Page 1

An Introduction to Logical Fallacies • Heisenburg's Uncertainty Principle Explained

REAS N May 2013

Art War the

of

A player's guide to diplomacy

PAGE 6

WHO DO YOU THINK YOU TRUST?

Secrets of the RIEMANN HYPOTHESIS

Why is "THE SETTLERS OF CATAN" So Fun?


CUBE

Art by: Booyabazooka

Can you solve the ultimate puzzle? Buy your’s at Rubiks.com


Contents The Art Of War

Follow Corin Wagen, expert Diplomacy player, on his path to victory. See the game through his eyes as he makes well-timed plans that lead him to his conquest. Even a strong player can learn a lot about Diplomacy from others’ experiences. What do you know about The Art of War?

6

Settling Catan

Many people enjoy games such as The Settlers of Catan, but what about these games make them so much fun to play?

12

Fickle Falsehoods

An explanation of logical fallacies, some of the most common errors in reasoning, and how to prevent them.

Prime Distribution

Learn about the history, purpose, application, and meaning of the Riemann Hypothesis and Riemann-Zeta Function.

Quantum Logic? What is the Heisenburg Uncertainty Principle? What does it mean for the world? Read the article to Find Out!

16

Δϰ ΔР ≥ ℏ/(2m)

18

Image Credit (Clockwise from Top Right): Niels Kornerup, Public Domain, Niels Kornerup, Michael Martinez, Michael Martinez

May 2013

14

Reason

Page 3


Contributors’ Page

The binary was taken from Cncplayer, the quasar was taken from NASA, and the comic was taken from XKCD.com. All photos were taken by Niels Kornerup and Ishan Shah

Niels Kornerup has always been a fan of logic. He first learned about formal logic in 8th grade, and has loved the subject ever since. This year, Niels is in AP computer science and is at the top of his class. At the beginning of his 9th grade year, Niels received his first introduction to Diplomacy and he became an expert ranked player in under a year. In his spare time, Niels likes to learn about concepts in science and math for Quiz Bowl, a team version of Jeopardy that he has played since 7th grade. As he says, “Learning new things is a fun way to spend pastime.” Michael N. Martinez is also a longtime fan of logic. He was first introduced to formal logic in Geometry class during 8th grade at Kealing Middle School. Michael uses his middle name because he shares his name with Michael E. Martinez, who he shares no relation with. Michael enjoys Quiz Bowl, in which he placed 11th in the nation individually, as well as sleeping and watching TV shows like Doctor Who and Star Trek. Michael also enjoys spending his time reading various types of books, and is in the pit in marching band where he plays various instruments such as the marimba. Ishan Shah has always been interested measure theory, sigma algebra, complex analysis and formal logic. In his free time, Ishan likes thinking about quantum phenomena, and reading books about measure theory. He has been interested in quantum since his first introduction in fifth grade. Ishan is a strong believer in the relation between math and the real world. This year Ishan is in the ninth grade at LASA and still finds interest in abstract mathematics and physics. He also loves to read about new mathematical concepts and research papers from sources such as Nature Magazine.

Letter from the Editors Reason is a magazine produced by students in LASA high school as part of an elective called Ezine. When our group first got together, we all found a common interest in math and science, and that is how we decided to create a logic magazine. Logic is the art of reason, so our group decided that Reason should be an appropriate name. Logic is a large part of our everyday lives, and Reason aims to provide its readers with everything relating to logic. We understand that we are amateurs and our work is not comparable to that of professionals, and we appreciate that you will spend your time to read our creation. Page 4

Reason

May 2013


XKCD

Bringing you comics every Monday, Wednesday and Friday


The art of war By: Niels Kornerup

War is a traitorous time, those who you think you can trust most will turn out to be your biggest threat. Backstab or be backstabbed, that is the way of war. If you keep it safe and make no enemies, you will find yourself victim to those who rose to the occasion and achieved success. Read on to learn about the ultimate game of strategy, negotiation and intrigue in World War One Europe (unlike the Chinese treatise), Diplomacy.

A “heat map� for the strategic value of territories in Diplomacy

Page 6

Reason

May 2013


Diplomacy 101 Diplomacy, a classic strategy board game, shares many characteristics with Risk, yet it involves no dice rolls. In the game you play as one of Europe’s seven major powers at the turn of the century (France, England, Germany, Austria, Turkey or Russia), and you make alliances with other players to take over Europe, only to backstab them when they rely on you the most. Each player plots their moves separately and then watches their plans unfold when moves get revealed. As Allan B. Calhamer, the game’s creator said, “Who do you trust”?

