Poll 1

(Midlertidig blogpost.)

Dere som er i mine forelesninger i Matte 1, svar på disse polls:

Tempo (kun i forelesningen, ikke snakk om video):
http://www.easypolls.net/poll.html?p=524ada90e4b07664a11a418b

Metoden (kun i forelesningen, ikke snakk om video):
http://www.easypolls.net/poll.html?p=524adb71e4b07664a11a418e

Videoopptakene:
http://www.easypolls.net/poll.html?p=524adc24e4b07664a11a418f

Prosent 1:
http://www.easypolls.net/poll.html?p=524adcdde4b07664a11a4191

Prosent 2:
http://www.easypolls.net/poll.html?p=524add63e4b07664a11a4192

Takk for hjelpen!

Understanding Thought part 3 Answers

So here is the answers I got to the challenge in the previous post. I will try to present them as faithfully as possible.

Dialogue with Person 1

Me: So, what does it usually take for you to change your mind?

1: A well thought out argument, or new information that I did not know before.

Me: Does it happen often, do you feel, that you have to change your mind?

1: No, rarely, but I remember this one time... So, all in all, I was given a new perspective from a very productive meeting with people who'd done their homework.

Dialogue with Person 2

Me: What does it usually take for you to change your mind?

2: I don't know. I guess it just happens when it happens.

Me: So when you go into a discussion, do you consider it likely that you will change your mind?

2: No, I never change my mind in a discussion. Sometimes I'm able to convince the other guy, sometimes not.

Dialogue with Person 3

Me: You say you change your mind when presented with good arguments. Let me take an exaple. If a random stranger and a known authority was having a disagreement, who would you listen to? Let us say that the discussion was about which food is healthy and what is not, and the random stranger had much better arguments. Who would you listen to?

3: Definately the authority.

Me: So what do you think is the most important factor when there is a complicated choice to be made, the authority or the arguments presented?

3: The arguments are relevant, but authority is the most important.

Dialogue with Person 4

Me: Is there any beliefs you have that are unchangeable, areas where you will never consider redesiding?

4: There are some things, like my atheism, that I have investigated for so long that I am now sure of.

Me: Let us take ateism as an example. If some entity were to float down from the heavens now, and tell you that it was God, and then continue to prove it to you by moving that mountain over there [we had a nice view where we were], what would you think?

4: I would look for any other explanation before re-evaluating my atheism.

Me: Does that include "I am hallucinating"?

4: Yes.

Me: So would you consider your belief about autheism to be a "holy" truth, excuse the phrasing, that no amount of evidence would ever change?

4: Yes, I think so.

Dialogue with person 5

Me: You said you would believe the scientist over the politician, why?

5: Well, the politician has a vested interest in the truth, and the truth is mostly what suits him, and, of course, the stuff that is undeniable. While the scientist, mostly, doesn't care what's true, he/she just wants to know.

Person 6 correspondence:

He decided to answer the specific questions I asked, in writing. See the previous post for the questions.

1. In the long term, as in months and years, I feel it happens automatically, but then again, I do a lot of thinking and reflection in my spare time. In the shorter term, I sort of imagine discussions as existing along a spectrum from exchange of opinions to a no holds barred duel. I think it's generally a good idea to try to keep the heat down, particularly if you have to reach an agreement - not only to win the opponent over more easily, but also to avoid committing yourself to a position you might not want to defend. I think this is more subtle than it might sound - changing your mind from A to B, even if you never really meant A in the first place, but that was somehow how he wound up misinterpreting you, is a lot harder than keeping your options open and hitting closer to B the first time around.
   
   
2. I don't really flag any of my beliefs read-only, but some of them are definitely more fundamental and support more load and are in that sense more resilient to change. I do have a sort of religious conviction that choosing to lead a reasonably selfless life is not actually a personal sacrifice, but counter intuitively the sort of optimal path to happiness. This is not a belief I try to challenge regularly, but nothing I would have trouble discussing either, so it probably doesn't count.
   
