tag:blogger.com,1999:blog-5687347459208158501.post6022042404178712405..comments2024-03-28T19:49:43.203-04:00Comments on The Philosopher's Stone: READING THE CRITIQUE PART FIVERobert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-5687347459208158501.post-76014512200087197112020-04-27T14:19:34.357-04:002020-04-27T14:19:34.357-04:00You are doing God’s work Dr. Wolff (that was an ex...You are doing God’s work Dr. Wolff (that was an expression; not an absolutism on the phenomenon of God:) I am not a philosophy major, but your Kant essays became the engine of my thesis on world literature. Your brilliant articulation of Kant’s solipsistic dilemma in ethics was especially useful. It helped me to articulate the struggle that an author encounters when creating his own universe and It’s moral structures. 🙏James Moseleyhttps://www.blogger.com/profile/14401164992029503314noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-17881187113175910322011-07-26T14:57:26.910-04:002011-07-26T14:57:26.910-04:00The Axiom of Countable Choice, although independen...The Axiom of Countable Choice, although independent of ZF, does not as intuitively unobvious as AC because it does not entail things like the Banach-Tarski Theorem or create problems for Lebesgue measurability. I think that Tarski's axiomatization of geometry, however, still does not assume even weaker versions of AC. My point was that by setting up the problem correctly, you will be able to prove it without a lot of set theoretical machinery. Exactly how much is a question that I don't know how to answer.J.Vlasitshttps://www.blogger.com/profile/10340794410334308312noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-31352268147745613972011-07-26T08:59:03.738-04:002011-07-26T08:59:03.738-04:00Even if you're only dealing with countable set...Even if you're only dealing with countable sets, some kind of choice principle is needed. There is an axiom of countable choice ("Every *countable* collection of nonempty sets has a choice function") but this, like its stronger counterpart, is independent of the rest of the axioms of set theory. (And of course, it wasn't on Euclid's or Kant's radar at all.)Michaelhttps://www.blogger.com/profile/12922719871297540449noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-60187818828363389302011-07-25T17:24:37.734-04:002011-07-25T17:24:37.734-04:00You are right that Tarski's axiomatization doe...You are right that Tarski's axiomatization does not assume the Axiom of Choice. Back in 1952, when I was a senior at Harvard, I was actually told to do my honors thesis on his monograph [by Hao Wang, a wonderful but rather incomprehensible logician.] I bombed out and instead wrote on Ryle's CONCEPT OF MIND, after which it was downhill for me all the way to political philosophy -- The Rake's Progress.Robert Paul Wolffhttps://www.blogger.com/profile/11970360952872431856noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-50986409367126088412011-07-25T15:57:10.206-04:002011-07-25T15:57:10.206-04:00I'm not sure that the Axiom of Choice is actua...I'm not sure that the Axiom of Choice is actually necessary here because I think you can run the proof so that the point of intersection is an algebraic number, in which case you will only be dealing with the countably infinite (an algebraic number is the root of a polynomial, e.g. squareroot(2). Cantor proved that these must be countable so the cardinality of the rationals plus the algebraics is also countable. It is only when you add the transcendental numbers like pi and e that you will reach uncountability.) I'm pretty sure that Tarski's axiomatization of Euclidean geometry does not assume the axiom of choice, although it does make assumptions about continuity.J.Vlasitshttps://www.blogger.com/profile/10340794410334308312noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-41554441705988975162011-07-25T05:58:49.776-04:002011-07-25T05:58:49.776-04:00Yes, but that is not a problem of intuition! You m...Yes, but that is not a problem of intuition! You might not like the axiom (and in this case, by the way, it's still not clear to me why the axiom of choice is necessary, but my ignorance is another question -- still, who said I needed to select the point -- I just know it's there, and then I say whereever it is, let's call it A), but the proof would still follow analytically from the definition, just with one extra axiom tossed in, no?<br /><br />I thought the point of the Kant is to show us you need something more than than axioms, definitions, and what is clearly contained within them analytically. The axiom of choice cannot be the entry point for the synthetic here, for the simple reason that one can name it as a na axiom, stipulate it, and then ignore it.summortushttps://www.blogger.com/profile/09870924799071871322noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-36831912603346712152011-07-25T04:40:16.478-04:002011-07-25T04:40:16.478-04:00Chris - Contrary to Prof Wolff, I heartily recomme...Chris - Contrary to Prof Wolff, I heartily recommend you study Euclid, if you at all have the opportunity. It is at times breathtakingly beautiful in its proofs and results, but, more importantly, it is probably also the most influential piece of technical writing in the Western tradition. For centuries Euclid's series of deductive proofs from self-evident axioms was taken as the