Nature of Mathematics (Exemplar)

Nature of Mathematics

There are two questions that I aim to clarify, in light of my own understanding, through this post. The first question is, what is an axiom?  An axiom is essentially to math, what a universal truth is to life.  For instance, one of these universal truths, though not often thought about or commonly used, is that the truth is always the truth, even if no one believes it.  So what this is saying is that, in life, if something happened and is true, even to the knowledge of only a single person, that true thing is, in fact, what actually happened.  This holds true whether everyone or no one else thinks that this "true thing" is true.  It is something that can be applied to any situation and must be accepted as real or fact by the person to whom it is true.  This expands to all of life, in fact.

What this looks like in mathematics is an axiom.  So an axiom is a truth about math, that we hold to be true regardless of the situation.  In mathematical terms, this truth would in actuality be reminiscent of a proof.   Though a proof must be proven to be accepted as true, an axiom (a statement about an aspect of math) is accepted as true without having to prove it....what the heck math!? Anyhow, I'm not sure who, but somewhere along the line some mathematician, who definitely was not bizarre in at least one way, decided that these statements didn't need to be proven.  Who'd have thought?

So, what in the world do we use these axioms for? Why would we need something that is accepted as true, without having been proven? Axioms are used for proving other mathematical statements which could not otherwise be proven.  That is, for many mathematical statements you come to a point in the proof of said statement where you have to accept at least something as true.  If you didn't then you would have to continue breaking down the statement forever...but seriously it would get frustrating or impossible to prove some things if you didn't accept some truth as true.  That truth is an axiom.  They are essentially the most basic statement that you can use in a proof that doesn't have to be proven itself. An example would be that if we could not assume that all right angles were congruent, we could not logically apply the pythagorean theorem to any triangle.  This would be in light of the fact that, if not all right angles are 90 degrees, then they could be other degrees.  Smaller or larger. A change in angle measure means a change in segment length on a triangle, which would be impossible to calculate with the pythagorean theorem, because the pythagorean theorem only works if there is a right angle.

That seems confusing. But a simple way to think of an axiom is this.  "I would not be at Grand Valley right now if I didn't apply, I wouldn't have applied if I didn't know it existed, I wouldn't have known it existed if...if...if....if...if....................axiom."  An axiom is the last in the line of the "ifs"...if you will.

Exemplar Additive:

For the exemplar of this weekly post I started off sort of unsure what I should do.  It seems that I pretty thoroughly covered what an axiom is and why it would be important to mathematics.  I even explained it in non-mathematical terms.  So as I began to think more about what I should do with this idea of axiom, I found myself starting to think of where axioms even came from.  As I researched this idea further, I found, as it turns out, that axioms came from ancient Greek findings in mathematics.  They basically decided that to deduce anything about math, there had to be a starting point.  These starting points came from logic.  If something was entirely logical, then there would be no way to dispute it.  For instance, if we look at the associative property of addition we see that it doesn't matter if we have 3 rocks in one and 4 in the other, in total there are 7 rocks.  It doesn't make a difference if you count the 3 or the 4 first, it still adds up to 7.  This is the kind of reasoning behind any axiom.  

After spending some time with this idea, I began to think about the concept of creating a new axiom.  Could it be done? If it could be done, would there be any point to it?  I began searching around for any literature on the subject and the best that I came up with was this 

http://math.stanford.edu/~feferman/papers/ASL2000R.pdf

There is a lot of filler talk in this article, but it does offer some good viewpoints on the matter that helped me to better form my opinion. I would recommend this read to anyone who has interest in thinking more deeply about the need or the logistics of creating new axioms.  The biggest question that I came into, as noted in the article as well, which axioms would need to be created and for which purposes?  Definitely food for thought.  

