square land math question
Locked
123
123
 
|
How about, "Points are maps from terminal objects?"
-- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
In reply to this post by Steve Smith
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?) -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
Re: square land math question
|
In reply to this post by jon zingale
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 <[hidden email]> wrote: How about, "Points are maps from terminal objects?" - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
In reply to this post by jon zingale
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?" -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
Re: square land math question
|
|
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 ↙↙↙ <[hidden email]> wrote: Maybe. But how do we handle things like reciprocals of infinities? Is 1/aleph0 the same as 1/aleph1? - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
In reply to this post by Frank Wimberly-2
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 :) -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
In reply to this post by Frank Wimberly-2
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. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
Re: square land math question
|
|
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 ↙↙↙ <[hidden email]> 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. - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
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. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
Re: square land math question
|
|
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 ↙↙↙ <[hidden email]> 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. - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
Zeno had several paradoxes, all intended to expose questionable assumptions. On Thu, Jul 23, 2020, 1:58 PM Frank Wimberly <[hidden email]> wrote:
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
Re: square land math question
|
|
In reply to this post by Frank Wimberly-2
Doesn’t that depend on how finely you can pick a nit> On 23 Jul 2020, at 12:28, Frank Wimberly wrote:
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
In reply to this post by gepr
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. > -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
|
|
In reply to this post by gepr
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?) > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
Nice challenge! ... Welllll, 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"? -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
Re: square land math question
|
|
"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 ↙↙↙ <[hidden email]> wrote: Nice challenge! ... Welllll, 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". Frank Wimberly
140 Calle Ojo Feliz Santa Fe, NM 87505 505 670-9918 - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
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. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
|
|
In reply to this post by Frank Wimberly-2
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) [hidden email] 505-453-4944 (cell) http://www.cs.unm.edu/~angel
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
Re: square land math question
|
|
In reply to this post by gepr
"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 ↙↙↙ <[hidden email]> 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. Frank Wimberly
140 Calle Ojo Feliz Santa Fe, NM 87505 505 670-9918 - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/ |
|
|
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. -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . 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/
uǝʃƃ ⊥ glen
|
«
Return to Friam
|
1 view|%1 views
| Free forum by Nabble | Edit this page |