Partner Wanted

I'm looking for a partner for my big idea. Read about it here.

I'm interested in a 50-50 partnership, and ideally the division of labor would be as follows:

• Me:
• Programming
• Partner:
• Administrative work (taxes and other paperwork)
• Shared
• Teaching
• Doing art with kids
• Constructing the arcade cabinet with kids
• Making sales pitches to schools
• Everything else

My ideal business partner would have education experience and a passion for innovation. No programming experience is necessary, but comfort with computers goes a long way. Being detail oriented is also a plus.

I would like to launch this business in New England, so if you live in the area and are interested, please email me at fegleynick (at) gmail (dot) com.

My Big Idea Pt.2

The Game

I substitute teach at a K-8 school, and for the last few months, I have been making a video game with the help of the art teacher and all her students. You can play it here. It's a work in progress, and at the time of my writing this, there are about 12 completed levels.

The kids come up with level themes and do some level design, and then the art teacher sends me their work and some instructions. I do some a lot of programming, and in the end each class gets their own level.

A sprite sheet for one of the levels

The kids then have the opportunity to send me feedback through the menu on the left of the game screen. And they do. A lot. Feedback is varied, and I include some of it here because it makes me smile:

The whole game was really cool, but when I got to the end, where the blue castle/tower is, I had no idea what to do!!!

maybe you should make the boats a little faster

This is really hard but really cool. I like the effects. Something you could work on is putting more features in.

i really likes the game! i just wish that when u get to the end of the level there is somethnh like that say congradulations or somethings

These are just some of the emails that students have sent me through the feedback form.  It's exciting to see students invested in a project.

The Machine

When I'm not working on the game, I'm working on my arcade machine:

My scratch-built, fully function prototype

This is just a prototype, so it's pretty ugly, but it's fully functional. The monitor is just an old (barely functional) CRT TV because it looks a whole lot better than an HD monitor.

The version in the picture is running off of a Raspberry Pi, but I recently switched to a Beelink Pocket. There's also an Arduino Micro in there that converts button and joystick presses into keyboard input.

The board I made to connect the Arduino with the computer.

 The Big Idea

What I would like to do is build arcade games with students professionally. Everything I've done to date has been for fun, but I don't have the time and financial stability to continue much longer without monetizing in some way.

I'm envisioning an artists in residence program that pushes into a school for a week. During that time, students would break into groups and each group would be responsible for some part of the game development:

• Story and Level Design
• In-game Art
• Digital Image Editing
• Cabinet Construction
• Wiring and Soldering
• Cabinet Art
• Sound Design

No one student will learn the whole process, but together the students will construct an arcade cabinet that they could play in the cafeteria/library/art room for years to come.

This is an ambitious project, but I believe that it has the opportunity to really change how students view themselves and the games they play.

Also, I'm looking for a partner.

Read where this idea came from in part one of this post.

My Big Idea Pt. 1

Mortal Kombat II

About four years ago I was working as a bartender in a small restaurant/resort. The owner had a few dated arcade games that I'd sometimes play after work. One day he came to me and offered to give me a broken Mortal Kombat II game, and I loaded it on a trailer, and transported it to my parents' garage.

MK II in my parents' garage.

Upon getting it home, the first thing I did was open it up and look inside.

It was amazing.

Before opening the machine, I had always assumed arcade games ran on magic. The act of opening it was one of the most empowering things I've done. Seeing how the wires connected, and realizing that I could (with some time and money) make my own, made me feel powerful. Games weren't magic, they were technology, and I could control that. I felt as though I had been let in on some cosmic secret, and I knew in that moment I would stop at nothing give others the same experience.

A couple of months later, I didn't have enough money to make some repairs on my car, and I reluctantly sold the MK II machine. I almost entirely forgot about the whole thing.

