To Infinity & Beyond

Phil wanted me to post the notes to my recent 2-part speech on the infinite which I gave to Toastmasters last week and the week before. My goal was to condense and present these math topics in under 7 minutes for a general audience to understand. It was a tough task, but I think I was modestly successful -

To Infinity and Beyond pt. 1 - Induction

            Our world is filled with finite things. Our minds can naturally conceive the very large, from grand nations like the United States, to giant multi-national corporations, to the very cosmos itself. On the other side of the spectrum, we can see the cells that make our body, we can manipulate the molecules of existence to create new substances and we can study atoms and subatomic particles. All of these things are finite. It is possible to count all of the stars in the sky and arrive at a number. Every transaction in a business can be accounted for. Every cell can make a miniscule percentage of our body at any given time. We can arrive at fixed numbers for these things.

            What is a struggle for us, what we can not intuitively grasp, is the concept of what is not finite. What is infinite? In this two part series of talks, we will examine infinity. In this talk we will uncover the infinitely large and in part two of this series, we will examine the infinitely small.

            Let us begin with a definition of infinity. Infinity is something which is not finite. Something that is finite is a collection of items which can be counted to a fixed number. For instance, there are 45 floors in this building.

            To begin to grasp the infinitely large, I’m going to ask a question - Is there a largest number? Can you think of a very large number? Well I can think of a bigger one. I’ll just add one to your number. If we extend the process of just adding one to the largest number for a limitless amount of time, we would never stop adding numbers. There are an infinite number of counting numbers.

            Mathematicians have built a method called “induction” to prove the existence of infinite things. Essentially, you begin by proving that the first case exists. In the following step, you assume that a fixed number of “n” cases exists. Then you prove that the following case, n + 1, exists. Since we have just proven that there are an infinite number of counting numbers, the proof holds for an infinite number of cases.

            We will return to induction in a moment. Now I would like to take you back to the year 1202 CE. An Italian mathematician, Leonardo Fibonacci, is faced with an interesting problem. If you start with a pair of rabbits, male and female, how many rabbits would you have at the end of a year?

            Mr. Fibonacci offered a solution to this problem in his book Liber Abaci. He makes the following assumptions:

            1) The male/female pair is observed from birth.

            2) Both rabbits mature after one month and the female gets pregnant.

            3) The pregnancy period lasts one month.

            4) A mature female always gives birth to one male and one female every month.

            5) No rabbits die.

            Month one, there is the initial pair of rabbits. Month two, the female is pregnant. Month three, the female gives birth. Month four, the initial pair give birth again and the second pair is pregnant. Month five, the first two pairs give birth and the third pair is pregnant for a total of five rabbits. As the problem develops, we arrive at the following sequence of numbers -

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

Thus, after 12 months, there are 144 pairs of rabbits or 288 rabbits.

            This sequence is named the Fibonnaci sequence and, as you can observe, it has the property that each subsequent number in the sequence is the sum of the prior two entries in the sequence [Exhibit 1]. This is due to the fact that our rabbits produce a pair of rabbits every month after two months of maturing and getting pregnant. My question to you is, how many numbers are in the Fibonacci sequence? I propose there are an infinite number of Fibonacci numbers.

Here is my proof -

            Begin by examining the first 3 entries in the sequence. [Exhibit 2] We see that the Fibonacci property holds in the first case.

            Now, assume that for a certain number of cases, n, the Fibonacci property holds. [Exhibit 3] If we reindex the sequence so that m = n +1, we discover [Exhibit 4] using the induction hypothesis. Returning to our previous index of n, we have proven the next Fibonacci number. Thus, by the induction principal, there are an infinite number of Fibonacci numbers.

            Lucky for us, Fibonacci’s assumptions about rabbits are not accurate, otherwise our universe would have filled with rabbits eons ago. However, we can walk away with the idea of infinity of process and that is what is at the core of induction. Next week, I invite all of you back for part two of this speech series, To Infinity and Beyond, where we will study the infinitely small, learn about the mathematical notion of limits and of how something as vast as the infinite can be tamed to a fix number. Thank you.

To Infinity and Beyond pt. 2 - Limits

            I would like to welcome you back to part two of my speech series “To Infinity and Beyond” where we are exploring the properties of the infinite. In our last session, we explored the world of the infinitely large and we discovered a method of defining the infinite - the infinity of process known as induction. Using the method of induction, we proved that there are an infinite number of Fibonacci numbers. In this speech, we will explore the world of the infinitely small.