The Role of The House of Wisdom (Exemplar)

The Role of The House of Wisdom

When examining the role of The House of Wisdom in mathematics I have found it to be very important to consider what the role of this place was in relation to anything.  First, The House of Wisdom was founded by a man by the name of Caliph Harun al-Rashid.  Now this is a guy who was pretty darn powerful in his day.  He was intellectually, politically, and militarily resourceful, which is the exact way that many modern liberals would describe the late "Dubblya Bush." Further his surname translates to "the just," "the upright" or "the rightly-guided," which just further demonstrates the correlation made above.  To get more to the point, and the value of the actual situation, this was a man who valued intellectuality and the discoveries that came along with it.  He founded The House of Wisdom to be a place where intellectuals of many different disciplines could come together to research, explore, and translate ideas in the areas of science, mathematics, astronomy, medicine, alchemy and chemistry, zoology, geography and cartography.  A good way to look at this place would be to think of it as a room where all of the best and brightest could come together to share their wisdom, learn from one another, and put their ideas into text of many languages, sort of like a coffee shop where people discuss the novels they are writing, but actually doing it. 

Exemplar Additive:

In order to better understand the importance of The House of Wisdom, it proves valuable to look at the history of it.  Though there is an incredible history behind and involving the house of wisdom, the point of this blog is mostly to give a brief history of it, not the full story (which can be found online and in several different books).  So what was The House of Wisdom exactly?  It was a library and a translation institute established somewhere later in the 8th century, which would continue to grow and receive information and translation for many, many years after its founding.  In terms of translation, what essentially was taking place, was the translation of pertinent texts from many different languages including Farsi, Aramaic, Syriac, Hebrew, Greek, Latin, Sankrit, and Devnagari into Arabic.  These translated texts were then circulated and distributed throughout the empire to enrich and further the knowledge from all over the world in Arabic speaking countries.  In light of this, contrary to the belief of many, The House Of Wisdom was not a place where people went and became wise by entering but rather, much like the libraries of today, housed information which could be accessed by different people. Further, The House of Wisdom was located in the city of Baghdad.  Was this just by chance? Certainly not. At that time, Baghdad was the center of the Islamic Empire and actually housed one of the world's first paper mills.  It makes sense then why a library was able to flourish the way it did.  When books and manuscripts can be made right there, why would you put the library elsewhere? Just food for thought. This fact really gave me a foundation for understanding how all of this knowledge was displayed and distributed, it didn't just happen by chance.  Lastly, The House of Wisdom was in Baghdad because of the power held by the residents.  This was not so much wealth and power that came from oil like we know today, but was due to the incredible intellectual presence located in the region.  So just like it made sense to make a library in the vicinity of a paper mill, it made sense to place an intellectual power house, among intellectual powerhouses, if you will.  

So where is The House of Wisdom today?  Unfortunately it was destroyed in 1258 when a Mongol invasion left the entire region almost completely decimated.  Books were destroyed, intellectuals were killed, and with them a wealth of knowledge passed away. The good news is that many of the texts were recovered in later years and of course, the spread of knowledge cannot necessarily be thwarted by an invasion.  It had already spread to many other places in the empire and eventually around the world. 
Anyhow, how does this apply to the current state of mathematics?  Given that the best and brightest were all gathered in this place, ideas could truly be explored and either proved or put to rest.  This was a place where mathematicians could collaborate with other mathematicians and intellectuals to discuss problem points with whatever it was that they were exploring.  I think that some of the greatest strides in mathematics came from collaboration and still do.  How many people did it take to prove or disprove any point of discrete mathematics?  Five line geometry? Any of Euclid's theorems?  It takes often takes many people to work through a given math problem and The House of Wisdom was a place where people could do that.  Not only did the work through the mathematics to come up with something that was valid and trustworthy, they translated those works into other languages so that the ideas could spread.  Without this type of translation, many modern math strategies and formulas and theorems would not have been used, or maybe not discovered in certain cultures until years later.  Maybe never.  The sharing of ideas is incredibly important to the spreading of knowledge.  The House of Wisdom is the hub of knowledge spread-ation, if you will.  Like a raptor chases down its prey, displays every aspect of predation and devours that prey, so the spread-ation of mathematics hunts down the minds of skeptics and devours them into parabola shaped smile of discovery.