Time Runs Out

Three years later. I was working as a substitute teacher (I still am) and I ended up in a fourth grade classroom. I saw a student drawing a picture, and I asked him if he could draw me a picture. The next week he gave me this masterpiece:

Time Runs Out

I knew that this picture wanted to be a video game more than anything, and that weekend I went home and made it one. You can play it here. It was one of my first attempts at writing a game in JavaScript, and looking back there's a lot I could do to improve it, but I still think it came out ok.

The next week I showed the student the game. He was ecstatic. I watched him play a game that he made, and saw a look on his face that reminded me of the time I opened the MK II game. For him, video games were just this little pocket of magic the existed on the internet or on an X-Box, but in that moment they became something he could do. He was empowered. It was awesome.

And so began my big idea.

My Big Idea Pt. 2. In which I explain my big idea and the progress I have made.

Comics in the Math Classroom

Here are some resources for people that went to my talk on Comics and Graphic Novels in the Math Classroom at this year's NEMATYC conference.



In the presentation, I made reference to the Kuleshov Effect


As well as this really great video about how we learn:



And here is a list of comics that I recommend:

Math

The Cartoon Guide to Calculus
The Cartoon Guide to Statistics
The Manga Guide to Calculus
The Manga Guide to Linear Algebra
The Mystery of the Prime Numbers

Science and Humanities

Wonderful Life with the Elements
The Manga Guide to Physics
The Cartoon Guide to Chemistry
Economix (Highly Recommended)
Action Philosophers

Webcomics

XKCD
Saturday Morning Breakfast Cereal
Abstruse Goose
Indexed
Spiked Math
Phd Comics

Biographies

Logicomix (Highly Recommended)
Fallout
Feynman
Suspended in Language

Comics Theory and History

Understanding Comics (Very Highly Recommended)
Making Comics
The Comic Book History of Comics

Ambiguity in Language: Purple People Eater

Well he came down to earth and he lit in a tree,
I said "Mr. Purple People Eater, don't eat me!"
I heard him say in a voice so gruff:
"I wouldn't eat you cuz you're so tough."

This is one of the first posts I wrote when I started this blog, but it never really felt finished it. So it sat un-posted and forgotten until recent events inspired me to finally post it. It still feels unfinished.


Purple People Eater is a song is about a "One eyed, one horned, flying purple people eater" that came to Earth to start a rock 'n roll band, and it goes a little something like this:

Most people imagine some sort of cycloptic, horned, purple monster that flies and eats people. This is a legitimate interpretation, but not the only one. Not even the correct one. Listen again to the verse that starts at 0:50:

I said Mr. Purple People Eater, what's your line?
He said, "Eating purple people, and it sure is fine."

According to these lyrics, the purple people eater eats purple people. This indicates that you or I should be safe (though this guy is pretty much screwed), and that Mr. Purple People Eater will most likely be pretty hungry on a planet without purple people (like ours). This is not how most people interpret it, including the creators of the 1988 Purple People Eater Movie:

The ambiguity stems from the fact that English is not associative and there is no convention for order of operations. Should the expression be evaluated from the right: purple (people eater), or from the left: (purple people) eater. [footnote 1] If we include the entire description, there are actually 5 possibilities:

Once you know to look for it, ambiguity is everywhere: Is a "big, bad dog catcher" a catcher of big, bad dogs, or a dog catcher that's big and bad? Is Dr. Seuss's book about ham and green eggs, or green eggs and green ham? Is Clifford a big, red dog, or a Big Red dog [foot note 2]? What does Candice mean when she asks for an X-Ray of a Kangaroo with three legs?

