A Mathematical Proof (for those who hate or fear math)

What follows is an example of a direct proof, the most basic form of proof. This post won’t be very interesting for people who already write proofs.

I

The photographs are of M.C. Escher’s Metamorphosis II, a woodcut print made during 1939-1940. 

★You don’t need to understand every detail. The ideal response is a spark of interest in mathematics. At minimum, you should know in advance what an integer is.

The integers: numbers that can be written without fractions or decimals.

Integers

symbol denoting the set of integers

The set of integers: {…, -3, -2, -1, 0, 1, 2, 3, … }

First, the proof is presented with zero explanation. Just note how short it is and how most of it consists of elementary arithmetic operations.

Afterwards, the proof is re-presented with excessive explanation.

Result: If x is an even integer, then 5x – 3 is an odd integer. 

Proof: Assume that x is an even integer. Since x is even, we can write x = 2k for some integer k. Then

5x – 3 = 5(2k) – 3 = 10k – 3 = 10k – 4 + 1 = 2(5k – 2) + 1

Since 5k – 2 is an integer, 5x – 3 is odd.

QED 

II

Now, taking it step by step:

Result: If x is an even integer, then 5x – 3 is an odd integer. 

Proof: Assume that x is an even integer. 

Direct proofs demonstrate that when the if-statement is true, then the then-statement is forced to be true. So the first step of a direct proof always assumes the if-statement is true.

Since x is even, we can write x = 2k for some integer k.

We don’t know what x is specifically, but we do know that x is a number divisible by 2 since x is even. k is an unknown, nonspecific integer, which is why we say in a vague way ‘for some integer k’. Side note: if x is an odd integer, we represent it as x = 2k + 1 for some integer k, since an odd integer has remainder 1 when divided by 2.

5x – 3 = 5(2k) – 3 = 10k – 3 = 10k – 4 + 1 = 2(5k – 2) + 1

5x – 3 is the expression whose oddness/evenness we’re ultimately concerned about. So we take the information from our last step, that we can write x = 2k, and we replace the x in 5x – 3 with our new representation of x, that is 2k. So

5x – 3 = 5*(2k) – 3

Now we simplify our new expression with arithmetic.

5x – 3 = 10k – 3

We change 10k – 3 to 10k – 4 + 1 since these expressions are equal, but 10k – 4 + 1 divided by 2 will have a remainder of 0 or 1 whereas 10k – 3 has a remainder of -1. According to our definition of odd numbers (represented with 2k + 1, where k is an integer) and even numbers (2k where k is an integer), we want a remainder 1 or 0.

Now we want to know if our expression (10k – 4 + 1) is odd or even, so we divide it by 2. If there is no remainder in dividing by 2, then our expression is even. Observe that there is a remainder:

5x – 3 = 10k – 4 + 1 = 2(5k – 2) + 1

Now, since (5k – 2) is an integer, 5x – 3 is odd. 

In the statement 5x – 3 = 2(5k – 2) + 1, the expression (5k – 2) is just some integer. We know this because k, 5, and -2 are integers and multiplying or adding integers results in an integer (see algebraic properties).

Since 5k – 2 is just some integer, we can reduce it to the unknown integer symbol k. In the equation, this reduces to 5x – 3 = 2(5k – 2) + 1 =  2k + 1, which is an odd integer. So we know that 5x – 3 is always equal to some odd integer when x is an even integer. So the result is true.

QED

QED just means the proof is done. It stands for the Latin phrase ‘quod erat demonstrandum’ which means ‘which had to be demonstrated’.

You can also indicate that your proof is done with this symbol ∎, which is called a tombstone.

III

Note that we started the proof by assuming the first part of the if-then statement is true. We assumed x is an even integer. Then we try to prove that the second part (5x – 3 is an odd integer) is true. Why this proves the result is true (even in cases when the first part is false) relies on simple, unintimidating logic.

