Galileo's Paradox
in [Math]
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From: cbrown on 4 Dec 2006 14:29 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. > 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. Cheers - Chas
From: Lester Zick on 4 Dec 2006 14:54 On Mon, 04 Dec 2006 12:43:06 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> Tony, I'm making some far ranging speculations here which I think are >> correct. But if I make some mistakes I hope you'll be patient. >> >Er, okay. > >> On Sun, 03 Dec 2006 00:28:04 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> Tony, let me see if I can provide an alternative line of reasoning to >>>> support my analysis. >>>> >>> Okely Dokums! >>> >>>> Over the past couple of years in addition to tautological analysis >>>> I've also provided analysis of angular mechanics in corrected form. >>>> And in that analysis I do make use of arithmetic combinations of >>>> finites and infinitesimals. In particular I take finites such as the >>>> radius of circles, r, and combine them with infinitesimal changes in >>>> radius, dr, showing that for any finite multiple of dr, such as ndr, >>>> the combination can change only infinitesimally such that r+ndr=r and >>>> r remains finitely constant. I don't know if you followed that >>>> discusion but the mechanics involved are identical to what you >>>> suggest. >>>> >>> Yes, I thought it was encouraging to see the notion of a mixture there, >>> though it didn't seem like it followed necessarily. It was more like, >>> there could be an infinitesimal change, that wouldn't be detected. In >>> any case, do go on... ;) >> >> It's more than a matter of detection. There are probably finite >> changes which couldn't be detected either. The problem is that r is >> finite and the changes infinitesimal. Different metrics entirely. >> > >But, isn't the change infinitesimal because the slice of time in the >rotation is infinitesimal? Does the radius really change, or is the >point simply accelerating in the diection of the center, without >changing velocity toward that point? Well the radius is finite and dr is infinitesimal but so is dt so the ratio dr/dt is finite. Centripetal v is therefore finite as well. Yet the radius really does change. It's just that changes in centripetal r produced by finite centripetal v have to be taken in combination with changes in finite r produced by a finite tangential v. Which renders the result rotation. In other words changes in centripetal r become infinitesimal because centripetal acceleration in combination with tangential v cause both to rotate in tandem with one another causing the time slice over which finite centripetal v acts to become infinitesimal instead of finite. Thus centripetal changes to r become rotation instead of finite changes to r. >>>> Now the problem for you and your idea of combining finites and >>>> infinitesimals arithmetically is that you can't combine finites and >>>> infinitesimals directly. >>> Oh, then no dr/dt for you tonight, young man. And after you brushed your >>> teeth already... >>> >>> In other words there is no way to say r+dr>r >>> >>> Except you just did it. >> >> Okay. You can say it but you can't demonstrate it. >> > >Then there is no way to say dr>0. > >x+y=y -> x=0 Not when you're doing finite algebra, no. X and y are both finite in your example. But the whole point of the calculus is not to do finite algebra. This is analogous to Zeno's paradox where the arrow can never arrive if we continuously subdivide the interval of travel. But that's only because we look at the problem in terms of finite algebra instead of a continuous calculus which treats infinitesimals instead. In other words to approach a continuum calculus using finite algebra one needs to continually subdivide the kind of calculation involved as finite intervals approach zero. So in saying for example that x+x+apples=2x+apples we're violating no mathematical principle; it's just that we're saying 2x+apples is not more than x+x+apples. I suppose we might also say that x+x+dr=2x+dr rather than trying to characterize x+x+dr as no more than 2x in finite terms. But the point I at least am trying to make is that without integration dr plays no direct role in finite algebra whatever its magnitude anymore than apples would play in the addition of finite non apple numbers. >>>> as you're trying to suggest because finites and infinitesimals don't >>>> lie together on a common line with the same metric. >>> Same line, different scale. >> >> Not quite, Tony. The issue is how you arrive at dr. The derivative of >> a straight line r with respect to itself is 0dr. You only find nonzero >> derivatives with respect to other metrics such as other lines or time. >> > >Well, the derivative of a line is 0 with respect to x where it's >horizontal. Do you really take derivatives of obhjects with respect to >themselves, or as one of their independent variables changes? Well the problem here is that x is your finite number line and any change is constant. I admit that the phrase "with respect to itself" is ambiguous since the slope of any line with respect to itself would be 0 dr. And usually what we mean is the slope