Journey Through Genius (exemplar)

So I've decided to take your idea of making this quote into a poster and run with it a little bit.  I think it would be cool to sort of get this going in the math department of Grand Valley.  Not just to show a guy hulking out at a gym (as pretty darn cool/hilarious that it is), but to sort of let students playfully display their favorite type of mathematics.  Maybe the professors could even put their favorite type of math or proof or equation on their doors?! That might be a little too much, but I really liked thinking through this idea a little more.  It took me a little while to find a picture that I thought would work well for this quote and then it took me even more of a little bit to figure out how to make this without photoshop installed on my computer.  I really like creating and drawing and art in general, whether it's designing a simple poster, drawing an info-graphic about Euler, or doing something a bit more serious, I wanted to make my interests more a part of my efforts in this class. So here you have it! The hulking man and the quote! The guy even has a nose bleed... golly.  Anyhow here's an example of my favorite math with the quote!

Multiplication In Other Countries (Exemplar)

Multiplication In Other Countries

One thing that I have found to be extraordinarily interesting is the topic of multiplication.  I got to thinking about what it really is and why we do it the way we do. It seems like everyone that I have every met has learned to do it the same way.  There are, of course differences in terminology and sometime the order that things are done, but for the most part it's the same.  You take your numbers and stack them up like a cake and then slowly work your way across, devouring each place value until you you get the very end (or beginning).  What you find yourself with then is an empty plate...or a number, depending on what we are still talking about.  Anyhow, I got to wondering if we all do multiplication that way because that's the only way to do it...that didn't seem logical, so there must be another way.  Then it came to me! A calculator...that's how most people do multiplication these days. I include myself in that blanket statement.  It honestly wasn't until my teacher assisting in a second grade class this past semester that I really had to retrain my brain to do multiplication like I learned at that age.

I couldn't possibly be building up this whole "how to do multiplication" thing, just to say that people use calculators, surely not.  So what then? Well it turns out that there is another way, a pretty cool way, in fact, that doesn't involve writing numbers at all.  At least in the way that we are used to.  From my understanding, this multiplication has it's origins in China.  So how do you do it?

Well basically it's like writing tallies.  Lets say you wanted to multiply 14 and 12, you would start by writing one tally and then four tallies, representing 14.  Then you would do the same thing for the 12 except crossing the lines horizontally over the 14 like this:

You then count up the intersections like the illustration below.  The red indicates the hundreds place, the blue added up represents the tens place and the green represents the ones place.  Of course you can do this same process with larger numbers, but then there is "carrying" involved.  But! I thought this was a really cool, visual way to teach students how to do multiplication differently.

Exemplar Additive:

One thought to consider when looking at these different methods of multiplication is, how might this look in a practical, educational setting?  Would there be any benefit to teaching this type of multiplication over other types?  Well to begin explaining what this might look like in a classroom, it would be helpful to know and look at the different types of learning that students fall into, or the umbrella of learning style under which they take refuge.  The explanation of these is a research paper in and of itself, so I am going to just say that this type of multiplication would in fact be a major benefit to any students who fall under the visual or bodily kinesthetic learning types.  Students who learn better from moving with learning, or seeing with learning would be able to step into a realm of understanding that they would be unable to in the traditional teaching of multiplication (memorizing facts, place value line ups, etc). Though younger students may not benefit from an explanation of why the line crossings work, they would benefit from being able to see what it is that they are counting up and how to place the numbers they had counted.  It works well because students can literally see a number of objects, rather than a set of numbers to be multiplied.  This approach also takes pressure off of students as they come into the stigmas associated with learning and doing multiplication.  It's not always numbers, it can be things or objects as well.  Even with larger numbers which can become more complicated, a teacher could provide students with pipe cleaners to use as their lines, which they can then physically move and count out.  Pretty slick for a student who is intimidated by numbers.

In terms of students who learn in a kinesthetic fashion, the pipe cleaners or other objects (maybe even large, movable objects in a gymnasium or on a playground) would allow them the movement that they require for valuable and meaningful learning.  Again, with younger students, the explanation of why these lines work may be above them, but the activities that are made possible with this form of multiplication would certainly not be. I would recommend this type of multiplication education for any students who fit into either of these learning styles.  This is actually a great way for teachers to diversify their teaching, even with students who learn more traditionally.  Seeing information presented in different ways can help to solidify a concept for any student. 