            In the 5^th century B.C.E., the Greek philosopher Zeno offered an interesting paradox involving the Greek demi-god Achilles entering a race against a turtle. Achilles gives the turtle a head start, since Achilles is so fast. The race begins. In order for Achilles to surpass the turtle, he has to cross the halfway point of the turtle. But the turtle has moved forward, so Achilles must cross the new midpoint, yet once that happens, there is a new midpoint. Zeno postulated that there are an infinite number of midpoints between the turtle and Achilles and therefore, Achilles could never surpass the turtle to win the race.

            We will return to this paradox later. In order to analyze the world of the very small, we need to define a number system that is smaller than 1. I offer the world of fractions, also known as the rational numbers. With this number system we can use any of the counting numbers in the upper portion of the fraction and all but zero in the lower part of the fraction (Zero is represented by a 0 in the upper part of the fraction). This is an ordered system of numbers where for any two variables x, y, we can say x < y or x > y or x = y. As the lower number, or denominator, decreases, the size of the fraction increases. As an example, 1/2 > 1/8. If we were to divide by zero, there would be no proper way to determine the number, so we claim that 1/0 is undefined.

            Since we’ve determined that as the denominator decreases, the size of the fraction increases, what happens as the denominator increases? From our previous study in induction, we know that this denominator can become infinitely huge. Is there some limit that the denominator can reach were it becomes too small to measure?

            As it happens, mathematicians have developed a concept to answer these questions. It is known as the limit. You’ll have to forgive me, but I will have to introduce this in math notation and then I will explain it. [Exhibit 1] Essentially, going back to Achilles and the turtle, if Achilles can not pass the turtle, at the very least, we can measure the distance between Achilles and the turtle. The distance function for rational numbers is absolute value and if the passing point of Achilles and the turtle is point zero, we can restate the limit this way [Exhibit 2].

            Here is another way to examine the limit concept. Suppose you gave me a pie. I claim I can slice the pie so that everyone in the world can have a slice. I could slice it this way [Exhibit 3]. As you see, each successive slice is smaller and smaller. Being a generous fellow, I’ll give you the first slice and take the second

            After so many slices of the pie, there comes a point know as “N” where each successive slice (“n”) will be smaller than the distance from the limit to the previous slice (“e”). Therefore, in an induction-like argument, the limit as “x” approaches infinity is the limit “L,” or in our case zero.

            Does the number of pies affect the limit of our slices? Consider what happens if we have 2 pies [Exhibit 4]. Thus the limit remains zero. This may be a bit extreme, but I’m going to prove that for any fixed number “c,” the limit remains the same [Exhibit 5].

            Returning to Achilles and the turtle, does Achilles ever pass the turtle? Rest assured he does. Assuming that Achilles and the turtle travel at constant speeds, “Ca” and “Ct,” and point “L” would be to point Achilles passes the turtle [Exhibit 6].

            Thank you for taking the time to learn about the infinite. I’ll be available after our meeting today for questions.

Depressing People

            Apparently I’ve been depressing people again lately. I don’t feel any more depressed recently than usual, but I suppose it is an effect I’m having on people. The oldest occurrence recently was when I got my results back from my leadership training interview, when I was told I was too negative, and my most recent occurrence was last night when I brought lyrics to a song I just wrote yesterday to a song my band was developing. After playing it, Tim said, “Great, now I feel depressed.”

            This supposed cloud I’ve had has followed me for a very long time. Before I just dismissed it as being broke most of the time and preferring to seclude myself rather than risk spending money. But now I think this is one of those fundamental tenants of who I am as a person - I’m a natural downer. I mean to be supportive, I really do, but my realism is just too much for people. Reality is meant to be challenged, not met with an “oh man, that sucks.”

Perhaps I just don’t get it.

            I think back to the film Journey on the Wild Coast and I think about the happy couple using their optimism to challenge reality. I think about what I heard from the leadership interview and I think, “Yeah, who wants to follow a downer? Who buys from a downer? How does a downer get the best out of anyone?” I’m going to ponder these questions since I believe there has to be an answer, for my own sake. Unbridled, unfounded optimism can make me pretty sick at times, to be honest.

