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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Surreal Arithmetic (Edited rerun)

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In my last post on the surreals, I introduced how the surreal numbers are constructed. It’s really fascinating to look back on it – to see the structure of numbers from 0 to infinity and beyond, and realize that ultimately, that it’s all built from nothing but the empty set!

Today, we’re going to move on, and start looking at arithmetic with the surreal numbers. In this post, I’m going to go through the basic definition of addition, subtraction, and multiplication of surreal numbers. Division will have to wait for a later post; division is quite a subtle operation in the surreals.

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Introducing the Surreal Numbers (Edited rerun)

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Late last summer, shortly after moving to ScienceBlogs, I wrote a couple of posts about Surreal numbers. I’ve always meant to write more about them. but never got around to it. But Conway’s book actually makes pretty decent train reading, so I’ve been reading it during my new commute. So it’s a good time to take a break from some of the other things I’ve been writing about, and take a better look at the surreal numbers. I’ll start with an edited repost of the original articles, and then move into some new stuff about them.

So what are surreal numbers?

Surreal numbers are a beautiful set-based construction that allows you to create and represent all real numbers in a simple elegant form that has the necessary properties to make them behave properly. In addition, the surreal number system allows you to create infinitely large and infinitely small values, and have them behave and interact in a consistent way with the real numbers in their surreal representation. And finally, it makes the infinitely large numbers as natural a part of the number system as any other number: there’s nothing about the construction of an infinitely large number that makes its construction any different from a perfectly reasonable real number like 1/3.

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More Stupid Tautology Arguments from Dr. Egnor

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So over at the DI whiners blog, Egnor is, once again, trying to pretend that he’s actually making a case for why evolution is irrelevant to antibiotic resistance. It’s really getting silly; he repeats the same nonsense over and over again, desparately doing the rhetorical version of sticking his fingers in his ears and shouting “La La La! I can’t hear you!”:

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Casey Luskin, Proud Idiot

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So, over at the DI’s media complaints department (aka evolutionnews.org), it appears that Casey Luskin has noticed how we SBers have managed to tear apart his buddy Dr. Egnor. Given that we did it so thoroughly, though, there’s no legitimate way to defend him. He’s repeatedly made incredibly idiotic statements, and many people have, quite rightfully, called him on the stupidity of his statements, the degree of ignorance that he’s demonstrated, and his astonishing arrogance as he spouts nonsense.

But since when have Casey and friends at the DI ever worried about doing the right thing? Or responding to any kind of argument in a legitimate way?

So for humour’s sake, I thought I’d address the part of Casey’s post that was directed at me.:

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