Maps by: Webdiplomacy, a free online source for playing diplomacy. The heat map is by Reason Editor Niels Kornerup

Moving and phase resolution

Each player can assign one of four moves to each of their units during each move phase; hold, move, support move or support hold. A unit that holds does nothing, but another unit can support hold it. A unit that moves will move to a specified bordering country. A unit that support moves will support a specified country to a destination bordering the supporting unit. A unit that support holds will help a bordering unit that either holds or support holds against an attack. When a unit moves into a territory, the unit succeeds if the attacking country has more units supporting it in than the defending territory has supporting the hold. In the event that two units try to move into a territory, the unit with the most supports succeeds. If both units have the same number of supports, nether unit move into the territory. A unit inside a captured country will need to retreat, a retreating unit will have the opportunity to move into any empty territory, that it borders. In the event that a unit cannot retreat or when a player wants to, the unit gets removed from play. A units support will fail if it gets attacked by another unit.

Fleets and convoys In addition to armies, one can also use fleets. Fleets can do everything that an army can, but they can only move into sea spaces and shoreline countries. When a fleet moves to one of three countries with two coastlines (Namely Spain, Bulgaria, and Saint Petersburg), it must move to the coastline that it borders. A fleet can also perform a convoy, where one or more fleets in a chain can transport an army to another land area. Each sea space that the army travels through in the convoy needs to convoy that unit to its destination or else the convoy will not go through.

In this example, the German fleet in Kiel tries to move to Denmark, but gets overpowered by the army in Sweden who received support into Denmark from the English Channel.

May 2013

Reason

In this example, the Austrian army in Moscow must retreat because the army in St. Petersburg received support into Moscow. The unit in Moscow then retreated to Warsaw.

Page 7


Army: Fleet:

Maps by: Webdiplomacy, a free online source for playing diplomacy.

The diplomacy board at the start of the game, showing fleets and armies in their starting positions.

The building phase

End of the game

At the end of every other retreat, any supply center (each territory with a dot on it) that you have a unit on becomes yours. Next, you count the number of supply centers you own and the number of units you have. If you have more supply centers than units, you get to build armies or fleets in each of your “homelands” (the supply centers that you started with) until you have less than or an equal number of units and supply centers. If you have more units than supply centers, you need to decide which of your units to remove until you have the same number of supply centers and units.

Decoding Diplomacy

You repeat the five phase rounds (consisting of a move, retreat, another move, another retreat, and a build) until someone has 18 supply centers, and that person wins. Commonly, players spend a good deal of time negotiating with each other between moves, and a game of diplomacy can take about 8 hours. If you want to play a game with people over a longer period of time, you should check out webdiplomacy.net. Read on to walk through a game of Diplomacy from Corin Wagen, an expert Diplomacy player’s point of view. The rules as they appear in the official rule book

In the next part, you will find the same visual representations for moves as Webdiplomacy, which you should know. A moving unit has a red arrow showing its path. The red arrow will have a cross if the move failed. A support moving unit has a yellow arrow that connects to a moving red arrow. A support holding unit has a green cross touching the unit it supports. A convoying unit has a blue arrow that connects to a moving red arrow. A pink arrow shows a unit’s retreat.

Page 8

Reason

May 2013


Diplomacy from a Pro’s Point of View I logon to VDiplomacy and get ready to play the

game, sitting at the dinner table with a cup of a hot, black tea by my side. All seven players have gathered, and the game begins in 5 minutes. The time slowly ticks to its end, and I refresh the page, finding that I will play this game as France.

Maps by: VDiplomacy, a free online source for playing diplomacy.

I have not had much success with France in the past, I think to myself. Despite my past record with France, I like playing countries in the western corner of the board, including France. I enjoy playing Diplomacy because it requires communication between players and does not leave anything to chance. There are many games that The first move has been made, and things look good. feature one of these two game elements; however, Diplomacy is one of the few games that successfully Everyone has made their moves, and the year ends. As I had feature both elements at the same time. hoped, Germany and England have not made any moves For spring 1901, I start the game by communicating with England, Germany and Italy; my three neighbors. England proposes an alliance to take out Germany and Germany also proposes an alliance against England. Perfect! I agree to both of them so that the two will engage in war and weaken each other out. As expected, Italy says that he plans to focus elsewhere and will not threaten me.

against me and I have many options. Germany managed to get three supply centers in his first year, a rare occurrence that never goes unnoticed. If a player grows too quickly like he did, his neighbors will normally form together to take him out and therefore he will not threaten me. England on the other hand, had a very strong opening. By taking Norway and Belgium, he exists in a good position to take away a lot of Germany’s supply centers and I will be his next target.

