Re: [FRIAM] square land math question

[glen]

Yeah that's an appropriate response to a child. You are boring so just, 
whatever.

On July 24, 2020 3:42:37 PM PDT, Frank Wimberly  wrote:
>This is my final comment on this topic.  Admitting points as squares
>makes
>these square covering problems uninteresting.  By placing the
>point-squares
>on the boundary you can cover a square with an arbitrary number of
>them.

-- 
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Re: [FRIAM] square land math question

[Frank Wimberly]

This is my final comment on this topic.  Admitting points as squares makes
these square covering problems uninteresting.  By placing the point-squares
on the boundary you can cover a square with an arbitrary number of them.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 6:07 PM Jon Zingale  wrote:

> Huh, that's fun. I love that my TI-86 correctly evaluates:
> (10+6√3)^(1/3) + (10-6√3)^(1/3) to 2, just saying :)
>
>
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Re: [FRIAM] square land math question

[Jon Zingale]

Huh, that's fun. I love that my TI-86 correctly evaluates:
(10+6√3)^(1/3) + (10-6√3)^(1/3) to 2, just saying :)



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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Ha! Speaking of π, this was hilarious:

https://www.youtube.com/watch?v=7LKy3lrkTRA

Apparently my TI-36X Pro is simply not as smart as the Casio FX-83.

On 7/23/20 3:40 PM, Steve Smith wrote:
> Let's change the value of Pi to 3.0 and deal with the resulting distortion of 
> space later.

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Re: [FRIAM] square land math question

[Steve Smith]

I have 8 chickens in my courtyard which is roughly 10.5x10.5 meters
(varas since this landscape was first surveyed by the Spanish).   Once I
showed them (when we first released them) that the grass in a .5x.5
meter (vara) square was tasty they proceeded to mow the entire 10.5x10.5
yard down nicely.   I don't know if this qualifies since the remaining
~400 squares of grass were not identical (mathematically) to the one I
introduced them to, but from my idiosyncratic point of view, I had
"mowed" the whole lawn by showing the chickens the one square?

Of course, the chickens didn't need showing and would have figured it
out for themselves (as they figured out how tasty virtually everything
in my gardens were too), and they left the longer, tougher grass-stems,
moving their focus to any and all tender shoots the grass root-clumps
decide to splurt out every day.

Have we agreed on what a "square" is and by whose "authority" we declare
that we are talking about the same description?  I don't think that nit
has been picked over yet.    And are we doing it in Cartesian or
spherical or ellipsoidal coordinates?   I'm not sure if there is a
conventional "ovoid "
coordinate system but my guess is that the chicken's would prefer
those.   And I don't think these chickens care for analytic or
computational paradigms, they just want to eat, play grab-ass with the
squirrels and jays invading their territory and lay single cells the
size of a chicken-egg for me to steal and treat as my personal property
to consume, sell, trade or gift.   I should have gone for the golden (or
palladium) chickens instead methinks...

If we can't even square a square, how can we expect to square a circle,
or more interestingly tessellate a sphere uniformly?   Let's change the
value of Pi to 3.0 and deal with the resulting distortion of space later.



Carry On,   

  - Sieze


On 7/23/20 3:41 PM, uǝlƃ ↙↙↙ wrote:
> We used to have this argument all the time about the apt use of relational 
> vs. OO databases. As in Ed's conception, the same square can be associated 
> with multiple locations. Then to update all the renderings of that 1 square, 
> say, change its color from red to blue, all you need do is change the object 
> and all its renderings change as a result. That's pretty handy.
>
> But what if you really did want multiple squares so that changing the color 
> of this square over here didn't change the color of that square over there? 
> You might want "square" to be a class but have color be an instance property 
> so you could change each square to a different color. Or you might even have 
> a concept of *scope* so that all  the squares in a neighborhood changed, but 
> no those far away ... or only the squares that are also rotated 90° 
> (invisibly) would change color, but those that haven't been rotated stay 
> whatever color they are.
>
> To my mind, computationalists tend to think like the latter (collections of 
> instances) whereas analysts tend to think like the former ("normalized" or 
> "unified"). I'm agnostic and like both teams. But when I see one team 
> winning, I tend to traitoriously jump from one side to the other.
>
> On 7/23/20 2:26 PM, Frank Wimberly wrote:
>> What?
>>
>> On Thu, Jul 23, 2020, 2:56 PM uǝlƃ ↙↙↙ > > wrote:
>>
>> Ha! No way. If that were true, then to mow my lawn, I'd only have to mow 
>> the little part in the corner and voilá all the other patches would also be 
>> mowed.
>>
>> On 7/23/20 1:52 PM, Frank Wimberly wrote:
>> > "is the same sized square, e.g. at {0.5,0.5}, the same square as the 
>> one at {10.5-10,10.5-10}" 
>> >
>> > If you agree that 10.5 - 10 = 0.5 then same square, different name.
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Re: [FRIAM] square land math question

[Frank Wimberly]

That's because I was trying to illustrate the
difference between the abstract mathematical definition and an
implementation suitable for computer graphics.  I had just asked Glen if he
grokked the difference and he said no.
---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 4:27 PM Angel Edward  wrote:

> You keep talking in terms of implementations rather than the abstract
> object.
>
> Here you say a square does not include information about its location but
> then you add the location in the class definition. In coordinate-free
> geometry, you have only three basic entities: scalars, points and vectors.
> You can use them to define all the standard geometric objects and write
> code purely in terms of these entities.
>
> Ed
> __
>
> Ed Angel
>
> Founding Director, Art, Research, Technology and Science Laboratory (ARTS
> Lab)
> Professor Emeritus of Computer Science, University of New Mexico
>
> 1017 Sierra Pinon
> Santa Fe, NM 87501
> 505-984-0136 (home)   edward.an...@gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On Jul 23, 2020, at 4:09 PM, Frank Wimberly  wrote:
>
> The mathematical concept of a point in R^2 is that a it is completely
> determined by the values of its coordinates.  Same coordinates, same
> point.  A square per se Is determined by the length of its side(s).  There
> is no information about it's location.
>
> If I were writing a Square class for a graphics application I would
> include two member variables:
>
> LocationOfLowerLeft point;
> LengthOfSide double;
>
> I haven't written code for years so beware.
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Thu, Jul 23, 2020, 3:58 PM uǝlƃ ↙↙↙  wrote:
>
>> No, I don't. What's the difference?
>>
>> On 7/23/20 2:46 PM, Frank Wimberly wrote:
>> > OK.  As long as you grok the difference between the mathematical
>> concept and the OO concept.
>>
>> --
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Spot on! And my cognitive disability prevents me from remembering who or where 
someone used that as an argument against the law of the excluded middle ... 
arguing for intuitionist logic. 

On 7/23/20 3:32 PM, Jon Zingale wrote:
> SDG is a rather cool example of where the point notion can be radically
> different than classically handled by Euclid. From the man himself, Anders
> Kock[1]:
> 
> "Euclid maintained further that R was not just a commutative ring,
> but actually a field. This follows because of his assumption: for any two
> points in the plane, either they are equal, or they determine a unique
> line.
> 
> We cannot agree with Euclid on this point. For that would imply that
> the set D defined by
> 
> D := [[x ∈ R | x^2 = 0]] ⊆ R
> 
> consists of 0 alone, and that would immediately contradict our
> 
> Axiom 1. For any g : D → R, there exists a unique b ∈ R such that
> ∀d ∈ D : g(d) = g(0) + d · b"
> 
> Gotta love Kock.
> 
> [1] Synthetic Differential Geometry: https://users-math.au.dk/kock/sdg99.pdf
> Also, the paper of his I am currently entrenched in to investigate further
> some
> ideas in the Instrumental Goal versus Evolutionary Function discussion:
> https://arxiv.org/pdf/1105.3405.pdf

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Re: [FRIAM] square land math question

[Jon Zingale]

SDG is a rather cool example of where the point notion can be radically
different than classically handled by Euclid. From the man himself, Anders
Kock[1]:

"Euclid maintained further that R was not just a commutative ring,
but actually a field. This follows because of his assumption: for any two
points in the plane, either they are equal, or they determine a unique
line.

We cannot agree with Euclid on this point. For that would imply that
the set D defined by

D := [[x ∈ R | x^2 = 0]] ⊆ R

consists of 0 alone, and that would immediately contradict our

Axiom 1. For any g : D → R, there exists a unique b ∈ R such that
∀d ∈ D : g(d) = g(0) + d · b"

Gotta love Kock.

[1] Synthetic Differential Geometry: https://users-math.au.dk/kock/sdg99.pdf
Also, the paper of his I am currently entrenched in to investigate further
some
ideas in the Instrumental Goal versus Evolutionary Function discussion:
https://arxiv.org/pdf/1105.3405.pdf



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Re: [FRIAM] square land math question

[Frank Wimberly]

Sorry.  I only took math courses in grad school until I was 29 years old
and at that time OO didn't exist as far as I know.  Databases were just
coming into prominence as an area of study.  The dissertations that were
published in my department the year I finished were all in database topics
except mine, which was in numerical analysis.  I did teach data structures
for many years.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 4:21 PM uǝlƃ ↙↙↙  wrote:

> I agree. I think Frank is simply prejudiced toward his way of thinking
> about math. Both relational (normalized) databases and OO databases can be
> mathematically well-founded. I don't know, but suspect, they're even dual.
>
> On 7/23/20 3:08 PM, Edward Angel wrote:
> > There really does not need to be a difference, Coordinate free geometry
> is much like vector analysis. You have the equivalent of axioms and I
> suppose if you so desire you can bring in formal proofs and all the other
> concepts you like. But what it does for me is give a unified view of linear
> algebra, odes and geometry as just different instantiations of the same
> objects and their methods.
>
> --
> ↙↙↙ uǝlƃ
>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

I don't think either of those are necessarily true. Math, like so many other 
things, is not a unitary thing that writes its definitions in stone for all 
time. Yes, a point can be defined that way. There are other definitions, some 
more general, some very different. And a square has alternate definitions, too. 
Just because you have 1 you like does not mean it can't be defined in a 
different way.

