I have a friend that is a math genius. He always blew us away in any math class in high school and went on to become a chemical engineer in college. We used to get into a lot of philosophical conversations about the universe, 4th dimensions and other abstract ideas like the
String Theory. He had a very scientific mind and liked to be able to understand things from a logical viewpoint. During college he would show me fascinating things that he could do with math, such as proving that 1+1=3. The deeper he got into math the more he was able to explain with it. While I cant claim to understand everything, it fascinates me how math really dictates so many things.
I recently started reading about
Kurt Gödel and his two incompleteness theorems. The idea is that no consistent system of axioms can fully prove every system. This means that although we can do countless things with math, there is actually proof that it cant do everything for us.
I started doing a little research because I wondered how it was that he could prove that. While it makes sense, his theorems proved this point mathematically. I used the BYU library to look up some articles on Gödel and found a couple that I thought would help me understand this idea. The first one is called
On Gödel's Second Incompleteness Theorem by Thomas Jech. It comes from the peer reviewed journal Proceedings of the American Mathematical Society, May, 1994
. It was a short explanation of the mathematical proof that Gödel used to show this theorem. It was interesting to see this put in mathematical terms, and it amazes me that he could come up with this. The second article I found was called An elementary exposition of godel's incompleteness theorem. This gave me more information on the principles and ideas behind the theorems.
I think math is fascinating. I think its even more fascinating to use math to prove that we cant use math for everything. What kinds of parallels does this have in our society? As we continue to develop computers, are we going to run into barriers that wont be possible to cross? My professor put up this chapter about what computers cant and will not ever be able to do. Its a fascinating point, but will someone figure out a way to make those things work? It'll be fun to see.
yeah, unfortunately, this weekend, I found out a computer harddrive cannot solve the problem of its being stepped on. :) No, I get your point. Fascinating how there really are unsolvable problems. I think we see so many problems computers can solve that we think they can do anything and get a bit spoiled by that when they can't.
ReplyDeleteI'm thinking about a story I heard by a woman who watched as her husband taught her 9 yo son to toil away in the garden with them every day. One day the garden was destroyed (some weather event I can't remember). That problem could not be solved. They just had to start over. She said her son grew in character from watching how her husband calmly dealt with the experience and learning how to deal with the loss of so much work himself. It's good they can't solve everything. I think it's a good moral lesson that some problems can't be solved.
p.s. I LOVE the fishy thing at the bottom of your blog. Did you make it?
ReplyDeleteHaha yeah the fish can be kind of distracting. It was just a blogger add-on. About your comment, they actually have made solid-state hard-drives now that can be stepped on :) Who can say what problems are unsolvable huh? Who knows, some of those problems we talked about in class may one day be solved.
ReplyDeleteI agree-unsolvable means I got tired, don't have enough information and need to come back later or have someone else look at it. How about Charles Babbage and his Engine. He did the creative conceptual work for his computing machine, but the technology of his time in the 1800's was limited. In 2008, someone else followed his creative blue print and created his computing machine. Unsolvable at one time because of limited resources/technology and solvable at another time. Thanks for the post!
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