LEFT: An X-Ray (of a Kangaroo with three legs)
RIGHT: An X-Ray of a Kangaroo (with three legs)

Is this strange ambiguity unique to English, or do other languages get bogged down by purple people eaters as well? I asked around, and here is a short summery of my results:

Language	Ambiguity?	Phrase
English	Yes.	Purple People Eater
Cantonese Chinese	No.	1. 紫色食人物 2. 食紫色人的物體
Mandarin Chinese	No.	1. 紫色食人者  2. 食紫色人者
Spanish [3]	Yes.	Comedor de gente purpura (Alternatively: Come gente purpura)
Greek	Yes.	Ο άνθρωπος που τρώει μωβ άνθρωποι
German	No.	1. Lila Menschenfresser 2. Der Ungeheuer frisst lila menschen 3. Der lila Ungeheuer frisst menschen
Albanian [4]	Yes.	Njerzit ngjyrë manushaqe ngrënës

-Nick

Footnotes:

[1] Here the parenthesis are being used to group words, not to indicate a parenthetical statement.

[2]

Hooray Puns!

[3] My Spanish speaking friend points out a bonus ambiguity: "Purpura" is neither masculine or feminine, so if "purpura" describes the eater, then it's unclear whether the eater is male or female.

[4] My Albanian friend's handwriting is not fantastic, so there may be some errors

Let's Get Mobius Up In Here

Here is an applet to explore the function $$\phi(z) = \frac{z - \lambda}{1 - z \bar{\lambda}}$$ for $| \lambda | \leq 1$. In the bottom left is the unit disc with Darth Vader super imposed on it. On the top is also the unit disc. For each point $|z| < 1$, the applet determines the color of $\phi(z)$ and and then colors $z$ that color.


The unit disc on the bottom right allows you to adjust $\lambda$. Just click anywhere in the unit disc and $\lambda$ will change appropriately.


Note that this demonstrates that $\phi$ maps the unit disc to itself. Click here for a proof.

Magic Squares

"Alright settle down, Tom, don't you dare throw that paper airplane. (Alright! We have a sub!) You all know the deal, sit down. Mrs. G. left instructions to have you do the Scholastic Math (I hate those!) but she also said that if I wanted, I could do something else (Yeah!).

"So here's the deal, everyone has to draw this grid on piece of paper...

"...and fill in each of the squares with the numbers 1,2,3,4,5,6,7,8, and 9. (Is this like Sudoku?) It's a little like Sudoku. (I hate Sudoku.) Ok, it's nothing like Sudoku. You have to put the numbers so that all the columns add to 15 and all the rows add to 15. If you're really bright (Well that leaves me out.) then try and get it so that the diagonals add to 15 as well.

"(This looks hard.) It is hard. (I suck at math.) That's ok, math is hard, just keep trying."

Eventually....

"(Oh, I think I have it!) Awesome! Now try for the four by four case.

"(What numbers do we use?) You have to use 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, and 16. (What do they add up to). I don't know. (You're not going to tell us what they add up to?) Nope. (Then it's impossible.) No it's not. (How can we make all the rows and columns add to the same number if you don't tell us what they have to add to?) Maybe you should try and figure it out. (How?) I don't know. Anyone have any ideas?

"(Add up all the numbers and then divide four.) That's an awesome idea. (Can we use calculators?) No. (But...) No. (But...) No. (You're mean.) Yep. (So we have to add up all the numbers by hand?) That's not how I would do it. (How would you do it?) I would start by adding all the numbers twice:

 "Why might I do that? (Because you're crazy?) Any other ideas? No? Well, what's 1+16? (17) and what 2+15? (17) and what's 3+14 (17) and what's the pattern here? (They're all 17!) And how many 17s are there? (16)

"(So the answer is 17 times 16?) (That's magic!) Not so quick. Remember that I added all of the numbers twice. (So the answer is 17 times 16 divided by 2?) Yes, that will give you the total. (Can we use calculators now?) Of course not. (So we have to do 17 times 16 by hand?) Not at all. Who can tell me how to simplify this?

"(Cancel the 2) And? (17 times 8... is that the answer?) Well we wanted to divide the answer by 4, remember?

"(So it's 17 times 2) What's that? (34!) Fantastic! Alright, somebody figure out the 4 by 4 case. (And then are you going to make us do the 5 by 5 case?) Yep."

Epilogue.

