#math rambling

OMG zero is amazing and important guys you don’t even know.

It’s one of those things a lot of people don’t even realize had to be invented–they think, oh, it’s obvious, but it’s not as obvious as all that. Early mathematical and counting systems didn’t have zero–after all, I don’t need to count how many bags of grain I have if I have none. The idea that we need to have a symbol for nothing in order to aid in counting somethings is actually pretty counterintuitive.

You can do very basic arithmetic without zero (although it’s harder–try multiplying with Roman Numerals), but to do anything beyond that you need place values. 

Place values are essentially the idea that a digit has a different meaning depending on where it is in the number. So the 2 in 208 means something different than the 2 in 28 (200 and 20, respectively). This may seem obvious, but it isn’t true in a system like Roman Numerals. V always means 5, no matter where it is in the number. 

In base ten our place values are multiples of ten (hence the name). From right to left, 10^0, 10^1, 10^2, 10^3, etc (or going left to right from the decimal point, 10^(-1), 10^(-2), 10^(-3), etc.). We then use our ten different digits to indicate how many 1s we have, how many 10s, how many 1,000s etc. This makes it possible to have a finite number of symbols and still be able to indicate any number possible. We can the line everything up all nice and neat and add or subtract.

But to do that we need to have a symbol that means “I have no 10s (or 1s, or 10,000s), move on” and thus we arrive at zero.

This system is elegant, compact, and allows all of modern math to exist. All of it. 

Zooming out a bit, the existence of zero enables us to make precise calculations about more abstract ideas, and also leads us to ask a lot of interesting questions, which leads to more and more useful systems.

For example, someone asks: “What happens if I subtract 1 from 0?” And so we invent negative numbers. That allows us to mathematically represent owing someone something, and then we ask how negative numbers interact with positive ones, and we get debits and credits and the math of balancing your checkbook. You’re welcome, accountants.

You also can’t have computers without zero and place values, because the entire concept of electronic computing is based on the fact that it is possible to represent base two with two states: on and off. One and zero. Thus the entire basis of computer language.

Our information systems and cognitive structures are prone to error so as we develop logic further and further we find inconsistencies and we need to find ways to work with them while maintaining coherence. a good idea if you don't want to rebuild formal logic from scratch. this is weak paraconsistency, classical logic holds and errors have to be handled but are, in the end, errors.

And then there's Strong Paraconsistency...

like just take some of the general ideas of strong paraconsistency:

Some contradictions may not be errors

Classical Logic is wrong in principle

Inconsistent theories can be true

There's a lot more nuance and detail here but that's the gist of it, you take logic and add some degrees of freedom.

I kinda see two arguments here for it,

A) you're interested in paraconsistent logic for its own sake, like how you can study non-euclidean geometric systems and not care about their truth, just playing with the rules and seeing what happens. this is lovely and I am 100% on board. we love making toy models and seeing what they do

B) Strong Paraconsistency matches the ground truth and that you can make inconsistent systems that are true and accurate representations of the world, which would then be inconsistent but coherent. using the analogy from before this would be more like studying non-euclidean geometry because we live in a non-euclidean (technically pseudo-euclidean) spacetime and it has real and visible effects on the physics we observe.

I can fully get behind A, building a system from alien rules and seeing how they interact is fascinating and a worthwhile past time. B I find to be a much harder sell, but it's also hard to comprehend foreign logic systems from inside one that is more restrictive. I look forward to digging into it more as I finish the book.

and for what it's worth the book only asks you to accept weak paraconsistency, but it doesn't dissuade you from considering the implications of strong either

y'know, this guy?

everyone loves when he gets in the corner. but is he garunteed to? Is there a way to know whether or not, given his slope, he will? I think I've cracked this question.

you see, consider the DVD logo a point. instead of it bouncing, let's just make it continue in a straight line, through the wall of a square grid. Each time it hits a "wall" , it /would/ have bounced. For instance, this shows us a nice little zig-zag before hitting a corner

This can be considered equivulant because every box just flips the angle. To turn the line back into the zig zag, we just flip the box along the edge it hit every time. Like folding paper. This line would look like this:

And so, this leads to a new question: What slope may never reach a whole-number coordinate? any m for mx which is rational will, since, with 'a' being any whole number(You could always make the ratio like this), in the statement (1/a)x when x=a, the whole thing is whole (since 1/a * a = 1) and x is whole since 'a' is and they're equal.

But, as numberphile explains in this video, this is not the case for pi , for a very similar reason as to why it's always the case for rational numbers.

Because pi is irrational, so are rational multiples of pi. If x is rational in mx and m is not, the whole thing will be irrational, and so if y=mx then y will be irrational. If x is a rational multiple of pi, making the whole thing rational, then x is irrational being a rational multiple of pi. If you want to make the DVD never bounce, make it's starting slope 1/pi, or some other irrational number.

By way of introduction: there are two sorts of people in the world with respect to Monopoly. Some people hate Monopoly when they are in the process of losing, and some people hate Monopoly all the time. I am in the first category.

If you are also in the first category - ie. you like winning Monopoly and are willing to apply basic arithmetic and logic in your quest to win it, there are a few things you've probably realized. In case you haven't, here's a quick primer.

anderson was a good professor too. he ran the boston marathon and came back and had class the next day. well, that and the class was fun as heck. one of the first serious proofs classes. excellent.