   
3. I started writing down a lot of completely irrelevant examples. I think what you really want here are examples of arguments or data which made me change my mind about something as some sort of discrete event. I have a really hard time giving any of these. I have a feeling that a lot of examples which could have gone here are sorted differently. Even when the catalyst is really something someone said, it tends to just remain as some sort of lingering memory, and some day you give it appears camouflaged as your own idea and you imagine you figured it out all on your own. For example, I don't think you'll hear somebody tell of their religious crisis when Bob told them you'd have been a Muslim if you were born in Pakistan - inside your own mind, it tends to appear as if you just had a private epiphany.

• If I have to give an example, I remember when my flatmate told me to watch Death Note because it was great, and I said it probably sucked, but he insisted, so why not, and for the first two episodes I was thinking I was right, Jesus, this is some stupid, contrived shit and then I watched the whole show in about five days.

4. I'm afraid 3 sort of swallowed everything I could have said here. I think presenting new data is always a good idea, though - if nothing else, it makes a lot more sense to say oh, I didn't know that, guess you're right than oh, I guess I was being stupid and your reasoning is better than mine. The Socratic dialogue can also be an eye-opener - being told what you ought to mean isn't very pleasant, but the whole if you mean A, do you also then mean B? can be a great way to discuss. Again - it is easier to find a good idea when you don't first have to abandon an old one which you might have invested some prestige in.
   
   
5. It sounds ridiculous to vaccinate yourself against ever changing your beliefs - it doesn't sound very likely that you currently are right about something but could be fooled into changing your mind to something wrong. If you ever want to discuss something with someone, it would be unfair to not be open to new ideas yourself. Altogether, though, I don't feel any urgent need to become either more or less open minded.

Understanding Thought part 3 Challenge

Surprise, I'm back.

I want to start this time by issuing a challenge to the readers. Send me an email or post an answer below. After a few weeks, I will post the answers in a new post anonymously.

I will also ask some of the people I meet during the summer.

How do you change your mind?

Some approaches to this question could be (inspiration only):

1. In general, what techniques do you have to enable yourself to change what you believe?
2. What elements are beyond change, so that you will not consider changing them?
• (Example I'm thinking of: Religion)
3. Can you describe an event when you changed your mind?
• Both small events like buying a phone, and bigger, like political view or religion
4. What kind of argument must be presented in a discussion for you to change your mind?
• Not just saying: "Yeah, whatever, OK"
5. Do you feel a need to be able to change your beliefs/truths about the world?

Understanding Thought part 2

Idea: Truth Projection

Let us start wth an example:

1. You have 2 apples and 3 pears
2. Someone asks you "How many apples do you have?"
3. You are only allowed to give a number in your answer.
4. What will you respond?

My answer here would be "3 apples" I think. Why?

Let us first agree on some things. If the rule in 3 wasn't there, you'd say "I have only 2 apples, but I do have 3 pears also". Perhaps someone is hungry, baking an apple cake, making apple juice or something else. Here it would be interesting to know about the pears.

The other thing we should easily agree on is that some answers are plain wrong: If you answer "1" or "0" that is (for all possible intents) wrong. If you answer "6 apples", "7 apples", or more, that is also wrong (for all possible intents of the question).

So why "3 apples"? I can picture a range of different scenarios (intents of the question). If the person asking only could eat apples and was hungry (think Death Note), then you would be slightly wrong. The same goes if the one posing the question was in the store and could easily have bought 1 more apple.

What if the one who asked wanted to make apple juice, and he thinks that pears are good too. Well, then 2 apples and 3 pears would make a lot more apple/pear juice than would 2 apples (150% more). On the other hand, 2 apples and 3 pears would make a bit more juice than 3 apples (67%).

In day-to-day life, this doesn't happen with apples and pears, as it is simple to give all the information. But you can never give all possible information, so we try to say what's relevant, and to not say stuff we think is irrelevant

This phenomenon appears clearly in the teaching of mathematics and physics.

Mathematics example: We normally teach that all (positive) functions can be integrated. Some teachers tell the students that there are some (positive) functions that cannot be integrated, but you would never teach what kind of functions these are, so in essence you say "Any function you will ever encounter is integrable". Then when you study some more you learn that there are lots of functions that cannot be integrated. Then you learn that these can actually be integrated, we just have to expand our method of integration a bit. Then you learn that even with this new kind of integration there are some super-strange functions that cannot be integrated.