“If I had moved to Portugal with my fleet, it would have cost me a turn, and the element of surprise”

Now that I have finished my negotiating for the turn, I decide to make my moves. I decide to try the Lisbon Leapfrog, an opening where France moves to Gascony, the Mid-Atlantic Ocean, and Spain. This opening gives me the option to retreat back to my homeland if Germany backstabs me by moving to Burgundy or if England takes the English Channel. If Germany holds true to his word, this opening gives me many options because I have remained neutral with all my neighbors. The moves go through, and things look good. England did not take the English Channel and Germany left Burgundy alone. As expected, Italy engages himself in combat with Austria and I can focus on the north. For fall 1901, I follow through on the Lisbon Leapfrog, convoying my army in Gascony to Portugal and leaving my army in Spain put. By making my fleet in the Mid-Atlantic Ocean convoy the army to Portugal, I made my fleet look busy, giving it an excuse to stay there.

Fall 1901

Spring 1902

Fall 1902

Wanting to take him out before he gets too strong and with his back turned to me, I decide that I must target him. To attack England, one usually needs to land units in Great Britain. Using one of my builds, I create a fleet In Brest so that I can create a convoy route into Liverpool. As I expected, this raised a red flag for the British and he asked that I keep true to my word and help him conquer the over powered Germans. I told him that I would not take the English Channel, and this seemed to satisfy him. Thinking that England would try and block me in the English Channel, I decide to take the Irish Sea and attack his back door. I moved my fleet in Brest to the Mid-Atlantic Ocean and I also moved the Mid-Atlantic Ocean to the Irish Sea, everything went as planned.

Spring 1903

Fall 1903

Spring 1904

Fall 1904


With my fleet in the English Channel, I have the power to take Belgium and do not hesitate in doing such. Afraid that he will bounce me again, I have my unit in Paris support me into Picardy. In England, I just keep my hold. I can’t really take London from him with his current defenses blocking me. Knowing that I have strong holds in the north, I look to my south. Italy appears to target Austria and Turkey, giving me the perfect opportunity to stab him. I build a fleet on the unprepared Italy’s back door, Marseilles. Distracted by Turkey and the dying Austria, Italy pays little attention to my move. Still paying attention to the north, I move into Ruhr, a key territory in the war between France and Germany, but most underrate its usefulness. With Ruhr under my control I think about taking Holland, but then I notice that Munich stands without any units in it. I check the rest of the board, and England seems determined to take out Germany. He took the Helgoland bight and tried to take Sweden, both anti-German moves. England realizes that I threaten him, and desperately tries to make me back down; offering me Belgium for peace. Knowing that England cannot stop me if I wanted it, I ignore this offer. For moves this turn I just pull off a convoy to Liverpool, taking the supply center without resistance. I also decide not to take Burgundy because that may anger Germany. The second year reaches an end, and things go how I expected them to. Now that I more or less have my English conquest secured, I should move on to the lowlands and North Sea. The North Sea, one of the most important spaces on the board, helps in the conquest of England and Scandinavia.

Munich is the center of the board, and it is very hard to win without, so I take this chance and seize it. At this point, I feel good. I can build more fleets in the south and deal with Italy and probably take the win. Turkey, the only one who can possibly overtake me, has had much success in the south. Having one less supply center than me and existing in a good position to take more from Italy, he could beat me to the 18th supply center! I know that if I attack him now, I [want] to attack him on my own terms, when I know I [can] defeat him. From here on out, I take Holland and finish off England with my northern fleets. Who could I trust? Nobody. That whole game seemed to be chaotic, and I was lucky to have few neighbors. With the game down to me, Italy and Turkey, the game really became a threeway war with no alliances.

Speed is of the essence now, I think to myself. I need to expand to my east and claim former Germany before anyone else does. I build two armies with the intent to send them east and steal Belgium from England. I don’t think I will need any more units to finish off the rest of Great Britain.

If you trust people too much, you’ll never win. People are a ruthless bunch in Diplomacy. Especially in this phase of the game, anyone who you place your trust in will take advantage of you.

