| Topic: |
Science > Physics |
| User: |
"Lester Zick" |
| Date: |
08 May 2007 11:28:48 AM |
| Object: |
Infinitesimal Arithmetic |
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 05:26:18 PM |
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On 24 May 2007 20:39:30 -0700, PD <TheDraperFamily@gmail.com> wrote:
nor
is dt
Might want to read up on that one too. dr might come before dt,
though, being alphabetically precedent.
So now all of a sudden dr and dt are infinitesimals? Or would that be
spoon feeding dr and dt?
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 08:47:33 PM |
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On May 25, 5:26 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 24 May 2007 20:39:30 -0700, PD <TheDraperFam...@gmail.com> wrote:
nor
is dt
Might want to read up on that one too. dr might come before dt,
though, being alphabetically precedent.
So now all of a sudden dr and dt are infinitesimals?
Now they are differentials, but not all of a sudden. They've been that
for quite some time. The differentials have infinitesimal measure, and
they've had that for quite some time, too.
Or would that be
spoon feeding dr and dt?
Not at all. You've asked a straightforward question, and gotten a
straightforward answer. See how easy that is?
~v~~
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 05:29:09 PM |
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On 24 May 2007 20:39:30 -0700, PD <TheDraperFamily@gmail.com> wrote:
But then again, that may not be what you're interested
in at all. It may be that all you're interested in is how prettily you
dance, and how esthetically you can randomly sprinkle jargon in
sentences.
Or you could just lick my stamen instead.
Ah, there you are, stooping to allusions and aspersions again. You
sure know how to wind up your brain when called to, don't you?
No but I sure know how to wind up yours.
~v~~
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 05:32:55 PM |
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On 24 May 2007 20:39:30 -0700, PD <TheDraperFamily@gmail.com> wrote:
But you haven't
asked anybody about the physics of circular rotation.
Funny I thought I just asked you.
No, you didn't. You uttered a nonsense sentence that included
sprinkled jargon that had nothing to do with the physics of circular
rotation. Didn't I say that already, too?
Is there anything you haven't said already except the explanations for
everything you've already said?
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 08:57:30 PM |
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On May 25, 5:32 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 24 May 2007 20:39:30 -0700, PD <TheDraperFam...@gmail.com> wrote:
But you haven't
asked anybody about the physics of circular rotation.
Funny I thought I just asked you.
No, you didn't. You uttered a nonsense sentence that included
sprinkled jargon that had nothing to do with the physics of circular
rotation. Didn't I say that already, too?
Is there anything you haven't said already except the explanations for
everything you've already said?
Well, if you're going to spend an entire Zick Blossom without even
asking for an explanation, then I don't know why you would expect
differently.
I don't know who taught you manners as a boy, Zick. As far as I know,
muttering "You haven't given me any cake as far as I can see," is not
really asking for cake. Shall we start on manners first, or do you
want to proceed directly to asking something about circular rotation?
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 05:32:00 PM |
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On 24 May 2007 20:39:30 -0700, PD <TheDraperFamily@gmail.com> wrote:
Sure I can explain the physics of circular rotation.
Which is obviously why you don't.
You haven't asked. Didn't I say that already?
You've said everything already. You never stop saying everything
already. I was just curious why you say everything over and over
except for the explanations for everything you say over and over.
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 08:50:23 PM |
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On May 25, 5:32 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 24 May 2007 20:39:30 -0700, PD <TheDraperFam...@gmail.com> wrote:
Sure I can explain the physics of circular rotation.
Which is obviously why you don't.
You haven't asked. Didn't I say that already?
You've said everything already. You never stop saying everything
already. I was just curious why you say everything over and over
except for the explanations for everything you say over and over.
I've already told you why. You haven't asked for them.
Here, let me help you.
Recite this as an exercise:
"I don't know anything about circular rotation. Could someone please
explain it to me?"
PD
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 05:23:23 PM |
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On 24 May 2007 20:39:30 -0700, PD <TheDraperFamily@gmail.com> wrote:
Lester believes that for a circle with radius r, that dr/dt <> 0.
Actually centripetal dr/dt>0.
Actually, this is wrong.
Oh I dunno.
That's apparent. Still doesn't change the fact that what you said is
wrong.
