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keep going for bonus #freetopiary artwork….

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War Machine:

War on Ourselves:



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Have you guys heard of bitcoin yet? Here’s the original paper: Bitcoin – A Peer to Peer Electronic Cash System. Or immerse yourself in bitcoin culture at the bitcoin wiki and the bitcoin forum. Bitcoins are currently trading around $9 $18 (6/6) at the largest bitcoin market, mt. gox (featuring dark pools and soon offering margin trading and options). You can get hyperanalytical comparing various international markets at bitcoin charts. Or watch live bitcoin transactions with a block explorer and bitcoin monitor. Right now the bitcoin economy is pretty small (eg. stores, classifieds) but you can still gamble at the bitcoin poker room or bitcoin vegas poker. Or buy drugs! For now… You can donate to your favorite hackers. It’s all up to you! Here’s some hyperbolic commentary:

If you want to get started, download the bitcoin software to obtain a wallet and mine some bitcoins. Then, send me some at: 14gFPynCLAGmhhmZcPaaXPf6vBt7NKaY56

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IMHO mathematics consists mathematical objects and their properties. We define these objects into existence and then use logic to prove theorems codifying their behavior. However, unless you major in mathematics in college, none of this is taught to us in school. Instead we’re taught to memorize theorems and use them for various computations. This was a useful skill before the invention of computers, but nowadays it’s mostly busywork. I believe that we should scrap the current high school curriculum and replace it with courses on logic and theoretical math. This would prepare students for a more sophisticated approach to the fields of computer sciencephysics, etc. I’ll be writing some basic proofs on this blog to give you a feel for what I think is important….

So, now that we developed an idea of sets, we should start classifying them. We’ll start by constructing the natural numbers.

= {0, 1, 2, 3, … }

At this point we only have sets to work with, so we define the empty set as our zero.

0 = ∅ = {}

We take the power set of the empty set and define this set as one.

1 = P(∅) = {∅}

We continue taking power sets to construct the natural numbers

2 = P(P(∅)) = {{∅},∅}

3 = P(P(P(∅))) = {{{∅},∅},{∅},∅}

4 = P(P(P(P(∅))))  …

This gives us a class of sets with a successor operation P which together satisfy the Peano axioms, the essential algebraic properties of the natural numbers. Ie. With this construction mathematicians are able to derive all the known properties of the natural numbers (eg. distributivity).

Even though I’ve ignored some technical details in this construction, this still seems like a lot of work to make the most basic numbers. However, this is an important construction in the foundations of mathematics. Sets are the raw material of math and the natural numbers have significantly more structure. For example, the naturals are well-ordered with unique predecessors. The naturals also admit the binary operations of addition and multiplication. We can’t talk about this structure in a vacuum and that’s why we build these numbers using power sets of the empty set.

We’ll continue from here to talk about the integers and the rational numbers…

Prev: Math I: Sets

Next: Math III: Functions

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