With the builds done, things look good for me. I intend to have Breast move to Picardy and I support Gascony to Burgundy, in fear that England’s army in Belgium may try and slow me down. I also have the Mid-Atlantic move to the English Channel, bringing it to the front line. England sees through my plan, and stops me in my attempt to take Picardy when he also attempts to move there. This annoys me and sets my plans back a whole turn, but I still have other options.

In the end I didn’t fight Turkey very much, and I won the race to 18 supply centers by a pretty good margin. Winning a game of Diplomacy shows you as a skillful player, both in your diplomatic skills and logical game planning. Communication is just as much about what you ‘say’ via units and actions than what you say via words. Ω

“People are a ruthless bunch in Diplomacy”

Spring 1905

Page 10

Fall 1905

Spring 1906

Fall 1906

Reason

Spring 1907

Fall 1907

Spring 1908

May 2013

Maps by: VDiplomacy, a free online source for playing diplomacy.

I win the race to 18 supply centers, leaving Turkey in second.


Want an Unlimited Supply of Energy? SPONSOR RESEARCH !

How it works: When electromagnetic radiation travels into the quasicrystal it makes the electrons inside the quasicrystal oscillate at the frequency of the light that hit. However, since the quasicrystal that we are dealing with behaves like a diode (a diode is a unit that only allows a current to move from the positive to negative sides of it) the electron can only move in one direction and thus they will not be able to complete the oscillation and will only be able to oscillate in one direction. This will generate a current which can be used for power. Electromagnetic radiation is travelling through all of space and while solar panels can turn the electromagnetic radiation emitted by the sun into usable energy most photovoltaic cells are only 2%-14%. This new Quasicrystal that behave like a diode can use all the frequence of the electromagnetic spectrum and is 90% efficient.

Go To Your Local University and sponsor Quasicrystal Research


S

g n C i l t a t ta n e

Today, many Americans play

many different board games, but many employ luck, rather than strategy, as the underlying mechanic. Some of the most prevalent games, such as Monopoly, are largely decided by the roll of a dice. Very few require the players to use logic in their decisions. One of these games, Diplomacy, a logic-employing board game based solely on decision, contains absolutely no luck. (Pg. 6) But some games played in America incorporate aspects of both strategy and luck. One such Page 12

By Michael Martinez

game is The Settlers of Catan. The Settlers of Catan, known as “Settlers,” or simply “Catan”, is a multiplayer board game played by millions. The game puts players on an island made of movable hexagonal tiles that signify different resources, and focuses on player-to-player interaction through trade, alliances, and embargoes. Players must collect ten Victory Points (VPs) by building towns, cities, roads and other things in order to win. The roll of a dice

Reason

determines gameplay, but in order to win, players must interact with each other. “German-Style” board games such as Settlers, are games that tend to avoid direct conflict. They generally have simple rules and short (less than an hour) playing times, as well as economic, rather than military, focuses. As the name would suggest, these types of games originated in Germany, but don’t attract as much attention in the U.S. as in Europe.

May 2013


German-Style games prevent one player from completely dominating until the very end, unlike more traditional, “American” games, players are almost never eliminated. Other, lesser-known German-Style games include Carcassonne, Puerto Rico, and Imperial. Many young people enjoy this game, such as Ethan Russo, a freshman at the Liberal Arts and Science Academy in Austin, Texas. “I like being able to antagonize people, and then being this iconoclast and placing embargoes on people.”

All Photos by Michael Martinez

A dice roll determines a large portion of the gameplay in Settlers, the amount rolled controls when each player to receives resources. It is nearly impossible for one player to have a reliable access to every resource, which encourages trading between players. Because of this, players cannot isolate themselves, and must rely on the “kindness” of others to receive victory points and, eventually, win. Additionally, the game is also balanced by the robber, a mechanic that allows players to stop a tile from producing whenever players roll a seven. The robber can be used to stop the players that are furthest ahead from winning easily, and can also “bribe” the player controlling the robber to so their resources aren’t stolen. These are only two game mechanics that force players to interact. Klaus Teuber (TOY-ber), the German game maker that created The Settlers of Catan, spent four years developing it, and finished in 1995. The game won many awards in Europe, such as the Spiel des Jahres, an award given to the best German-made game of the year, in 1995. “Catan” eventually spread to the U.S., where it became wildly popular, selling over 600,000 copies

May 2013

in 2008. Expansions to the game have added features such as shipping routes and support for a greater number of players, but the underlying objective of the game, to explore the world around you, remains the same.