Do we get a clue as to why or are you still stuck on stupid?
No, and no. I am, however, stuck on not spoonfeeding you.
So in modern math not explaining your opinions is now called spoon
feeding?
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
25 May 2007 08:44:57 PM |
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On May 25, 5:23 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 24 May 2007 20:39:30 -0700, PD <TheDraperFam...@gmail.com> wrote:
Lester believes that for a circle with radius r, that dr/dt <> 0.
Actually centripetal dr/dt>0.
Actually, this is wrong.
Oh I dunno.
That's apparent. Still doesn't change the fact that what you said is
wrong.
Do we get a clue as to why or are you still stuck on stupid?
No, and no. I am, however, stuck on not spoonfeeding you.
So in modern math not explaining your opinions is now called spoon
feeding?
No, nothing of the sort. Not explaining your *opinions* is what some
people call reserve or wisdom. Not explaining truth when it is readily
available elsewhere with just the barest effort is called declining to
spoonfeed. Catering to the whims and whines of a squawling infant is
called spoon feeding.
PD
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
23 May 2007 01:58:08 PM |
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On 22 May 2007 20:26:17 -0700, Math1723 <anonym1723@aol.com> wrote:
Finitely yes infinitesimally no.
Why not infinitessimally?
Well I'm not quite sure where we are in the conversation at
this point but I have yet to notice a finite change to
finite r produced by a finite number of infinitesimals in
circular rotation. Probably my fault; I just haven't looked
hard enough.
I don't follow. What is r
r is a finite
and what are these "infinitessimals" you
are talking about?
I wish I knew better how to explain the concept in exhaustive
mechanical terms but the best I'm able to do at present is dr.
Do you perhaps mean "points" on the circle? (A
geometric point has length 0, not infinitessimal.)
No I definitely don't mean points on a circle because the whole issue
I'm trying to deal with in this context is how to mechanize circular
rotation to begin with. All these "points on a circle" are mathemtical
approximations to curvilinear motion in any event. And that motion is
mechanized by means of linear velocity and transverse acceleration.
~v~~
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
23 May 2007 10:47:31 AM |
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Lester Zick wrote:
On 22 May 2007 11:18:03 -0700, Math1723 <anonym1723@aol.com> wrote:
On May 21, 7:23 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Finitely yes infinitesimally no.
Why not infinitessimally?
Well I'm not quite sure where we are in the conversation at this point
but I have yet to notice a finite change to finite r produced by a
finite number of infinitesimals in circular rotation. Probably my
fault; I just haven't looked hard enough.
~v~~
It's true that a finite number of infinitesimal changes does not
constitute a finite change, but an uncountable number of infinitesimal
increments can.
01oo
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
23 May 2007 02:28:34 PM |
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On Wed, 23 May 2007 11:47:31 -0400, Tony Orlow <tony@lightlink.com>
wrote:
Lester Zick wrote:
On 22 May 2007 11:18:03 -0700, Math1723 <anonym1723@aol.com> wrote:
On May 21, 7:23 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Finitely yes infinitesimally no.
Why not infinitessimally?
Well I'm not quite sure where we are in the conversation at this point
but I have yet to notice a finite change to finite r produced by a
finite number of infinitesimals in circular rotation. Probably my
fault; I just haven't looked hard enough.
~v~~
It's true that a finite number of infinitesimal changes does not
constitute a finite change, but an uncountable number of infinitesimal
increments can.
And despite the pretentions of modern mathematikers the only way to
achieve that is not by counting uncountable countables but through
definite integration of infinitesimals between finite limits, Tony.
~v~~
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| User: "M.Kane Jeeves" |
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| Title: Re: Infinitesimal Arithmetic |
21 May 2007 02:54:44 AM |
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On Mon, 21 May 2007 00:14:11 +0200, G. Frege wrote:
On Sun, 20 May 2007 15:02:11 -0700, Lester Zick <dontbother@nowhere.net>
wrote:
On 20 May 2007 11:49:40 -0700, (Daryl
McCullough) wrote:
In the usual treatment of infinitesimals, r+i is *never* equal to r
except in the case i=0. So that's not a good definition of
infinitesimal.
Well, Daryl, it's true that r+i=r is not a completely satisfactory
definition, because it doesn't say what i actually is.
