Mathematics – Invented or Discovered

During my time as a student in my MTH495 Capstone class, one of the major themes has been the title of this blog post.  Is mathematics something that exists, and we are just discovering it?  Or is it a purely man made invention, something that doesn’t exist?  This question really pulls at the fibers of our whole study through this program at GVSU.  What is the nature of math?  We have had many in-depth discussions in class and I have spent a good amount of time pondering this on my drive home everyday.  One could say that if math is purely an invention, then certainly it fits extraordinarily well into our real world!  Example;  we have put rockets on the moon using mathematics.  Therefore math must be something that exists and we certainly are discovering it.  As put by Einstein,

““How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

While this is a strong argument, I will contend, in my opinion, that math is a man made invention, and that it doesn’t technically “exist”.  To the point the argument that I mentioned previously, I would agree, it does seem to fit perfectly in the real world, but that is not enough to convince me that there are “mathematical laws to the universe”.  However, I would say that there are laws of physics that exist, and are not invented.  If you throw a rock up in the air, it falls back to the earth, right?  Gravity exists, it is not a man made invention.  Then, setting up a few equations using some of the discovered laws of physics, we can use calculus to determine many things about the motion of the rock.  However, the more we study the laws of motion the more we see how our models are not 100% accurate.  We discover things such as air resistance and spin, that affect projectile motion.  As good as we are at figuring out models that predict reality, we will never be able to be perfect.  We have simply invented this system of mathematics in an attempt to understand things.

Sources:
https://www.huffingtonpost.com/derek-abbott/is-mathematics-invented-o_b_3895622.html

Cantor & Infinity – Blog Post 3

Infinity, an idea used so frequently in mathematics, yet still so mysterious and abstract. It is so full of mystery that as I sit here thinking about what to write about, the more I realize I am unsure of what it really is!  As a more-amateur mathematician i’m sure I used the word “number” to describe it, but after being scolded more times than I can remember, I know better. Now, as a less-amateur mathematician I find myself asking questions such as “is there a mathematical definition of infinity?”, and “are there different sizes of infinity?”  The second question is one that was discussed in my math Capstone class and is a question that I find very interesting.  You have two infinite sets, is it possible that one is larger than the other?  It turns out that the accepted answer is yes, you can have different sizes of infinite sets.  Ergo there are different sizes of infinity!  Nuts.  Lets take a look at two infinite sets.  A = {1,2,3,4,5,6,…} B = {2,4,6,8,10,12,…}.  The question asked was then, which set is bigger.  The following conversation ensued:

Oliver: I think “A” is bigger
Grant: Why?
Oliver: Because for every element in “B” I can give you two elements in “A”, so “A” is                 twice as big as “B”.
Grant: Well for every element in “A” I can give you an element in “B”, so I think                         they’re the same size.

It turns out Grant was right.  The idea of the size or Cardinality of sets of infinity has to do with the existence of a bijective function between the two sets.  As there exists a bijective function f: A->B with f(x)=2x, A & B are said to be equal in size.  When you think about it in this regard, it makes sense.  Further, it turns out that Cardinality can be broken up into three different groups.  Finite, Countably Infinite, and Uncountably Infinite. They can be defined by this:
(1) Any set of numbers with a cardinality less than that of the natural numbers is Finite.
(2) Any set of numbers with the same cardinality as the natural numbers is said to be Countably Infinite.
(3) Any set of numbers with a cardinality greater than the natural numbers is said to be Uncountably Infinite.
It seems that the we can define a cardinal number by defining it as a certain number set.  So how does one go about proving that a set is uncountable? Georg Cantor came up with a way aptly called Cantor’s Diagonal Argument that was introduced in a proof published by Cantor in 1891.  In this proof he proved that there are infinite sets that cannot be put into one-to-one (bijective) correspondence with the natural numbers. In other words, there are infinite sets larger than the set of natural numbers.  Cantors set went like this.  He first thought about the set T of all the infinite sequences of binary digits.  He then creates an example of an enumeration of the elements s of T.

s1 = {0,0,0,0,0,0,0,….}
s2 = {1,0,1,1,0,1,0,….}
s3 = {0,0,1,1,0,0,0,….}
s4 = {0,0,0,1,1,0,1,….}
s5 = {0,1,0,0,0,1,0,….}.

Then he creates an additional s = {1,1,0,0,1,…} in which the nth digit of s is the complement of the nth digit of sn.  In other words, each digit is the opposite the digit of the diagonal shown bold below:

s1 = {0,0,0,0,0,0,0,….}
s2 = {1,0,1,1,0,1,0,….}
s3 = {0,0,1,1,0,0,0,….}
s4 = {0,0,0,1,1,0,1,….}
s5 = {0,1,0,0,0,1,0,….}.

This guarantees that s is different than every enumeration in sn.  Thus we can say that there will always exist an s which is an element of T that is different than any sn in the enumeration.  The proof then goes by contradiction.   Assume T is countable (the same size as the natural numbers). Then T can be written as an enumeration.  However we created an s in T that is not in the enumeration, thus T is not countable. So it’s true, the set of all the infinite sequences of binary numbers is larger than the set of natural numbers.

In closing, it seems bizarre to me that there are different sizes of infinity considering that infinity is not even a number, but, I guess that’s how math goes, for every question answered, there seem to be infinitely many more that take its place.