Below: The five resource cards in The Settlers of Catan, from top; wool, brick, ore, lumber, and wheat.

“If you want to win you have to look at the bigger scheme of things, and where you’re gonna go, and how you’ll win all your victory points,” says John

“See, the way I play Catan is like, turn to turn. What looks best, what would make somebody most mad” DiCarlo, another freshman at LASA. “You want to look ahead to the big picture.” Catan, like Chess, takes little time to learn, and a long time to master. The board changes every playthrough, and players can employ many different strategies, making it impossible to develop a universal way to win. The flexibility of Catan makes it a game that doesn’t get boring after only a few rounds. “See, the way I play Catan is like, turn to turn. What looks best, what would make somebody most mad,” Russo, a regular player of Settlers, said. “I’m not out to win. I’m just out to have fun.” Ω

Above: The box of The Settlers of Catan, and how the game is packaged.

Reason

Page 13


A Derivation of Planck's constant from a Classical Perspective By: Ishan Shah

Introduction to Planck's Constant:

Ever since Max Planck solved the ultraviolet catastrophe, he wanted to find a way to derive the "meaningless" Planck's constant. However, his motives were in vane. After Planck died the scientific community took Plack's constant for granted wand treated it like a random constant that was derived form experiment.

What is Planck's Constant:

Art by: JHausuer

Planck's constant is a constant that is used in quantum mechanics as the constant for many equations. Planck's constant was found experimentally during the ultraviolet catastrophe. Planck's constant was the experimental constant that defined the integer multiples of the energy quantization and then began to define many more quantized systems. Planck's constant is approximately 6.62607*10-34 joule seconds.

The Derivation

Lets start by defining the equation the scientist Frank Znidarsic came up with when working with quantum transition. The equation is Vt=fλ where Vt which is equal to 1094000m/sec (a number proved through experimentation with superconductors being stimulated through an external induced radio frequency) which is the speed of the quantum transition, f is the frequency of the photon that induced the quantum transition, and λ is the wavelength of the photon during the quantum transition (it is smaller during the quantum transition due to the fact that the wave function of the photon collapses during this event). The next equation we will use is C=ε0A/D. This equation is a generally known equation and can be found in any physics or electrical engineering textbook. In this equation C is capacitance, ε0 is a constant known as the permittivity of free space, A is the area of each cooperator and D is the distance between the capacitor. You might

be thinking where the capacitor in all in all of this, well the capacitor is the photon. Now lets think how this is possible. The photon is an oscillating periodic wave kind of like a sine of cosine wave, the photon's wave function is always oscillating between the charges positive and negative. The two parts of the wave can be considered capacitors separated by a distance of λ/2. When the photon's wave function collapse due to the quantum transition the wave function will collapse into a piratical that has an area of λ2. The last equation that we will use can also be found in any physics or electrical engineering textbook. The equation is E=Q2/(2C) where E is energy, Q is the strength of a given charge (in our case this would be the strength of the charge of an electron), and C is capacitance.

NOW LETS DERIVE PLANCK'S CONSTANT

Vt=fλ, therefor λ=Vt/f; C=ε0A/D, where A= λ2 D= λ/2;There for C=2ε0λ; Now lets plug in: λ=Vt/f into C=2ε0λ, we get C=(2ε0Vt)/f; Now lets plug in C=(2ε0Vt)/f into E=Q2/(2C), we get E=(Q2/ (4ε0Vt))f; Now lets plug in: E=hf, which is the equation that was used to solve the photo electric effect, intoE=(Q2/ (4ε0Vt))f; hf=(Q2/ (4ε0Vt))f, when we solve for h we get h, or Plancks constant, is equal to Q2/ (4ε0Vt) {h=Q2/ (4ε0Vt) }; and Q2/ (4ε0Vt) just happens to be the value for Planck's constant.