Of the three seventh grade classes I taught (about twenty students a piece), three students managed to solve the 4 by 4 magic square. One student skipped recess to work on it. Many students did not take to this assignment. One girl in particular was straight-up pissed. (Things were thrown.) Eventually though, she figured it out and was elated.

Guess the Function

I substitute teach middle school. Anyone that's ever subbed (especially middle school) knows that one of the cardinal rules of subbing is to not allow any downtime. (I have a story involving downtime in a middle school classroom that ends with a student shouting, "No Brad, you're stapling your face wrong!") Because of this, I'm always prepared with a math game or activity to occupy my students.

My favorite game (and often their's too) is Guess the Function. The rules are to guess the function. Allow me to elaborate:

First I write down a function on a piece of paper. The difficulty of the function varies with the skill level of the class, but usually I start with a linear function.

Next I generate inputs (sometimes I do this by rolling dice, sometimes I ask students to shout numbers out, and sometimes I just pick numbers myself).

Then I write the number and the result of applying the function to that number on the board. Students then have the option of guessing my function, or waiting to see the result of my function on more numbers.

I give one point for right answers and penalize two points for wrong answers.

For example: suppose my function is f(x) = x + 2. I would start by writing 2 → 4 on the board (because f(2) = 4). Now bold students might start guessing, but that's a bad idea. With the available information, they might guess f(x) = 2x or f(x) = x², and they'd lose two points. After everyone has either guessed or passed, I would write another pair of numbers on the board (for example 3 → 5).

Some notes:

• I use arrow notation (3 → 5) instead of function notation (f(3) = 5) because I usually play this game with students that haven't been exposed to function notation.
• I usually have students play this in teams of four or five.
• Usually, sixth graders are much better at this game than seventh and eighth graders. Typically, seventh and eighth graders tell me that they "haven't learned this yet," whereas sixth graders don't seem to know that they don't know.
• Mathematicians will point out that for any finite collection of points, there is literally an infinite number of functions that will fit the points. This is true, but I like to think of this game as a game about psychology (what am I thinking?) as much is a game about math.
• Younger students often give English descriptions of rules (like plus two or times itself), whereas older students are comfortable with describing things algebraically (like x+2 or x²).
• I use the scoring system +1 for right answers and -2 for wrong answers because I want to discourage wild guessing. (Recall that as a sub, my goal is primarily to keep a class under control-- wild guessing descends in to chaos rather quickly.)
• When I play this game, I announce that I'm writing down a rule (not a function) because most of my students don't know what a function is.
• Examples of functions that I would use include x+1, 2x, 2x+5, x², x(x-1), 2^x, x/2, 1/x, 10 - x...

I love this game. One of my favorite things is that since it's not a part of the curriculum, the students that "aren't good at math" actually do quite well. This is because the psychological hurtles they have when facing math class aren't in place when they're just playing some dumb game the sub made up.

Merry Christmath

MATLAB:

x = -4:.01:4;
a = 1.5;
p = @(x) (1+abs(x)./x)/2;
tree = {@(x) (10-a*abs(x)).*p(x+2).*p(2-x)
         @(x) 7*p(x-.6).*p(2-x) + 7*p(x+2).*p(-.6-x)
         @(x) (8-a*abs(x)).*p(x+3).*p(3-x).*(1-p(-x+1/a).*p(1/a+x))
         @(x) (6-a*abs(x)).*p(x+4).*p(4-x).*(1-p(1.7-x).*p(1.7+x))
         @(x) 3.5*p(x-1.7).*p(3-x) + 3.5*p(-x-1.7).*p(3+x)
         @(x) 0
         @(x) p(x+1.1).*p(-x+1.1).*(5*abs(x)-5.5).*(p(x-.9)+p(-x-.9))
         @(x) (p(x-1)-1).*p(x+1)};
hold on
for i = 1:length(tree)
   plot(x,tree{i}(x),'g.');
end
red_ornaments = [1 2 4 5 2 3 6 9 4 3 2 2 1];
cyan_ornaments = [1 .2 3 7.5 5 .5 6 1 .6 4 1];
plot(linspace(-3,3,length(red_ornaments)),red_ornaments, 'ro');
plot(linspace(-3,3,length(red_ornaments)),red_ornaments, 'r*');
plot(linspace(-3,3,length(cyan_ornaments)),cyan_ornaments, 'co');
plot(linspace(-3,3,length(cyan_ornaments)),cyan_ornaments, 'c*');
hold off
axis([-6, 6, -2, 10]);


There are certainly easier ways to plot this. I just wanted to have fun with absolute values.