I really like defining square in terms of right angles, myself. That allows me 
to know what my woodworking friend is asking for when he asks me to hand him 
the square. He has many different squares of different side lengths (one of 
which is shorter than the other!), made of different materials, having 
different widths, etc. But one thing is constant, they all have that 90° angle 
staring you in the face.

On 7/23/20 3:09 PM, Frank Wimberly wrote:
> The mathematical concept of a point in R^2 is that a it is completely 
> determined by the values of its coordinates.  Same coordinates, same point.  
> A square per se Is determined by the length of its side(s).  There is no 
> information about it's location.


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Re: [FRIAM] square land math question

[Angel Edward]

You keep talking in terms of implementations rather than the abstract object. 

Here you say a square does not include information about its location but then 
you add the location in the class definition. In coordinate-free geometry, you 
have only three basic entities: scalars, points and vectors. You can use them 
to define all the standard geometric objects and write code purely in terms of 
these entities.

Ed
__

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home) edward.an...@gmail.com
505-453-4944 (cell) http://www.cs.unm.edu/~angel

> On Jul 23, 2020, at 4:09 PM, Frank Wimberly  wrote:
> 
> The mathematical concept of a point in R^2 is that a it is completely 
> determined by the values of its coordinates.  Same coordinates, same point.  
> A square per se Is determined by the length of its side(s).  There is no 
> information about it's location.
> 
> If I were writing a Square class for a graphics application I would include 
> two member variables:
> 
> LocationOfLowerLeft point;
> LengthOfSide double;
> 
> I haven't written code for years so beware.
> 
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz, 
> Santa Fe, NM 87505
> 
> 505 670-9918
> Santa Fe, NM
> 
> On Thu, Jul 23, 2020, 3:58 PM uǝlƃ ↙↙↙  > wrote:
> No, I don't. What's the difference?
> 
> On 7/23/20 2:46 PM, Frank Wimberly wrote:
> > OK.  As long as you grok the difference between the mathematical concept 
> > and the OO concept.
> 
> -- 
> ↙↙↙ uǝlƃ
> 
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

I agree. I think Frank is simply prejudiced toward his way of thinking about 
math. Both relational (normalized) databases and OO databases can be 
mathematically well-founded. I don't know, but suspect, they're even dual.

On 7/23/20 3:08 PM, Edward Angel wrote:
> There really does not need to be a difference, Coordinate free geometry is 
> much like vector analysis. You have the equivalent of axioms and I suppose if 
> you so desire you can bring in formal proofs and all the other concepts you 
> like. But what it does for me is give a unified view of linear algebra, odes 
> and geometry as just different instantiations of the same objects and their 
> methods.

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Re: [FRIAM] square land math question

[Frank Wimberly]

The mathematical concept of a point in R^2 is that a it is completely
determined by the values of its coordinates.  Same coordinates, same
point.  A square per se Is determined by the length of its side(s).  There
is no information about it's location.

If I were writing a Square class for a graphics application I would include
two member variables:

LocationOfLowerLeft point;
LengthOfSide double;

I haven't written code for years so beware.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 3:58 PM uǝlƃ ↙↙↙  wrote:

> No, I don't. What's the difference?
>
> On 7/23/20 2:46 PM, Frank Wimberly wrote:
> > OK.  As long as you grok the difference between the mathematical concept
> and the OO concept.
>
> --
> ↙↙↙ uǝlƃ
>
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Re: [FRIAM] square land math question

[Edward Angel]

There really does not need to be a difference, Coordinate free geometry is much 
like vector analysis. You have the equivalent of axioms and I suppose if you so 
desire you can bring in formal proofs and all the other concepts you like. But 
what it does for me is give a unified view of linear algebra, odes and geometry 
as just different instantiations of the same objects and their methods.

Ed
___

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home) an...@cs.unm.edu 

505-453-4944 (cell) http://www.cs.unm.edu/~angel 


> On Jul 23, 2020, at 3:46 PM, Frank Wimberly  wrote:
> 
> OK.  As long as you grok the difference between the mathematical concept and 
> the OO concept.
> 
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz, 
> Santa Fe, NM 87505
> 
> 505 670-9918
> Santa Fe, NM
> 
> On Thu, Jul 23, 2020, 3:41 PM uǝlƃ ↙↙↙  > wrote:
> We used to have this argument all the time about the apt use of relational 
> vs. OO databases. As in Ed's conception, the same square can be associated 
> with multiple locations. Then to update all the renderings of that 1 square, 
> say, change its color from red to blue, all you need do is change the object 
> and all its renderings change as a result. That's pretty handy.
> 
> But what if you really did want multiple squares so that changing the color 
> of this square over here didn't change the color of that square over there? 
> You might want "square" to be a class but have color be an instance property 
> so you could change each square to a different color. Or you might even have 
> a concept of *scope* so that all  the squares in a neighborhood changed, but 
> no those far away ... or only the squares that are also rotated 90° 
> (invisibly) would change color, but those that haven't been rotated stay 
> whatever color they are.
> 
> To my mind, computationalists tend to think like the latter (collections of 
> instances) whereas analysts tend to think like the former ("normalized" or 
> "unified"). I'm agnostic and like both teams. But when I see one team 
> winning, I tend to traitoriously jump from one side to the other.
> 
> On 7/23/20 2:26 PM, Frank Wimberly wrote:
> > What?
> > 
> > On Thu, Jul 23, 2020, 2:56 PM uǝlƃ ↙↙↙  >   > >> wrote:
> > 
> > Ha! No way. If that were true, then to mow my lawn, I'd only have to 
> > mow the little part in the corner and voilá all the other patches would 
> > also be mowed.
> > 
> > On 7/23/20 1:52 PM, Frank Wimberly wrote:
> > > "is the same sized square, e.g. at {0.5,0.5}, the same square as the 
> > one at {10.5-10,10.5-10}" 
> > >
> > > If you agree that 10.5 - 10 = 0.5 then same square, different name.
> 
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

No, I don't. What's the difference?

On 7/23/20 2:46 PM, Frank Wimberly wrote:
> OK.  As long as you grok the difference between the mathematical concept and 
> the OO concept.

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Re: [FRIAM] square land math question

[Frank Wimberly]

OK.  As long as you grok the difference between the mathematical concept
and the OO concept.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 3:41 PM uǝlƃ ↙↙↙  wrote:

> We used to have this argument all the time about the apt use of relational
> vs. OO databases. As in Ed's conception, the same square can be associated
> with multiple locations. Then to update all the renderings of that 1
> square, say, change its color from red to blue, all you need do is change
> the object and all its renderings change as a result. That's pretty handy.
>
> But what if you really did want multiple squares so that changing the
> color of this square over here didn't change the color of that square over
> there? You might want "square" to be a class but have color be an instance
> property so you could change each square to a different color. Or you might
> even have a concept of *scope* so that all  the squares in a neighborhood
> changed, but no those far away ... or only the squares that are also
> rotated 90° (invisibly) would change color, but those that haven't been
> rotated stay whatever color they are.
>
> To my mind, computationalists tend to think like the latter (collections
> of instances) whereas analysts tend to think like the former ("normalized"
> or "unified"). I'm agnostic and like both teams. But when I see one team
> winning, I tend to traitoriously jump from one side to the other.
>
> On 7/23/20 2:26 PM, Frank Wimberly wrote:
> > What?
> >
> > On Thu, Jul 23, 2020, 2:56 PM uǝlƃ ↙↙↙  geprope...@gmail.com>> wrote:
> >
> > Ha! No way. If that were true, then to mow my lawn, I'd only have to
> mow the little part in the corner and voilá all the other patches would
> also be mowed.
> >
> > On 7/23/20 1:52 PM, Frank Wimberly wrote:
> > > "is the same sized square, e.g. at {0.5,0.5}, the same square as
> the one at {10.5-10,10.5-10}"
> > >
> > > If you agree that 10.5 - 10 = 0.5 then same square, different name.
>
> --
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>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

We used to have this argument all the time about the apt use of relational vs. 
OO databases. As in Ed's conception, the same square can be associated with 
multiple locations. Then to update all the renderings of that 1 square, say, 
change its color from red to blue, all you need do is change the object and all 
its renderings change as a result. That's pretty handy.

But what if you really did want multiple squares so that changing the color of 
this square over here didn't change the color of that square over there? You 
might want "square" to be a class but have color be an instance property so you 
could change each square to a different color. Or you might even have a concept 
of *scope* so that all  the squares in a neighborhood changed, but no those far 
away ... or only the squares that are also rotated 90° (invisibly) would change 
color, but those that haven't been rotated stay whatever color they are.

To my mind, computationalists tend to think like the latter (collections of 
instances) whereas analysts tend to think like the former ("normalized" or 
"unified"). I'm agnostic and like both teams. But when I see one team winning, 
I tend to traitoriously jump from one side to the other.