Physics example: When you learn about electrons on a small scale (quantum mechanics), you learn that they work the way waves on the water works. You can have one big "up" wave meeting a big "down" wards wave, and cancel out. When a wave hits a small opening, it expands out after this opening. But electrons do not work as waves on the ocean; they can be collapsed, they have more directions than just "up" and "down" (they have a kind of imaginary "left" and "right" also).

Often the whole point of being pedagogical is to be able to give the best possible Projection of what you considers to be True, down onto the 'plane' of the listeners understanding.

Understanding Thought part 1

- Preliminaries

There's nothing like reading a book every day for two weeks, so I'm a bit behind. I have a dozen half-finished blog posts that, for some reason, never seems to get past the initial stage into something half-finished (i.e. ready to post).

This series/collection I will start on now is a rethinking of how I blog. I will take one big issue and go with it for several months. I would very much like to be able to start with the beginning and just give a clear and concise presentation, but this is impossible as I don't know this stuff. What I will do is present my finding when researching the research and ideas on the subject.

My goal is to post at least once in each calendar month. The purpose is to understand how we think, how the processes of thinking works, what we do differently, and the difference between intuitive and analytic processes.

Before we start this adventure we need to agree on some things and some definitions of words.

I read an Internet-poll today that asked "Do you believe in God" with alternatives "Yes" and "No". I could write a book about how limited this model of yes/no is, but let's instead look at what I will use as definitions (with minimum time in parenthesis):

A Thought – A stray thought, random combination of ideas or words (2 seconds)

An Idea – An insight that you think may be fruitful to investigate (10 minutes)

Theory – An understanding of the world that you consider to be important and true, and from which you can draw conclusions about actions and results (descriptive and/or normative) (1 hour)

Active Belief – Something you, personally, use to decide what you do, which actions to take. A theory where you follow the conclusions.

I understand some natural progress between these four: Though -> Idea -> Theory -> Active Belief.

So let me try to describe the differences here. You are not allowed to believe in a though or an idea, they are independent objects of study. You are allowed to 'have faith in an idea', think it is a very good idea, when working with it, but it is not something ready for a true/false discussion yet. You can believe/unbelieve in a theory (and, of course, in an active belief). I will use the word unbelieve when you believe the opposite of the statement. This gives us 3 modes: belief, uncertainty, unbelief.

What should the possible answers to "Do you believe in God" be using these words? (After any clarifications you may need; it is, after all, a rather ambiguous question.)

- I am uncertain (I think this was in the original poll)

- In Theory Yes

- In Theory No

- In Active Belief Yes

- In Active Belief No

I am uncertain is typical agnostic. In Theory No is typical 'I don't see any reason to believe it', but someone who is too lazy to withdraw from the state church; someone who does nothing about it. In Active Belief No is a typical atheist, someone who tries to remove the state church, someone who argues that there should be no mandatory religion course in school (or if it is, then it should contain humanism and other big religions equally). These people are certain that there is no God, so why should we spend time on Him/Her/They?

Most people I know who call themselves christian are 'In Theory Yes', they say "we believe", go to church, marries in church, and does everything religious that is considered normal in the culture/society. If I am of this group, and I think abortion is OK (within the set limits), and one night God comes to me in a dream and says "Abortion is wrong" (and I have a religious experience), what will I do? Nothing/Ask for proof of His existence. So this belief is not active, I do not take actions based on it. This group writes God with capital G, and Him with capital H, no because they are afraid to be disrespectful to God, but to other religious people. One of the things that makes this group uncomfortably misunderstood by atheists (Active Belief No) is that they tend to believe strongly things that seem very contradictory. For example, they know that monotheism was invented about 3000-5000 years ago, that there are a thousand different gods (and it is improbable that you should choose the one in your state religion as The God), that the earth is round and goes around the sun, that the exodus from Egypt is more of a folk tale than significant history, that life on earth is a million years old.

Last, but not least, we have the 'In Active Belief Yes'. They can start an argument by "The bible says ..." or "The pope said, ex cathedra, that ...", they can spend resources converting others to the faith, and they can feel bad when their children are not properly religious because (in many religions) they will end up in (some sort of) hell (eternally?).