Cayley Tables!

Long ago I posted about a mathematical structure called a group. In particular I described the group S₃. If you would like a refresher, read this post. This post picks up where that one left off.

S₃ can be thought of as the set of rotations and reflections on equilateral triangle. Doing that gives the following table called a Cayley Table:

*	I	R1	R2	F1	F2	F3
I	I	R1	R2	F1	F2	F3
R1	R1	R2	I	F2	F3	F1
R2	R2	I	R1	F3	F1	F2
F1	F1	F3	F2	I	R2	R1
F2	F2	F1	F3	R1	I	R2
F3	F3	F2	F1	R2	R1	I

Here's the thing though: This table is too busy to read. Sure, if you wanted to know what R1*F1 was, you could look it up (F2), but it's hard to take in all this information at once. Basically, it's hard to see. 

But you know what's easy to see? Color!

*	I	R1	R2	F1	F2	F3
I	I	R1	R2	F1	F2	F3
R1	R1	R2	I	F2	F3	F1
R2	R2	I	R1	F3	F1	F2
F1	F1	F3	F2	I	R2	R1
F2	F2	F1	F3	R1	I	R2
F3	F3	F2	F1	R2	R1	I

Two simplifications are in order: First, the top row and far left column are redundant, so we can dispense with them. Secondly, the labels are more distracting than helpful, so lets dispense with them as well. This leaves us with just a square grid:

The Cayley Table for S₃

This allows us to more easily see symmetries (the whites are symmetrical along the main diagonal) as well as asymmetries (notice the pattern of blues, greens and purples in the top right corner vs the same colors in bottom left). 

As I mentioned, the group we have been considering is called S₃. There are other groups too:

The Cayley Table for S₄

The Cayley Table for S₅

The Cayley Table for S₆

All of the pictures above are of what mathematicians call Symmetric Groups. But not all groups are symmetric groups. For example, there are the Alternating groups:

The Cayley Table for A₄

The Cayley Table for A₅

The Cayley Table for A₆

And the Cyclic groups:

The Cayley Table for Z₆₀

The Cayley Table for Z₆₀ with the elements arranged by their order

Hopefully I will post some explanations of these groups soon. In the mean time, try your luck with Wikipedia.

The groups are generated by a Python script I wrote. The Pictures are generated by Processing. The list of all possible ways four people can stand in line was generated by Matlab. Altogether there's about 450 lines of code going into these pictures. Anyone who wants access to this code is welcome to leave a comment.

Myth of the Right Answer Redux

Part 1 here.

A classmate brought this riddle to a study group I'm in:

You have eight pills. One of them is poisonous. The poisonous pill weighs slightly more than the others, but otherwise they appear to be identical. You have access to a scale, but you may only use the scale twice (for some reason). How do you determine which pill is the poisonous one?

The solution produced by our group (four undergraduate math majors) is as follows:

Satisfied with this solution, the other members of my study group were ready to move on. This is the myth of the right answer.

We are programmed since elementary school to find "the" answer and move on to our next assignment. Each quiz, riddle, puzzle, and problem is simply an obstacle to overcome in the ongoing mission to meet our teachers', principals', parents', and professors' approval. Why should the authority figure determine when our problem is solved?

