From: kirby urner
Date: Thu, Dec 17, 2015 at 1:16 PM
Subject: Re: Presenting misconceptions detrimental to learning ...
To: The Physics Learning Research List
On Thu, Dec 17, 2015 at 12:43 PM, [ JD ] wrote:
I'd like to riff off this paragraph a little in order to draw a distinction between "misconceptions" and "alternative conceptions".
My approach to the latter w/r to non-Euclidean alternatives is to start with Karl Menger's 'Modern Geometry and the Theory of Relativity' [1] wherein he suggests a "geometry of lumps" in which points, lines and planes are distinguished not by "dimension" (Karl was a dimension theorist) but by shape (topological characteristics).
I combine this with an alternative model of multiplying two numbers, where the two lengths A, B are posited at 60 degrees in alignment with unit-area triangles, such that A x B is the area closed off by the origin O, and segment AB. [2]
This model leads to an analogous treatment of 3rd powering (or multiplying any 3 numbers) that goes to a tetrahedron, not a cube. "Three to the third power" is not a cube but a tetrahedron. This is well known from the 1970s writings of geodesic dome architect Buckminster Fuller. [3]
Given the pronounced 4ness of the tetrahedron (4 windows, 4 points), and no distinction in dimension twixt points, lines and planes ("infinitely thin or small" e.g. "depth without height or width" are handled by a concept of "subfrequency" instead), we get a "pre-frequency" Platonic world that is considered "4D" (tetrahedron = concept of "container" i.e. that with inside/outside concave/convex aspects), with Time / Energy added as additional dimensions. [4]
That's all an opening into a philosophical investigation of what's permissible, in terms of having different namespaces. Are these moves "allowed"? It's a different way of talking. I cast it as "Martian Math" to acknowledge it's alien, but not necessarily "wrong".
Challenging the efficacy and utility of Euclidean geometry is not the point. Pointing to another way of connecting the dots that likewise holds water is the point. One needs a limber mind to tackle slippery concepts (like "dimension") and getting too stuck in a rut is not a good way to stay limber.
Kirby
[1] in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474
[2] https://youtu.be/2B1XXV2Eoh8
[3] http://www.rwgrayprojects.com/
[4] https://en.wikipedia.org/wiki/
Date: Thu, Dec 17, 2015 at 1:16 PM
Subject: Re: Presenting misconceptions detrimental to learning ...
To: The Physics Learning Research List
On Thu, Dec 17, 2015 at 12:43 PM, [ JD ] wrote:
In particular, many of the most important concepts (in physics and elsewhere) are so fundamental that they simply cannot be defined in terms of anything more fundamental. One place where you see this with particular clarity is in Euclidean geometry, where the fundamental objects -- points, lines, and planes -- are emphatically and explicitly *not* defined. The words acquire meaning from how they are used, and not otherwise.
I'd like to riff off this paragraph a little in order to draw a distinction between "misconceptions" and "alternative conceptions".
My approach to the latter w/r to non-Euclidean alternatives is to start with Karl Menger's 'Modern Geometry and the Theory of Relativity' [1] wherein he suggests a "geometry of lumps" in which points, lines and planes are distinguished not by "dimension" (Karl was a dimension theorist) but by shape (topological characteristics).
I combine this with an alternative model of multiplying two numbers, where the two lengths A, B are posited at 60 degrees in alignment with unit-area triangles, such that A x B is the area closed off by the origin O, and segment AB. [2]
This model leads to an analogous treatment of 3rd powering (or multiplying any 3 numbers) that goes to a tetrahedron, not a cube. "Three to the third power" is not a cube but a tetrahedron. This is well known from the 1970s writings of geodesic dome architect Buckminster Fuller. [3]
Given the pronounced 4ness of the tetrahedron (4 windows, 4 points), and no distinction in dimension twixt points, lines and planes ("infinitely thin or small" e.g. "depth without height or width" are handled by a concept of "subfrequency" instead), we get a "pre-frequency" Platonic world that is considered "4D" (tetrahedron = concept of "container" i.e. that with inside/outside concave/convex aspects), with Time / Energy added as additional dimensions. [4]
That's all an opening into a philosophical investigation of what's permissible, in terms of having different namespaces. Are these moves "allowed"? It's a different way of talking. I cast it as "Martian Math" to acknowledge it's alien, but not necessarily "wrong".
Challenging the efficacy and utility of Euclidean geometry is not the point. Pointing to another way of connecting the dots that likewise holds water is the point. One needs a limber mind to tackle slippery concepts (like "dimension") and getting too stuck in a rut is not a good way to stay limber.
Kirby
[1] in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474
[2] https://youtu.be/2B1XXV2Eoh8
[3] http://www.rwgrayprojects.com/
[4] https://en.wikipedia.org/wiki/