On 7/23/20 2:26 PM, Frank Wimberly wrote:
> What?
> 
> On Thu, Jul 23, 2020, 2:56 PM uǝlƃ ↙↙↙  > wrote:
> 
> Ha! No way. If that were true, then to mow my lawn, I'd only have to mow 
> the little part in the corner and voilá all the other patches would also be 
> mowed.
> 
> On 7/23/20 1:52 PM, Frank Wimberly wrote:
> > "is the same sized square, e.g. at {0.5,0.5}, the same square as the 
> one at {10.5-10,10.5-10}" 
> >
> > If you agree that 10.5 - 10 = 0.5 then same square, different name.

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Re: [FRIAM] square land math question

[Frank Wimberly]

What?

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 2:56 PM uǝlƃ ↙↙↙  wrote:

> Ha! No way. If that were true, then to mow my lawn, I'd only have to mow
> the little part in the corner and voilá all the other patches would also be
> mowed.
>
> On 7/23/20 1:52 PM, Frank Wimberly wrote:
> > "is the same sized square, e.g. at {0.5,0.5}, the same square as the one
> at {10.5-10,10.5-10}"
> >
> > If you agree that 10.5 - 10 = 0.5 then same square, different name.
>
> --
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>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Ha! No way. If that were true, then to mow my lawn, I'd only have to mow the 
little part in the corner and voilá all the other patches would also be mowed.

On 7/23/20 1:52 PM, Frank Wimberly wrote:
> "is the same sized square, e.g. at {0.5,0.5}, the same square as the one at 
> {10.5-10,10.5-10}" 
> 
> If you agree that 10.5 - 10 = 0.5 then same square, different name.

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Re: [FRIAM] square land math question

[Frank Wimberly]

"is the same sized square, e.g. at {0.5,0.5}, the same square as the one at
{10.5-10,10.5-10}"

If you agree that 10.5 - 10 = 0.5 then same square, different name.

On Thu, Jul 23, 2020 at 2:47 PM uǝlƃ ↙↙↙  wrote:

> Well, we're talking about sub-squares, not just any old reduction. So,
> this would be the reductions where both elements of the tuple are reduced
> by the same scalar. But, more importantly, is the same sized square, e.g.
> at {0.5,0.5}, the same square as the one at {10.5-10,10.5-10}? I think most
> people would say they're different squares even if they have the same
> reductions (area, circumference, etc.). So, by extension, an infinitesimal
> closest to zero ("iota"?) is different from one just above, say, 10 even if
> they're the same size.
>
> Along those same lines, I think an alternative answer the kid could've
> given was to set the origin of the original square in the middle of the
> square, then say that any square with corners at
> {{x,x},{-x,x},{-x,-x},{x,-x}} where x less than ½ the length of the
> original square would cut into 2 squares. Where the original answer the kid
> gave used an alternate definition of "square" than what Cody was using,
> this uses yet *another* definition of "square", one that's more agnostic
> about the space inside the square's borders. Is a square picture frame a
> square? Or just a set of 4 sticks wherein the squareness property is
> emergent? [pffft]
>
>
> On 7/23/20 1:20 PM, Frank Wimberly wrote:
> > Good point, Steve.  There are infinitely many ways of resolving a
> vector.  E.g. (1, 1) = (1, 0) + (0, 1/2) + (0, 1/4) + (0, 1/4) etc.
>
>
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-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Well, we're talking about sub-squares, not just any old reduction. So, this 
would be the reductions where both elements of the tuple are reduced by the 
same scalar. But, more importantly, is the same sized square, e.g. at 
{0.5,0.5}, the same square as the one at {10.5-10,10.5-10}? I think most people 
would say they're different squares even if they have the same reductions 
(area, circumference, etc.). So, by extension, an infinitesimal closest to zero 
("iota"?) is different from one just above, say, 10 even if they're the same 
size. 

Along those same lines, I think an alternative answer the kid could've given 
was to set the origin of the original square in the middle of the square, then 
say that any square with corners at {{x,x},{-x,x},{-x,-x},{x,-x}} where x less 
than ½ the length of the original square would cut into 2 squares. Where the 
original answer the kid gave used an alternate definition of "square" than what 
Cody was using, this uses yet *another* definition of "square", one that's more 
agnostic about the space inside the square's borders. Is a square picture frame 
a square? Or just a set of 4 sticks wherein the squareness property is 
emergent? [pffft]


On 7/23/20 1:20 PM, Frank Wimberly wrote:
> Good point, Steve.  There are infinitely many ways of resolving a vector.  
> E.g. (1, 1) = (1, 0) + (0, 1/2) + (0, 1/4) + (0, 1/4) etc.


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Re: [FRIAM] square land math question

[Angel Edward]

In geometry, I find it better to think in terms of objects. A point is an 
object that has a location, dimension 0 (no measurable property) and no other 
properties; a line segment is an object with one dimension, has dimension one,  
and is defined by two points and so on. For each object, we have a set of 
functions. A point has no functions defined for it. When you say a point is an 
n-tuple in R^n you are talking about the representation of a point in some 
space, not the geometric object. To get back to Cody’s original question. From 
a geometric perspective, a sequence of two dimensional objects (the squares), 
which can be scaled,  cannot turn into a point which is a different object  
type.

Here’s a somewhat different geometric view of why you have to be wary of what 
the kid claimed. Suppose I start with a unit square. I divide it evenly in both 
directions to get four equal squares. I then throw away two diagonally opposite 
squares so I have half the original area. However, if I follow the edges I the 
distance between the opposite vertices is still 2. As you repeat this 
construction, the area of total of all the 2^n squares goes to zero but the 
distance along the edges between the original opposite vertices remains as 2. 

We can’t say this construction converges to a line connecting the two original 
vertices since we just showed it has a length two not sqrt(2). Or does it since 
if we add up the diagonals of all little cubes they do sum to sqrt 2. It gets 
even more interesting if we remove only one of the subcubes each time and add 
up the perimeters of all the subcubes thus creating an object than in the limit 
has no area but an infinite perimeter. Fractal geometry has nice definition of 
dimension that cover these issues.

Ed
__

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home) edward.an...@gmail.com
505-453-4944 (cell) http://www.cs.unm.edu/~angel

> On Jul 23, 2020, at 2:20 PM, Frank Wimberly  wrote:
> 
> "While a point and a vector in R^n might be described by the same tuple,
> dividing the numeric elements of the tuple does not "partition" the
> point..."
> 
> Good point, Steve.  There are infinitely many ways of resolving a vector.  
> E.g. (1, 1) = (1, 0) + (0, 1/2) + (0, 1/4) + (0, 1/4) etc.
> 
>   
> 
> On Thu, Jul 23, 2020 at 2:09 PM uǝlƃ ↙↙↙  > wrote:
> Nice challenge! ... Wel, the original question was basically how Cody 
> might respond to the kid's suggestion that a point is a square with no area. 
> My suggestion to Cody would be to answer the kid with a discussion about the 
> actuality or potentiality of infinity ... or intermediately, distinguishing 
> between *definitions* of "square".
> 
> And if you define define a square geometrically, then it makes complete sense 
> that there is no arealess square. But there are OTHER ways to define a 
> square. And since this kid already pulled out a sophisticated mathematical 
> argument, it's useful and interesting to see how far that kid can go.
> 
> You're free to hem and haw about the foundations of math and which foundation 
> you like better than another. But the point of discussing the extent of a 
> point was to answer the kid's challenge. Answering a bright kid with "because 
> Euclid says so" is not all that useful. >8^D
> 
> On 7/23/20 1:00 PM, Steve Smith wrote:
> > Can you illuminate us as to what treating the *location* of a point as a
> > *quantity* and demonstrating that the quantity can be divided
> > arithmetically adds to the meaning of a point? 
> > 
> > While a point and a vector in R^n might be described by the same tuple,
> > dividing the numeric elements of the tuple does not "partition" the
> > point, it merely scales the vector which is quite useful, but I'm not
> > sure if in any way doing so has any meaning that could be construed as
> > having "divided" the point?
> > 
> > I think Euclid's geometry is pretty "standard math"?
> 
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> -- 
> Frank Wimberly
> 140 Calle Ojo Feliz
> Santa Fe, NM 87505
> 505 670-9918
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Re: [FRIAM] square land math question

[Frank Wimberly]

"While a point and a vector in R^n might be described by the same tuple,
dividing the numeric elements of the tuple does not "partition" the
point..."

Good point, Steve.  There are infinitely many ways of resolving a vector.
E.g. (1, 1) = (1, 0) + (0, 1/2) + (0, 1/4) + (0, 1/4) etc.