Dreams

            It doesn’t help that I’ve had some bad dreams this week. On Monday, I dreamt that I was parked on Clinton Ave. and Lake St. in Minneapolis and I was coming out to my car after dark and there were 3 kids sleeping in my car with trashbags filled with their clothes and things. I felt bad turning them out, but I had to do it since it was my car. A pair of cops drove by with their lights on and just watched, not getting involved. Then a pair of masked thugs started coming from across Lake St. with guns to rob me. I pointed and told the cops to stop them, but all the cops said was, “Oh this is gonna be good.” I ducked behind my car and found a handgun under the car. As the robbers approach, I started shooting. I killed one instantly, wounded the other, then got shot myself.

            In my dream last night, I spent a very pleasant day with one of my ex’s, Kennesha. When I woke up, I remember what I told myself when I broke up with her: This is either the greatest risk of my life or my greatest regret.

So yeah, downer shit.

Mpls Int’l Film Festival

            Today marks the end of the Minneapolis International Film Festival. I decided to pony up the $60 and get a 6 movie pass (me + guest, if I chose). I have to admit, I was impressed with the films that I saw. Here are the films that I saw -

Kill List (UK)

            I read the reviews of the films before I saw them so that I could avoid the typical indy-film turkeys of the “coming of age” and “angsty” and “art” film. The first film I saw was a good ol’ fashion action.

            The story follows a pair of old, washed up hitmen going on their last hit. They were Iraq war vets (although part of me thought it was more Gulf War I than II). The first third of the film was following the family, the second third was gruesome action as they made their hits, but the final third of the film, what can I compare it to? It was like watching a zombified Blair Witch Project, complete with druidic death squads.

            If that’s not enough to make you see this film, than I don’t know what to say. I was shaking at the end of this one and I was afraid I would have druidic death squad nightmares.

Kong Kurling (King Curling) (Norway)

            When I saw that there was a comedic film about curling in the film festival, it totally leapt to the top of my “must see” list. I took my nerdy, German friend Raija to this one and we both had a total blast. It was so funny, from the dialogue, to the acting, to the facial expressions, down to the air fight scene between one of the curlers and a group of bird watchers.

            The plot is not entirely original, but it was still a charming film. A champion curler loses his grip on reality and is thrown in a mental ward. After some years, he is released, but discovers his mentor is on his death bed, needing a lung transplant. The only way he can get the organ he needs is to become the national champion in curling, due to a special multi-million dollar prize. It’s a total “get the gang back together film,” which causes him to lose his wife in the process (with hilarious consequences).

            There is a nail biting finish that had me literally cheering in the theater at the end of the film. The theater was packed at the time and Raija and I were forced to sit in the very front, yet we still had a fantastic time.

Journey on the Wild Coast (USA)

            When I read the review of this film, I knew that at last, I finally had found a film that both my mother and I could enjoy. Unfortunately, mom had already made plans, so I went to the movie with my Aunt Cindy instead. While wheel chair bound, we watched as a married couple put together a pack and walked and kayaked from Seattle to the Aleutian Islands.

            I imaged daunting challenges and amazing vistas, which the movie delivered. What I didn’t expect was how uplifted I felt by the couple. Nothing could get them down. Every challenge was an opportunity to be positive and to laugh and it made me appreciate the importance of how a positive attitude can combat the depths of despair.

            Towards the end of the journey, the woman got pregnant. Pregnant and hungry, the couple arrived at their final destination. The had encountered hungry bears, deprivation, and deadly ice flows. Another thing that shocked me was how abosolutely awful the weather was all the time. It was constantly sleeting, snowing, raining, windy and cloudy. I could never have survived it!

Endless Night (France)

            The review told me that this would be an action film that would take place entirely within a nightclub. Since this was my last film of the festival, I decided to bring Andrew. He moaned when he realized it had subtitles (he can’t read), but he told me afterwards, he thought it was one of those movies that wouldn’t make sense even if he had been able to read the subtitles. In a way, he is right.

            The movie opens well enough with car chase robbery of drugs and a murder. The robbers are cops and one of them gets ID’d and his son kidknapped. The rest of the film is all about him trying to recover his son from the drug baron’s nightclub fortress while the baron tries to get his drugs back.