Skimpression

For my "skimpression” I went through Euler: The Master of Us All.  I found this book to be much like the other book that I read for this class Journey Through Genius.  I say this due to the fact that this book focuses not just on the mathematics itself, but focuses also on the history and culture surrounding the situation.  Obviously this book is focused just on Euler, mentioning other pertinent mathematicians as necessary, while my first book addressed many different mathematicians and theorems.  I really liked that this book outlined the culture, history, and context of Euler because it helps to better explain the revolutionary nature of his mathematical discoveries.   On the other hand of the equation, pun definitely intended, this book does a really great job of explaining the results that Euler came up with along with the theories and culture and history and context.  The book doesn’t just skim through the results either.  I really value that this book goes to great depths to actually explain the reasoning behind the consequences of each part of the proofs and results.  For instance, there is an explanation of a proof that involves eggs and omelets.  There is nothing that helps me to understand concepts better than analogies and this book gives those exact type of descriptions.   Overall this book does a great job of explaining, so to speak, the man, the place, the history, the results of Euler. 

As far as my recommendations for this book, I would recommend it to anyone who has an interest in mathematics.  A person who loves not only the pure math itself, but someone who values the history behind a matter.  That is, someone who wants to know not just what a car does, but who made it and how they did it, given the economical climate of the time.  This person would want to know what type of gravel caused someone to fall on their bike.  Was it crunchy? Was it big enough to roll? Also, on a more serious/not so serious note.  I think that this book would be great for anyone who learns from analogies, or comparing new information to known information in a way that is plain and simple.  This book does a great job of explaining things in a way, and outlining how Euler thought, in a way that is incredibly understandable for this type of person.  

The one critique I would give for this book is not so much a critique of the way it was written or of the content, but rather is a recommendation for readers.  This book is not necessarily something that, at least I, would sit and just read right through.  It is a book that, while offering a lot of quality information, also requires a lot of close attention and studying, if you will.  So my one beef with this book is that there is too much good information for one sitting...which sounds pretty ridiculous, actually, but shapes the way one reads the book. 

In conclusion, I would say that this is a great, informative read that displays the information in a coherent, understandable, and exciting way.  Definitely would like to see this book in more classrooms! 

Modeling Data

I recently decided to dive into what modeling data looks like, or more specifically, what can it look like?  To do this, I experimented for a while with my graphing calculator to see what I could draw using some simple equations.  I figured that since the equations that I was using were very simple, they would give me a good idea of the, sort of, bottom line of what is possible on the high end of data modeling.  So these are just a few examples of what I came up with.

y = 10sin(cos(tan(x)))

y = 10sin(cos(4x))

y = 90sin(550x)

 y = 10cos(sin(4x))

After experimenting with these "bottom line" examples of data modeling.  I began to think about the top end.  If images like these can be made from simple formulas, what can be made from more complicated formulas?  The most complicated formulas?  I actually got to thinking that there must be some way to model literally anything based on formulas.  I got to thinking of what this might look like and thought, CGI!!! So I did a little digging into this and found something really interesting.

 http://www.nytimes.com/2010/12/30/movies/30animate.html?pagewanted=all&_r=0

This link takes you to a page that outlines, in brief, how the creators of the movie "Tangled" actually programmed and used formulas to get the hair to do what it was supposed to.  They talk about how this type of "data modeling" is absolutely necessary to get the realistic effect that many CGI and animated movies display.  This also fits in with the fractals that we talked about that are essential to creating and modeling the realism that is inherent in the world around us.  So from crude pictures of flowers to intricate modeling of a princess' hair, modeling data is the basis.  Pretty cool. 

Role of Our Number System

Today I was discussing the nature of number systems with a former seminary professor from Erskine Seminary.  At first it seems strange that I would be discussing such a "mathematical" idea with a seminary professor, however, the idea is not as strange as it seems.  What one will find through a discussion like this, that is, a discussion about number systems with a non-math person, is that number systems stretch far beyond mathematics.  This is exactly what I discovered through my conversation with this former professor.  The two main ideas that came out of this conversation were that: There is reasoning behind each number system, and, The uniqueness of each number system.