On Thu, Jul 23, 2020 at 2:09 PM uǝlƃ ↙↙↙  wrote:

> Nice challenge! ... Wel, the original question was basically how Cody
> might respond to the kid's suggestion that a point is a square with no
> area. My suggestion to Cody would be to answer the kid with a discussion
> about the actuality or potentiality of infinity ... or intermediately,
> distinguishing between *definitions* of "square".
>
> And if you define define a square geometrically, then it makes complete
> sense that there is no arealess square. But there are OTHER ways to define
> a square. And since this kid already pulled out a sophisticated
> mathematical argument, it's useful and interesting to see how far that kid
> can go.
>
> You're free to hem and haw about the foundations of math and which
> foundation you like better than another. But the point of discussing the
> extent of a point was to answer the kid's challenge. Answering a bright kid
> with "because Euclid says so" is not all that useful. >8^D
>
> On 7/23/20 1:00 PM, Steve Smith wrote:
> > Can you illuminate us as to what treating the *location* of a point as a
> > *quantity* and demonstrating that the quantity can be divided
> > arithmetically adds to the meaning of a point?
> >
> > While a point and a vector in R^n might be described by the same tuple,
> > dividing the numeric elements of the tuple does not "partition" the
> > point, it merely scales the vector which is quite useful, but I'm not
> > sure if in any way doing so has any meaning that could be construed as
> > having "divided" the point?
> >
> > I think Euclid's geometry is pretty "standard math"?
>
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-- 
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140 Calle Ojo Feliz
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505 670-9918
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Nice challenge! ... Wel, the original question was basically how Cody might 
respond to the kid's suggestion that a point is a square with no area. My 
suggestion to Cody would be to answer the kid with a discussion about the 
actuality or potentiality of infinity ... or intermediately, distinguishing 
between *definitions* of "square".

And if you define define a square geometrically, then it makes complete sense 
that there is no arealess square. But there are OTHER ways to define a square. 
And since this kid already pulled out a sophisticated mathematical argument, 
it's useful and interesting to see how far that kid can go.

You're free to hem and haw about the foundations of math and which foundation 
you like better than another. But the point of discussing the extent of a point 
was to answer the kid's challenge. Answering a bright kid with "because Euclid 
says so" is not all that useful. >8^D

On 7/23/20 1:00 PM, Steve Smith wrote:
> Can you illuminate us as to what treating the *location* of a point as a
> *quantity* and demonstrating that the quantity can be divided
> arithmetically adds to the meaning of a point? 
> 
> While a point and a vector in R^n might be described by the same tuple,
> dividing the numeric elements of the tuple does not "partition" the
> point, it merely scales the vector which is quite useful, but I'm not
> sure if in any way doing so has any meaning that could be construed as
> having "divided" the point?
> 
> I think Euclid's geometry is pretty "standard math"?

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Re: [FRIAM] square land math question

[Steve Smith]

Glen -

Can you illuminate us as to what treating the *location* of a point as a
*quantity* and demonstrating that the quantity can be divided
arithmetically adds to the meaning of a point? 

While a point and a vector in R^n might be described by the same tuple,
dividing the numeric elements of the tuple does not "partition" the
point, it merely scales the vector which is quite useful, but I'm not
sure if in any way doing so has any meaning that could be construed as
having "divided" the point?

I think Euclid's geometry is pretty "standard math"?

- Steve

> Well, as I tried to point out, I have a tough time understanding nonstandard 
> math. The actuality of infinities seems to have been handled by Cantor and 
> infinitesimals seem to have been fully justified by Conway and Robinson. But 
> I don't understand much about *how* they built up that infrastructure.
>
> Whether the output of division is different from its input or identical to 
> its input doesn't prevent me from applying the function. As I said, it's 
> similar to 1. If I divide X by 1, I get X. So, X is clearly "divisible", even 
> if it has no "parts" ... whatever "part" might mean ... to you or Euclid. >8^D
>
> On 7/23/20 9:48 AM, Steve Smith wrote:
>> Can you unpack that in the light of Euclid's definition of a point, to whose 
>> authority I presume Frank was deferring/invoking.
>>
>> I'm curious if this is a matter of dismissing/rejecting Euclid and his 
>> definitions in this matter, or an alternative interpretation of his text?
>>
>> αʹ. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν. 1. A point is that of which there is 
>> no part
>>
>> I'm always interested in creative alternative interpretations of intention 
>> and meaning, but I'm not getting traction on this one (yet?)
>


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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

So, apparently, 1/ω ≠ 1/(ω+1) in surreal numbers. But if I understand 
correctly, which is unlikely, we still don't have a definition of integration 
for surreal numbers. So, I'd hesitate to rely on that as an authority. I now 
wonder if all infinitesimals have the same size in the hyperreals? And even if 
they have the same size, are they the *same number*?

In my ignorance, it seems like we have 2 examples with which to form a (perhaps 
false but useful) dichotomy:

https://en.wikipedia.org/wiki/Nonstandard_analysis, where it seems like 
infinitesimals are distinguishable and 
https://en.wikipedia.org/wiki/Synthetic_differential_geometry, where they are 
not (or not all of them ... or ... something). I have a lot of homework to do, 
I guess.


On 7/23/20 10:40 AM, uǝlƃ ↙↙↙ wrote:
> Thanks for putting in a little more effort. So, in your definitions, 1/aleph0 
> = 1/aleph1. That's tightly analogous, if not identical, to saying a point is 
> divisible because point/2 = point. But before you claimed a point is 
> indivisible. So, if you were more clear about which authority you were citing 
> when you make your claims, we wouldn't have these discussions.
> 
> On 7/23/20 10:35 AM, Frank Wimberly wrote:
>> I am aware of the hierarchy of infinities.  Aleph0 is the cardinality of the 
>> integers.  Aleph1 is the cardinality of the power set of the integers which 
>> is the cardinality of the real numbers (that's a theorem which is easy but I 
>> don't feel like typing it on a cellphone keyboard).  Aleph2 is the 
>> cardinality of the power set of aleph1, etc.
>>
>> In my definition of 1/infinity, assume infinity means aleph0.  But I believe 
>> it works for any infinite number.  That last word is important.
> 

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Re: [FRIAM] square land math question

[Barry MacKichan]

Doesn’t that depend on how finely you can pick a nit>

On 23 Jul 2020, at 12:28, Frank Wimberly wrote:

> points are indivisible.  Pardon the tone of authority.
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Re: [FRIAM] square land math question

[Eric Charles]

Zeno had several paradoxes, all intended to expose questionable
assumptions.

On Thu, Jul 23, 2020, 1:58 PM Frank Wimberly  wrote:

> A lot of it has to do with using a cell phone keyboard and not wanting to
> get too technical here.  But maybe Jon is right about "the List can take
> it."
>
> I should have said that aleph(n) is the cardinality of the power set of a
> set with cardinality aleph(n-1).  That's slightly different from what I
> said before.
>
> Frank
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Thu, Jul 23, 2020, 11:40 AM uǝlƃ ↙↙↙  wrote:
>
>> Thanks for putting in a little more effort. So, in your definitions,
>> 1/aleph0 = 1/aleph1. That's tightly analogous, if not identical, to saying
>> a point is divisible because point/2 = point. But before you claimed a
>> point is indivisible. So, if you were more clear about which authority you
>> were citing when you make your claims, we wouldn't have these discussions.
>>
>> On 7/23/20 10:35 AM, Frank Wimberly wrote:
>> > I am aware of the hierarchy of infinities.  Aleph0 is the cardinality
>> of the integers.  Aleph1 is the cardinality of the power set of the
>> integers which is the cardinality of the real numbers (that's a theorem
>> which is easy but I don't feel like typing it on a cellphone keyboard).
>> Aleph2 is the cardinality of the power set of aleph1, etc.
>> >
>> > In my definition of 1/infinity, assume infinity means aleph0.  But I
>> believe it works for any infinite number.  That last word is important.
>>
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Re: [FRIAM] square land math question

[Frank Wimberly]

A lot of it has to do with using a cell phone keyboard and not wanting to
get too technical here.  But maybe Jon is right about "the List can take
it."

I should have said that aleph(n) is the cardinality of the power set of a
set with cardinality aleph(n-1).  That's slightly different from what I
said before.

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 11:40 AM uǝlƃ ↙↙↙  wrote:

> Thanks for putting in a little more effort. So, in your definitions,
> 1/aleph0 = 1/aleph1. That's tightly analogous, if not identical, to saying
> a point is divisible because point/2 = point. But before you claimed a
> point is indivisible. So, if you were more clear about which authority you
> were citing when you make your claims, we wouldn't have these discussions.
>
> On 7/23/20 10:35 AM, Frank Wimberly wrote:
> > I am aware of the hierarchy of infinities.  Aleph0 is the cardinality of
> the integers.  Aleph1 is the cardinality of the power set of the integers
> which is the cardinality of the real numbers (that's a theorem which is
> easy but I don't feel like typing it on a cellphone keyboard).  Aleph2 is
> the cardinality of the power set of aleph1, etc.
> >
> > In my definition of 1/infinity, assume infinity means aleph0.  But I
> believe it works for any infinite number.  That last word is important.
>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Thanks for putting in a little more effort. So, in your definitions, 1/aleph0 = 
1/aleph1. That's tightly analogous, if not identical, to saying a point is 
divisible because point/2 = point. But before you claimed a point is 
indivisible. So, if you were more clear about which authority you were citing 
when you make your claims, we wouldn't have these discussions.

On 7/23/20 10:35 AM, Frank Wimberly wrote:
> I am aware of the hierarchy of infinities.  Aleph0 is the cardinality of the 
> integers.  Aleph1 is the cardinality of the power set of the integers which 
> is the cardinality of the real numbers (that's a theorem which is easy but I 
> don't feel like typing it on a cellphone keyboard).  Aleph2 is the 
> cardinality of the power set of aleph1, etc.
> 
> In my definition of 1/infinity, assume infinity means aleph0.  But I believe 
> it works for any infinite number.  That last word is important.

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Re: [FRIAM] square land math question

[Frank Wimberly]

Glen,

I am aware of the hierarchy of infinities.  Aleph0 is the cardinality of
the integers.  Aleph1 is the cardinality of the power set of the integers
which is the cardinality of the real numbers (that's a theorem which is
easy but I don't feel like typing it on a cellphone keyboard).  Aleph2 is
the cardinality of the power set of aleph1, etc.