So it's great that we found the algorithm to solve this problem with two weighs. But why stop there? Some related questions:

• Given n pills, what is the minimum number of weighs required to finding the poison pill?
• Given n pills that can be weighed with w weighs, is there an alternate weighing scheme that can find the poison pill  in exactly w weighs?
• What if we don't know if the poison pill is heavier or lighter (only that it weighs a different amount)?
• What if every pill weighs a different amount, but the poison pill is still heavier than all the others?
• What if there are two poisoned pills
• Given n pills, m of which are poisoned, how many weighs are required to find the poisoned pills?
• What if the scale can only hold two or fewer pills at a  time?
• Given n pills, one of which is poisoned, and a scale that can only hold k or fewer pills, how many weighs does it take to find the poisoned pill?
• Given n pills, m of which are poisoned, and a scale that can only hold k or fewer pills, how many weighs does it take to find the poisoned pills?
• Given n pills, m of which are poisoned, and a scale that can only hold exactly k pills, how many weighs does it take to find the poisoned pills?
• Suppose we would settle for knowing which is the poisoned pill with probability p, what is the minimum number of weighs?

None of my classmates asked these questions, they were satisfied with just having the answer. To be clear: my classmates are not stupid. In fact, they're all quite bright. But they (we) have been programmed to find the answer, to report the answer, then to forget the question. Somewhere in all of this answer-fetishism we have forgotten how to ask questions. We have lost our curiosity.

A good question is more interesting than a satisfying answer. Why then do we let other people ask all the questions? 

First World Problems

That is one really ugly red car.

Three Page Comic

I really hope I caught all the spelling errors. I probably didn't.

Note Taking in Math Class

I'm a math undergrad with a real passion for mathematics (enough to try to maintain a blog about it). As of today, I'm taking four math classes, have a gpa of 3.9, and spend a good amount of my time tutoring. Also, I don't take notes.

I'm not sure what place note taking has in a math classroom. The books that I spent more than $100 a piece on (way too much!) cover all the same content that my classmates' notes do. If I forget the definition of the Laplace Transform, then I don't need notes to look back on, I have a book. Failing that, I have Wikipedia, Wolfram's Math World, and Paul's Online Math Notes. If I need to have problem worked out, I have Khan Academy or any other of a variety of YouTube videos. My school, and I imagine every other school in the world, is packed with thousands of books, many of which are about differential equations. If I need help with Laplace Transforms, or any other topic in math, I have a plethora of sources to reference. Why then, on top of all that, should I take notes?

Further, I think taking notes in math class have negative consequences. Sometimes I look around and notice my classmates too absorbed in their note taking to actually be paying attention in class. The professor might add a bit of interesting information verbally, and my classmates are often too busy copying what's on the board to hear it. Also, there is the problem of divided attention: if your attention is being put into your notes, you are not working on comprehending the material. I assume that these students go back over their notes at a later time and try to make sense of the material then, but that has to be very tough when there is no professor to offer insights.

So why do so many students take notes? I suspect it's because of years of programming by high school and elementary school teachers. I also think it's because students have developed skills that are appropriate for other classes (note taking is very valuable in English or history class) and mistakenly believe that those good habits will translate to their math classes. I think all of this negatively impacts math students' education.

Wrong vs. Not Helpful

In solving a problem, students inevitably make mistakes. This is how we learn. However, there is a huge difference between being wrong and being not helpful.

Wrong

Suppose a student is asked to solve the following for x:

Seeing the 2 next to the x, a student may try to divide both sides by 2. This can be done correctly, but let's assume that the distributive rule is momentarily forgotten:

This leads to the incorrect answer of x = 1, whose falsity can be demonstrated by substituting 1 in for x in the original equation (which would give 6 = 10).

Not Helpful

Now imagine that the student takes the same equation, and subtracts 2x from both sides, giving:

This is not wrong, subtracting 2x is not a violation of any mathematical rule, but neither is it helpful. A student who attacks a problem like this may still be in need of assistance, but a different type of assistance from before.