First, our conversation started with the origins of the current number system that we use here in the United States.  Though it is up for some dispute, it is pretty generally accepted that our number system spawned from ancient Arabic numbering systems.  Other than origins, is there a meaning behind the number system and, more importantly, what is the meaning?  What came up in our conversation is that there is, in fact, a meaning behind our current number system.  Although the numbers that most of us write daily do not display the following characteristics, they find their purpose in the traditional style of these numbers.  There is a meaning! What is it?  So we see, even in this common typeface that the number one, 1, is not just a line, but has a small tic or downslope at the top.  It turns out that this is not simply the product of a sloppy writing, greasy teenager, but rather, is the remnant of the intentional meaning.  Traditionally, the number 1 is written more like this ^, minus the superscript. We find actually that many countries still write the number 1 this way (Germany and other European countries). The traditional 2 is written more like Z, the traditional 3 is written more like a sideways W.  Anyhow, what we notice about these numbers is vertices! Yes! As it was originally written, for the most part, the number of vertices or changes in direction signified which number it was. a 1 has one vertex, a 2 has
2 vertices, etc.  This is something that I, and I assume, many other people, even mathematicians, have not thought about.  What does this have to do with the role of a number system?

The answer to the previous question came, for me, in the furthering of our discussion about these unique properties in each number system.  This will also serve to outline the role of our current number system.  This discussion was not as in depth as you might think, in fact, it was pretty basic.  Sometimes the most prolific understandings of an idea or concept.  The main topic of our conversation revealed itself in Roman numerals.  The professor outlined that in his current career as a superintendent, he works with young students teaching Greek from time to time and, in relation to that conversation, talked about Roman numerals with the students.  On the one dollar bill, he explained, 1776 is printed on the bill in Roman numerals.  He simply explained that the number is nearly 9 characters long.  The students, naturally found this very confusing. We then talked about how incredibly difficult it would be to do any type of meaningful mathematics using that number system.  It really brings into perspective, the incredible mathematical achievements by ancient mathematicians using number systems that were, basically horrible. What does this mean for the role of our number system? It means that it makes mathematics more accessible, meaningful, and more quickly progressing than ever before.  Without our current, concise, and easily understood numbering system, mathematics would be slowwwww.  It would be like doing internet research with dial up.  Or eating rice with a single chopstick.  Our current number system effectively eliminates the single chop stick rice eating of math.  

Journey Through Genius

The book that I read for our "book club," if you will, is Journey Through Genius by William Dunham.  First of all, I would like to say that this is a great book.  It is absolutely filled with information about not only some of the greatest proofs of all time, but also about the mathematicians themselves, the culture in which they lived, and the era in which they studied.  That being said, I also want to add that this is not a book that you read for leisure in the evenings to relax.  I would say that it is almost more of a book to read in spurts, not all at once.  I say that because, though the information is extremely interesting, it would take an incredible amount of time to read through the entire book and actually have made sense of it all.  I really loved the fact that the contents laid out the specific sections dedicated to particular proofs, theorems, and mathematicians.  This allows the reader to skip around to the proofs that might be the most interesting to them at a particular time and truly dive in.  That's the thing about this book.  It's not about a joyful afternoon swim, but a rapids of facts and explanations that seem crazy to "dive into," but once you come out at the end of any of these rapids, you find yourself with a smile on your face.  This is the beauty of mathematics at its finest. As described by the author, mathematics is not simply about practical application, memorization, or finite use.  Rather, mathematics is like a Rembrandt a Picasso, a masterpiece.  Something that is not only practical but visually and logically pleasing.  Logic, in fact, is noted in this book as being one of the only prevailing pieces of history.  Science changes and is replaced.  Medical practices from centuries ago are naive attempts at best of something great, but the logic behind a tried, tested, and true proof of a theorem prevails over the centuries.

I would recommend this book to anyone who has an interest not just in the application of a theorem or how the proof is derived, but how these things are affected by the culture of the region, era of time, and specific personalities of the mathematicians who discovered them.  If you are someone who is looking for a to the point, concise, and application geared explanation of some of these proofs, I would say this is not the book you are looking for.  This book is not so much about learning a theorem or proof as it is discovering the theorems and proofs. So one might ask, "What is the difference between learning and discovering?"  Well the answer to that is quite simple.  There is a certain appreciation and fulfillment that comes from discovering something, while learning offers merely the satisfaction of a good grade, a praising note, or a gold star.  The gratification in learning spawns from the reward, whereas the gratification in discovery spawns not only from the reward, but also the journey, hence the title of the book, Journey Through Genius.  It is not titled, Genius Explained, How To Be a Genius, or Guess What This Theorem is Used For! And that is for a very good reason.

