Against Pi Day

I don՚t want to poop on anybody՚s celebration of Pi Day, but while pi itself is certainly something to be wondered at and celebrated, the fact that an arbitrarily numbered point on the human calendar aligns with the decimal approximate representation of something transcendental and timeless does not excite me in the least.

What I՚m saying is, we live in a universe where pi pops up everywhere, so every day (defined by a rotation of our pi-manifesting spherical planet) should be pi day.

Portraits of Unicode Characters, #1 of a series: ⧜ Incomplete Infinity

The Unicode character set is a huge but barely explored extension of what is essentially now the common linguistic matrix of the human race.  It contains every conceivable glyph and symbol, including some rather strange ones that seem to have been inserted in a spirit of philosophy, comedy, or both. This series will examine some of these and what is being done with or could be done with them.

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Incomplete Infinity 

In mathematics, at least, it is an absurd concept.  Or a redundant one ‒ as Cantor showed, any infinite set is incomplete in a sense, because you can always construct a larger one.  But transfinite numbers aside, it doesn't seem to make much sense to talk of an incomplete infinity. A set is finite or it is infinite; there is no in-between.  Very large is not close to infinity and if you take anything away from infinity you are still left with infinity. So there doesn't seem to be anything for "incomplete infinity" to refer to.

But from a non-mathematical perspective, this symbol is just a perfect, its form and name a masterpiece of  compacted metaphysical irony. It is a representation of the human condition - partaking of the infinite, cognizant of the infinite, often driven mad by the infinite, yet irredemiably incomplete and finite.  No matter how much we accumulate, the gap between our finite selves and the infinite is never closed, except perhaps in death. 

Birds, frogs, pasta, and the ultimate nature of reality

Things I'm reading instead of what I should be reading: Max Tegmark's new paper, The Mathematical Universe, which expands on his earlier work that I've mentioned before.

The External Reality Hypothesis: There exists an external physical realitycompletely independent of us humans.

The Mathematical Universe Hypothesis: Our external physical reality is an abstract mathematical structure.

This is one of those "so crazy it might be true" theories. And for me, it seems to talk about some of the more abstruse issues in theoretical physics from the math up, in a way that makes them almost comprehensible.

When considering such examples, we need to distinguish between two different ways of viewing the external physical reality: the outside view or bird perspective of a mathematician studying the mathematical structure and the inside view or frog perspective of an observer living in it.

...If history were a movie,the structure would therefore correspond not to a single frame of it but to the entire videotape. Consider the first example above, a world made up of classical point particles moving around in three-dimensional Euclidean space under the influence of Newtonian gravity. In the four-dimensional spacetime of the bird perspective, these particle trajectories resemble a tangle of spaghetti...To the frog, the world is described by Newton's laws of motion and gravitation. To the bird, it is described by the geometry of the pasta, obeying the mathematical relations corresponding to minimizing the Newtonian action.

[under Quantum Field Theory]...If the bird sees such deterministic frog branching, the frog perceives apparent randomness.

I like those bird and frog terms and will adopt them for my own scruffy thoughts on the nature of subjective and objective views of the universe (I may leave out the pasta though).

Science is the process whereby frogs laboriously attempt to take the perspective of birds.

Bad math award

From the SF Chronicle:

On the back of each Mega Millions ticket is a suggestion to "play responsibly," but on the front are numbers, and Tuesday at 8 p.m. a combination of those digits is expected to be worth $355 million..."I realize I don't have a chance, but nobody's got a chance. So the way I look at it, I have a 50-50 chance -- either I win it or someone else wins it," reasoned Barrie Green, 60, after buying a single ticket Monday afternoon at the Merritt Restaurant and Bakery near his home in Oakland.

I like that "reasoned".

"Good luck, sir," said cashier Weida Han, who chose not to explain to Green that his odds of winning -- and being able to quit his job driving cars to auctions -- are 1 in 175,711,536.

The Anthropic Cosmological Principle meets Computational Complexity; Hilarity Ensues

Scott Aaronson, the second-most amusing person in string theory, demonstrates how to solve NP-complete problems, about halfway through a talk:

But what could NP-hardness possibly have to do with the Anthropic Principle? Well, when I talked before about computational complexity, I forgot to tell you that there's at least one foolproof way to solve NP-complete problems in polynomial time. The method is this: first guess a solution at random, say by measuring electron spins. Then, if the solution is wrong, kill yourself! If you accept the many-worlds interpretation of quantum mechanics, then there's certainly some branch of the wavefunction where you guessed right, and that's the only branch where you're around to ask whether you guessed right! It's a wonder more people don't try this.

This is a pretty obvious idea once you hear it (those are always the best ones. As Huxley was suppoed to have said upon reading The Origin of Species, "How extremely stupid not to have thought of that").

Aaronson takes this as evidence that anthropic arguments are invalid, since he's fairly attached to the idea that NP-hard problems are hard. I tend to agree, there has always been something Panglossian about anthropic cosmology. In fact...you could implement Dr. Pangloss using Aaronson's procedure. If you are convinced that the world is not as good as it could be, you kill yourself. Then the versions of you left alive are perforce inhabiting the best of all possible worlds.

A map of mathematics

This paper Is “the theory of everything” merely the ultimate ensemble theory? by Max Tegmark is extremely interesting in many respects, but aside from its way-out metaphysics, I really appreciated this diagram showing common mathematical structures and their relationships. When I actually was trying to do math years ago, this is something I wished for and occasionally tried to construct myself.

Alas, I've mostly forgotten most of my math (and I've never really been able to grasp modern physics) but this table lets me fantasize about picking it up again -- at least I would be able to figure out where to start, what bit depends on which other bit, and where it all ultimately leads. Although apparently category theory is really where it's at in physics (or was 10 years ago) and that doesn't even appear in the diagram.

The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically “real” world.