In my definition of 1/infinity, assume infinity means aleph0.  But I
believe it works for any infinite number.  That last word is important.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 11:26 AM uǝlƃ ↙↙↙  wrote:

> Again, you're making unjustified claims. This argues that all infinities
> are the same and leaves someone to stew in their juices about whether
> infinities are actual or potential. If they're potential, then 1/∞ is
> *undefined* and we only *approach* 0. If they're actual, then 1/∞ is an
> actual number and we can compare it's size to other very small numbers.
>
> I think most mathematicians these days, accept the actuality of
> infinitesimals and some might be larger or smaller than others in the same
> way that some infinities are larger than others.
>
> Talking the way you're talking sweeps Cody's question under the rug
> without answering it.
>
> On 7/23/20 10:21 AM, Frank Wimberly wrote:
> > 1/infinity is the limit of 1/x as x goes to infinity, which is zero.
>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Again, you're making unjustified claims. This argues that all infinities are 
the same and leaves someone to stew in their juices about whether infinities 
are actual or potential. If they're potential, then 1/∞ is *undefined* and we 
only *approach* 0. If they're actual, then 1/∞ is an actual number and we can 
compare it's size to other very small numbers.

I think most mathematicians these days, accept the actuality of infinitesimals 
and some might be larger or smaller than others in the same way that some 
infinities are larger than others.

Talking the way you're talking sweeps Cody's question under the rug without 
answering it.

On 7/23/20 10:21 AM, Frank Wimberly wrote:
> 1/infinity is the limit of 1/x as x goes to infinity, which is zero.  

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Re: [FRIAM] square land math question

[Jon Zingale]

Frank,

I will send my regards. Because of the kinds of conversations that
occasionally heat up around ideas like electron wave-particle duality, I
feel that it is important to include definitions that extend to more general
concepts. This list can take it :)



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Re: [FRIAM] square land math question

[Frank Wimberly]

1/infinity is the limit of 1/x as x goes to infinity, which is zero.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 11:16 AM uǝlƃ ↙↙↙  wrote:

> Maybe. But how do we handle things like reciprocals of infinities? Is
> 1/aleph0 the same as 1/aleph1?
>
> On 7/23/20 10:02 AM, Jon Zingale wrote:
> > How about, "Points are maps from terminal objects?"
>
>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Maybe. But how do we handle things like reciprocals of infinities? Is 1/aleph0 
the same as 1/aleph1?

On 7/23/20 10:02 AM, Jon Zingale wrote:
> How about, "Points are maps from terminal objects?"


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Re: [FRIAM] square land math question

[Frank Wimberly]

OK with me.

Unlike you, Jon, I don't assume my reader is a graduate level
mathematician.  Did you see my discussion of infinite series?  That was
approximately sophomore level.  When Cody said that limits were a
mysterious or magical concept to him I could have launched into a set of
formal definitions but I restrained myself.

Regards to Sarah and Tycho,

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 11:02 AM Jon Zingale  wrote:

> How about, "Points are maps from terminal objects?"
>
>
>
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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Well, as I tried to point out, I have a tough time understanding nonstandard 
math. The actuality of infinities seems to have been handled by Cantor and 
infinitesimals seem to have been fully justified by Conway and Robinson. But I 
don't understand much about *how* they built up that infrastructure.

Whether the output of division is different from its input or identical to its 
input doesn't prevent me from applying the function. As I said, it's similar to 
1. If I divide X by 1, I get X. So, X is clearly "divisible", even if it has no 
"parts" ... whatever "part" might mean ... to you or Euclid. >8^D

On 7/23/20 9:48 AM, Steve Smith wrote:
> Can you unpack that in the light of Euclid's definition of a point, to whose 
> authority I presume Frank was deferring/invoking.
> 
> I'm curious if this is a matter of dismissing/rejecting Euclid and his 
> definitions in this matter, or an alternative interpretation of his text?
> 
> αʹ. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν. 1. A point is that of which there is 
> no part
> 
> I'm always interested in creative alternative interpretations of intention 
> and meaning, but I'm not getting traction on this one (yet?)


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Re: [FRIAM] square land math question

[Jon Zingale]

How about, "Points are maps from terminal objects?"



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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Yes! That is of interest. I've been trying to understand a claim I've heard 
that *actual* infinities are required for full 2nd order math. I.e. potential 
infinities (which I suppose are necessary for intuitionism and/or 
program-as-proof) limit the 2nd order operators you can use.

I shouldn't be surprised that the Church got involved. Thanks.

On 7/23/20 9:47 AM, Prof David West wrote:
> maybe of interest:
> 
> In the 1630s, when the Roman Catholic Church was confronting Galileo over the 
> Copernican system, the Revisors General of the Jesuit order condemned the 
> doctrine that the continuum is composed of indivisibles. What we now call 
> Cavalieri’s Principle was thought to be dangerous to religion. 
> 
> Why did the Church get involved in evaluating the “new math” of indivisibles, 
> infinitesimals, and the infinite?  The doctrine of indivisibles was on the 
> side of Galileo. Besides opposing the Church about whether the earth went 
> around the sun, Galileo treated matter as made of atoms, which are physical 
> indivisibles. Bonaventura Cavalieri, who pioneered indivisible methods in 
> geometry, was among Galileo’s followers. Furthermore, Catholic theology owes 
> much to Aristotle’s philosophy, and Aristotle, arguing for the potentially 
> infinite divisibility of the continuum, had explicitly ruled out both 
> indivisibles and the actual infinite. So it is no wonder that Jesuit 
> intellectuals opposed using indivisibles in geometry.


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Re: [FRIAM] square land math question

[Steve Smith]

On 7/23/20 10:47 AM, Frank Wimberly wrote:
> In R2 a point is an ordered pair.  How can (1,1) be decomposed into
> other points.
>
> I am correct, goshdarnit.  When I was about 9 I said that word in the
> presence of my Southern Baptist grandfather.  He said, "Say Goddamit. 
> It means the same thing and it sounds better."

My grandfather taught me "Gauldarnitt!" at age 5 while teaching me how
to lose at checkers.   His daughter (my mother) was not impressed.  I
had no clue what the utterance meant, and it probably took me many years
to associate it with the profane analog...   but I *did* know it had
some kind of arcane power.  

... ShuckeyDarn



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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Well, at least in this post, you *try* to define things such that you'd be 
right. Although normally considered a rhetorical fallacy, programming into the 
premises the conclusion you seek is a perfectly reasonable thing to do in math. 
As long as you actually *do* it ... make the definitions, then your assumed 
conclusions will be just fine.

On 7/23/20 9:47 AM, Frank Wimberly wrote:
> In R2 a point is an ordered pair.  How can (1,1) be decomposed into other 
> points.
> 
> I am correct, goshdarnit.  When I was about 9 I said that word in the 
> presence of my Southern Baptist grandfather.  He said, "Say Goddamit.  It 
> means the same thing and it sounds better."
> 
> On Thu, Jul 23, 2020 at 10:34 AM uǝlƃ ↙↙↙  > wrote:
> 
> Ha! I can't pardon the tone because the authority is simply wrong. 
> Besides, asserting such things with no justification is not merely a tone.
> 
> On 7/23/20 9:28 AM, Frank Wimberly wrote:
> > points are indivisible.  Pardon the tone of authority.
> >
> >
> > On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙    >> wrote:
> >
> >     But a *relevant* question for me is whether or not you can divide 
> an infinitesimal point into an infinity of points? My *guess* is that a point 
> divided an infinite number of times is like a power set and is a greater 
> infinity than the point, itself. But I still haven't read a book I bought 
> awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read 
> many of the English intros and such and a few of the proofs ... but Whew! 
> It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up 
> and just cherry-pick sections that I only kindasorta understand by analogy at 
> this point. At least with math papers I don't feel like such a failure when I 
> give up on reading it ... another way papers are better than books!


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Re: [FRIAM] square land math question

[Steve Smith]

Glen -


> Ha! I can't pardon the tone because the authority is simply wrong. Besides, 
> asserting such things with no justification is not merely a tone.

Can you unpack that in the light of Euclid's definition of a point, to
whose authority I presume Frank was deferring/invoking.

I'm curious if this is a matter of dismissing/rejecting Euclid and his
definitions in this matter, or an alternative interpretation of his text?

αʹ. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν. 1. A point is that of which there
is no part

I'm always interested in creative alternative interpretations of
intention and meaning, but I'm not getting traction on this one (yet?)

- Steve

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Re: [FRIAM] square land math question

[Prof David West]

maybe of interest:

In the 1630s, when the Roman Catholic Church was confronting Galileo over the 
Copernican system, the Revisors General of the Jesuit order condemned the 
doctrine that the continuum is composed of indivisibles. What we now call 
Cavalieri’s Principle was thought to be dangerous to religion. 

