Nature of Mathematics: Infinity

The other day in class when we had the discussion of what zero divided by zero is: zero, one, infinity... etc. At some point during this discussion, I believe professor Golden briefly mentioned something about how there are some mathematicians who don't think infinity should be included in math. For some reason or another this really intrigued me, so I decided to research more on my own about infinity and the problems along with it. While investigating, I came across an article from Quanta magazine, which I wanted to share with you through this blog. Just a disclaimer, I'm definitely not an expert on this subject and some of still confuses me, however I'd like to share my thoughts and knowledge before and after reading this article and the subsequent research I did on the topics presented by it. I'll start by briefly summarizing the article and then move on toward my views on the debate.

Today in mathematics, the "ZFC" or Zermelo-Fraenkel set theory with the axiom of choice is the standard form of axiomatic set theory and the most common foundation in mathematics (Princeton.edu). From my understanding its basically the most accepted concept of fundamental laws of mathematics and set theory. However, there are some holes in it, namely infinity. One of these problems that has brought up much debate is the "continuum hypothesis" (Wolchover).

The continuum hypothesis states: "There is no set whose cardinality is strictly between that of the integers and the real numbers." (Wikipedia). The problem is that with the "ZFC" there is no way to prove if this hypothesis is true or false. In 1940, Kurt Godel showed that the continuum hypothesis could not be disproved, whereas in 1960, Paul Cohen showed that it couldn't be proven either, both assuming the ZFC axioms to be true (Wikipedia). This is where to two differing theories presented come in to play. First there is the forcing axioms which say the continuum hypothesis is false and then there is the inner model axiom, "V=ultimate L," which says the continuum hypothesis is true (Wolchover).

The V=ultimate L model, from my understanding, says that you must construct some set L that is equal to V, the universe of sets. In this constructed L, you must start with the empty set and use it to build bigger and bigger sets until encompassing all of them. In this manner, the resulting universe of sets would prove the continuum hypothesis to be true (Wolchover). Hugh Woodin, a mathematician at the university of California, Berkeley, found that such an L could, in theory, be constructed. However, such an L has not actually been made yet (Wolchover).

On the other hand, the forcing axiom looks not to create a new model that encompasses everything, but rather expand and simply fill some of the holes of the existing model. Stevo Tordorcevic, one of the leading activists for this method, likens it to the idea of complex numbers (Wolchover), which I thought made it really easy to understand. Just like how complex numbers fill gaps in our understanding of the number system by adding on to it, the forcing axioms work in the same way.

Before reading this article, I honestly had no idea that this problem with infinity existed, or any problem with it for that matter. That's why I thought it was so interesting when it was mentioned in class. After doing some research, I found out that are actually quite a few problems with infinity. For instance, infinity has two distinguishable types, actual infinity and potential infinity. The difference between the two is also quite intriguing but for fear of digressing I'll leave it at that.

So on to my thoughts about the article. After learning about both sides of the argument, I think I'm more inclined to agree with the forcing axioms. The sole reason for this is because the forcing axioms have concrete results that deal with the problems we have now. Natalie Wolchover, the author of this article, puts it in to words much better than I could by saying, "while V=ultimate L is busy building a castle of unimaginable infinities, forcing axioms fill some problematic potholes in everyday mathematics." I've always enjoyed the more applied part of mathematics so perhaps that is why the forcing axioms appeal more to me. However, the idea that one day a set could be found that encompasses and fixes everything is also fairly appealing. As of right now, its only a matter of opinions on which one you like more until anything can be further proven, so I'd like to ask which side you would chose ?

Doing Math: Archimedes' Stomachion

When I was younger, I had an obsession with puzzles. I would spend hours at a time putting puzzles together or playing with tangrams online. So when there was an option to try Archimedes' stomachion as part of work for this class I was stoked. I was fairly confident this wouldn't be too difficult, but soon found out how wrong this assumption was. If you don't know what the stomachion is, its basically a rectangle that has been cut into 14 distinct pieces and the goal is to put it back together.

I spent hours working on this but ultimately was never able to complete it. To start out, I just randomly started placing some of the bigger tiles into the corners and trying to put the smaller ones around them. Obviously, this strategy didn't really work too well. Here was my best attempt just randomly placing the tiles anywhere.

For a while, I thought the top left corner was right but I wasn't making any progress so decided to try a different approach. Instead of trying to fill in the rectangle, I started to focus on the individual pieces. The long light blue one in the bottom right corner was a real pain to find a decent spot for, so I literally spent about 20 minutes trying to find out the other piece that went along the top of it. As you can see the darker blue triangle worked pretty well.  I then spent the next hour or so doing the same thing for each of the other larger more annoying pieces like the big purple piece. Then I started to focus on filling out the sides of the entire rectangle. This was about the time my second big progress came when I filled out half of the rectangle.

At this point I was 100% positive I had gotten that half the rectangle correctly tiled and thought I would be done with the puzzle in no time. Not long after I came as close as finishing it as I possibly could without actually doing so.

Words cannot describe how disappointed I was when I found out that the last little piece didn't fit. At this point I finally caved and decided to look up the solution. As it turns out, Bill Cutler found out there are 17,152 different solution and 536 truly distinct ones. After finding out there are so many I'm a little upset I couldn't find one of them. However, it amazes me that Archimedes, or anyone, could have created something that seems so simple on the outside, but when actually analyzed is so much more complicated.

What is Math?

When asked what math is, my inner mathematician wants to come up with some rigid and precise definition for it.  However, thinking about the question a tad more carefully, I realize that perhaps there isn't one.  I could say math is a way of explaining what happens around us in a logical and numerical way.  While sure this is a fine definition of math, there is also so much more to math than just numbers and logic.  New discoveries in mathematics are occurring all the time to describe anything and everything about the world, and with these the definition of math is growing as well.  So for me, the best way I could define math is by likening it to an infinite series, how mathy of me.  Just like with the next term in the series, each new discovery broadens the scope of mathematics and as a result the definition becomes that much different than before.

Perhaps you could see from my definition of math that I am a big fan of models and examples so, I think math milestones would somewhat resemble how we learn math as kids.  First you start recognizing numbers and amounts, like hey four apples is more than 2.  At this point I still didn't fully grasp concepts such as addition or subtraction but that did come next.  After that, it was most likely multiplication and then division.  Meanwhile, also at about this time, or maybe a little before, I'd say kids would start learning about simple geometry like shapes at this time.  Geometry is sort of weird for me as I'd say its easier to learn than algebra, the beginnings of it at least, however its very possible geometry didn't become a thing until after multiplication was discovered.  Finally you go into algebra and calculus and everything else.