Oliver Crosby

Sources:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
https://en.wikipedia.org/wiki/Cardinality

Blog Post 2

For my book I read Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, by Keith Devlin.  The author recounts his journey to Europe and Italy and along the way tells of his discoveries about Fibonacci.  The story paints a portrait of a brilliant mathematician whose work not only drastically advanced Mathematics, but drastically advanced civilization.  In this post I will talk briefly about the biographical details of Fibonacci’s life, but focus more on his work and some of the more fascinating tidbits I got from text.  Fibonacci was born in Pisa, Italy around 1170 as Leonardo Pisano (Leonardo of Pisa).  Fibonacci is actually a nickname and the contraction of “filius Bonacci”, meaning “Son of Bonacci”.  His father, Bonacci, was merchant, in the trade hub of the western world.  Fibonacci then spent several years of his youth in Northern Africa.  This is where he learned the invention that would make him renown.  After this Fibonacci went back to Italy where he remained for his life.  His death is unknown, but it is speculated to have been in the 1240s in Pisa.

The invention that Fibonacci learned in Africa, was a new way to write and calculate numbers.  Being the son of a merchant, Fibonacci applied the hindu-arabic system to bookkeeping, showing the practical applications of the new system.  He then wrote the book Liber Abaci, a detailed textbook on how to it works, as well as many story problems showing the various applications.  This book was hand copied over and over and was used to teach basically all of Europe the system.  Before this, merchants were using roman numerals and finger counting methods to keep track of their business.  Using the numbers 0-9 and place counting as well as the use of an abacus led to growth of accounting and trading in Europe.  This whole topic was very interesting to me as it shows how mathematics drastically changed the world.  It really emphasizes to me that at it’s core, mathematics is a tool that we use to advance and grow.  Mathematics is such an abstract concept yet it’s applications are so real, it just takes a mind like Fibonacci to realize these applications and the potential it has to change the world.  Fibonacci also must have had an incredible work ethic and a very strong feeling about this topic as he put his entire life’s work into it.  I will end this blogpost with an example of a problem and solution in Liber Abaci given by the author.

A certain man buys 30 birds which are partridges, pigeons, and sparrows, for 30 denari. A partridge he buys for 3 denari, a pigeon for 2 denari, and 2 sparrows for 1 denaro, namely 1 sparrow for 1/2 denaro.  It is sought how many birds he buys of each kind.

You can start by assigning x,y, and z to the number of partridges, pigeons, and sparrows, and you can come up with the equations:

x+y+z=30     (the total number of birds is 30)

3x+2y+1/2z=30 (the price of the birds is 30).

The author notes that two equations with three variables, seems unsolvable at first, but then hints that x,y, and z must be non-zero positive whole numbers. By doubling the second equation you get:

x+y+z=30

6x+4y+z=60.

Subtracting the first equation from the second gives us

5x+3y=30.

The trick now is that since 5 divides the first and the third term, it also must divide y. Therefore, y must be 5, 10, 15, … etc.  But as x is positive and non zero, y cannot be 10 or larger, thus y is 5.  It follows that x=3 and z=22. Tricky!

The hard step for me to understand was deducing that y must be a multiple of 5.  Thinking about it though, it makes sense, the first term has to be one of 5, 10, 15, 20, … and the last term must also be 5, 10, 15, 20, 25, 30, …  So it makes sense that you must add in multiples of 5 to the first term to get to the third term.  I believe this is an example of modular arithmetic!

 

 

Blog Post 1.

One of the most interesting things we have covered so far is the bit about squaring numbers and what it actually means to square something.  The method of solving quadratics in which you literally “complete the square” physically was very enlightening to me.  It made the process of what I have been doing for years make much more sense.    For those who haven’t used this method, the method works by equating a quadratic to a few rectangles whose sides are the coefficients and the variables. To solve the quadratic you physically complete the square.  However since its an equation, for everything you do to the square, you must do to the quadratic.

Another interesting topic was learning about the “2 to 1 machine” or the “3 to 1” machine and how that eventually led to the base number system.  It is really odd to think of numbers in any other base than 10, but once you realize the possibilities it makes me curious as to why we landed on ten as standard?  One theory that I found is that it is because we have ten fingers, which makes sense.

It is quite impressive to learn that these methods were invented thousands of years ago and now they are learned but every child in school.  I’d be interested to see what mathematics looks like in another 2,000 years.  For all I know, it could be in a completely different number system, completely unrecognizable!

 

-Oliver Crosby

Zeroeth Blog Post- What is Math?

What is math? Math is the study of numbers and anything related to numbers.  So what is the importance of studying numbers, and why do we do it?  I think their are many answers to this question, but mostly, I think that math is a tool that helps us solve problems, improve our way of life, and ensure our survival.  If you think in terms of Darwin and Evolution, math is a tool that helped us survive, and that is why it still exists today.   Math has further been used with science and physics to help explain things about the universe and attempt answer deeper questions like, where did we come from, what is our origin?  These are my deep thoughts on the nature of math.

The Five Greatest Math Discoveries

1. Numbers – The discovery that you can actually quantify things is the genesis of mathematics, so I think that it should be counted as the most important discovery in math.
2. Zero – The epiphany that it is quantifiable to have nothing of something is pretty crazy if you think about it.  The debate on whether or not zero is a natural number is always a fun one to have. I believe zero deserves a place on the five greatest math discoveries.
3. Negative Numbers – The concept of having  less than zero of something is also a pretty wild concept.  I’m sure whoever discovered them was laughed at.  They don’t even exist in reality.  You can’t have negative three apples.  What are these mysterious things?
4. Rational Numbers –  The discovery that you can have a part of something and not the whole thing is certainly one of the greatest discoveries in math.
5. Irrational numbers – The fact there are numbers that can’t even be represented by only using ratio’s must have opened the doors to a whole new world to mathematicians.