Why did the Church get involved in evaluating the “new math” of indivisibles, 
infinitesimals, and the infinite?  The doctrine of indivisibles was on the side 
of Galileo. Besides opposing the Church about whether the earth went around the 
sun, Galileo treated matter as made of atoms, which are physical indivisibles. 
Bonaventura Cavalieri, who pioneered indivisible methods in geometry, was among 
Galileo’s followers. Furthermore, Catholic theology owes much to Aristotle’s 
philosophy, and Aristotle, arguing for the potentially infinite divisibility of 
the continuum, had explicitly ruled out both indivisibles and the actual 
infinite. So it is no wonder that Jesuit intellectuals opposed using 
indivisibles in geometry.

davew

On Thu, Jul 23, 2020, at 10:34 AM, uǝlƃ ↙↙↙ wrote:
> Ha! I can't pardon the tone because the authority is simply wrong. 
> Besides, asserting such things with no justification is not merely a 
> tone.
> 
> On 7/23/20 9:28 AM, Frank Wimberly wrote:
> > points are indivisible.  Pardon the tone of authority.
> > 
> > 
> > On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙  > > wrote:
> > 
> > But a *relevant* question for me is whether or not you can divide an 
> > infinitesimal point into an infinity of points? My *guess* is that a point 
> > divided an infinite number of times is like a power set and is a greater 
> > infinity than the point, itself. But I still haven't read a book I bought 
> > awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read 
> > many of the English intros and such and a few of the proofs ... but Whew! 
> > It's almost exactly like Alexandrov's "Combinatorial Topology". I've given 
> > up and just cherry-pick sections that I only kindasorta understand by 
> > analogy at this point. At least with math papers I don't feel like such a 
> > failure when I give up on reading it ... another way papers are better than 
> > books!
> > 
> 
> -- 
> ↙↙↙ uǝlƃ
> 
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Re: [FRIAM] square land math question

[Frank Wimberly]

In R2 a point is an ordered pair.  How can (1,1) be decomposed into other
points.

I am correct, goshdarnit.  When I was about 9 I said that word in the
presence of my Southern Baptist grandfather.  He said, "Say Goddamit.  It
means the same thing and it sounds better."

On Thu, Jul 23, 2020 at 10:34 AM uǝlƃ ↙↙↙  wrote:

> Ha! I can't pardon the tone because the authority is simply wrong.
> Besides, asserting such things with no justification is not merely a tone.
>
> On 7/23/20 9:28 AM, Frank Wimberly wrote:
> > points are indivisible.  Pardon the tone of authority.
> >
> >
> > On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙  geprope...@gmail.com>> wrote:
> >
> > But a *relevant* question for me is whether or not you can divide an
> infinitesimal point into an infinity of points? My *guess* is that a point
> divided an infinite number of times is like a power set and is a greater
> infinity than the point, itself. But I still haven't read a book I bought
> awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read
> many of the English intros and such and a few of the proofs ... but Whew!
> It's almost exactly like Alexandrov's "Combinatorial Topology". I've given
> up and just cherry-pick sections that I only kindasorta understand by
> analogy at this point. At least with math papers I don't feel like such a
> failure when I give up on reading it ... another way papers are better than
> books!
> >
>
> --
> ↙↙↙ uǝlƃ
>
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>


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Frank Wimberly
140 Calle Ojo Feliz
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Re: [FRIAM] square land math question

[Steve Smith]

> So, we’ve finally come to the essential question:
>
>  
>
> How many points can dance on the head of a point?
>
We've come full circle again...

https://friam-comic.blogspot.com/2017/10/truthiness-games.html

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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

Ha! I can't pardon the tone because the authority is simply wrong. Besides, 
asserting such things with no justification is not merely a tone.

On 7/23/20 9:28 AM, Frank Wimberly wrote:
> points are indivisible.  Pardon the tone of authority.
> 
> 
> On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙  > wrote:
> 
> But a *relevant* question for me is whether or not you can divide an 
> infinitesimal point into an infinity of points? My *guess* is that a point 
> divided an infinite number of times is like a power set and is a greater 
> infinity than the point, itself. But I still haven't read a book I bought 
> awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read 
> many of the English intros and such and a few of the proofs ... but Whew! 
> It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up 
> and just cherry-pick sections that I only kindasorta understand by analogy at 
> this point. At least with math papers I don't feel like such a failure when I 
> give up on reading it ... another way papers are better than books!
> 

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Re: [FRIAM] square land math question

[thompnickson2]

So, we’ve finally come to the essential question: 

 

How many points can dance on the head of a point?

 

Nick 

 

Nicholas Thompson

Emeritus Professor of Ethology and Psychology

Clark University

 <mailto:thompnicks...@gmail.com> thompnicks...@gmail.com

 <https://wordpress.clarku.edu/nthompson/> 
https://wordpress.clarku.edu/nthompson/

 

 

From: Friam  On Behalf Of Frank Wimberly
Sent: Thursday, July 23, 2020 10:28 AM
To: The Friday Morning Applied Complexity Coffee Group 
Subject: Re: [FRIAM] square land math question

 

points are indivisible.  Pardon the tone of authority.

 

 

On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ mailto:geprope...@gmail.com> > wrote:

But a *relevant* question for me is whether or not you can divide an 
infinitesimal point into an infinity of points? My *guess* is that a point 
divided an infinite number of times is like a power set and is a greater 
infinity than the point, itself. But I still haven't read a book I bought 
awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read 
many of the English intros and such and a few of the proofs ... but Whew! It's 
almost exactly like Alexandrov's "Combinatorial Topology". I've given up and 
just cherry-pick sections that I only kindasorta understand by analogy at this 
point. At least with math papers I don't feel like such a failure when I give 
up on reading it ... another way papers are better than books!

On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> And it's similarly degenerately trivial to divide a point into 2 points.


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-- 

Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918

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Re: [FRIAM] square land math question

[Frank Wimberly]

points are indivisible.  Pardon the tone of authority.


On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙  wrote:

> But a *relevant* question for me is whether or not you can divide an
> infinitesimal point into an infinity of points? My *guess* is that a point
> divided an infinite number of times is like a power set and is a greater
> infinity than the point, itself. But I still haven't read a book I bought
> awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read
> many of the English intros and such and a few of the proofs ... but Whew!
> It's almost exactly like Alexandrov's "Combinatorial Topology". I've given
> up and just cherry-pick sections that I only kindasorta understand by
> analogy at this point. At least with math papers I don't feel like such a
> failure when I give up on reading it ... another way papers are better than
> books!
>
> On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> > And it's similarly degenerately trivial to divide a point into 2 points.
>
>
> --
> ↙↙↙ uǝlƃ
>
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> FRIAM-COMIC 
> http://friam-comic.blogspot.com/
>


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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

But a *relevant* question for me is whether or not you can divide an 
infinitesimal point into an infinity of points? My *guess* is that a point 
divided an infinite number of times is like a power set and is a greater 
infinity than the point, itself. But I still haven't read a book I bought 
awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read 
many of the English intros and such and a few of the proofs ... but Whew! It's 
almost exactly like Alexandrov's "Combinatorial Topology". I've given up and 
just cherry-pick sections that I only kindasorta understand by analogy at this 
point. At least with math papers I don't feel like such a failure when I give 
up on reading it ... another way papers are better than books!

On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> And it's similarly degenerately trivial to divide a point into 2 points.


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Re: [FRIAM] square land math question

[uǝlƃ ↙↙↙]

I'm surprised EricC didn't say "it all depends on the definition of 'square'". 
I regard a point as a degenerate square (also a degenerate sphere, cube, etc.). 
It's the same sort of object as the empty set or an identity like 0 (for +) or 
1 (for *).

If all we need for a square is an object with 4 sides of the same length, then 
a point is clearly a square and the kid is correct. But Cody's also correct. 
You can't divide a finite square into TWO finite squares. But you can divide it 
into an infinity of infinitesimal squares. And it's similarly degenerately 
trivial to divide a point into 2 points.

That's the beauty of math, all you need for the object is for it to satisfy its 
definition. All that excess meaning y'all are piling onto "square" and its 
vernacular referent is irrelevant. If you stopped using the word "square" and 
called it XYZ, then you'd be freer to see its membership.

On 7/23/20 8:40 AM, Frank Wimberly wrote:
> The point is there is no way to partition a square into two squares.
> 
> 
> On Thu, Jul 23, 2020, 9:17 AM Frank Wimberly  > wrote:
> 
> Right.  When its area reaches zero it's not a square.  That is, there is 
> only one square then.
> 
> 
> On Thu, Jul 23, 2020, 9:10 AM Edward Angel  > wrote:
> 
> Why would you call the limit of the increasing smaller squares a 
> “square”? Would you still say it has a dimension of 2? It has no area and no 
> perimeter. In fractal geometry we can create objects with only slightly 
> different constructions that in the limit have a zero area and an infinite 
> perimeter. 
> 

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Re: [FRIAM] square land math question

[Frank Wimberly]

The point is there is no way to partition a square into two squares.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 9:17 AM Frank Wimberly  wrote:

> Right.  When its area reaches zero it's not a square.  That is, there is
> only one square then.
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Thu, Jul 23, 2020, 9:10 AM Edward Angel  wrote:
>
>> Why would you call the limit of the increasing smaller squares a
>> “square”? Would you still say it has a dimension of 2? It has no area and
>> no perimeter. In fractal geometry we can create objects with only slightly
>> different constructions that in the limit have a zero area and an infinite
>> perimeter.
>>
>> Ed
>> ___
>>
>> Ed Angel
>>
>> Founding Director, Art, Research, Technology and Science Laboratory
>> (ARTS Lab)
>> Professor Emeritus of Computer Science, University of New Mexico
>>
>> 1017 Sierra Pinon
>> Santa Fe, NM 87501
>> 505-984-0136 (home)   an...@cs.unm.edu
>> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>>
>> On Jul 23, 2020, at 9:03 AM, Frank Wimberly  wrote:
>>
>> p.s.  Zeno's Paradox is related to
>>
>> 1/2 + 1/4 + 1/8 +...
>>
>> = Sum(1/(2^n)) for n = 1 to infinity
>>
>> = 1
>>
>> (Note:  Sum(1/(2^n)) for n = 0 to infinity
>>
>> = 1/(1 - (1/2)) = 2)
>>
>> ---
>> Frank C. Wimberly
>> 140 Calle Ojo Feliz,
>> Santa Fe, NM 87505
>>
>> 505 670-9918
>> Santa Fe, NM
>>
>> On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly  wrote:
>>
>>> Incidentally, people are used to seeing limits that aren't reached such
>>> a  limit as x goes to infinity of 1/x = 0.  But there are limits such as
>>> limit as x goes to 3 of x/3 = 1.  The question of the squares is the latter
>>> type.  There is no reason the area of the small square doesn't reach 0.
>>>
>>> On Wed, Jul 22, 2020 at 7:36 PM Eric Charles <
>>> eric.phillip.char...@gmail.com> wrote:
>>>
 This is a Zeno's Paradox styled challenge, right? I sometimes describe
 calculus as a solution to Zeno's paradoxes, based on the assumption that
 paradoxes are false.