First we make the decisions, then we make the reasons. I don't know what this is called, but it is a very strong fallacy. One of the modes in which I think is deciding on an option, then writing down any arguments I can come up with. Then deciding on the other alternative, and writing down any arguments for that. Then I try to read the arguments with an open mind. This method is a result of how our brains (at least mine) work – it is a lot easier to come up with arguments after taking a standpoint. I really need a good name for this fallacy; the fact that the reasons you present are not the reasons that made you decide.

Example: Let's say I drive at a speed of 110 on the highway with speed limit 100. Perhaps that is because it is unsafe to drive at a speed of 120? But more commonly it is because you get an expensive speeding-ticket. 

The most important thing here is to stop lying to yourself. Later you can consider telling the truth (surprisingly often this is embarrassing, or you'll come across as very frank). I will do my best to note when I make this fallacy.

Other stuff:

- I will use a numbering for later reference.

- Note that I have allowed room for nonscientific theories with these assumptions. You could make up untestable hypotheses, like "praying only helps the faithful" etc. Even though you can't do a proper double blind, there is usually some way to do a statistical test if you allow for weaker conditions.

- Surprise is a good thing when we search the scientific literature.

Infinity in an Hour

- How it Feels to study Pure Mathematics

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

- William Blake

I have a master in pure mathematics (it also includes a substantial amount of applied mathematics and physics). People often wonder what is it that we do? And they visualize the most complicated mathematics they know, and they try to take it to the next level mentally. But it is very hard to try to understand something you know nothing of. People who only have a minimum of mathematical knowledge guess "Oh, so you multiply really large numbers", those who had math in high school/college guess "Oh, so you differentiate and integrate difficult functions?", those who have a university education where they needed advanced mathematics guess "Oh, so you solve hard differential equations on difficult grids/spaces?". Today I will endeavor to give a picture of how I feel when doing pure math.

For inspiration, I recommend the following videos. After we have seen them, we will try to do some of this mathematics. Note that we mention Cantor several times.

http://video.google.com/videoplay?docid=-8492625684649921614&hl=en

(Part 2 if you're interested)

http://video.google.com/videoplay?docid=-1663091361786740235

Pure mathematics is that of unlimited abstraction and precision. We define everything as precisely as humanly possible, and we try to make everything into abstract concepts. Understanding something in pure mathematics is often like walking a tightrope over an abyss, no room for small deviations or imperfect intuition.

A lot of the problems I think about when working, you would need 4 years of university education just to be able to ask. But now I will use some examples that you need almost no mathematics education to understand.

First we will look at sizes of infinity (this is very close to the paradoxes talked about in the videos above). To talk about size, we have to define it, and size of what?

Definition A set contains elements, and an element is contained in a set.

Example {1,2,3,5} is a set containing the elements 1, 2, 3, and 5.

Example 2 {a, b, car, boat, 99} is a set containing the elements a, b, car, boat and 99.

Definition 3 The size of a set is the number of elements it contains

Example 4 The size of {1,2,3,5} is 4

Example 5 The size of {a, b, car, boat, 99} is 5

Using these definitions, we could have defined a set to be infinite size if it has no finite size. But to be able to speak of different infinity sizes, we have to use a more fine tuned definition.

Definition 6 Two sets are equal if there is a bijection between them

Definition 7 A bijection is a function which maps all the elements of a set to the elements of another set, and where no two elements are mapped to the same.

To understand this last definition, you should see some examples. So look at wikipedias page bijection.

For more information, see function and Bijection-Injection-Surjection.

Now we can state the first question

Question 1 Are there as many even (positive) numbers as there are (positive) numbers in total?

Answer 1 (with proof) Yes. Let E={2, 4, 6, 8, 10, ...} be the set of even numbers, and let N={1, 2, 3, 4, 5, ...} be the set of all numbers. Then we can construct a function, f, from E to N by f(x)=x/2. This function is a bijection. Hence the size of E is the same as the size of N.

How can this be? Clearly the set N contains the set E, and then some, how can they be equal? Well we just proved that they were. The only thing we can conclude form the fact that N contains E is that the size of E is smaller or EQUAL to the size of N.

What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite.

