Olaf Beyersdorff, Deputy Director of Research and Innovation
University of Leeds, Leeds, LS2 9JT
Bounded Arithmetic (Samuel R. Buss)
Univalent Foundations of Mathematics, Intuitionism and more generally constructive mathematics
First email: 22 August 2018
RE: Computational simplicity quick evolved into computational complexity. Where’s the logic? What are we doing?
Dear Prof. Dr. Olaf Beyersdorff:
Would applying base-2 notation to the Planck base units render useful information?
I hope so.
Unwittingly, back in 2011 in a New Orleans high school, we backed into the tetrahedron (and the octahedron within it) and have not yet re-emerged! We were studying the interior geometries of the tetrahedron by dividing each edge in half and connecting those new vertices. We discovered the four half-sized tetrahedrons in each corner and the octahedron in the middle. Applying the same process with our octahedron, we discovered the six half-sized octahedrons in each corner and the eight tetrahedrons in each face, all sharing a common centerpoint. We also found the four hexagonal plates surrounding that centerpoint and quickly tiled the universe with them. We also found the plates of squares, triangles, and other configurations.
It was all quite fascinating, but our adventure was just starting.
We continued to divide the edges in half for both objects, going deeper and deeper inside. We had Zeno in mind. By the 45th step inside (of course, on paper only), we were in the range of the sizes measured within particle physics. We called it the CERN-scale. In another 67 steps within, we were down in the Planck scale. That was a total of 112 steps from our little 2.5″ tetrahedron down to the all those dimensionless constants that make up the Planck base units. We rather quickly learned that what we were doing was applying base-2 notation by using the Planck base units for scaling. We then learned about Kees Boeke’s base-10 work in 1957 (also in a high school).
And, rather quickly, we thought about multiplying by 2. That was the biggest surprise.
In just 90 steps or doublings going out, we were at the Hubble’s measurement of the approximate size of the universe. A NASA scientist helped us with that calculation. Many others did as well. But three years later, we finally added Planck Time to the scale and it blew us away. The two scaled so well together. But then, we thought, “Of course they do!”
In another year, we added Planck Charge and Planck Mass and that was totally challenging. We are still working on those numbers and have a long ways to go. But, in the process, we have talked with some of the finest people on the planet. Yet, we know what we have done is quite idiosyncratic and naive because nobody is critical, yet everybody is reluctant to engage such simplicity.
Do we trust simple logic and math?
The implications are just too strange. With those 202.34 notations or doublings or steps, we have a very unique map of the universe. But, is it useful? Or, is it just a collection of useless numbers? What does a retired high school teacher do? What would you do with this odd-duck collection of simple math and simple logic? Just forget it?