Let P be a statement and Q be another statement. P ⟹ Q means that ‘P implies Q’ or ‘if P, then Q’.

This is a truth table for implication:

implication

In the proof above, P is ‘x is an even integer’ and Q is ‘5x – 3 is an odd integer’. P ⟹ Q is: If x is an even integer, then 5x – 3 is an odd integer. In the truth table above, if P is false, then regardless of whether Q is true or false,  P ⟹ Q is true. If P is true and Q is true, then again P ⟹ Q is true. Consequently, we only need to prove that it never happens that P is true and Q is false. Otherwise, our result P ⟹ Q isn’t true.

I learned most of what I know about proof writing from Mathematical Proofs by Chartrand et al. It was helpful in the transition from calculus/linear algebra to higher level math classes. However, it certainly doesn’t require calculus or linear algebra. With this book, you can start from scratch as long as you know some algebra.

Proofs

I assure you that the other types of proofs (especially contradiction and induction) are more fun and interesting than direct proofs. Furthermore, there are much more elegant, relevant, beautiful, significant, interesting results and theorems than this one. The main point of using this proof as an example was to demonstrate that it isn’t out of anyone’s grasp to understand and to write proofs.

I used to think math was dry, uncreative, lifeless, sterilized, and monotonous. I only started studying it (beyond a high school level) regularly and seriously a little over a year ago. Since then I’ve found that mathematics can be fascinating, beautiful, meaningful, even romantic. I’ve felt some of the same captivation and passion while writing proofs that I feel while writing poems.

IV

2 Comments

Filed under Uncategorized

Being or Nothingness

Image

I started reading The Psychopath Test at a bookstore and became immediately enthralled by the mysterious events introduced in the beginning. Ronson describes how intellectuals around the world began receiving anonymous envelopes from Gothenburg, Sweden containing a manuscript titled Being or Nothingness. The academics were of multifarious disciplines including technology, computer science, science, religion, psychology, literature, anthropology, and philosophy. This is the front cover of the manuscript:

Image

(Photograph by Levi Shand, who uploaded the entire manuscript)

There were many interesting quirks about the manuscript. B or N is 42 pages, but every other page is blank. The quality was remarked on by some recipients as ‘quite beautifully done’ and expensive looking.  On the envelope was written by hand: ‘Will tell you more when I return!’ The handwritten address numbers were written in an American, rather than European style. (By the way, you can buy Being or NothingnessI was tempted to until I found Levi Shand’s uploaded pages.)

So these intellectuals started posting about the mysterious manuscript online and others responded to them. They discussed the details of the package and manuscript, the meaning of the text and its dissemination, and why they thought they were chosen as recipients. I found these forums extremely interesting. One reason for my interest was that there were some posts suspected to be written by the author of the manuscript himself. Some of these suspected comments, all from the username MR, were:

“That is the key question, isn´t it – what does DD stand for? Ever since I read “The Fabric of Reality” I have been waiting for David Deutsch´s next book to see if he can carry ‘the unification’ futher – to a true unification. So I wonder, Could DD stand for David Deutsch? Could it be that a true ‘theory of everything’ cannot include it´s originator and the unusual method of distributing the book is just a consequence/part of the theory itself? Maybe DD just wants a little recognition! Daniel Dennet´s writings don´t seem compatible with the core theme of BON.”

and

“I am sorry for pestering everyone but I can´t let go. A true ‘theory of everything’ not only has to include its originator but also a copy of the theory itself, leading to infinite regress. BON creates the illusion of including its originator and through the loop created by ‘the letter to R’ it does contain a copy of itself, regressing infinitely. The book doesn´t seem to be a theory of anything but if I am correct then BON challenges the implications of Russell’s paradox – I wish some logician would step up and formulate this in terms of set theory.”

finally:

“I go back and forth – one moment I am convinced that BON is a ‘theory of everything’ only to conclude that it is a ‘theory of nothing’ a couple of hours later. I haven´t slept for more than 48 hours and cannot think straight anymore. After posting this note I plan to burn the book, drink two bottles of wine and when I wake up I will stay away from computers till things settle down.”

