Many would state that some theorems
are simpler than others. The problem that many run into is measuring the
simplicity of any given theorem. In chapter 42 of The Colossal Book of Mathematics it explains simplicity and how
important it really is in the context of math and logic. Through the works of
many mathematicians, including Albert Einstein, it became obvious that people
want to take the simplest approach to any desired result and that adding something
that is unnecessary creates unwanted complications. There are two main points
of conflict in this chapter: one, the measurement of simplicity and two, discovering
new information about a theorem making it less simple.
There
is no defined way to measure simplicity. After all, you cannot just take a
ruler and measure the difficulty of a given theorem. The chapter explains that
the best way to determine the simplicity of a theorem is to look at its
maturity. A very new theorem may be a lot more complicated because there are
potentially hundreds of different ways to “simplify” the theorem. Though it may
sound contradictory the more opportunities there are to simplify a theorem, the
more complicated it is. Though theorems start out this way, as they mature they
become more simple until a theorem cannot be simplified anymore. Basically what
chapter 42 says is that the newer a theorem is the more complicated it is and
after a theorem has been simplified repeatedly it takes a simpler form.
The
chapter also goes on to discuss data and how simplicity can overrule data. For
example, if there are plots of data on a graph it makes more sense to use a
linear line to represent the basic trend of the line than to connect all of the
points and create a random, non-linear line without a simple equation such as
x=2y. Nelson Goodman demonstrates that using the most basic, primitive concepts
is the best way to keep a theorem simple. For example, making up a new word to
describe something that is already generally known as something else makes a
theorem more complicated.
Towards
the end of the chapter the writer used proofs about shapes inscribed inside of
circles that were written by readers. I found it interesting that there could
be multiple proofs written about one concept and it showed how complicated a
theorem can be before it takes its simplest form. Nothing in this chapter took
me by surprise, but on the last couple of pages of the chapter there is a very
complicated looking formula that is meant to define inscribed lines in a circle.
All
in all, Chapter 42’s discussion of simplicity showed how important it is to
essentially “dumb things down” until it cannot be simplified anymore. After all
of the new information has been found for simplifying a problem it can become a
theorem that makes sense and is easy to understand without having unnecessary
junk thrown in with it. In conclusion, this chapter explained the true importance
of simplicity and how to read into it.
I found chapter 42 to be very interesting. As I was reading the title I wondered wha simplicity was and how there could be so much information on it to write about, but when you look at it from a different perspective it seems to make more sense in the context of math. Charlotte then goes on to discuss how a new theorem can actually come with many complications because of how many times it could be fixed to become more simple. Since new theorems tend to be more complicated, older theorems that have been simplified over and over again tend to be more simple because they have been stripped to the bare minimum. This is the closest way to measure the simplicity of a theorem because you cannot take a physical measurement as Charlotte explains. She also discussed how important it is to make sure a theorem does not have extra, unnecessary information along with it because it’s human nature to stay away from things that are not the most direct path. Lastly, she explains how simplicity can sometimes override data in statistics for example. In the blog she explained how linear trends make more sense than random lines, I think this is because that would not really demonstrate the general trend for data. Overall, she did a great job!
ReplyDeleteChapter 42 had an interesting take on how the author viewed the concept of simplicity. The author describes that measuring simplicity isn't easily obtainable and I would have to agree with him. The author, Martin Gardner, uses examples of Albert Einstein, Galileo and Newton who are some of the greatest minds to ever live. What these great minds view as "simple" could be considered extremely complex to not so famous mathematicians and scientists. This leads to simplicity being a subjective term because something may be simply to one person but extremely difficult to another. Gardner uses this example on page 555, where he states, "Moreover, a theory may seem simple to one scientist because he understands the mathematics and complicated to another scientist less familiar with the math". I think it would've been good to get your opinion on this phenomena as I believe it is a major part of explaining why simplicity is so hard to calculate.
ReplyDeleteYou also covered how a new theorem is most complex in its early stages of completion and gets simplified as it matures. Why do you think this occurs? Would you agree that as a theorem "matures" and more and more experts analyze and simplify it, the concept of the theorem becomes more understandable and thus is able to be simplified? Take for example a new type of game that comes out. At first you don't understand the rules or how it works, but as you play more and other people explain the game to you in various ways, you are now able to not only play the game but explain the game yourself with your own thoughts and ideas. I find this is how theorems become more "simplified" as they get older. As more experts analyze the theorem and give their opinions and make their simplifications, the theorem becomes easier to understand for everyone else. Then a new expert will look at the theorem with a better understanding and will be able to apply his/her thoughts and simplifications to it and the pattern repeats. That is why I believe these theorems get "simplified" as they mature.
I think you did a good job in summarizing the material from the book and giving examples for us as readers to look at. However, I think it would've been beneficial to give your own opinion more often on the matters you were discussing and explaining why you believed these opinions to be true.
I found this chapter to be rather interesting. I thought it was interesting to connect what we know now in a math or science class but actually see that the equations we know and use can actually be more complicated than we expected. The example of x=2y on page 556 is one of the simplest equations we have for a line. In figure 42.1, the two possibilities that the data could represent was where I thought the idea of simplicity really comes into play. The function in part (b) is clearly more complicated than the other but x=2y would make more sense since it is a prediction equation which is what you mentioned about simplicity overruling data. I thought that the explanation for combinational theory could have been better explained in the book but I also found it interesting how complicated it can be to define the lines in a circle rather than just the basic radius and diameter we all know. Like Molly said, it is easier to stay off the path that over complicates that we already know but it is interesting to see how simplicity can be a solution for defining which of say two complicated theorems or laws is the closest to being correct. Simplicity can make such complicated theorems easy for everyone, not just scientists or mathematicians, to understand. That’s where I think it is extremely beneficial.
