The beauty of the two-color sphere theorem Ronny Desmet 10 December 2015

Whitehead’s philosophy Whitehead’s philosophy is “a critique of pure feeling” The psyche is a temporal series of occasions of experience Each occasion of experience is a creative synthesis of feelings of past occasions Each feeling has objective content (what is felt, the objective pattern) and subjective form (how it is felt, the emotional pattern) Each creative synthesis is a progressive integration of feelings controlled by their emotional pattern But it is not merely subjective, because the emotional pattern of each of its initial feelings is conformally derived from the objective pattern

Whitehead: Beauty & Beautiful How does an occasion of experience become subjectively qualified by Beauty? Each occasion is a unification, a unity amid diversity. More is needed to produce Beauty of subjective form: variety of detail with effective contrast (minimal ‘anesthesia’ and ‘aesthetic destruction’). But Beauty is not merely subjective. It is also objective. The objective content may be termed Beautiful by reason of its conformal contribution to the Beauty of the subjective form of the complete occasion.

Mozart’s Jupiter Symphony The secret of the beauty of its last movement is a four note motif or thematic melody that allows the four other melodies to express themselves contrapuntally Reduction to the motif = simplificiation (anesthesia); arbitrarily mingle the competing melodies = discordance (aesthetic destruction) Farmer: “Beauty can save us from our smallness and lack of vision”

Euler’s formula … is like the four note motif of Mozart’s Jupiter symphony … unifying a diversity of mathematical melodies (the melodies of the natural numbers, the real numbers, the complex numbers, and the exponential functions)

Whitehead: Beauty is background dependent “There are two sides to aesthetic experience. In the first place, it involves a subjective sense of individuality. It is my enjoyment. … In the second place, there is the aesthetic object which is identified in experience as the source of the subjective feeling.” BUT the idea “that there is a definite object correlated to a definite subjective reaction” is “a violent abstraction.” Each occasion of experience has “an infinitude of relations” with the occasions of its antecedent universe. “In our experience there is always the dim background from which we derive,” and which is “hardly touched by consciousness.” Consequently: beauty is background dependent!

Gian-Carlo Rota: mathematical beauty is background dependent The beauty of a piece of mathematics is dependent upon schools and periods. A theorem that is in one context thought to be beautiful may in a different context appear trivial.  …  Many occurrences of mathematical beauty fade or fall into triviality as mathematics progresses. However, given the historical period and the context, one finds substantial agreement among mathematicians as to which mathematics is to be regarded as beautiful.  …  In other words, the beauty of a piece of mathematics does not consist merely of the subjective feelings experienced by an observer. The beauty of a theorem is an objective property on a par with its truth.  …  Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context dependent.

Intro to the two-color sphere theorem A two-colored sphere – No big deal

Intro to the two-color sphere theorem Triplets of orthogonal symmetry-axes

The two-color sphere theorem Take any triplet of orthogonal symmetry-axes: X, Y, and Z Paint the two intersection points of X with the sphere in green Paint the four intersection points of Y and Z with the sphere in red Take any such set not yet taken and proceed similarly, again and again It is impossible to paint the whole sphere in green and red according to these instructions … SO WHAT? Unimportant, insignificant, … no experience of beauty

The 33 symmetry-axes of Perres

Proof (case 1)

Proof (cases 2, 3, …)

Proof (final case)

Alternative determination of 33 axes

Experiencing the beauty of the two-color sphere theorem by taking into account its quantum mechanical background Rota: “The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory.”

Intro to the Kochen-Specker theorem: the quantum world Spin is a property in the world of elementary particles (e.g. electrons are spin ½ particles and photons are spin 1 particles) When measuring the component of a spin 1 particle in a certain direction, the result will always be 0 or 1 or -1 When measuring the squared components of a spin 1 particle in three orthogonal directions, the result will always be (1,0,1) or (0,1,1) or (1,1,0) = the spin-axiom

The Kochen-Specker theorem The spin-axiom implies that the spin-components of a spin 1 particle do not exist prior to spin- measurement!

Reductio ad absurdum proof of the Kochen-Specker theorem Take a spin 1 particle and suppose its spin- components in all directions exist prior to any measurement = supposition Then the spin-axiom implies that for each set of 3 orthogonal directions this particle has one pre-existing spin-squared-component that is 0 and two that are 1 = consequence This consequence contradicts the two-color sphere theorem = contradiction QED

Metaphor to explain the Kochen- Specker theorem: Poire Williams

The Poire Williams metaphor The spin-components in all directions of a spin 1 particle cannot exist prior to any measurement, for else I can do what is impossible to do according to the two-color sphere theorem The size-components in all directions of the pear in the Poire Williams bottle cannot have existed prior to bottling, for else someone must have been able to do what is impossible to do, push a full-grown pear through a small bottleneck without crushing it

Compare the measurement process with the following growth process:

The circle has been closed Beauty is no pre-existing property but one that can emerge in the process of becoming of an occasion of experience Spin is no pre-existing property but one that can emerge in the process of measuring of an elementary particle Full-grown size is no pre-existing property but one that can emerge in the process of growth of the pear in the Poire Williams bottle

To conclude I hope that by highlighting the important role of the two-color sphere theorem in the wonderful world of quantum mechanics, the reader has experienced agreement with Rota’s claim that the beauty of a theorem is best observed by presenting it as the crown jewel of a theory Q&A … (cf.