Ronson writes, ‘What seemed obvious was that a brilliant person or organization with ties to Gothenburg had devised a puzzle so complex that even clever academics like them couldn’t decipher it’ (Ronson 11).

Levi Shand’s role in the mysterious events was finding a box of Being or Nothingness manuscripts. He saw the sticker on them that said: ‘Warning! Please study the letter to Profesor Hofstadter before you read the book. Good luck!’ Shand, an Indiana University student, delivered the books to Professor Hofstadter, a cognitive science profesor at his school.

One of the people Ronson was working with suggested that Levi Shand didn’t exist, citing the fact that his name was an anagram for ‘live hands’ (as in the Escher painting). She theorized that Hofstadter created Being or Nothingness as an intellectual endeavor. Ronson then contacted Hofstadter, asking him about the conspiracy. Hofstadter denied any involvement with the creation of Being or Nothingness, except to say that he had received dozens of copies and some cryptic Swedish postcards. Hofstadter conjectured that whoever was behind the conspiracy was abnormal, obsessive, and likely insane. Ronson’s realization occurred then: ‘Yes, there was the missing piece of the puzzle, Douglas Hofstadter was saying, but the recipients had gotten it wrong. They assumed the endeavor was brilliant and rational because they were brilliant and rational, and we tend to automatically assume that everybody else is basically just like us’ (31).

Ronson describes how he went to Sweden to find the purported author, a psychiatrist named Petter Nordlund (pseudonym for Per Norfeldt). Nordlund was named as the English translator for Being or Nothingness in a Swedish library archives. When Ronson finally meets Nordlund, he soon concludes that he is the man behind the conspiracy. ‘[Nordlund] had a big, kind, cryptic smile on his face, and he was wringing his hands like a man possessed’ (32).

This story was the impetus for The Psychopath Test. Ronson marvels over the impact of one man’s off centered mind on the rest of the world: ‘Disparate academics, scattered across continents, had become intrigued and paranoid and narcissistic because of it. They’d met on blogs and message boards and had debated for hours, forming conspiracy theories about shadowy Christian organizations’ (34).

Image

(Photograph/photocopy by Levi Shand)

The events of Being or Nothingness form only a small portion of Ronson’s book. The next chapters more specifically investigate madness in the form of psychopathy. The parts about B or N were most interesting to me, but the rest of the book does contain some interesting information (as well as some uninteresting information). For example, the book raised some questions for me about whether people think that the state should preserve individual freedom at the cost of incarcerating innocent, but dangerous people in order to protect the wellbeing of society. There’s plenty more to say about the book and its thesis, which I found ultimately dissatisfying (except insofar as Nordlund’s reappearance), but that can be a post for another day.

I think part of my dissatisfaction with the rest of the book was that I wanted more stories of fascinating, nonstandard individuals like Nordlund. I found the story of Being or Nothingness extremely compelling for a number of reasons. For one, I like ‘postmodern literature’ which is often characterized by multiple, unreliable narrators. An obvious example of this House of Leaves. I also like nonlinear narrative and hypertext as in ‘The Garden of Forking Paths’ and ‘Continuity of Parks’ by Julio Cortázar (in Blow-up and Other Stories). More broadly, I’m drawn to texts that deviate from the conventions.

Joe K writes, ‘This I have regretted many times. The manuscript has haunted me ever since’ (K 6) and ‘Should the text resemble what its cover implies it to be, reading it could be dangerous’ (3). This reminds me of Johnny Truant describing his experience with Zampanó’s manuscript. There is no evidence that The Navidson Record, the subject of Zampanó’s manuscript, exists; similarly Joe K’s subject, “Sir Arthur Conan Doyle’s long lost manuscript ‘Being or Nothingness’ commonly referred to as ‘The Giant Rat of Sumatra'” seems to be made up (I haven’t done enough research to sound more certain than that) (2). In The Adventure of the Sussex Vampire, Sherlock Holmes says, ‘the giant rat of Sumatra, a story for which the world is not yet prepared’, which is the only time I’ve found Sir Arthur Conan Doyle refer to the giant rat of Sumatra.