or derivative of some line with respect to some other line such as the x axis. But here the difficulty is that we have no other x axis. The finite number line you specify winds up being the x axis. And the derivative or slope of that finite number line has to be zero. The derivative or slope of one line can only be non zero with respect to some other. It can't have a non zero derivative with respect to itself. >>>> In the case of angular mechanics this is also true. However I provide >>>> a common metric for them by definitely integrating a finite velocity, >>>> dr/dt, between 0 and dt which provides a finite dr of infinitesimal >>>> magnitude. >>> Uh, what? A finite dr of infinitesimal magnitude? What makes it finite? >> >> The definite integral between 0 and dt. Just because dt is an >> infinitesimal doesn't mean the interval between 0 and dt is an >> infinitesimal (although I'd like some more time to consider this >> aspect of the problem as I'm not entirely happy with what I'm saying >> here.) >> > >Yeah, I think you need to think about that statement. It doesn't fill me >with glee either. Sorry. No problem, Tony. I said I was going out on a limb with some of this material. I think I'd like to qualify what I say above by drawing a distinction between being an infinitesimal and being of infinitesimal magnitude. In other words when I refer to the integral between 0 and dt the magnitude doesn't change but the integration renders the result finite. >>> In other words you can't provide an arithmetic sum for >>>> finites and infintesimals directly without first providing a common >>>> finite metric for them through definite integration of some kind. >>>> >>> Yeah. That's what IFR's about. The line, Man. That's the common metric. >> >> The problem though, Tony, is what I pointed out above, the production >> of dr. If you take the derivative of a straight line r with respect to >> r you get 0dr not ndr or 00dr. >> > >I don't understand. You take a derivative of a formula with respect to >one of the variables in the formula. You don't take the derivative of >the formula with respect to itself. Do you? Do you take the derivative >of a square with respect to x^3? You can take the third derivative, with >the right unit, and get 8, the number of points. But.... ? Look at it this way. In what you propose you only have one finite number line and that line is supposedly straight. There is no other line with respect to which to take a derivative. So if you maintain dr is a constituent part of that finite number line I see no way to prove that without taking the derivative of the line with respect to itself. >>> _______________________________________________________________________ >>> >>>> This is how we can know arithmetic combinations of finites of finite >>>> magnitude and finites of infinitesimal magnitude. Mathematically >>>> modern mathematikers incorrectly analyze the same problem in the >>>> reciprocal terms of n/dr instead of ndr and wind up with various kinds >>>> of 00 they like to pretend follow the finites on a common real number >>>> line. However this makes the proper analysis of angular mechanics >>>> impossible unless one takes r to be an infinite and ndr to be finite. >>>> >>>> In any event I hope this clears up my perspective on analysis of the >>>> arithmetic combination of finites and infinitesimals. >>> Actually I got lost at the end there. Infinitesimals are things that, if >>> you multiply them together, they disappear. *poof* >> >> Not sure exactly what you're getting at here, Tony. If you want to >> combine finites and infinitesimals you have to first produce the >> infinitesimals then you have to produce some finite interval by >> definite integration between limits. Only then is some combination >> possible. When I say the definite integral of dr/dt between 0 and dt >> in the context of angular mechanics is finite but of infinitesimal >> magnitude it's only because I'm integrating dr/dt with respect to an >> infinitesimal dt not dr. (In any event this seems to be the way it >> works out at the moment.) >> >> ~v~~ > >Alright. Hopefully all will work out in the end, eh? Well I think we're doing pretty well at the moment, Tony. At least you're asking the right questions. ~v~~
From: Eckard Blumschein on 4 Dec 2006 14:59 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. > > Bob Kolker
From: Bob Kolker on 4 Dec 2006 15:54 Eckard Blumschein wrote: > > You are right if you consider each time the complete set. Neither the > integers not the rationals are actually complete. So both complete sets Really? Tell me an integer or rational that is not in the set of integers or rationals? Which ones did we miss? Bob Kolker
From: Bob Kolker on 4 Dec 2006 15:55
Eckard Blumschein wrote: > On 12/1/2006 7:04 PM, MoeBlee wrote: > > >>The Cartesian plane is the set of ordered pairs of real numbers. > > > Doesn't coordinate transform e.g. into circular coordinates require > fictitious real numbers? Here engineers might have a better feeling for > the categorical difference between number and continuum. All numbers are ficticious so call a number or a set of numbers ficticious is redundant and conveys no information. Bob Kolker > |