Nonsense. r+i=r exactly "says" what i actually is: i is zero, in
symbols: i = 0. Got it?!
F.
Are you Gottlob Frege? I loved your Begriffsschrift, eine der
arithmetischen nachgebildete Formelsprache des reinen Denkens.
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| User: "Ross A. Finlayson" |
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| Title: Re: Infinitesimal Arithmetic |
20 May 2007 03:23:39 PM |
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On May 20, 11:49 am, (Daryl McCullough)
wrote:
HMSBeagle says...
On 8 May 2007 18:58:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 9 mayo, 00:55, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
In the usual treatment of infinitesimals, r+i is *never* equal to
r except in the case i=0. So that's not a good definition of
infinitesimal. The whole original point of infinitesimals (Newton's
version) was to define derivatives:
f'(x) = (f(x+i) - f(x))/i
where i is infinitesimal. If x+i is always equal to i, then f'(x)
turns out to be zero.
--
Daryl McCullough
Ithaca, NY
Consider a "linear" function on the non-negative reals with constant
derivative i.
Then, between zero and one, the evaluated integral is 1/2 i, on [1,2],
3/2 i, [2,3], 5/2 i, etcetera. Then, for each unit interval between
naturals n and n+1 it is (2n+1)/2 i, or (1/2 + n) i. Then, the sum of
those for n from 0 to infinity is oo/2 (+1/2), half an infinity, where
for each value of f where f(n) < 1/2 there is exactly one value of m
n such that f(m) + f(n) = 1/2.
If the constant derivative was i^2, (1/oo)^2, then the integral over
the non-negative reals would evaluate to 1/2.
(This is where, it seems that the integral of i n defined on the
naturals is 1, that the integral of i^2 r defined on the non-negative
reals is 1/2.)
That's basically about a symmetry argument about a point halfway to
infinity, that as f(n) = (i n) goes to 1, that the average value over
all values of n is 1/2, so the integral is oo/2, divided that by oo
yields 1/2.
Then, for some least positive real iota i, \( \int_{r=0}^{\infty}
\frac{1}{\infty^2} r, dr = 1/2 \).
I'm not talking here about spurious results along the lines of oo + oo
= oo, instead oo + oo = 2 oo.
Did Newton get into that?
Then, there's a notion that the integral on [0,oo) of 1/oo^2 x dx is
not one half, but instead, 3/2.
It's very easy to construct examples of arithmetic on infinite
(infinitesimal) quantities that leads to ridiculous and obviously
false results back in the finite. Where that is so, what is the
direction to look for rules of arithmetic on infinital quantities that
not only do preserve our regular notions of arithmetic on finite
quantities, but also give novel means in deriving finite quantities
from symmetrical and other arguments in infinital/infinitary
quantities?
Where are the "unusual" treatments of infinitesimals that would extend
intuitive knowledge of them (infinitesimals vis-a-vis infinities), and
not just rehash IST/hyperreals etcetera in non-answers, giving
analytical results?
Ross
--
Finlayson Consulting
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| User: "Jonathan Hoyle" |
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| Title: Re: Infinitesimal Arithmetic |
22 May 2007 10:39:48 PM |
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On May 20, 4:23 pm, "Ross A. Finlayson" <r...@tiki-lounge.com> wrote:
Consider a "linear" function on the non-negative reals with constant
derivative i.
Then this function is simply f(x) = i*x for some constant i.
Then, between zero and one, the evaluated integral is 1/2 i, on [1,2],
3/2 i, [2,3], 5/2 i, etcetera. Then, for each unit interval between
naturals n and n+1 it is (2n+1)/2 i, or (1/2 + n) i.
Correct so far.
Then, the sum of those for n from 0 to infinity is oo/2 (+1/2), half an infinity, where
for each value of f where f(n) < 1/2 there is exactly one value of m
Incorrect. This series converges only when i=0 and diverges for all
other i.
If the constant derivative was i^2, (1/oo)^2, then the integral over
the non-negative reals would evaluate to 1/2.
Incorrect. "If the constant derivative was i^2", then the function
would simply be f(x) = (i^2)*x for some constant i, and the integral
over the non-negative reals would diverge for all non-zero i.