 The solution, while clever, doesn't' work if we assert either of the
 following:

 A) When the small-square reaches the limit it stops being a square (as
 it is just a point).

 B) You can never actually reach the limit, therefore the small square
 always removes a square-sized corner of the large square, rendering the
 large bit no-longer-square.

 The solution works only if we allow the infinitely small square to
 still be a square, while removing nothing from the larger square. But if we
 are allowing infinitely small still-square objects, so small that they
 don't stop an object they are in from also being a square, then there's no
 Squareland problem at all: *Any *arbitrary number of squares can be
 fit inside any other given square.



 ---
 Eric P. Charles, Ph.D.
 Department of Justice - Personnel Psychologist
 American University - Adjunct Instructor
 


 On Tue, Jul 21, 2020 at 7:59 PM cody dooderson 
 wrote:

> A kid momentarily convinced me of something that must be wrong today.
> We were working on a math problem called Squareland (
> https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
> It basically involved dividing big squares into smaller squares.
> I volunteered to tell the kids the rules of the problem. I made a
> fairly strong argument for why a square can not be divided into 2 smaller
> squares, when a kid stumped me with a calculus argument. She drew a tiny
> square in the corner of a bigger one and said that "as the tiny square 
> area
> approaches zero, the big outer square would become increasingly 
> square-like
> and the smaller one would still be a square".
> I had to admit that I did not know, and that the argument might hold
> water with more knowledgeable mathematicians.
>
> The calculus trick of taking the limit of something as it gets
> infinitely small always seemed like magic to me.
>
>
> Cody Smith
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>>>
>>>
>>> --
>>> Frank 

Re: [FRIAM] square land math question

[Frank Wimberly]

Right.  When its area reaches zero it's not a square.  That is, there is
only one square then.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 9:10 AM Edward Angel  wrote:

> Why would you call the limit of the increasing smaller squares a “square”?
> Would you still say it has a dimension of 2? It has no area and no
> perimeter. In fractal geometry we can create objects with only slightly
> different constructions that in the limit have a zero area and an infinite
> perimeter.
>
> Ed
> ___
>
> Ed Angel
>
> Founding Director, Art, Research, Technology and Science Laboratory
> (ARTS Lab)
> Professor Emeritus of Computer Science, University of New Mexico
>
> 1017 Sierra Pinon
> Santa Fe, NM 87501
> 505-984-0136 (home)   an...@cs.unm.edu
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On Jul 23, 2020, at 9:03 AM, Frank Wimberly  wrote:
>
> p.s.  Zeno's Paradox is related to
>
> 1/2 + 1/4 + 1/8 +...
>
> = Sum(1/(2^n)) for n = 1 to infinity
>
> = 1
>
> (Note:  Sum(1/(2^n)) for n = 0 to infinity
>
> = 1/(1 - (1/2)) = 2)
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly  wrote:
>
>> Incidentally, people are used to seeing limits that aren't reached such
>> a  limit as x goes to infinity of 1/x = 0.  But there are limits such as
>> limit as x goes to 3 of x/3 = 1.  The question of the squares is the latter
>> type.  There is no reason the area of the small square doesn't reach 0.
>>
>> On Wed, Jul 22, 2020 at 7:36 PM Eric Charles <
>> eric.phillip.char...@gmail.com> wrote:
>>
>>> This is a Zeno's Paradox styled challenge, right? I sometimes describe
>>> calculus as a solution to Zeno's paradoxes, based on the assumption that
>>> paradoxes are false.
>>>
>>> The solution, while clever, doesn't' work if we assert either of the
>>> following:
>>>
>>> A) When the small-square reaches the limit it stops being a square (as
>>> it is just a point).
>>>
>>> B) You can never actually reach the limit, therefore the small square
>>> always removes a square-sized corner of the large square, rendering the
>>> large bit no-longer-square.
>>>
>>> The solution works only if we allow the infinitely small square to still
>>> be a square, while removing nothing from the larger square. But if we are
>>> allowing infinitely small still-square objects, so small that they don't
>>> stop an object they are in from also being a square, then there's no
>>> Squareland problem at all: *Any *arbitrary number of squares can be fit
>>> inside any other given square.
>>>
>>>
>>>
>>> ---
>>> Eric P. Charles, Ph.D.
>>> Department of Justice - Personnel Psychologist
>>> American University - Adjunct Instructor
>>> 
>>>
>>>
>>> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson 
>>> wrote:
>>>
 A kid momentarily convinced me of something that must be wrong today.
 We were working on a math problem called Squareland (
 https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
 It basically involved dividing big squares into smaller squares.
 I volunteered to tell the kids the rules of the problem. I made a
 fairly strong argument for why a square can not be divided into 2 smaller
 squares, when a kid stumped me with a calculus argument. She drew a tiny
 square in the corner of a bigger one and said that "as the tiny square area
 approaches zero, the big outer square would become increasingly square-like
 and the smaller one would still be a square".
 I had to admit that I did not know, and that the argument might hold
 water with more knowledgeable mathematicians.

 The calculus trick of taking the limit of something as it gets
 infinitely small always seemed like magic to me.


 Cody Smith
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 FRIAM Applied Complexity Group listserv
 Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
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>>>
>>
>>
>> --
>> Frank Wimberly
>> 140 Calle Ojo Feliz
>> Santa Fe, NM 87505
>> 505 670-9918
>>
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
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> FRIAM-COMIC 

Re: [FRIAM] square land math question

[Edward Angel]

Why would you call the limit of the increasing smaller squares a “square”? 
Would you still say it has a dimension of 2? It has no area and no perimeter. 
In fractal geometry we can create objects with only slightly different 
constructions that in the limit have a zero area and an infinite perimeter. 

Ed
___

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home) an...@cs.unm.edu 

505-453-4944 (cell) http://www.cs.unm.edu/~angel 


> On Jul 23, 2020, at 9:03 AM, Frank Wimberly  wrote:
> 
> p.s.  Zeno's Paradox is related to
> 
> 1/2 + 1/4 + 1/8 +...
> 
> = Sum(1/(2^n)) for n = 1 to infinity
> 
> = 1
> 
> (Note:  Sum(1/(2^n)) for n = 0 to infinity
> 
> = 1/(1 - (1/2)) = 2)
> 
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz, 
> Santa Fe, NM 87505
> 
> 505 670-9918
> Santa Fe, NM
> 
> On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly  > wrote:
> Incidentally, people are used to seeing limits that aren't reached such a  
> limit as x goes to infinity of 1/x = 0.  But there are limits such as limit 
> as x goes to 3 of x/3 = 1.  The question of the squares is the latter type.  
> There is no reason the area of the small square doesn't reach 0.
> 
> On Wed, Jul 22, 2020 at 7:36 PM Eric Charles  > wrote:
> This is a Zeno's Paradox styled challenge, right? I sometimes describe 
> calculus as a solution to Zeno's paradoxes, based on the assumption that 
> paradoxes are false. 
> 
> The solution, while clever, doesn't' work if we assert either of the 
> following: 
> 
> A) When the small-square reaches the limit it stops being a square (as it is 
> just a point). 
> 
> B) You can never actually reach the limit, therefore the small square always 
> removes a square-sized corner of the large square, rendering the large bit 
> no-longer-square. 
> 
> The solution works only if we allow the infinitely small square to still be a 
> square, while removing nothing from the larger square. But if we are allowing 
> infinitely small still-square objects, so small that they don't stop an 
> object they are in from also being a square, then there's no Squareland 
> problem at all: Any arbitrary number of squares can be fit inside any other 
> given square. 
> 
> 
> 
> ---
> Eric P. Charles, Ph.D.
> Department of Justice - Personnel Psychologist
> American University - Adjunct Instructor
>  
> 
> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson  > wrote:
> A kid momentarily convinced me of something that must be wrong today. 
> We were working on a math problem called Squareland 
> (https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p
>  
> ).
>  It basically involved dividing big squares into smaller squares. 
> I volunteered to tell the kids the rules of the problem. I made a fairly 
> strong argument for why a square can not be divided into 2 smaller squares, 
> when a kid stumped me with a calculus argument. She drew a tiny square in the 
> corner of a bigger one and said that "as the tiny square area approaches 
> zero, the big outer square would become increasingly square-like and the 
> smaller one would still be a square". 
> I had to admit that I did not know, and that the argument might hold water 
> with more knowledgeable mathematicians. 
> 
> The calculus trick of taking the limit of something as it gets infinitely 
> small always seemed like magic to me. 
> 
> 
> Cody Smith
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam 
> 
> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com 
> 
> archives: http://friam.471366.n2.nabble.com/ 
> 
> FRIAM-COMIC http://friam-comic.blogspot.com/ 
>  
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam 
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> 
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> 
> FRIAM-COMIC http://friam-comic.blogspot.com/ 
>  
> 
> 
> -- 
> Frank Wimberly
> 140 Calle Ojo Feliz
> Santa Fe, NM 87505
> 505 670-9918
> -  . -..-. . -. -.. -..-. .. ... -..-.  

