I've gotten a lot of e-mail about this particular pigcast so I'm posting here to answer some of your questions.

When the pig was saying things like 1 = 2 or 1 < 2 he was not making mathematical assertions or statements. He was using shorthand for tasking or task methodology. Go back to your game theory and remember how any given task that can be completed by a single person responds as more people are added.

For instance a certain task might involve moving 400 boxes from point A to point B where one person can carry only one box at a time. This task responds linearly as more people are added, ie. adding a second person means the speed-of-completion happens twice as fast, or it would take one person twice as long to complete than two. Adding a third person will increase the speed-of-completion another unit, and so on. These increases are quasi-linear (if we neglect fractional units) up to 400 people. Once you have 400 people, you can complete this task in 1 unit time and adding more, no matter how many more, will not decrease that time. So we can summarize the task properties of this task by saying:

linear

2 > 1 (two people finish the task quicker than one)

1 + 1 = 2 (adding a second person increases speed-of-completion twice as fast)

induction: N <= 400, N = 1 + (N - 1)

induction: N > 400, N + 1 = N

When he was referring to a "2 = 1" task he means a task where adding people does not increase the speed-of-completion or mean it will get completed sooner. For instance, the above example with only a single box. Or, getting a message from point A to point B by running only (if you assume everyone tires and runs at the same speed). The intention isn't to model probability here. It's to begin to model things in a parallel fashion rather than a serial one, which is useful in things like computing once we are beginning to move toward a multi-processor platform. A "2 = 1" task is one which must be completed serially or 'in order', or where one step cannot be started until the previous step is finished.

singular

2 = 1

induction: any N, N + 1 = N

You also have synergistic (and negative synergistic) tasks. Anti-synergistic tasks are ones where two people given the same task take longer or are less efficient than a single person. For instance say there is a single seller on e-bay who has an item up for auction that you want. You bid on that item a few times and continue to outbid anyone else who bids on it. This is an anti-synergistic task. The more people who want and bid on the item, the harder it is for any given bidder to get it. Now only a total idiot in this situation would call up his friend and tell him to bid on the same item in an attempt to get it and then send it to him once the auction is completed. This would only make it even harder to get the item. These tasks most often have to deal with things like market-driven limited supply actions.

anti-synergistic

2 < 1

1 + 1 < 1

induction: any N, 1 + N < 1, N + 1 < 1

Logarhythmic tasks are similar to linear ones, only the increase decreases with each addition of a person. For instance, if one person completes the task at a rate of 1x, two people might complete it at 1.5x, 3 people at 1.75x, 4 people at 1.875x, 5 people at 2x, and so on.

logarhythmic

2 > 1, 3 > 2, 4 > 3, but decreasing

induction: still any N, N + 1 > N, but (N + 2) - (N + 1) < (N + 1) - (N)

Synergistic tasks are ones where the rate increases exponentially depending on how many people involved.

synergistic

1 + 1 > (2) 1

1 + 1 > 2

induction: any N, (N + 2) - (N + 1) > (N + 1) - (N)