With fiction, I don’t want everything to be clear and comprehensible at first glance. I want to be pushed to think and to make unexpected connections. That’s infinitely more rewarding than being given the proof already solved.

Another reason for my intense interest was that I had been intending to read Gödel Escher Bach for awhile and the connection to Douglas Hofstadter was intriguing. I still haven’t read GEB, but I’m planning to start it again today or tomorrow if all goes optimally. With an impression of Hofstadter as a highly curious person, I was a little surprised by his denunciation of Being or Nothingness. Jon Ronson gives context for this reaction and I suppose if I was a tremendously successful thinker and writer, I too might deem such attempts to catch my attention uninteresting… yet at an essential level, I like trying to solve puzzles, codes, and mysteries. Or maybe I just like getting mail.

Lastly, I also find intelligence, especially in conjunction with madness or wild nonconformity, to be especially interesting. The ideas produced by such a person can be staggering. It’s clear that Nordlund is highly intelligent. The orchestration of the conspiracy was done skillfully. Enshrouded in mystery, the text produced an abundance of discussion and reaction. It was an ingenious way to get a lot of people (who didn’t receive the manuscript) to want to read his ideas. A less intelligent (or more unhinged) crazy person would wander the streets, raving to whoever came by.

With that, I’ll end with a concluding quote from The Psychopath Test: ‘in fact our unhappiness and our strangeness, our anxieties and compulsions, those least fashionable aspects of our personalities, are quite often what lead us to do rather interesting things’ (Ronson 230).

P.S. Analysis of B or N deserve a post of its own!

Work Cited

K, Joe. Being or Nothingness. 2007.

Ronson, Jon. The Psychopath Test. New York: Penguin Publishing, 2011. Print.

Leave a comment

Filed under Uncategorized

How to Think for Yourself: Act 1

What follows in no way pretends to be an authority on the crucially important subject of thinking for oneself. These are preliminary, rough ideas. You should think for yourself when it comes to thinking about thinking for yourself. This storm of ideas begun in a conversation with a meta individual who will be introduced eventually, as he is one of my intellectual influences. I’ll refer to him for now as G.

