i><br /><br />Yes...<br /><br /><i>These occur every two integers.</i><br /><br />...and yes...<br /><br /><i>Thus, the set of all integers is twice as large as the set of even integers.</i><br /><br />...and no! For a <i>finite</i> interval of integers that would indeed be correct. And because your intuition is geared to finite quantities, your intuition suggests that. But when you are dealing with infinite quantities, this is not so, nor has any reason to be so. This kind of mathematics is not <i>irrational</i> at all -- it is just very, very counterintuitive.Joseph W.http://www.blogger.com/profile/09480728887840887200noreply@blogger.com