ReplyDeleteI think the main concepts behind simplicity come from what we understand. For some people, history might be simpler than math, and vice versa. Albert Einstein put it really well when he said our experience helps up simplify things more. Topics seem to be easier to understand the more we hear, practice, or see them. In science, it seems as if we take a very complex theorem and try to make it simpler. And although one equation might be simple when we first look at it, there is an entire backbone of knowledge behind it. I find that part of this chapter extremely fascinating. Everything seems simple, but there are different ways to get to the same answer, meaning there is different complexities behind everything we do. The way mathematics and physics reach a conclusion is very similar in relations with simplicity. Both make simply paper doodles to help make models of geometrical figures. We have to know the basics of something before we can expected to know the complex stuff. This chapter was really interesting because it made me think about how complex simplicity really is.
ReplyDeleteIn chapter 42, I was shocked at how much they could write about the topic "simplicity". As I read through it became more personal to me and made a whole lot more sense. As a person I tend to ask a ton of follow up questions to understand an idea fully. I can't just take someones answer and be okay with it, I have to make sure it is true and makes sense to me. I also tend to over analyze stuff and that is what this chapter kind of touched base on in a way. All of the most known and used math theorems were at some point more complicated then they are now, and that is because mathematicians would over analyze and try to make it more simple and just a better theorem in overall. The thing I found most intriguing was just all of the Albert Einstein quotes. It showed how he was a successful mathematician in how he would want things as simple as possible and how he thought math ruled everything in the world and I think that's what made him such a great philosopher. Overall simplicity is very complex, everything around you now in the 21st century may be simple to you but back then it was very complex. That shows how working out, proving, disproving, and over analyzing anything helps make it more simple.
ReplyDeleteIn chapter 42, I was shocked at how much they could write about the topic "simplicity". As I read through it became more personal to me and made a whole lot more sense. As a person I tend to ask a ton of follow up questions to understand an idea fully. I can't just take someones answer and be okay with it, I have to make sure it is true and makes sense to me. I also tend to over analyze stuff and that is what this chapter kind of touched base on in a way. All of the most known and used math theorems were at some point more complicated then they are now, and that is because mathematicians would over analyze and try to make it more simple and just a better theorem in overall. The thing I found most intriguing was just all of the Albert Einstein quotes. It showed how he was a successful mathematician in how he would want things as simple as possible and how he thought math ruled everything in the world and I think that's what made him such a great philosopher. Overall simplicity is very complex, everything around you now in the 21st century may be simple to you but back then it was very complex. That shows how working out, proving, disproving, and over analyzing anything helps make it more simple.
ReplyDeleteI think one reason why everyone who commented on this post “found the chapter very interesting” is because we, as humans, are infinitely complex, therefore making the concept of simplicity a more difficult idea. Especially in our society where we are essentially brainwashed by the media to constantly consume and want more and more, true simplicity is usually never really attained (specifically in our contemporary, first world, developed country and ones of similar statuses). Its perplexing to me how we don’t currently have a way to accurately measure simplicity, let alone completely define it, yet there are projections (maybe more like hopes) to discover such a way in the nearing future. Furthermore, it seems simplicity has been studied and dissected for centuries, yet to no avail has any scientist or mathematician come to. I liked the image of Johnny Hart’s B.C. comic strips in which a caveman is trying to invent a wagon wheel, beginning with a square shape, then a triangle, arriving at this form because of the significant difference in simplicity between the number of corners in the square and triangle. It’s even more perplexing to me that simplicity has the ability to sometimes overrule the data of the most complex and intricate theories. Knowing this, in my opinion, really solidifies the truth in the saying that “less is more”.
ReplyDeleteChapter 42 is not one that caught my attention right away, but after reading your summary I decided it would be worth looking into. After reading this chapter I was actually dumbfounded on how easy to understand it was, based off the title I thought there was no way they would title a chapter Simplicity then actually make it easy to understand. In my personal opinion simplicity is one of the most important things to new mathematicians today. What I mean by this is, if we simplify known equations and formulas of the past it will make it easier for mathematicians and scientists today to understand mathematics of the past and progress the mathematics in the present. I really enjoyed the amount of quotes there were in this chapter one of my favorites was "we are apt to "fall into the error" of thinking that nature is fundamentally simple "because simplicity is the goal of our quest".". I enjoyed this quote in particular because it points out the error in peoples thinking. All in all you did a really solid job of explaining this chapter and keeping it simple. one thing I would have like to see is a bit more of an explanation on the equation at the end however I could not have explained that myself so who am I complain.
ReplyDeleteChapter 42 is not one that caught my attention right away, but after reading your summary I decided it would be worth looking into. After reading this chapter I was actually dumbfounded on how easy to understand it was, based off the title I thought there was no way they would title a chapter Simplicity then actually make it easy to understand. In my personal opinion simplicity is one of the most important things to new mathematicians today. What I mean by this is, if we simplify known equations and formulas of the past it will make it easier for mathematicians and scientists today to understand mathematics of the past and progress the mathematics in the present. I really enjoyed the amount of quotes there were in this chapter one of my favorites was "we are apt to "fall into the error" of thinking that nature is fundamentally simple "because simplicity is the goal of our quest".". I enjoyed this quote in particular because it points out the error in peoples thinking. All in all you did a really solid job of explaining this chapter and keeping it simple. one thing I would have like to see is a bit more of an explanation on the equation at the end however I could not have explained that myself so who am I complain.
ReplyDelete