Image

  1. Acknowledge that much of what you think and how you act is a product of the system. Conformists socialize conformists. People lied to become liars. This is the first movement: understanding the cycle of obedience and groupthink.
  2. Start consuming ideas about epistemology, rationality, logic, the scientific method, mathematics, and related topics. These subjects inform the thinker about the nature of thought, bias, knowledge, truth and falsehood. It’s important to have some basic ideas about how true beliefs are formed.
  3. Don’t be content with what you currently know, even if you’re an expert. Every thinker needs to keep updating with new information, research, and insight.
  4. Become aware of the norms of thought, belief, and behavior that govern your life. Start the search and analysis of ideas and assumptions that have been programmed into you since you were a child. Question why you hold these ideas and whether you actually have good reason to keep holding onto them. As your perspective deviates further from the starting point, keep returning to these ideas to see if they’re still valid.
  5. Confront the beliefs you hold most closely. Ask yourself why you have these beliefs and test if they withstand the pressure of evaluation and increased perspective. Is there evidence to support your belief? Is that evidence valid? What are the counterarguments?
  6. Abandon arrogance regarding any quirkiness of positive abnormality you’ve exhibited in the past. It isn’t enough to rebel in socially acceptable ways—you need to acknowledge this and thereby, concede that your historical rebellions have been weak and do not deserve the label of rebellion.
  7. Actively read and evaluate the validity of the ideas of others. Learn how you learn and how you formulate your views on the world. Continue to challenge your beliefs and assumptions.
  8. Question your actions. Observe yourself as if from a third person perspective: what rules and laws, written and unspoken, do you follow? Why? Is your obedience driven by habit, fear, or a need to feel socially accepted? Analyze your reasons and see if you consciously agree with your conditioned mind. If you think your reasons are valid, try to defend them with reasoned argument. Then attack those reasons. Write all of this down.
  9. Challenge yourself to break codes, rules, and laws that you follow. It doesn’t have to involve anything illegal. Be careful if it does. The point is to challenge yourself to break out of the box of normality and conformity. Start small and see how it feels. Write down your thoughts about it. Changing your behavior can lead you to new insights about yourself and new thoughts about your ways of thinking.
  10. Avoid media that encourages groupthink and promotion of ill-researched, ill-considered ideas. This includes a lot of journalism, online communities, and websites where a system of upvoting is in place. Such websites like Imgur encourage the adherence of the individual to certain norms to maintain the interests and status quo of the group.
  11. Spend less time with conforming individuals and try to find others with solid, interesting ideas who think on a high level. It is highly instructive to find those who will challenge your ideas in an informed, reasoned way and who will let you challenge their ideas.
  12. Don’t make the mistake of innovating against conformity, society, and groupthink by adhering to the doctrine of some religious figure, by adapting the conventions of people you find eccentric and interesting. It’s okay to experiment with some of their ideas and behaviors. It is okay to steal some of their ideas too (don’t plagiarize them!). However, it is not okay to not create your own ideas about who you are, how you act, and your reasons and ideas. You will get stuck in another box of not being able to think for yourself if you don’t invent and experiment with your own original ideas.
  13. That being said, it is highly necessary to read, to learn, and to expose yourself to as many great, interesting ideas as possible. An individual grows much faster in a community of other individuals thinking at a higher level. You want to have good ideas! Thinking in a vacuum isn’t conducive to good ideas. You must take great thoughts and inspiration from the world because it’s impossible to derive all of them yourself. But don’t be satisfied with mere absorption. Encountering others’ ideas is crucial, but you must think about them and challenge them. Let them spur you to further ideas. Always, the emphasis is further ideas: never be satisfied with the ideas of others. Use them, but don’t let them be a stopping point for you.
  14. Read literature on groupthink and conformity. Read psychological studies and literature. Some obvious example starting points are Milgram’s experiment on obedience, Asch conformity experiments, the Stanford prison experiment, 1984, Steal this Book, and Fahrenheit 451. One blog that has been invaluable in my own process of learning how to think is the rationality blog Less Wrong, specifically the sequences by Eliezer Yudkowsky.
  15. Never stop. Don’t be satisfied with your amount of deviation. The goal is to get to a state where you are changing, improving, inventing constantly. If you’ve learnt how to think for yourself, but you stop thinking, then you’ve lost. Never stop thinking. ‘If you are thinking, you are winning’.
  16. Challenge yourself more and more. Keep breaking rules that you have considered and deemed stupid. When doing something illegal, don’t be stupid. You are still living in a community of obedient group members, even if you aren’t thinking in their community any longer. Be clever. Be careful. Don’t forget that you are still fallible. Don’t get overconfident about the rightness of your ideas, especially when your wellbeing is contingent on society’s notion of rightness (i.e. if you don’t want to go to jail).
  17. Figure out what you want to do. Figure out who you want to be. Recognizing and deflecting the influence of conformist thinking allows you to do what you as an individual want to do with your life. You’re no longer constrained to the narrowness of possible options and ideas put forth by society as the only ones. Seeing the potential for your creation and innovation is the freedom to break through the walls of the limited, inelastic reality posited by society.

Image

Leave a comment

Filed under Uncategorized

Double-headed serpent turquoise mosaic

Double-headed serpent turquoise mosaic

Leave a comment

March 3, 2013 · 9:34