My Point

This distinction is obvious to educators. It is not, however, necessarily obvious to students. And it needs to be made obvious to students. There is a rampant misconception of math that there is a right way and a wrong way of doing math, and if you're not not doing it correctly, you're wrong. (I previously wrote about that here.) Sometimes you're doing something useless that is not necessarily wrong. And that is ok.

I think a good way to get this distinction across is by way of analogy. Consider chess: there is a notable difference between a wrong move (moving a rook diagonally) and a foolish one (exposing yourself to checkmate).

Six Study Tips for Math

Math is Hard. It really, really is. There are thousands of tips on how to succeed in math class, here are some that I think are especially important.

Create Your Own Problems

Solving someone else's math problems demonstrates and ability to follow rules. Creating your own problems shows an understanding of the subject. It also forces creativity, an essential (and often overlooked) skill in mathematics.

Compare and contrast problems

Post game Analysis is so underrated. After doing 5 or 6 problems, ask yourself: which of these was the hardest? The easiest? What did all the problems have in common? What made them unique?

Don't Do Your Homework All at Once

If you decided to start training for a marathon, you wouldn't start by going outside and just running for four hours straight. Instead, you would run a little each day until you got into shape. Math is no different. Study or do homework for 20 minutes to half hour at a time. Then read a book, or make dinner, or call a friend... whatever. Just don't sit and do math for hours on end, it's counterproductive.

Do math in your downtime

Play math games; practice your arithmetic. I like factoring numbers in my head (the time on a digital clock, licence plates), it keeps me sharp, and it strengthens my ability to visualize difficult problems.

Read the Textbook

Yes, I know it's poorly written, all text books are. And no, you're not going to understand after one read through. But reading the text will at least familiarize yourself with the vocab and notation.

Later, when getting help (in class, from a tutor, from a friend) reflect on what you read. Why didn't you understand it when you read it? Do you understand it now? Over time you will learn to read math books.

The Internet is Awesome

Read the Wikipedia article on what ever it is you're studying. Look at the pictures. Google the subject, or watch lectures on YouTube. The more versions of a lesson you get, the more likely you are to gain insight.

Math: Discovered or Invented?

In my previous post, I offered up two classes of truths: Discovered Truths (The Earth is round) and Created Truths (Darth Vader is Luke Skywalker's Dad). Then I asked what class mathematical truths (2+2=4) belong to. Here are some possible answers:

Math is Discovered

Math, like science, is something we discover. Before there was anyone to count them, one dinosaur and one dinosaur made two dinosaurs. Before we had a word for 50, it was still the sum of two squares in two different ways (7²+1² = 5²+5² = 50).

Math is Created

Math is an art and, like the arts, is a creation. We create the truths in math. Mathematical objects like triangles and the number 17 are created by mathematicians, thus any truths about them are constructed.

Math is A Priori

Unlike the arts, which are dependent on our imaginations, or the sciences, which are dependent on the physical universe, math is A Priori, independent of everything. If there were no physical universe, 1+1 would still equal 2 (though there would be no way to express it). Math does not fit into the categories of "Discovered Truths" and "Invented Truths," Rather, they get their own category: A Priori Truths.

There is No Truth

Truth is a convenient fiction. Language is just a series of meaningless sounds (or symbols) that are used to invoke specific behaviors in others (or oneself). But just because those sounds/symbols succeed in creating reactions in others, doesn't mean they actually mean anything. There is no meaning therefore, there is no truth [footnote 1].

All Truths Are Created

We create our language(s), we express all of our truths with language, therefore we create all of our truths. Any "truth" of science (or math) is a construct of our minds and hence a created truth.

All Truths Are Discovered

Every possible linguistic expression is essentially a number (See Jorge Borges, Library of Babel), all numbers exist independently of us, so any truth expressible in language exists independently of us. More on this soon.


[1] I am very much aware of the irony of using language to express the meaninglessness of language.