            From there, the movie really goes off the reservation. It turns out the robber was trying to expose crooked cops from the robbery. Mick Jagger arrives as an ineffective body guard for the buyers of the drugs and just runs around in the film for who knows why.

            There are, like, 5 fight scenes that take place in the same kitchen, including one between the robber and the crooked cop where he beats him with kitchen drawers until he passes out, drags the body through the kitchen to a stairwell, climbs the stairs, gets attacked from behind from the same cop, chucks him over his shoulder so that the crooked cop falls 2 stories onto his back on a bunch of steps AND STILL this same cop creates a car accident later in the film at 40 mph with him in the back seat in order to kill the people who would expose him. He fails. On so many levels…

            This is one of those films where push comes to shove goes to slap until late in the film when the guns appear and people start to die. Yet, why doesn’t anyone kill anyone else for most of the film? Why do people still stay in this club after the gunshots and the fights in the kitchen and restaurant? Do people just not notice blood in their salad?

            All in all, I had fun with the film festival and I suspect that I will return next year.

Turned Down for Leadership

            As you may recall from my April 18^th post entitled “Midweek 16, 2012,” I had dedicated myself to applying for a position in the leadership program. I had dry cleaned my suits, product my hair, and prepared for interviews. The results are in and I was not admitted. All of my co-workers who heard of it are besides themselves in disbelief whereas I have adopted a different tack.

            To be honest (some could say, “in retrospect”), I was really using the application process as a sounding. People around me say that I’m more confident and more influential and better spoken and all those la-dee-dah-dee-dahs of social standing, but I still see myself the same way. Sort of like how, on your birthday, you really don’t feel that extra year, but over the span of years a change may be noticeable. I wanted to see if I could be considered a leader yet.

            Gratefully, the rejection was treated as a coaching opportunity. My application packet was pristine, but they wanted to see me look up to a business leader, as oppose to my true inspiration of Nelson Mandela. The interviewers were remarkably impressed by my dress and grooming, so far as to call it “perfect.” The soft spots were in the interviews. I was too negative, despite the fact that I didn’t think so. As you readers know all too well, the darkness can be incredibly, and laughably, dense. However, when things go dark, I need to do a better job of injecting optimism (example: here is what I learned from failure: blah-ti-dah…).

            Job interviews, after all, are all about optimism. People want to look forward to something, so make them look forward to hiring you. Optimism and enthusiasm, like the common cold, are contagious. Instead of giving them an opportunity to wash their hands of you, sneeze right in their Goddamn faces and infect them with who they want to hire - you.

I just didn’t quite make it.

Alas, to Plan B (which, as the application dragged on, should have been Plan A): Seek new positions.

What? Leave the bank? Are you daft? Heaven forefend!

No, I shant leave the bank.

            Having observed the leadership program for nearly 3 years, it is a time intensive volunteer exercise with rich rewards in experience, mentorship access, and tutorials. At a price tag of $FREE, even I have to admit it is an outstanding growth opportunity which generally results in promotion.

But as my Grandfather always tells me, “God helps those who help themselves.”

            The world is not as it was three years ago. The era of foreclosures is declining. People are going back to work. Banks, pinched on the nose by government, are being dragged through loss mitigation hoops to modify loans so that homeowners who have been paying their bills, will continue to pay their bills. Greedy investors have come to the rescue and are buying properties in bulk. The great unraveling of the securitized mortgage debt swindle hasn’t taken place, so I’m content to believe that lawmaking (and court) activism is too weak to change the way we do business. Furthermore, I am content to believe that what I consider to be a “swindle” is unenlightened and misguided. (Aren’t the repentant soooo virtuous?)

            It is upon this faith of recovery that I hope to extend my hand upon the next wrung of real estate mortgage - Mortgage-backed Securities. My bank happens to be in the top 5 players when it comes to mortgage-backed securities and I foresee a renewed spur in growth. As people go back to work, they want mortgages. As more mortgages accrue, more are bundled and sold.

Who wants to sell them? *raises hand*

Who wants to explain them? *raises hand*

Worst case scenario, who would enjoy doing the arithmetic calculations of minutiae for these securities? *sticks-hand-under-elbow to raise hand higher*

Its time to make moves. I’ll close with this inspirational quote -

When a river encounters a rock, does it stop?

It goes around, does it not?