Calculations with infinite

If you use a modern computer program, it often has inf (infinite) as a kind of number. It will normally give the following computations:

inf + 1000 = inf

inf – 1700 = inf

inf + inf = inf

inf*1000 = inf

inf*inf+inf^inf = inf

1/inf = 0

1/0 = inf

inf-inf = NAN

inf/inf = NAN

(-1)^inf = NAN

Here NAN means Not A Number. That is because you are not allowed to do these operations, the result is undefinable (if you chose a definition you would end up with a contradiction). The problem with inf-inf is that we don't know which inf is "largest". In some applications you may get the answer 0, in others 31, in yet others you may get inf.

Countable and Uncountable infinity

The size of the set Z (all finite numers), or the set N (all positive finite numbers), is called countable infinity (the sizes of Z and N are equal). It is easy to prove that the set of all fractions, the rational numbers Q, also has countable size (see http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).

But what about the set R of real numbers (all numbers, including pi and the square root of 2, but not including imaginary numbers), is this also the same size as Z and N and Q? No. This proof is quite deep, and took me several days to understand (several years ago). If you want a challenge, see wikipedia's page on Cantors diagonal argument.

The Length of the Rationals (the set of fractions Q)

There is another very much used notion of size. This is what we use for integration, and to avoid confusing it with the size of sets from before, we call this new thing for length.

Definition The length of an interval on the real line, is the right endpoint minus the left endpoint.

Example We write [-3,7] for all the numbers between -3 and 7 including -3 and 7. The length of this interval is 7-(-3) = 10.

We can generalize this concept of length to other sets than intervals, for example to the union of intervals. Not surprisingly we get:

Theorem If one set is contained in an interval, the length of the set is smaller or equal to the length of the interval.

Supertheorem: The size of Q is countably infinite, but the length of Q is 0 (on the real line).

Proof: We have already seen that the size of Q is countably infinite. What about its length? Well, write Q as a sequence Q={q1, q2, q3, q4, ...} where all qi are fractions. Let K>0 be any arbitrary number (for example 0.000000001). Then the first fraction, q1, is contained in an interval of length K, namely [q1-K/2, q1+K/2]. The second fraction is contained in an interval of half that length (namely [q2-K/4, q2+K/4]). The third fraction q3 is contained in an interval with half that length again. Let us sum up this:

q1 contained in an interval of length K

q2 contained in an interval of length K/2

q3 contained in an interval of length K/4

q4 contained in an interval of length K/8

q5 contained in an interval of length K/16

...

So Q must be contained in the union, which will be an set with size smaller than (smaller because some of the intervals may overlap):

K + K/2 + K/4 + K/8 + K/16 + ... = 2K

How to calculate this? This is what we call a geometric series.

What do we now know? Q is contained in something of length 2K (you can choose any K>0). Hence the length of Q is smaller or equal to 2K (you can choose any K>0). The only possibility is that the length of Q is 0. QED.

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Comment by Nok er Nok:

I'll keep beating the same horse as always, I guess. We have spent some time discussing these questions and generally agree, but I can't help but object to the following paragraph. I have mostly the same knee-jerk reaction to it as the Newcomb's paradox, Hangman's paradox, etcetera. On the whole, I'll even illustrate my point with an XKCD strip (gasp!),  http://xkcd.com/169/   , though I'm not sure if the miscommunication is intentional or not.

"What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite."

This is most certainly a paradox. The mathematical formalism you describe and which mathematicians use is consistent and useful, yes, that's not what I'm trying to deny. Using that formalism to claim there are -as many- even numbers as natural numbers, however, is dishonest; using layman's terms in that manner -creates- the paradox. Nobody protests against the bijection-wizardry which is firmly belonging to alternate math-dimension. They do, however, have issues with mathematicians redifining common English words to mean something entirely different, and then marveling at how counterintuitive it is - particularly when they a few moments later set the trap by mixing real world example, impossible mathematical constructs, normal English and indistinguishable mathematical definitions.


Let's deconstruct Hilbert's Hotel to start with, and let's pretend we are not familiar with the mathematical formalism for dealing with infinities. Hilbert then claims the following:

 - hotel with infinite number of rooms
 - infinite number of guests at the hotel
 - such that all rooms are occupied, each by a single guest
which is all fine and dandy, but sets up for the counterintuitive conclusion
 - the hotel can still house another group of guests, exactly as large as the number which currently occupies every single room

Now, if you look at that without consulting your English<->Mathematics dictionary, you will surely conclude that this is perfect nonsense. Somebody is pulling a cheap parlour trick on you, one of these words have to mean something else that is appears. You can easily construct equally (more?) valid arguments than Hilbert presents, to show that the conclusion is impossible. For instance, it is easy to visualise that no room will be left unoccupied after you swap any two guests between their respective rooms, any number of times, and even though Hilbert does this an infinite number of times, this shouldn't change anything.