What Is Your Math Philosophy?

http://www.gocomics.com/calvinandhobbes/2011/05/31/

Truth is a tricky thing. Some things are true because we say so, others are true independent of us. Lets call these two classes of truths Created Truths and Discovered Truths. Some examples of Created Truths:

• In chess a bishop moves diagonally.
• The ninth word in Never Gonna Give You Up is rules.
• The main character in the above comic strip is named Calvin.
• Darth Vader is Luke Skywalker's father.
• Peter Parker is Spiderman.

All of these things are true because someone said so (the creator(s) of chess, Rick Astley, Bill Watterson, George Lucas, and Stan Lee).

Some examples of Discovered Truths:

• The Earth revolves around the sun.
• Water and ice have the same molecular structure.
• Your eye color is determined by your genes.
• Fire burns wood.
• Iron is attracted to magnets.

Each of these is true independent of us. No one decided that the Earth revolves around the sun, it just does.


Now consider Mathematical Truths. Some examples:

• 2 + 2 = 4
• There is no largest prime number
• 1729 can be written as the sum of two cubes in two different ways
• The graph of y = x²-2x+3 has a minimum at (1,2).
• The area of a circle with radius r is πr²

Are these truths discovered or created?  More on this soon.

David the Gnome

My hat is conical!

When I was very young I watched a show called David the Gnome. As you might guess, the show was about David and he was a gnome. Not a terribly creative title, but when I was young I was captivated.

The Wikipedia article on David the Gnome says that the American version was actually a dubbed Spanish version that was based on a book written by a Dutch author. So apparently, in addition to being a gnome, David was an ambassador to the UN.

In one of David's adventures, he came across a chicken with six chicks. Tragically, the chicken could only count to three, and the chicks kept getting lost without the mother hen even knowing. Although a negligent parent, you have to give the bird bonus points for being able to count at all. [footnote 1]

David, recognizing the problem as a serious one, taught the mother that she should arrange her chickens in two groups of three. The mother could count to three, and she could do it twice, thus she could keep track of all her baby chicks. [footnote 2]

I think that the simple brilliance of this might have been lost on even the creators of the show. David’s poultry grouping insight alludes to many fundamental concepts in mathematics. The fact that every whole number can be represented with a unique product of its prime factors (like 6 = 2×3) is called the fundamental theorem of arithmetic. Had the mother had a prime number of chicks (like five or seven) David’s solution wouldn’t have been so simple because neither 5 nor 7 can be written as a product of smaller numbers.

The chicken could even keep track of 12 chicks, though it's a little trickier. You couldn't put the chicks in 2 rows of 6 (mama chicken can't count to 6), nor could you put them in 3 rows of 4 (4 is still too big), but you could put the chicks in three 2 by 2 groups:

You could also organize a peep of 18, or 27. If you're clever, you can do 16 too.

In addition to the fundamental theorem of arithmetic, David alluded to a tool often employed by mathematicians: when confronted with a difficult problem, reduce it to a problem that’s already been solved. Rather than teach  the mama chicken how to count higher, most likely an impossible task, he had the chicken do the something simple twice.

David applied simple and elegant mathematical thinking to a life threatening situation and his solution is profound, yet straightforward enough for an animated chicken to understand. He's my hero.

-Nick

Footnotes
[1] There is actually some anecdotal evidence that some birds can count as high as three. (Chapter 1, fourth paragraph)
[2] I like to imagine that before David came across the chicken, there were at least a dozen more chicks, all dead now due to negligence.

Fret Spacing on a Guitar

As you look down the neck of a guitar, the frets get closer and closer. What is the rule for deciding how close the frets should be?

Preliminary Information:

1. A string halved in length vibrates at twice the frequency (physics).
2. An octave is the interval between one musical pitch and another with half or double its frequency (Wikipedia article on octaves).
3. There are 12 notes in an octave (music theory).

To go up one octave, you have to halve your string. To go up another octave you must quarter your string. In general, to go up t octaves you need to have a string of length 

To have our length function, l, go up by twelfths of an octave (notes) rather than octaves, we need to adjust it to be: