M Theory Lesson 112
Recall that a lattice is a poset such that each pair of elements has a least upper bound and greatest lower bound. In topos theory we also assume that the whole lattice has a largest element 1 and smallest element 0. For example, in Set the lattice of subsets of a set $S$ has union (join) and intersection (meet). The greatest element is $S$ and the zero is the empty set.
Consider ordering a set of numbers by size. The property of the rational numbers that leads to the reals is the following: consider the set of rationals $q$ defined by $q^2 < 2$. This set is bounded above in $\mathbb{Q}$ but contains no least upper bound in $\mathbb{Q}$. Only the real numbers have the property that this least upper bound exists. The interval $[0,1]$ in the (extended) reals is an example of a complete lattice, as is the power set example above.
Considering Vect as a quantum logic category, it is still true that lattices of subspaces are complete, even for the rationals! In logos theory the logic of lattices may be weakened. Notions of meet and join are still useful but may not be required to exist for all objects. A ternary analogue of lattice is something else entirely.
Consider ordering a set of numbers by size. The property of the rational numbers that leads to the reals is the following: consider the set of rationals $q$ defined by $q^2 < 2$. This set is bounded above in $\mathbb{Q}$ but contains no least upper bound in $\mathbb{Q}$. Only the real numbers have the property that this least upper bound exists. The interval $[0,1]$ in the (extended) reals is an example of a complete lattice, as is the power set example above.
Considering Vect as a quantum logic category, it is still true that lattices of subspaces are complete, even for the rationals! In logos theory the logic of lattices may be weakened. Notions of meet and join are still useful but may not be required to exist for all objects. A ternary analogue of lattice is something else entirely.
posted by Kea | 8:40 AM
3 Comments:
Hi Kea,
I noticed the following, sort of related to the lattice.
Consider constructing this 2D geometric representation that appears to unify concepts of Pythagoras, Archimedes, Newton and Einstein:
An [obtainable?] ideal of E=mc^2 [from Einstein] when m=1:
At the origin (0,0), construct a circle with a radius representing c as a vector.
All rays from the origin may be considered as vectors of c.
Focus only upon the x and y axes.
In all four quadrants, draw the hypotenuse from the end of each vector c on the x to the y axis, to construct a circumscribed square.
Note that each hypotenuse is equal to c*2^(1/2) from Pythagoras.
Each hypotenuse may represent a scalar relative speed.
Circumscribe the original circle within a square where both the x and y axes bisect the edges, each of length 2c, of this square.
This is a step in how Archimedes attempted to determine the value of PI by maximizing the perimeter of a regular polygon within and minimizing the perimeter of a regular polygon outside the circle.
In Newton’s equation: F=ma=(1/2)mv^2
From the above construction, with m=1: (1/2)*(c*2^(1/2))^2=c^2.
This is apparently the energy E obtained by Einstein when m=1.
Yet the maximum relative speed from -x to x or -y to y is 2c.
If Einstein had used this value then maybe E would equal 4c^2.
This appears to suggest that c may be “constant extrema” rather than merely a constant.
The constructions allow for these areas:
square c^2 < square 2*c^2 < circle 2*PI*c^2 < square (2c)^2=4c^2
Please correct any errors..
Doug, I guess you realise that we have been talking about quantised speeds with only a 'local' constant c. That is, a heirarchy of Planck constants associated with dimension, where you are considering d=2 instead of the usual d=1. So your sqrt(2)c is analogous to Carl's (and Louise's) sqrt(3)c and not analogous to the usual c, which happens along a line.
Hi Kea,
1 - Thanks for the comment.
Although I intended to use my construction only for 2D, by accident rather than by design, I think that two 1D perspectives are nested within.
Namely the y-axis and more conventionally the x-axis are in themselves one dimensional.
This may be why Newton used 1/2 of the x-axis since if both halves were used, the opposite directions would tend to cancel.
2 - I do not see how Einstein could have used only 1D for E=mc^2 when this equation appears to correspond to orthogonal relative velocities [or speeds]
3 - As I become more familiar with mathematical terminology, it may be that there is only one degree of space with 3 constraints resulting in 3 or 4 dimensions of space? These may be the more common x, y, z directions with trajectory as the fourth?
4 - I suspect that Planck's constant h may also be a "constant extrema". Although h-bar [reduced Plank or Dirac constant] is easier to use in calculations, the sense of periodicity is lost.
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