Monthly Archives: April 2007

Surreal Division (A weak post)

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Coming back from games to numbers, I promised earlier that I would define
division. Division in surreal numbers is, unfortunately, ugly. We start with
a simple, basic identity: if a=b×c, and a is not zero, then c=a×(1/b). So if we can define how to take the reciprocal of a surreal number, then division falls out naturally from combining it the reciprocal with multiplication.

This is definitely one of my weaker posts; I’ve debated whether or not to post it at all, but I promised that I’d show how surreal division is defined, and I don’t foresee my having time to do a better job of explaining it in a reasonable time frame.. So my apologies if this is harder to follow than my usual posts.

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Todays tautology: "Egnor writes idiotic things" and "All people who write idiotic things are idiots", therefore "Egnor is an idiot"

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Apparently, Michael Egnor just can’t get enough of making himself look like an idiot. His latest screed is an attack on me, for criticizing his dismissal of evolution as a tautology.

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George Shollenberger Returns to Prove his Innumeracy

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A couple of weeks ago, I revisited George Shollenberger, the creator the alleged “First Scientific Proof of God”, and commented on his pathetic antics on amazon.com, trying to explain just why no one had bothered to post a single review of his book. (If you’ll
recall, according to George, it’s because everyone is too busy considering the impact that his proof is going to have on their activities.)

Normally, I wouldn’t revisit a two-bit crank like George after such a short interval, but he showed up in the comments again to specifically point at a post he made on his own blog, which he claims justifies his position that all of mathematics needs to be reconsidered in light of his supposed proof.

And it’s just too silly to pass up.

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From Surreal Numbers to Games

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Today we’re going to take our first baby-step into the land of surreal games.

A surreal number is a pair of sets {L|R} where every value in L is less than every value in R. If we follow the rules of surreal construction, so that the members of L sets are always strictly less than members of R sets, we end up with a totally ordered field (almost) – it gives us something essentially equivalent to a superset of the real numbers. (The reason for the almost is that technically, the surreals form a class not a set, and a field must be based on a set. But for our purposes, we can treat them as a field without much trouble.)

But what happens if we take away the restriction about the < relationship between the L and R sets? What we get is a set of things called games. A game is a pair of sets L and R, where each member of L and R is also a game. It should be obvious that every surreal number is also a game – but there are many more games than there are surreal numbers, and most games are not surreal numbers.

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