<Rant snipped>
Where are the "unusual" treatments of infinitesimals that would extend
intuitive knowledge of them (infinitesimals vis-a-vis infinities), and
not just rehash IST/hyperreals etcetera in non-answers, giving
analytical results?
They are whereever you want them to be. Begin by defining your
assumptions, and let these axioms generate the theorems you want. To
my knowledge though, Non-Standard Analysis is the only location where
infinitessimals have been consistently defined.
Regards,
Jonathan Hoyle
Eastman Kodak
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
23 May 2007 11:03:38 AM |
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Jonathan Hoyle wrote:
On May 20, 4:23 pm, "Ross A. Finlayson" <r...@tiki-lounge.com> wrote:
Consider a "linear" function on the non-negative reals with constant
derivative i.
Then this function is simply f(x) = i*x for some constant i.
Then, between zero and one, the evaluated integral is 1/2 i, on [1,2],
3/2 i, [2,3], 5/2 i, etcetera. Then, for each unit interval between
naturals n and n+1 it is (2n+1)/2 i, or (1/2 + n) i.
Correct so far.
Then, the sum of those for n from 0 to infinity is oo/2 (+1/2), half an infinity, where
for each value of f where f(n) < 1/2 there is exactly one value of m
Incorrect. This series converges only when i=0 and diverges for all
other i.
Not incorrect. Ross is saying the same thing you are but more
specifically. Standard mathematics considers a divergent series simply
that. Ross is comparing this sum to a standard infinite sum, a standard
infinite unit. Where you have a standard infinite unit I, such as the
number of reals in the unit interval, then you also have a standard
infinitesimal unit i, the space that each of those reals occupies in
that interval, the multiplicative inverse of I. I*i=1.
If the constant derivative was i^2, (1/oo)^2, then the integral over
the non-negative reals would evaluate to 1/2.
Incorrect. "If the constant derivative was i^2", then the function
would simply be f(x) = (i^2)*x for some constant i, and the integral
over the non-negative reals would diverge for all non-zero i.
<Rant snipped>
Where are the "unusual" treatments of infinitesimals that would extend
intuitive knowledge of them (infinitesimals vis-a-vis infinities), and
not just rehash IST/hyperreals etcetera in non-answers, giving
analytical results?
They are whereever you want them to be. Begin by defining your
assumptions, and let these axioms generate the theorems you want. To
my knowledge though, Non-Standard Analysis is the only location where
infinitessimals have been consistently defined.
Regards,
Jonathan Hoyle
Eastman Kodak
Hi Jonathan -
Do you think it unreasonable to consider an uncountable unit I as the
number of reals in (0,1], and an infinitesimal unit i as its reciprocal?
Doesn't this marry measure and count, the infinite with the finite, and
infinitesimal, in a relatively consistent way?
Tony Orlow
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| User: "Ross A. Finlayson" |
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| Title: Re: Infinitesimal Arithmetic |
23 May 2007 08:33:27 AM |
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On May 22, 8:39 pm, Jonathan Hoyle <jonho...@mac.com> wrote:
On May 20, 4:23 pm, "Ross A. Finlayson" <r...@tiki-lounge.com> wrote:
Consider a "linear" function on the non-negative reals with constant
derivative i.
Then this function is simply f(x) = i*x for some constant i.
Then, between zero and one, the evaluated integral is 1/2 i, on [1,2],
3/2 i, [2,3], 5/2 i, etcetera. Then, for each unit interval between
naturals n and n+1 it is (2n+1)/2 i, or (1/2 + n) i.
Correct so far.
Then, the sum of those for n from 0 to infinity is oo/2 (+1/2), half an infinity, where
for each value of f where f(n) < 1/2 there is exactly one value of m
Incorrect. This series converges only when i=0 and diverges for all
other i.
If the constant derivative was i^2, (1/oo)^2, then the integral over
the non-negative reals would evaluate to 1/2.
Incorrect. "If the constant derivative was i^2", then the function
would simply be f(x) = (i^2)*x for some constant i, and the integral
over the non-negative reals would diverge for all non-zero i.
<Rant snipped>
Where are the "unusual" treatments of infinitesimals that would extend
intuitive knowledge of them (infinitesimals vis-a-vis infinities), and
not just rehash IST/hyperreals etcetera in non-answers, giving
analytical results?