Th

s:

o M d el Pr d r a d n e a se t S nt e QUARKS! U

U

U

U U B U U C U UC U B B C U U B C S C B DB S C C S CC D DS C D T D S C S C T D T DT U D T UB T DDUT T D T T B U TU S U DD TB B CB T B U S S C T T UD CU B S B B BD T C S D C BB S U T SB C C C TU S U DC U D DB C C D S C UC C B U B U S C D D C BT T D C T U C C D C T C D T T U B U D TB T B C S S C U T S B D U T B D B CT S BT UD C U U S D B S T T C C D D C U C B S CBU B U DC CUC D B T U S S C C T U B T C U C D D T T D U B BU BC B S C S C D BT T T C D S C C D S B T S C B U D D D T C D BT S U T C D D T BT D C CU T B T S B T C D C B T BS U S D T DCU C B B U C C C C


Prime Distribution

By: Ishan Shah

ζ(x)=∑ 1/(n^x)=∏ 1/(1-p^(-x)) P Prime

n=1

This is the Riemann Zeta Function in all its glory. This is expressed in both product and summ form allowing us to iunderstand how this relates to the distribution of prime numbers.

Want to win a million dollars? Well you can, if you prove the Riemann Hypothesis. Even after decades of work, the Riemann Hypothesis, a part of a series of mathematical problems, known as the Millennium Prize Problems, which are a set of mathematical problems that if proved, the prover gets a million dollars, still has not been proved. The Riemann hypothesis, proposed by Bernhard Riemann in 1859, has puzzled mathematicians ever since it was proposed. The Hypothesis is part of a branch of mathematics called Analytic Number Theory and relates to the distribution of prime numbers as can be seen by just looking at the function in its product form. The Hypothesis is an analysis of the Riemann Zeta Function all of whose zeros are less than or equal to 1. The Zeta Function’s trivial Page 16

zeros are negative even numbers. The Hypothesis states that all of the Riemann Zeta Function’s non trivial zeros have a real component of 1/2. The Millenium Prize Problems also contain the Continuum Hypothesis, a hypothesis that was dubbed undetermined after it was figured out that it can not be proved using the rules of set theory. The Hypothesis was proposed during the time in which mathematics was being formalized. At first many mathematicians believed that arithmetic was the basis for mathematics, but later on logic was dubbed the basis of mathematics. When this occurred, mathematicians integrated the rules of formal logic into their Reason

own problems and books were written on the subject matter. the Master Catalog paradox. This paradox stated: a librarian is making a catalogue of all the books in her library. She thinks of putting the name of the catalogue in the catalogue, but ultimately decides not to. She then sends the catalogue to a master librarian who has the job to make a master catalogue of all the catalogues that don’t have their name. He realizes that the master catalogue doesn't have its name in it and thus it must go in it, but if he takes it in he must take it out because then it would not be in the set of catalogues that don’t have their names in them. But if he take it out, then it must go because now it is a part of the set. This is when mathematicians realized that the only way to stop these recurring paradoxes was to formalize May 2013


The picture of the Riemann Critical Line: http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/RiemannCriticalLine.svg/800px-RiemannCriticalLine.svg.png The Riemann Zetaq Function equation: Made by Ishan Shah

because it is intrinsically beautiful and interesting in itself. Certainly, most types of math a mathematical theory can often precede its application. The standard example of this situation, involves something called "Differential Geometry." It is a type of math that had no applications during the time that it was first studied, but later, after Einstein discovered This is an image of the real and complex parts of the Riemann Zeta Function graphed on a plane. This model helps us understant what Riemann HypothGeneral Relativity, people esis states and halps us understand the basic idea of the Riemann Zeta Func- realized that the principles tion. of Differential Geometry are exactly what one needs, to study gravity under Einstein's model,” mathematics. Many people think that since says Dr. Paige. Ever since the the Hypothesis is a hypothesis, mathematics formalized after Mathematicians then began to mathematicians should have paradoxes occurred in set theory, try to formalize mathematics an opinion. However, that is mathematics has lost its meaning and set theory. During this time not always the case. Dr. Paige, when being compared to the arithmetic began to be defined in who is a mathematics teacher real world. However this change terms of formal logic. When this at the Liberal Arts and Science was made in order to solve many began to occur, mathematicians Academy, says “Yes I tend to think recurring paradoxes. began to write books on numbers the Hypothesis is true (and I think and number theory. They also most mathematicians tend to Many mathematicians think that started to derive arithmetic think this way),”. The Hypothesis the use of complex analysis, a very concepts. During this time is a mathematical concept that elementary concept, can’t prove analytic number theory took rise needs proving.” says Dr. Paige. the Hypothesis, and that new and the Hypothesis was proposed. Mathematicians have to prove methods must be used to prove it. the Hypothesis, and not form an However, other mathematicians Bernhard Riemann also came up opinion. think that complex analysis can with a number of different topics prove the Hypothesis. Right now, swwuch as the Riemann sums, an we can not identify the correct approximation used in calculus The Hypothesis exists as a party because it has not been to approximate the area under a hypothetical mathematical proven. curve. Riemann also worked with concept, that would probably not manifolds, and with continuous have any physical application or Many proofs of the Hypothesis and discrete systems. would have application to physics have been attempted; however, or any other science. “I think a the submitted proofs have all Our society has a very low key point to realize is that most contained mistakes. Till this awareness rate outside the mathematicians don't do their date no one has ever proved the mathematical community when work for applications. They Hypothesis, but the one who it come to the Hypothesis. Those study may have applications, but does prove will receive a million who are aware about it do not that is not why mathematicians dollars. know much about it. care about it. Another thing to keep in mind is that thematics May 2013

Reason

Page 17


By Michael Martinez

broken rules. A fallacy occurs when this happens, and these errors in reasoning range far and wide, making it very easy to use one by mistake. These nine simple fallacies will help you better understand and avoid these errors. For a more complete list of fallacies, visit the Online Encyclopedia of Philosophy.

Sweeping Generalization: Making a generalization that disregards all exceptions. Example: Killing people is a crime. Soldiers kill people. Therefore soldiers are criminals.

Hasty Generalization: Making a general rule from too little data. Example: All apples that I have ever seen are red. Therefore all apples are red.

R.I.P.

R.I.P.

Johnny

Not Johnny

“He Died of Plague”

“He Died”

Affirming the Consequent: Inferring the converse of a statement from the statement itself. Example: If you have the Plague, then you will die. Johnny is dead. Therefore Johnny had the Plague.

Page 18

Denying the Antecedent: Inferring the inverse of a statement from the statement itself. Example: If you have the Plague, then you will die. Johnny does not have the Plague. Therefore Johnny is not dead.

Reason

Red Herring: Drawing attention away from addressing a claim instead of analyzing its content. Example: Claim: Humans can go to the moon. Argument: Neil Armstrong says humans can go to the moon, so humans can go to the moon.

? Begging the Question: Supporting a conclusion with premises that assume that conclusion. Example: People become addicted to smoking because smoking is addictive.

May 2013

Art By (clockwise from top left): Own work, Gigillo83(Wikimedia), Gervais et Boulart, Mohamed Ibrahim, Own Work, Own Work

Logic is a very big and complex subject that contains many easily


OFF

Art By (clockwise from top left): Own Work, Rursus (Wikimedia), Deluge (Wikimedia), Wikipedia, Wikipedia, Raphael

False Cause: Incorrectly assuming something is the cause of something else. Also called Non Sequitur, Latin for “It does not follow.” Example: I turned off the lights, therefore it is dark.

Straw Man: Misinterpreting an opponent’s position to more easily argue your position. Example: Michael: I like doing math. Ishan: If all you did was math, you would starve.

A FEW MORE FACTS ABOUT FALLACIES

May 2013

MATHEMATICAL FALLACIES Mathematical Fallacies are a completely

1. Fallacies were first categorized by Aristotle, a Greek philosopher, in his “Sophistical Refutations” 2. The above examples are called Formal Fallacies. There other types of fallacies include verbal and deductive fallacies. 3. Some fallacies have very interesting names, such as the Texas Sharpshooter Fallacy (finding patterns from random results) or Reductio ad Hitlerum (disputing an argument by involving Hitler).

No True Scotsman: Modifying an argument to exclude a counterexample. Example: Isaiah: All Scots like to eat haggis. Thomas: My grandfather was Scottish, and he hated haggis. Isaiah: Well in that case, all true Scots like to eat haggis.

different type of logical error. Errors in reasoning like other fallacies, they occur only in mathematical proofs and can lead to nonsensical and incorrect results.

Some Examples: When someone makes an obvious error in their reasoning but the answer turns out correctly, it is called a howler. Another type of mathematical fallacy occurs when a person uses radicals incorrectly. Aristotle

Reason

OR Page 19


Quantum Logic?

By: Ishan Shah

THE QUANTUM WORLD is indeed a very strange place; however, the fundamental concept that defines a lot of the behaviors of the quantum particles, the Heisenberg uncertainty principle, has some very basic logic behind it. The Heisenberg, the pillar of quantum mechanics, only requires normal Newtonian logic for its expliunation.