At the beginning of this blog I said that this book is not a leisurely read, but I have come to the conclusion that leisure, much like beauty, is in the eye of the beholder.  For many mathematicians, like those mentioned in this book (Euclid, Archimedes, Euler, etc) these things may be one in the same.

The Role of The House of Wisdom

When examining the role of The House of Wisdom in mathematics I have found it to be very important to consider what the role of this place was in relation to anything.  First, The House of Wisdom was founded by a man by the name of Caliph Harun al-Rashid.  Now this is a guy who was pretty darn powerful in his day.  He was intellectually, politically, and militarily resourceful, which is the exact way that many modern liberals would describe the late "Dubblya Bush." Further his surname translates to "the just," "the upright" or "the rightly-guided," which just further demonstrates the correlation made above.  To get more to the point, and the value of the actual situation, this was a man who valued intellectuality and the discoveries that came along with it.  He founded The House of Wisdom to be a place where intellectuals of many different disciplines could come together to research, explore, and translate ideas in the areas of science, mathematics, astronomy, medicine, alchemy and chemistry, zoology, geography and cartography.  A good way to look at this place would be to think of it as a room where all of the best and brightest could come together to share their wisdom, learn from one another, and put their ideas into text of many languages, sort of like a coffee shop where people discuss the novels they are writing, but actually doing it.

Anyhow, how does this apply to the current state of mathematics?  Given that the best and brightest were all gathered in this place, ideas could truly be explored and either proved or put to rest.  This was a place where mathematicians could collaborate with other mathematicians and intellectuals to discuss problem points with whatever it was that they were exploring.  I think that some of the greatest strides in mathematics came from collaboration and still do.  How many people did it take to prove or disprove any point of discrete mathematics?  Five line geometry? Any of Euclid's theorems?  It takes often takes many people to work through a given math problem and The House of Wisdom was a place where people could do that.  Not only did the work through the mathematics to come up with something that was valid and trustworthy, they translated those works into other languages so that the ideas could spread.  Without this type of translation, many modern math strategies and formulas and theorems would not have been used, or maybe not discovered in certain cultures until years later.  Maybe never.  The sharing of ideas is incredibly important to the spreading of knowledge.  The House of Wisdom is the hub of knowledge spread-ation, if you will.  Like a raptor chases down its prey, displays every aspect of predation and devours that prey, so the spread-ation of mathematics hunts down the minds of skeptics and devours them into parabola shaped smile of discovery.

Multiplication In Other Countries

One thing that I have found to be extraordinarily interesting is the topic of multiplication.  I got to thinking about what it really is and why we do it the way we do. It seems like everyone that I have every met has learned to do it the same way.  There are, of course differences in terminology and sometime the order that things are done, but for the most part it's the same.  You take your numbers and stack them up like a cake and then slowly work your way across, devouring each place value until you you get the very end (or beginning).  What you find yourself with then is an empty plate...or a number, depending on what we are still talking about.  Anyhow, I got to wondering if we all do multiplication that way because that's the only way to do it...that didn't seem logical, so there must be another way.  Then it came to me! A calculator...that's how most people do multiplication these days. I include myself in that blanket statement.  It honestly wasn't until my teacher assisting in a second grade class this past semester that I really had to retrain my brain to do multiplication like I learned at that age.

I couldn't possibly be building up this whole "how to do multiplication" thing, just to say that people use calculators, surely not.  So what then? Well it turns out that there is another way, a pretty cool way, in fact, that doesn't involve writing numbers at all.  At least in the way that we are used to.  From my understanding, this multiplication has it's origins in China.  So how do you do it?

Well basically it's like writing tallies.  Lets say you wanted to multiply 14 and 12, you would start by writing one tally and then four tallies, representing 14.  Then you would do the same thing for the 12 except crossing the lines horizontally over the 14 like this:

You then count up the intersections like the illustration below.  The red indicates the hundreds place, the blue added up represents the tens place and the green represents the ones place.  Of course you can do this same process with larger numbers, but then there is "carrying" involved.  But! I thought this was a really cool, visual way to teach students how to do multiplication differently.