They are whereever you want them to be. Begin by defining your
assumptions, and let these axioms generate the theorems you want. To
my knowledge though, Non-Standard Analysis is the only location where
infinitessimals have been consistently defined.
Regards,
Jonathan Hoyle
Eastman Kodak
IST, Nelson's Internal Set Theory, which is Robinson's NSA's
hyperreals in set theory, has been proven _coconsistent_ with ZFC,
Zermelo-Fraenkel set theory with choice, not consistent by itself. So
that is not entirely correct.
For, if you could prove ZFC consistent, it would be incomplete (with
respect to statements about the real numbers), as a finitely-
axiomatized non-null theory.
Then, it wouldn't be making some true statements about the real
numbers.
How about Conway surreal infinitesimals? Are they not infinitesimals,
some of the non-zero ones?
If there is the constant infinitesimal then Vitali type results about
the existence of non-measurable sets in the reals don't hold in the
same way. Structure is preserved in that way. Consider where the
infinite and infinitesimal terms are linearly dependent.
I wonder, in what ways would you consider IST and NSA any different
from each other? Basically IST posits a field tree which NSA uses to
contains non-real halos, except halos also are everywhere in a way
contiguous, or (thus) their points are. IST has the fluents and
fluxions.
No, nothing I wrote there is incorrect. You're just using the
standard definition of diverge and converge. Consider the definition
of the derivative in terms of an infinitesimal from two posts ago.
That's standard. The limit of the derivative for the free variable of
change is not finite and not zero. For example: rise over run,
slope, the derivative.
Partitioning the unit interval into infinitely many equal sized
partitions, and calling those real numbers, basically has via
properties of well-ordering the reals that using it in the theory
enables bookkeeping on the partitions. It's like division by zero.
They're degenerate intervals, because, they're intervals that only
contain the one number, for example as we were discussing about well-
ordering the real numbers. Yet, in terms of finite approximations as
are actually used in the applications, the logical number of
partitions is the Nyquist frequency of twice the product of the
frequencies of the events. That way, the least cross cancellative
terms are preserved, anti-principal component analysis on the large
systems. Any sample of the real numbers is probably _not_ the number,
they're the continuum, it's all quite simple.
Really the point of that is the approximation correctors with the
polydimensional points, for example in these sensical non-real
functions that I describe. In physics terms, Heaviside was called
unorthodox, in for example the application of the step and impulse
function as indicator series in digital logic, where the actual N-body
component simulations readily synthesize.
In that way, real numbers seem consistent. They are quite consistent,
how could they not be?
Ross
--
Finlayson Consulting
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 12:23:36 PM |
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On 8 May 2007 18:58:11 -0700, Tonico <Tonicopm@yahoo.com> wrote:
On 9 mayo, 00:55, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
************************************************************************
Ok! So r being a finite cardinal (and thus just about the same as a
natural number), then r + i = r means i = 0 ==> so you call
transfinite what we, most of all other weird mortals, call
"zero"....why then call it transfinite? Beats me....but fine.
You might want to consider a refresher in ESL, Tonico.
~v~~
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| User: "Tonico" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 02:18:58 PM |
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On 9 mayo, 20:23, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 18:58:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 9 mayo, 00:55, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
************************************************************************
Ok! So r being a finite cardinal (and thus just about the same as a
natural number), then r + i = r means i = 0 ==> so you call
transfinite what we, most of all other weird mortals, call
"zero"....why then call it transfinite? Beats me....but fine.
You might want to consider a refresher in ESL, Tonico.
******************************************************
Hehe...:>) ....I really don't know what ESL stands for, but yes: I was
thinking that I, precisely I, was needing of a refresher in
whatever....lol!
Regards
Tonio
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 06:08:49 PM |
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On 9 May 2007 12:18:58 -0700, Tonico <Tonicopm@yahoo.com> wrote:
On 9 mayo, 20:23, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 18:58:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 9 mayo, 00:55, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
************************************************************************
Ok! So r being a finite cardinal (and thus just about the same as a
natural number), then r + i = r means i = 0 ==> so you call
transfinite what we, most of all other weird mortals, call
"zero"....why then call it transfinite? Beats me....but fine.
