Galileo's Paradox
in [Math]
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From: Virgil on 4 Dec 2006 18:14 In article <45745E74.5070401(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 9:46 PM, Virgil wrote: > > In article <456FEEEF.5070409(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/30/2006 1:32 PM, Bob Kolker wrote: > >> > Eckard Blumschein wrote: > >> > > >> >> > >> >> > >> >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> >> to justify his basic idea. > >> > > >> > What sort of evidence? Surely not empirical evidence. Mathematics done > >> > abstractly has no empirical content whatsoever. > >> > > >> > Bob Kolker > >> > >> Serious mathematicans have to know the pertaining confession. Dedekind > >> wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit > >> beizubringen". In other words, he admitted being unable to furnish any > >> mathematical proof which could substantiate his basic assumption. > > > > That conclusion assumes something not in evidence, that no one else has > > been able to do what Dedekind said he had not done. > > Yes. It is about as impossible as to get younger. > > > >> Consequently, any further conclusion does not have a sound basis. > >> Dedekind's cuts are based on guesswork. > > > > So are everyone else's equally based on "guesswork", as without ASSUMING > > something, one cannot deduce anything. > > The ideas by Archimedes and Aristotele, by Galilei and Spinoza, and by > many others have proven very fruitful in each case, they started with a system of assumptions, which, if all those assumptions were made explicit, in mathematics would be called an axiom system. But none of them made all their assumptions explicit.
From: Virgil on 4 Dec 2006 18:16 In article <4574755B.4070507(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 4:56 PM, Tony Orlow wrote: > > Eckard Blumschein wrote: > > >> In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b. > > Generalize where possible. Why is this not true in the infinite case? > > 2*oo is not larger than oo. Infinity is not a quantum but a quality. Which "infinity" is that?
From: Virgil on 4 Dec 2006 18:18 In article <45747E16.6020904(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 12:57 PM, Bob Kolker wrote: > > Eckard Blumschein wrote:> > >> Serious mathematicans have to know the pertaining confession. Dedekind > >> wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit > >> beizubringen". In other words, he admitted being unable to furnish any > >> mathematical proof which could substantiate his basic assumption. > >> Consequently, any further conclusion does not have a sound basis. > >> Dedekind's cuts are based on guesswork. > > > > Dedikind cuts are well defined objects which have exactly the algebraic > > properties one wishes real numbers to have. So it is quite sensible to > > identify the cuts with real numbers. > > > > We can define addition, mutliplication, subtraction and division for > > cuts and the cuts satisfy the postulates for a an ordered field. > > Furthermore every set of cuts (identified with real numbers) with an > > upperbound has a least upper bound. Bingo! Just what we want. > > No wonder. Exactly this selfdelusion was the intention of dedekind. Which is a great improvement on the manifold self delusions of EB.
From: David Marcus on 4 Dec 2006 19:12 Eckard Blumschein wrote: > On 11/30/2006 4:41 AM, zuhair wrote: > > Six wrote: > > > and I think it was the > > idea before Cantor showed that there can be infinite sets of different > > sizes, > > He did not! Not! Not! He just misinterpreted uncountable as mor than > countable. Is incorrect more than correct? Are you saying that the cardinality of the reals is not greater than the cardinality of the integers? -- David Marcus
From: cbrown on 4 Dec 2006 20:57
Lester Zick wrote: > On 4 Dec 2006 11:29:33 -0800, cbrown(a)cbrownsystems.com wrote: > > > > >Lester Zick wrote: > >> On 3 Dec 2006 11:22:56 -0800, cbrown(a)cbrownsystems.com wrote: > >> > >> >Tony Orlow wrote: > >> >> cbrown(a)cbrownsystems.com wrote: > >> >> > Tony Orlow wrote: > >> >> > > >> >> > <snip> > >> >> > > >> >> >> Anyway, ala Leibniz, each object IS the set of properties which it > >> >> >> possesses, so any two objects with the exact same set of properties are > >> >> >> the same object. > >> >> > > >> >> > But this begs the question: what do we mean, exactly, by a SET of > >> >> > properties? What exactly are we trying to say when we say "This set of > >> >> > properties is the same as this other set of properties"? > >> >> > > >> >> > Cheers - Chas > >> >> > > >> >> > >> >> Well, what we really mean is that there is a set of universal > >> >> properties, each of which is a set of values, and that each object is > >> >> defined by a set of values, one from each set of property values, such > >> >> that any two distinct objects differ in at least one property value. Was > >> >> that specific enough? > >> > > >> >Well, you used the term "set" four times in your above definition of > >> >what we mean by a "set". That's why I said "this begs the question, > >> >what do we mean, exactly, by a set of properties?". > >> > > >> >There's something that we intuitively seem to think of as a "set"; but > >> >unless such a thing is carefully defined, we end up with the > >> >contradictions of naive set theory: > >> > >> I think the more basic question is whether non naive sophisticated set > >> "theory" represents all of mathematics... > > > >Of course it doesn't; "all of mathematics" is an extremely broad range > >of discourse. > > So when mathematikers conflate mathematical ignorance with set > "theory" ignorance they are being extremely overly broad? > Not all of Italian cooking involves sauteeing things in olive oil; however it is somewhat bizzare for someone to claim to be a knowledgeable Italian cook without knowing how to sautee things in olive oil. > >> and whether it has any pre > >> emptive prerogative to the definition of terms such as "cardinality" > >> etc. in mathematics generally? > >> > > > >The reason why "cardinality" has the general definition you refer to is > >because it is generally useful to have a term with that definition. We > >could call it anything; but "cardinality" is the word used for this > >term, for mostly historical reasons. > > Oh I don't much care what we call it. Mathematikers are the ones who > insist definitions are only abbreviations. And I agree empirical > utility is a cachet of modern mathematical significance. I just don't > agree that empirical definitions are particularly true. > So be it then. Each to their own. Cheers - Chas |