Re: [FRIAM] square land math question

[Frank Wimberly]

p.s.  Zeno's Paradox is related to

1/2 + 1/4 + 1/8 +...

= Sum(1/(2^n)) for n = 1 to infinity

= 1

(Note:  Sum(1/(2^n)) for n = 0 to infinity

= 1/(1 - (1/2)) = 2)

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly  wrote:

> Incidentally, people are used to seeing limits that aren't reached such a
> limit as x goes to infinity of 1/x = 0.  But there are limits such as limit
> as x goes to 3 of x/3 = 1.  The question of the squares is the latter
> type.  There is no reason the area of the small square doesn't reach 0.
>
> On Wed, Jul 22, 2020 at 7:36 PM Eric Charles <
> eric.phillip.char...@gmail.com> wrote:
>
>> This is a Zeno's Paradox styled challenge, right? I sometimes describe
>> calculus as a solution to Zeno's paradoxes, based on the assumption that
>> paradoxes are false.
>>
>> The solution, while clever, doesn't' work if we assert either of the
>> following:
>>
>> A) When the small-square reaches the limit it stops being a square (as it
>> is just a point).
>>
>> B) You can never actually reach the limit, therefore the small square
>> always removes a square-sized corner of the large square, rendering the
>> large bit no-longer-square.
>>
>> The solution works only if we allow the infinitely small square to still
>> be a square, while removing nothing from the larger square. But if we are
>> allowing infinitely small still-square objects, so small that they don't
>> stop an object they are in from also being a square, then there's no
>> Squareland problem at all: *Any *arbitrary number of squares can be fit
>> inside any other given square.
>>
>>
>>
>> ---
>> Eric P. Charles, Ph.D.
>> Department of Justice - Personnel Psychologist
>> American University - Adjunct Instructor
>> 
>>
>>
>> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson 
>> wrote:
>>
>>> A kid momentarily convinced me of something that must be wrong today.
>>> We were working on a math problem called Squareland (
>>> https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
>>> It basically involved dividing big squares into smaller squares.
>>> I volunteered to tell the kids the rules of the problem. I made a fairly
>>> strong argument for why a square can not be divided into 2 smaller squares,
>>> when a kid stumped me with a calculus argument. She drew a tiny square in
>>> the corner of a bigger one and said that "as the tiny square area
>>> approaches zero, the big outer square would become increasingly square-like
>>> and the smaller one would still be a square".
>>> I had to admit that I did not know, and that the argument might hold
>>> water with more knowledgeable mathematicians.
>>>
>>> The calculus trick of taking the limit of something as it gets
>>> infinitely small always seemed like magic to me.
>>>
>>>
>>> Cody Smith
>>> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
>>> FRIAM Applied Complexity Group listserv
>>> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
>>> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
>>> archives: http://friam.471366.n2.nabble.com/
>>> FRIAM-COMIC http://friam-comic.blogspot.com/
>>>
>> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
>> FRIAM Applied Complexity Group listserv
>> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
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>> archives: http://friam.471366.n2.nabble.com/
>> FRIAM-COMIC http://friam-comic.blogspot.com/
>>
>
>
> --
> Frank Wimberly
> 140 Calle Ojo Feliz
> Santa Fe, NM 87505
> 505 670-9918
>
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Re: [FRIAM] square land math question

[Frank Wimberly]

Incidentally, people are used to seeing limits that aren't reached such a
limit as x goes to infinity of 1/x = 0.  But there are limits such as limit
as x goes to 3 of x/3 = 1.  The question of the squares is the latter
type.  There is no reason the area of the small square doesn't reach 0.

On Wed, Jul 22, 2020 at 7:36 PM Eric Charles 
wrote:

> This is a Zeno's Paradox styled challenge, right? I sometimes describe
> calculus as a solution to Zeno's paradoxes, based on the assumption that
> paradoxes are false.
>
> The solution, while clever, doesn't' work if we assert either of the
> following:
>
> A) When the small-square reaches the limit it stops being a square (as it
> is just a point).
>
> B) You can never actually reach the limit, therefore the small square
> always removes a square-sized corner of the large square, rendering the
> large bit no-longer-square.
>
> The solution works only if we allow the infinitely small square to still
> be a square, while removing nothing from the larger square. But if we are
> allowing infinitely small still-square objects, so small that they don't
> stop an object they are in from also being a square, then there's no
> Squareland problem at all: *Any *arbitrary number of squares can be fit
> inside any other given square.
>
>
>
> ---
> Eric P. Charles, Ph.D.
> Department of Justice - Personnel Psychologist
> American University - Adjunct Instructor
> 
>
>
> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson 
> wrote:
>
>> A kid momentarily convinced me of something that must be wrong today.
>> We were working on a math problem called Squareland (
>> https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
>> It basically involved dividing big squares into smaller squares.
>> I volunteered to tell the kids the rules of the problem. I made a fairly
>> strong argument for why a square can not be divided into 2 smaller squares,
>> when a kid stumped me with a calculus argument. She drew a tiny square in
>> the corner of a bigger one and said that "as the tiny square area
>> approaches zero, the big outer square would become increasingly square-like
>> and the smaller one would still be a square".
>> I had to admit that I did not know, and that the argument might hold
>> water with more knowledgeable mathematicians.
>>
>> The calculus trick of taking the limit of something as it gets
>> infinitely small always seemed like magic to me.
>>
>>
>> Cody Smith
>> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
>> FRIAM Applied Complexity Group listserv
>> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
>> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
>> archives: http://friam.471366.n2.nabble.com/
>> FRIAM-COMIC http://friam-comic.blogspot.com/
>>
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
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> archives: http://friam.471366.n2.nabble.com/
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>


-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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Re: [FRIAM] square land math question

[Eric Charles]

This is a Zeno's Paradox styled challenge, right? I sometimes describe
calculus as a solution to Zeno's paradoxes, based on the assumption that
paradoxes are false.

The solution, while clever, doesn't' work if we assert either of the
following:

A) When the small-square reaches the limit it stops being a square (as it
is just a point).

B) You can never actually reach the limit, therefore the small square
always removes a square-sized corner of the large square, rendering the
large bit no-longer-square.

The solution works only if we allow the infinitely small square to still be
a square, while removing nothing from the larger square. But if we are
allowing infinitely small still-square objects, so small that they don't
stop an object they are in from also being a square, then there's no
Squareland problem at all: *Any *arbitrary number of squares can be fit
inside any other given square.



---
Eric P. Charles, Ph.D.
Department of Justice - Personnel Psychologist
American University - Adjunct Instructor



On Tue, Jul 21, 2020 at 7:59 PM cody dooderson  wrote:

> A kid momentarily convinced me of something that must be wrong today.
> We were working on a math problem called Squareland (
> https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
> It basically involved dividing big squares into smaller squares.
> I volunteered to tell the kids the rules of the problem. I made a fairly
> strong argument for why a square can not be divided into 2 smaller squares,
> when a kid stumped me with a calculus argument. She drew a tiny square in
> the corner of a bigger one and said that "as the tiny square area
> approaches zero, the big outer square would become increasingly square-like
> and the smaller one would still be a square".
> I had to admit that I did not know, and that the argument might hold water
> with more knowledgeable mathematicians.
>
> The calculus trick of taking the limit of something as it gets
> infinitely small always seemed like magic to me.
>
>
> Cody Smith
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> archives: http://friam.471366.n2.nabble.com/
> FRIAM-COMIC http://friam-comic.blogspot.com/
>
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Re: [FRIAM] square land math question

[Frank Wimberly]

Off the top off my head.  As long as the small square isn't of zero area
the larger square isn't a square.  When the smaller square reaches area
zero there is only one square.

What do you think?
---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Tue, Jul 21, 2020, 5:59 PM cody dooderson  wrote:

> A kid momentarily convinced me of something that must be wrong today.
> We were working on a math problem called Squareland (
> https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p).
> It basically involved dividing big squares into smaller squares.
> I volunteered to tell the kids the rules of the problem. I made a fairly
> strong argument for why a square can not be divided into 2 smaller squares,
> when a kid stumped me with a calculus argument. She drew a tiny square in
> the corner of a bigger one and said that "as the tiny square area
> approaches zero, the big outer square would become increasingly square-like
> and the smaller one would still be a square".
> I had to admit that I did not know, and that the argument might hold water
> with more knowledgeable mathematicians.
>
> The calculus trick of taking the limit of something as it gets
> infinitely small always seemed like magic to me.
>
>
> Cody Smith
> -  . -..-. . -. -.. -..-. .. ... -..-.  . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> archives: http://friam.471366.n2.nabble.com/
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>
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