You might want to consider a refresher in ESL, Tonico.
******************************************************
Hehe...:>) ....I really don't know what ESL stands for,
English as a Second Language.
but yes: I was
thinking that I, precisely I, was needing of a refresher in
whatever....lol!
Certainly not in replying to posts you don't reply to.
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
08 May 2007 05:11:49 PM |
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On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
Let's see. That would be i=0.
I supose that by transfinite
you mean an infinite cardinal, and r is a finite one, thus I + r is a
well defined cardinal, but what is an "infinitesimal" one, and how do
you define the sum of such cardinals?
r+i=r
By the way, Tonico, I missed your reply to my last post to you. I'm
sure it was just an oversight so I reprint here for your convenience:
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 12:21:54 PM |
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On 8 May 2007 15:11:49 -0700, PD <TheDraperFamily@gmail.com> wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
Let's see. That would be i=0.
It might if i weren't infinitesimal.
I supose that by transfinite
you mean an infinite cardinal, and r is a finite one, thus I + r is a
well defined cardinal, but what is an "infinitesimal" one, and how do
you define the sum of such cardinals?
r+i=r
By the way, Tonico, I missed your reply to my last post to you. I'm
sure it was just an oversight so I reprint here for your convenience:
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 12:42:29 PM |
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On May 9, 12:21 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 15:11:49 -0700, PD <TheDraperFam...@gmail.com> wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
Let's see. That would be i=0.
It might if i weren't infinitesimal.
Which you've just defined as whatever satisfies r+i=r.
So, let's see, something that satisfies r+i=r nevertheless does not
satisfy r+i=r.
We seem to be back to your not(not) -- or something like that.
I supose that by transfinite
you mean an infinite cardinal, and r is a finite one, thus I + r is a
well defined cardinal, but what is an "infinitesimal" one, and how do
you define the sum of such cardinals?
r+i=r
By the way, Tonico, I missed your reply to my last post to you. I'm
sure it was just an oversight so I reprint here for your convenience:
~v~~- Hide quoted text -
- Show quoted text -
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 06:07:13 PM |
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On 9 May 2007 10:42:29 -0700, PD <TheDraperFamily@gmail.com> wrote:
On May 9, 12:21 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 15:11:49 -0700, PD <TheDraperFam...@gmail.com> wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
Let's see. That would be i=0.
It might if i weren't infinitesimal.
Which you've just defined as whatever satisfies r+i=r.
And is infinitesimal.
So, let's see, something that satisfies r+i=r nevertheless does not
satisfy r+i=r.
More to the point does not satisfy you.
We seem to be back to your not(not) -- or something like that.
What not(not)? Anything like is(is)?
I supose that by transfinite
you mean an infinite cardinal, and r is a finite one, thus I + r is a
well defined cardinal, but what is an "infinitesimal" one, and how do
you define the sum of such cardinals?
r+i=r
By the way, Tonico, I missed your reply to my last post to you. I'm
sure it was just an oversight so I reprint here for your convenience:
~v~~- Hide quoted text -
- Show quoted text -
~v~~
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| User: "PD" |
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| Title: Re: Infinitesimal Arithmetic |
09 May 2007 07:55:08 PM |
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On May 9, 6:07 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 9 May 2007 10:42:29 -0700, PD <TheDraperFam...@gmail.com> wrote:
On May 9, 12:21 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 15:11:49 -0700, PD <TheDraperFam...@gmail.com> wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
On 8 May 2007 11:19:11 -0700, Tonico <Tonic...@yahoo.com> wrote:
On 8 mayo, 19:28, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
**********************************
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
Let's see. That would be i=0.
It might if i weren't infinitesimal.
Which you've just defined as whatever satisfies r+i=r.
And is infinitesimal.
Ah, well done. Sort of like defining a square to be a quadrilateral
that is square.
So, let's see, something that satisfies r+i=r nevertheless does not
satisfy r+i=r.
More to the point does not satisfy you.
We seem to be back to your not(not) -- or something like that.
What not(not)? Anything like is(is)?