Lets begin this analysis by stating that the newtonian physics states the equation p=mv and thus states that light has very little momentum and can not affect any object (light has very little mass, this is known from the equation E=√(p2c2+m2c4 ) which Figure 1

shows the actual relationship between mass and energy of a particle with momentum and since light has energy it also has some mass). However, in a world in which the photons are almost as large as the electrons the photon’s momentum matters. In the quantum

world momentum is calculated through the equation p=h/λ=hk/ (2π) (Figure 1) where k is the wave number.

Art Work By: Ishan Shah

Light has momentum

Figure 3

Figure 2

Time

Figure 2 and 3 This demostrates the Hisenberg uncertianty principle. It demonstrates that as the wave length of the photon used to mesure the particle increases the change in momentum of the particle decreases. The diagram also demonstrates that as the wave length of the photon used to mesure the particle decreases the change in momentum of the particle increases .

Radio Gamma

Page 20

Reason

May 2013


Now that we know how to calculate the momentum of a photon, we can see how this quantum phenomena works. Any good photographer knows that the shorter the wavelength of light, the more precise the measurement. For example, we could measure the position of a golf ball better if we used gamma ration than if we used radio. In classical mechinecs the lightest measure can be made, because the light behaves like a wave and thus the amplitude of the wave also matters. However, in a world in which energy is quantized and light behaves like a particle, the amplitude does not matter, only the frequency does. And thus if a gamma photon

interacted with an electron, then the position of the electron could be known very precisely. However then it would excite the electron and change its momentum greatly. If we used a radio photon then the momentum of the electron would not change much but the position would not be known as precisely. Lets start by analysing the inequality: ΔxΔp≥h/(4π). This inequality indicates to us that the more we know about the moment or the change in momentum, Δp, of a quantum particle the less we know about its position or its change in position, Δx (Figure 2 and 3). The h is just planck’s constant, constant first used in the energy quantiztion

Figure 4

Δx

Figure 4 We can think of the change in positon as the diameter of the circle that encloses the area that the particle could be in.

equation. Lets now observe why this phenomena occurs. When Werner Heisenberg thought about this experiment, he first started to think about the fact that the measuring device used in this experiment would be photons which are small enough to be considered quantized. He thought that since the photon’s frequency now matters a lot more than the amplitude of the light and that the individual frequency matters a lot more than the frequency of the photon should be the main concern during this measurement.

Figure 5 This only occuyrs in quantum mechinics because in quantum mechinics the photons are allmost as big as the partical being mesured, in addition in quantum mechinics the photons behave as particals and not waves and thus the amplitude (the number of photons used) does not matter as much and the frequency of the photons matters a lot more.

Figure 5

Photons

Electoron

Through this analysis we can now understand the quantum phenomena: the Heisenberg uncertainty principle and through this we can also deduce why particles are represented as a wave of potential and why we can never know a quantum particle's exact location. May 2013

Reason

Page 21


CONWAY’S Everyone’s

G O L ame

f

ife

Favorite

Cellular

Automaton

Since 1970

Learn

More

At Conwaylife.com


Prisoner’s dilemma

By: Niels Kornerup

You and your partner in crime are caught and are put on trial with some weird rules: 1) You and your partner each get a chance to testify. 2) If neither of you testify, you each are sentenced with four years. 3) If both of you testify, you each are sentenced with six years. 4) If one of you testify, that person will get out free and the other is sentenced for 10 years.

Art by: Christopher X Jon Jensen and Greg Riestenberg

Now, it is always better for the individual to testify; however, it is better for the whole for the individual to be silent. Obviously, the best strategy is to testify because you can never trust your partner; however, the best possible outcome then becomes impossible. The game really becomes one of psychology, can you trust the other player or are they going to backstab you? You will feel horrible if you are the one who gets out free, but you can’t trust that the other person will be as sympathetic towards you. Multiple countries own nuclear weapons, which have the capacity to wipe out other countries that have Nuclear weapons. If two countries are at war, they both want to get rid of the other. If both of them have Nuclear weapons, they both have the capacity to wipe out the other. If nether side fires, the population of both sides will remain intact. If one side fires at the other, that side will win the war and the other’s population will be exterminated. If both sides fire at each other, both populations will be exterminated and both sides will lose.

May 2013

Reason

Page 23


What Are Birds? We Just Don’t Know.

Profile for LASA Ezine

Reason  

Using Premises to Reach Conclusions

Reason  

Using Premises to Reach Conclusions

Profile for lasaezine
Advertisement