Cantor Confusion
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From: Jonathan Hoyle on 15 Dec 2006 10:53 > Be that as it may Aristotle's work on logical inference and semantics > still stands up rather well. This is over just about 2400 years. Too bad > Aristotle's physics was not of the same quality. If it were, we would be > traveling around in Star Ships and not jet airplanes. True. A lot of Aristotle's physics was based upon limitations to the thinking of his day. One particular error that I thought he should have avoided was his concept that force = mass * velocity, instead of as Newton defined it as mass * acceleration. I suppose it was easy to mistake, as riding on a horse at a faster, but constant, velocity would give you the sensation of more "force", when in fact this is merely air resistance. Similarly to the prevailing belief that heavier objects fell at a faster rate than lighter objects. One could hardly blame Aristotle for these blunders, since no one else was able to figure these out back then either. However, I truly believe that Aristotle was enough of a genius that he could have, had he taken to the time to pursue it, corrected these errors and truly get physics started. Unfortunately, his reputation in other areas made the ancients take Aristotle's word as dogma in this, and no one questioned it for centuries. One wonders how the world would be today if we could get those centuries back. :-/
From: Bob Kolker on 15 Dec 2006 11:30 Jonathan Hoyle wrote: > > True. A lot of Aristotle's physics was based upon limitations to the > thinking of his day. One particular error that I thought he should > have avoided was his concept that force = mass * velocity, instead of > as Newton defined it as mass * acceleration. I suppose it was easy to Aristotle had no clear idea of either velocity or acceleration. Else he would not have concluded that heavier objects fall faster than light objects. > mistake, as riding on a horse at a faster, but constant, velocity would > give you the sensation of more "force", when in fact this is merely air > resistance. Similarly to the prevailing belief that heavier objects > fell at a faster rate than lighter objects. > > One could hardly blame Aristotle for these blunders, since no one else > was able to figure these out back then either. However, I truly He didn't check. Of his dictum heavier objects fall faster than lighter objects, a six year old kid could have dropped a heavy rock and a light rock at the same time from the same height and put stop to that canard. Aristotle did not feel the necessity for empirically corroberating his conclusion. Like many Greeks, he believe that reasoning from "self evident" principles was sufficient. Therefore I fault him. Except for lenses Aristotle had every contrivence and technology that was available to Galileo. Greek shipwrights could have made straight smooth wooden ramps to be used as inclined planes. The Greeks had Egyption drip clocks at their disposal. So it wasn't lack of tools or technology that prevented Aristotle from checking. It was attitude. > believe that Aristotle was enough of a genius that he could have, had > he taken to the time to pursue it, corrected these errors and truly get > physics started. Unfortunately, his reputation in other areas made the > ancients take Aristotle's word as dogma in this, and no one questioned > it for centuries. One wonders how the world would be today if we could > get those centuries back. :-/ We would be a millenium ahead or extinct. Aristotle's followers cost us over a thousand years of progress. Bob Kolker
From: William Hughes on 15 Dec 2006 11:31 Han de Bruijn wrote: > William Hughes wrote: > > > Han de Bruijn wrote: > > > >>William Hughes wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>William Hughes wrote: > >>>> > >>>>>Han de Bruijn wrote: > >>>>> > >>>>>>Let's repeat the question. Does there exist more than _one_ concept of > >>>>>>infinity? Isn't unbounded the same as infinite = not finite = unlimited > >>>>>>= without a limit? Please clarify to us what your "honest" thoughts are. > >>>>> > >>>>>You are *way* in deficit on clear answers. Try answering > >>>>>the following question with yes or no. > >>>>> > >>>>> Is there a largest natural number? > >>>> > >>>>No. > >>> > >>>I there an unbounded set of natural numbers? > >> > >>Suppose you mean "Is". What does it mean that a set is unbounded? > > > > An unbounded set of natural numbers is a set of natural > > numbers that does not have a largest element. > > Please answer yes or no. > > No. Does the "potentailly infinite set" of natural numbers exist? A potentially infinite set is a function on sets that takes on the values true and false. The potentially infinite set of natural numbers takes on the value true for a set containing only natural numbers, and false for any other set. (i.e. is there a way of recognizing a set of natural numbers?) - William Hughes
From: Jonathan Hoyle on 15 Dec 2006 11:39 > Current axiom systems are almost exclusively associated with Set-like > theories. Completely untrue. You always have axioms, whether you formalize them or not. Any time you assume something, you are postulating it is true. Let's go to something basic, one of the simplest laws of logic: hypothetical syllogism. If given A->B and A are true, you know B is true. This is essentially the logical axiom (A->(B->A)). You're using this all the time (or at least I hope you are). You might say, "I am using hypothetical syllogism because it's correct, not because it's an axiom!" The whole point of axioms is to systemitize what we believe is true. By believing it's true, you implicitly call it out as an axiom, whether you like it or not. "What's the point? It's a waste of time!" Tell that to the followers of Naive Set Theory last century. Euclidean and Non-Euclidean Geometries differ on a single axiom. You can spout off all you want about "axioms are unnecessary". But you use them. Everyday. Both in mathematics and in real life. You are just choosing to pretend you don't have them. By the way, why are you so fearful of listing your basic assumptions? Could it be that you realize that by doing so, it would show that you hold contradictory beliefs?
From: Lester Zick on 15 Dec 2006 13:39
On Fri, 15 Dec 2006 07:32:34 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Virgil wrote: >> >> >> Actually, we today have a number of improvements on Aristoteles logic, >> and Euclid's axiom system had to be revamped by Hilbert to bring it up >> to modern standards. But they have both stood the tests of time >> remarkably well. > >Aristotelean logic was sound as far as it went. The main improvements to >logic has been broadening and deepening the subject. Most important the >embedding of formal logical systems in metatheories has enabled us to >see the ultimate limits of logic. For example the Goedel Incompleteness >Theorems. > >This is an approach that Aristotle (at his level of technique and >understanding) could not have dreamed of. > >Be that as it may Aristotle's work on logical inference and semantics >still stands up rather well. This is over just about 2400 years. Too bad >Aristotle's physics was not of the same quality. If it were, we would be >traveling around in Star Ships and not jet airplanes. And too bad Aristotle's approach to syllogistic inference doesn't produce the truth he hoped it would yield. The best we can say for it is that it produces a series of truisms which in turn produce the empirical approaches to math and science which belabor the present. That mathematics and science rely on it even today after two millenia stands in mute testament not to its universal validity but to the fact there is no other better formal system of logical inference available. ~v~~ |