PD
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| User: "" |
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| Title: Re: Infinitesimal Arithmetic |
18 May 2007 02:22:14 PM |
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On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
In any algebra with the properties
(1) for all x: x + 0 = x = 0 + x
(2) for all x: x + -x = 0 = -x + x
(3) for all x, y, z: (x + y) + z = x + (y + z)
it follows from r + i = r that
-r + (r + i) = -r + r = 0 by (2)
-r + (r + i) = (-r + r) + i by (3)
(-r + r) + i = 0 + i by (2)
0 + i = i by (1)
Therefore, i = 0.
If (1), (2) and (3) are true, the only infinitesimal is 0. Since
you're obviously not referring to a trivial situation such as this,
then you're stating that either (1), (2) or (3) is false. You didn't
say which, therefore you didn't answer the question or even begin to
address it.
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| User: "HMSBeagle" |
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| Title: Re: Infinitesimal Arithmetic |
19 May 2007 11:21:59 PM |
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On 18 May 2007 12:22:14 -0700, wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
In any algebra with the properties
(1) for all x: x + 0 = x = 0 + x
(2) for all x: x + -x = 0 = -x + x
(3) for all x, y, z: (x + y) + z = x + (y + z)
You are confusing finite cardinal arithmetic with transfinite
arithmetic over commutative fields.
it follows from r + i = r that
[snip]
It doesn't matter what follows.
-r + (r + i) = -r + r = 0 by (2)
-r + (r + i) = (-r + r) + i by (3)
(-r + r) + i = 0 + i by (2)
0 + i = i by (1)
Therefore, i = 0.
If (1), (2) and (3) are true, the only infinitesimal is 0. Since
you're obviously not referring to a trivial situation such as this,
then you're stating that either (1), (2) or (3) is false. You didn't
say which, therefore you didn't answer the question or even begin to
address it.
blah blah blah
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
18 May 2007 06:38:03 PM |
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On 18 May 2007 12:22:14 -0700, wrote:
On May 8, 4:55 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Uh? What is an "infinitesimal" for you?
It's whatever satisfies r+i=r.
In any algebra with the properties
(1) for all x: x + 0 = x = 0 + x
(2) for all x: x + -x = 0 = -x + x
(3) for all x, y, z: (x + y) + z = x + (y + z)
it follows from r + i = r that
-r + (r + i) = -r + r = 0 by (2)
-r + (r + i) = (-r + r) + i by (3)
(-r + r) + i = 0 + i by (2)
0 + i = i by (1)
Therefore, i = 0.
Thanks for nothing. Even I could have figured that out.
If (1), (2) and (3) are true, the only infinitesimal is 0.
Not exactly. The only finite cardinal which satisfies r+i=r is zero.
Since
you're obviously not referring to a trivial situation such as this,
then you're stating that either (1), (2) or (3) is false.
Not so much false as irrelevant. I'm talking infinitesimal arithmetic
not finite cardinal arithmetic.
You didn't
say which, therefore you didn't answer the question or even begin to
address it.
Thanks for telling me what I meant. Next time you start a thread I'll
be happy to explain what you meant.
~v~~
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| User: "Don Stockbauer" |
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| Title: Re: Infinitesimal Arithmetic |
08 May 2007 11:36:12 AM |
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On May 8, 10:28 am, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
I love you.
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
08 May 2007 12:38:21 PM |
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On 8 May 2007 09:36:12 -0700, Don Stockbauer
<don.stockbauer@gmail.com> wrote:
On May 8, 10:28 am, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
I love you.
That makes two of us.
~v~~
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| User: "" |
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| Title: Re: Infinitesimal Arithmetic |
08 May 2007 06:37:06 PM |
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On May 8, 12:28 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
It's even more curious than idiot mathematicians
who understand that l+l=l, do not understand
that i+i=i, and r+r=r.
It's like the pope told them that Turing was
a Jesuit, but the random-zoids forgot to mention that
neither (GoedelvTuring)^(Goedel->Turing)
was a sub-contractor with AT&T.
~v~~
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| User: "" |
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| Title: Re: Infinitesimal Arithmetic |
22 Jun 2007 09:26:23 PM |
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On May 8, 9:28 am, Lester Zick <dontbot...@nowhere.net> wrote:
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
Point nine repeating is not one. Add an infinitesimal 1/10^infinity
and it equals One. This is a nonzero infinitesimal of the highest
order.
Burt Schwarzkauf
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