In this episode:

- Zeno’s beach bungalow and why he can’t come out to play
- Do train tracks
*actually*touch at infinity? How do YOU know? Have you been there? - Watch out for the Hordes!!
- Welcome to the Hilbert Hotel … you can check in any time you like, blah, blah, blah …
- What do you do with an infinite number of mathematicians? Run!
- Explaining jokes at the Unit Interval, a great neighborhood bar.
- Is one type of infinity BIGGER than another kind?
- Counting ALL the Real Numbers, “… 2, ah ah … 3, ah ah … pi, ah ah!”
- The Power Set: the enemy of the X-Men.
- Fractals, Hippies, and gettin’ trippy
- Is Nick a quantum human?

It’s taken a while to get this episode up. Our original plan of having a weekly or bi-weekly schedule was more than ambitious than we’d expected.

But, never fear, we haven’t stopped – only slowed down to make room for those pesky things in life like work, school, and family!

]]>Oh, and we pre-record a lot of these, so even if we are planning on using your idea, it may take a while to incorporate it. All part of the process. Thanks for your patience!

OK, in this episode:

- How math can turn you into a fortune teller.
- What is a “precise falafel”?
- Is an infinite number of Nick’s a good or a bad thing? I say good!
- $7 bucks is a lot of money. Just sayin’
- Expected value of using your Quantum Superpowers to play the lottery.
- The primate brain’s pattern recognition is both kick-ass, and dumb as hell.
- Nick: “Get out while you can, monkey!”
- The reason Las Vegas is not a Not for Profit city.
- Chess or Poker, that is the question.
- Natufian tribes, genes, and humpin’.

Please feel free to comment in the comments section about what you think worked and what you think didn’t. Your opinions will help to make this an even better show!

In This Episode:

- All about Combinatorics – the Mathematicians fancy way of “counting”.
- The magical musical and production styling’s of Keith Schreiner make Tom and Nick seem “almost” respectable.
- Tom invents the natural numbers in under 3 minutes.
- 5 primates are standing in a line … can you come up with a punch line?
- Is the factorial [5! = 5x4x3x2x1] just a way for mathematicians to justify yelling?
- How many ways can you cage 2 primates out of 5? And is this illegal?
- The importance of watching Stargate SG1.
- Why is your genome smarter than your computer?
- If you went far enough on a space ship, would your toothpaste taste like rye bread?
- Apparently Nick ‘Horton’ has an alter-eg0, Nick ‘Lion’. It does sound more manly.

This program was produced by Keith Schreiner who’s other work you can find here.

We’ve got a new episode recorded, that I’ll have edited and up on the site by tomorrow at the latest.

To tide you over I’ve compiled a list of Game Theory books that we love. we’ve been getting a surprisingly large number of emails asking if Tom and I can recommend any great books on the subject. Well, hell yes we can!

I’m gonna put them into a couple of categories.

OK, I’m going to be totally honest here. I don’t usually like popular game theory books. They come off as silly (often). I know how that must sound coming from me of all people, but hey, that’s just the way it is.

However I DO like a lot of books that are implicitly game theoretic. That is, they use game theoretic thinking without being all that clear about the fact that they are using the types of reasoning that is inherit to the subject.

Here we go:

The Armchair Economist by Steven Landsburg is just funny. In fact, Landsburg is himself a game theorist and a rather accomplished one at that. But, this book is all about everyday life and how economic reasoning (read: game theoretic reasoning) can help make sense of some of the big (well, small) questions we all have about day to day existence.

Landsburg went at it again with this book on more of the same subjects as his last.

My favorite is this one:

by Herbert Gintis. It’s mostly just a huge collection of problems. The only math you need is high school algebra for many of the problems, but having some calculus and linear algebra wouldn’t hurt.

by Philip Straffin. This book is very introductory and has very little “math”. It’s small and easy to read. It’s the first game theory book I read, and it still holds up as a undergrad level intro.

Anatol Rapoport has made massive contributions to game theory in Political Science. This book is a small intro book with a bit more math that Straffin’s book. I found it helpful in my first term.

Maynard-Smith changed the face of evolutionary research by introducing the field to game theory. This is book that did it. It also serves as a great intro to people who aren’t as mathy.

Below are a couple books that you might be into. They aren’t exactly intro books, but they aren’t “upper level” book either. Instead they are speculative, or scientific, and approach the subject with a sense of possibility.

This book is about a totally new subject in the social sciences called “behavioral game theory” (which we’ll eventually do a podcast on). It is not math, it is science. Instead of looking at the theory of games and the consequences of that theory, they aim to test the theory. They put people in rooms and actually have them play games and see what happens. How closely to people stick to the strategies that game theory would predict them to use? Very interesting!

This book is also by Herbert Gintis, and is a work of philosophy as much as it is a work of game theory. He thesis is that game theory is to the social and behavioral sciences what calculus is to the physical sciences. That is, this is his grand unifying theory of the behavioral sciences. And I totally agree with him. Read it, and be converted.

]]>- Tom and Nick are not as happy with Pi Day as you’d expect.
- Why do Hippies like Pi so much?
- Where does pi come from, and why do we care?
- Is mounting a Ferris Wheel on a Flat-Bed Truck a good idea?
- How many digits of pi can YOU recite? I’ll bet not 69,000!
- Nick and Tom give you back 23 hours and 40 minutes of your life … ish.