gold standard for scientific explanation - a look at almost any page of Descartes or Hobbes will make that clear. Also, not everybody has gotten the memo yet about the painful lessons we learnt with logical positivism, that such a project could never describe our empirical sciences. Euclid's model and the issues that arise from it (basically, it's either tautologies or false) is an excellent object lesson in perhaps the most seductive method of investigation and its failure.Marinushttps://www.blogger.com/profile/13492009758043047531noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-2057841908578529272011-07-25T04:35:58.517-04:002011-07-25T04:35:58.517-04:00Summortus - it's one of those peculiar discove...Summortus - it's one of those peculiar discoveries that pure mathematics allows us that in fact you <i>can't</i> prove that from merely the definition of line and circle - you also need something called <a href="http://en.wikipedia.org/wiki/Axiom_of_choice" rel="nofollow">the axiom of choice</a>. This is because, since the amount of points on the circle is infinite, it is by no means obvious that you are able to select <i>the</i> point of intersection between the circle and the line. To use a beautiful analogy from Bertrand Russell, to do so requires the same skill as being able to choose a matching pair of socks out of a collection of infinitely many. Even if there is a sock to match the one you have already chosen, it is by no means certain that you'll pick its partner out in a finitely long time (which is the same as saying that perhaps you'll never find it). This is like the line intersecting the circle case where you have the point you suspect the line is equal to some x^2.y^2 = r^2 (one sock of a pair) and are looking for the point where the circle is equal to that same line (the second sock).<br /><br />The axiom of choice is commonly assumed uncritically by mathematicians, but it is by no means obviously true.Marinushttps://www.blogger.com/profile/13492009758043047531noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-41341218623514225152011-07-24T21:38:38.563-04:002011-07-24T21:38:38.563-04:00This may be silly (as maybe I am just pushing the ...This may be silly (as maybe I am just pushing the intuitiv obviousness back a step), but it seems to me one can prove that a line from the center of a circle to a point outside the circle will intersect with the circle from any reasonably strong set of geometric axioms (I don't have Euclid's handy...). <br /><br />If a circle is the set of points equidistant from the center, then we know that any point of distance r (=radius) from the center will be on the circle.<br /><br />Now, by hypothesis, the line connecting the center of the circle to the point outside the circle has length more than r. And by the definition of a line segment (which I would loosely take to be a continuous subset of the points (x,y) that satisfy "y=ax+b" for a,b=constant, there'll be some point on that line segment which is length r from the center -- the continuity condition here being crucial, as it guarantees that if (letting x,y for the center be 0,0, is the x1,y1 such that the distance measure between the center and x1,y1 is greater than r, there'll be a x2,y2 with a distance measure equal to r<br /><br />Thus, by the definition of a line segment and the definition of a circle and the stipulation that the point is outside the circle, you can analytically prove that the line and the circle intersect...<br /><br />I think, in my experience, all of the experiences you're describing as "just look at it" in math either a) are false and illusory or b) can actually be rigidly and analytically proved. But I may be missing something...summortushttps://www.blogger.com/profile/09870924799071871322noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-92130968600349109652011-07-24T14:23:58.443-04:002011-07-24T14:23:58.443-04:00Quite true about Hume [and Kant, as it happens].
...Quite true about Hume [and Kant, as it happens].<br /><br />Don't bother with Euclid [save for antiquarian purposes.] Much better to study modern analytic geometry, which is quite elegant.Robert Paul Wolffhttps://www.blogger.com/profile/11970360952872431856noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-50932856242386098802011-07-24T12:35:13.326-04:002011-07-24T12:35:13.326-04:00Professor,
I must say I rather regret slacking off...Professor,<br />I must say I rather regret slacking off in HS and entirely blowing off my geometry course. Is Euclids Elements approachable for a basic virgin to geometry to begin restudying the matter?<br /><br />Also I was reading some of Beattie and was quite impressed and horrified at one aspect I found in his book. Horrified that Hume was such an irrational racist, and impressed that Beattie rather took him to task for it!Chrishttps://www.blogger.com/profile/08250295324149056708noreply@blogger.comtag:blogger.com,1999:blog-5687347459208158501.post-56262615689058927012011-07-24T11:12:10.715-04:002011-07-24T11:12:10.715-04:00After reading your post, I decided to see if an on...After reading your post, I decided to see if an online version of your article is available. Turns out it is (under the title "Kant's Debt to Hume via Beattie") at <br /><br />http://www.jstor.org/pss/2708003Michaelhttps://www.blogger.com/profile/12922719871297540449noreply@blogger.com