#### The Mathematical Impossibility of Compromise. A multidimensional abstraction.

Posted by Doug_Huffman 1 month ago to Science

From The Mathematical Impossibility of Compromise by Alan ‘Uncle Al’ Schwartz

“[ ... ] Create a multidimensional compromise space. Choose a variable, normalize it on a scale of 1 through zero to -1 inclusive. Love thy neighbor, kill thy neighbor. Racial integration, racial segregation. Keep all that you earn, the State confiscates all that you earn. Every school kid gets automatic straight As in everything, Darwinian social evolution. United Nations, mean Mother Green. Sister Boom-Boom, Vlad the Impaler.

Each normalized line is assigned its own dimension and is appended orthogonal to all prior lines. The zero centers of all lines coincide as a common origin. N variables then create a filled N-dimensional unit hypersphere, a closed N-hyperball, with each line being one of its diameters (each variable being wholly independent of all the others). The closed N-hyperball embraces every possible combination and magnitude of overall policy as the totality of single points on its surface and in its volume. Points near the center are maximally effete compromise. Points near the surface are the most draconian events possible.

A hypercube with edge length 2 and volume 2N circumscribes about any unit N-hypersphere. The ratio of our N-hyperball's volume to its circumscribed hypercube monotonically decreases as the dimension N, the number of independent variables, increases. As a problem's complexity increases the closed N-hyperball of all possible solutions - of any kind - monotonically shrinks toward a zero volume point. The results are inescapable:

As N, the number of variables, increases the number of compromise choices of all kinds shrinks toward zero. A large multi-variable problem has asymptotically no solutions at all much less a menu of nice compromise solutions.

By N=10, 89.3% of the volume is within 20% radius of the surface. Almost every choice is a draconian imposition.

By N=20, 98.8% of the volume is within 20% radius of the surface, and the overall volume - the summation of choices available - is essentially zero. There is no compromise possible. The few choices remaining are fanatic.[ ... ]”

See the Something Awesomely Strange, Somebody Should Look link at the bottom of the page. The EP test occupied us for some years. I volunteered to introduce Uncle Al at his Nobel Prize ceremony. Alas ...

“[ ... ] Create a multidimensional compromise space. Choose a variable, normalize it on a scale of 1 through zero to -1 inclusive. Love thy neighbor, kill thy neighbor. Racial integration, racial segregation. Keep all that you earn, the State confiscates all that you earn. Every school kid gets automatic straight As in everything, Darwinian social evolution. United Nations, mean Mother Green. Sister Boom-Boom, Vlad the Impaler.

Each normalized line is assigned its own dimension and is appended orthogonal to all prior lines. The zero centers of all lines coincide as a common origin. N variables then create a filled N-dimensional unit hypersphere, a closed N-hyperball, with each line being one of its diameters (each variable being wholly independent of all the others). The closed N-hyperball embraces every possible combination and magnitude of overall policy as the totality of single points on its surface and in its volume. Points near the center are maximally effete compromise. Points near the surface are the most draconian events possible.

A hypercube with edge length 2 and volume 2N circumscribes about any unit N-hypersphere. The ratio of our N-hyperball's volume to its circumscribed hypercube monotonically decreases as the dimension N, the number of independent variables, increases. As a problem's complexity increases the closed N-hyperball of all possible solutions - of any kind - monotonically shrinks toward a zero volume point. The results are inescapable:

As N, the number of variables, increases the number of compromise choices of all kinds shrinks toward zero. A large multi-variable problem has asymptotically no solutions at all much less a menu of nice compromise solutions.

By N=10, 89.3% of the volume is within 20% radius of the surface. Almost every choice is a draconian imposition.

By N=20, 98.8% of the volume is within 20% radius of the surface, and the overall volume - the summation of choices available - is essentially zero. There is no compromise possible. The few choices remaining are fanatic.[ ... ]”

See the Something Awesomely Strange, Somebody Should Look link at the bottom of the page. The EP test occupied us for some years. I volunteered to introduce Uncle Al at his Nobel Prize ceremony. Alas ...

SOURCE URL: https://www.mazepath.com/uncleal/comprom.htm

Edit add: I'm smiling at the beauty of the function, not the fact it seems to indicate we are doomed unless a whole lot of rational choices can be made.

It is one of my favorites of his essays and the source of lots of daydreams.

That is an unwarranted assumption.I have long been a fan of Ayn Rand / Atlas Shrugged / Objectivism / Brandon / various academics, and only recently discovered Galt’s Gulch On-line.

I live in my ‘Galt’s Gulch’ - a monocultural isolated rural Island - not with an illusion field but with a stiff ferryman’s tariff. 54246

I am in the process of withdrawing from other special interest forums, and have paid GGO-l membership. - A commitment of sorts.