Continued fraction recipe for pie at Math World

Recipe for Boysenberry Pie:

]]>If you haven’t read it already, Tom was interviewed at Technoccult.com with Klint Finley about mathematics, punk rock, and unicorn anatomy.

In it Tom proves that he clearly is an Orangutan.

]]>- What’s wrong with the “medal count” ordering in the Olympics?
- Is it really all that surprising that Canada is good at Curling?
- quasi ordering vs partial ordering vs total ordering – who cares?
- Nick, leave the carrots out of this!
- Dealing with our Daddy issues with an ancestor semi-lattice
- Are you your own ancestor?
- For that matter, is a sandwich equal to 5 dollars?
- Discovering the quasi-order of joy-points

- Geometry on Acid: The category of Topological spaces and continuous maps.
- Tom has a silly putty reason for everything.
- The continuity of film.
- Did Xeno predict calculus?
- Ms. PacMan also loves donuts.

Here’s a video of a coffee cup continuously deforming into a donut. Mmm … donuts.

]]>The main underlying concept is that of a partial order. I’m going to use < as the symbol for an arbitrary partial order in what follows.

Say we’ve got a set, and we’d like to talk about an ordering on that set. “Xavier is behind Yolanda in line,” for example. We could write that as

x < y

Now, if Xavier is behind Yolanda in line, and Yolanda is behind Zelda in line, we’ve got

x < y and y < z

and, you know from your many experiences standing in line, that x < z must also be true.

In general, a partial order is any ordering which satisfies 3 properties:

1) The ordering is reflexive.

Reflexive means that x < x. (Think of a reflection – x looking back at itself.) Our use of < doesn’t allow that, since Xavier is not behind himself in line. But we could modify that; let’s re-define x < y to mean “x is not further ahead of y in line”. It’s verbally clunky, but now we can use any theorems we can find about general partial orders, and it is true that Xavier is not further ahead of Xavier in line, so our re-defined partial order is reflexive.

2) Transitive

A transitive ordering means that the following is true for any x, y, z in the set we’re working with:

If x < y, and y < z, then x < z.

This is true for <, so we’re good.

That’s it! You’re already familiar with one partial ordering, which we might call “less than or equal to” and designate with the symbol <=. It is true that, say,

3 <= 3

because while 3 is not less than 3, it IS equal to 3. Transitivity is pretty obvious: 3 <= 5, 5 <= 7, and of course 3 <= 7.

And the relation is anti symmetric. That is, IF x<y and y<x then x = y.

If an order satisfies the first two properties, we call it a quasi-order. If it also satisfies the last one, then it is a partial order.

Given two elements x and y of a set X that has a partial order <, the least upper bound (lub) of x and y is the “smallest” element of X that is “at least as big” as both x and y.

For our people in line example, the lub is really easy. Suppose Xavier < Yolanda. Recall that by our redefinition, Yolanda < Yolanda. So

lub(Xavier, Yolanda) = Yolanda

because Xavier is no further ahead in line than Yolanda, and Yolanda is no further ahead in line than Yolanda.

In the standing-in-line partial order, it’s really easy to compute the lub, because it’s just the “biggest” of the given pair.

Our line happens to be not only a quasi order, but a partial order. A total order has all of the above properties, but also has the following one. Every element is comparable. That is, either x<y or y<x. So, total orders are often called linear orders for obvious reasons.

Every total order is a partial order and ever partial order is a quisi order, but not every partial order is a total order, etc.

What started me on this post was suddenly thinking of two orderings that I think will give us two interesting and primate-y partial orders (with some tweaking).

Given two animals x and y, let the least upper bound of x and y be their most recent common female ancestor. So, the lub of you and your sibling is your mom. We can generalize this to species without screwing things up too much. The least upper bound of you and a gorilla is a female animal living 7 million years ago.

I’ve phrased this in terms of least upper bounds without really defining the ordering, but it seems obvious that there’s a partial order in here somewhere.

Let x < y denote “I am willing to trade you an x for a y.” I’m happy to trade you an x for an x, because it’s the same object. And surely, if you would trade x for y, and you would trade y for z, then you should trade x for z. (If you wouldn’t, you are in danger of being screwed by the money pump.) So the trade ordering is reflexive and transitive, therefore it is a partial ordering.

However, it is NOT a partial ordering! To see this, consider

$5 < sandwich

Remember, this means “I am willing to trade you $5 for the sandwich.” Perhaps I’m not that hungry right now, and so it currently is true that

sandwich < $5

because I am willing to make you a sandwich if you give me $5. However, it is NOT true that a sandwich IS $5.

Numbers are totally ordered. If x <= y and y <= x, then x = y. But as per the above example, goods and services are NOT totally ordered. So what I want to know is: What weird effects occur from projecting the non-total partial ordering of goods to the total partial ordering of currencies?

[ETA: I am totally wrong! A partial order is antisymmetric; it's preorderings (or quasiorderings) which are not antisymmetric. In partial orders, any two elements may simply not be related -- it is possible to have neither a < b nor b < a be true.

We'll hash this out on the podcast next Friday!]

]]>- Cooperative vs. Non-cooperative games
- OR Combinatorial vs. Procedural Games
- Does Curious George belong on the show “Shark Tank”?
- The most hardcore competitive market ever devised
- How rapacious can Tom get?
- Is Nick totally worthless?