The parlour trick here is, of course, that occupied does not mean occupied at all, it has to do with bijections, and infinity does not mean a number you can increment arbitrarily many times, it is instead some mathematical construct with such properties that it cannot possibly have anything to do with any actual hotel. You might say the point of Hilbert's Hotel is that infinity cannot be treated as just any large number, you claim that our intuitions are not prepared to deal with infinity, but I strongly disagree. Hilbert's Hotel only shows that the mathematical lingo he ends up translating to 'occupied' and 'infinity' has nothing to do with a normal understanding of these words.

Taken as a story to accompany the mathematical formalism, to illustrate how it handles infinities, cardinality and size as something to do with bijections, it does an okay job. Without that context, it is not the slightest bit clever or enlightening, but just a load of gibberish. Hilbert's Hotel says -nothing- -whatsoever- about how hotels of arbitrary size work; it intentionally mixes mathematical formalism with a real world example which it then -fails to describe-!


Of course, all of this comes from the same sort of people who with a straight face will call f(x) = constant  an increasing function - and a decreasing one, at the same time. Nevermind that increasing is a code word for non-decreasing, which you cannot know without consulting your Google-translate English<->Mathspeak, or being familiar with the tradition of inclusive definitions in mathematics.

I think inclusive definitions are useful, and I'm not quite decided on whether using somewhat familiar but inaccurate and misleading terms is better than inventing new, arbitrary ones. However, I'm certainly not going to give mathematicians any credit for clever paradoxes which does nothing but illustrate that the mathematicians themselevs do not understand that their redefined words cannot be inserted into common English prose without appropriate and careful translation.


As an endnote, I feel fairly certain that it would be very possible to develop a formalism in which the natural numbers, the even numbers, the prime numbers and so on and so forth were -not- the same size. Of course, these alternate defintions would not develop fruitfully into integration and cardinality, like the current one does, but this alternate mathematics would be able to present the exact same Hilbert Hotel and the exact opposite conclusion; the hotel -cannot- accomodate even one more guest, much less another infinity of them. Or perhaps they would balk the moment you suggested that every one of the infinite number of rooms is currently occupied. And if this is true, it should be all the more obvious why Hilbert's Hotel is, indeed, a paradox of sorts.  

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Morality from Evolution

-Prisoners dilemma, tit for tat, and cheating.

Today I want to look at how the (Darwinian) Theory of Evolution can model morality from purely egoistic assumptions.

Let us quickly recap the assumptions of Darwin:

1. There are more offspring than can survive (there would be exponential growth if everyone survived) (survive means: have offspring, not survive forever).
2. There are differences between offspring.
3. These differences are heritable (if your father is taller than average, we expect you to be – this only needs to be statistically true)

The conclusion then follows:

Those traits we see after many generations are the traits that is well suited to survive the environment (survive still means: have offspring). Note also that you cannot develop a complicated trait without there being several stages of beneficial traits leading up to this complicated trait.

In morality we have a principle known as "tit for tat". Tit for tat is: "Do unto another as another has done unto you". If Sam helps you, then the next time you help Sam. If Sam steals from you, then you steal from Sam.

We will discuss how this principle (tit for tat) can follow from the Theory of Evolution and purely egoistic assumptions. First let us define our environment/setup/problem, what is known as the iterated prisoner's dilemma.

The prisoner's dilemma has its name for a reason, but for reasons of clarity of exposition I will present it differently. It goes as follows. You and Joe find some food in the jungle (think about our ancestral environment). You and Joe get two options: Fight or Cooperate. You can choose different actions. Results are as follows:

1. Both cooperate: Both get 3 (units of) food
2. Both fight: Both get 1 food
3. A fights while B cooperates: A gets 4 food, B gets 0 food.

Note how this is supposed to model how food is ruined (the prey escapes etc) when someone fights (in case 2 we loose 4 food, in case 3 we loose 2 food). We assume that the choice you and Joe picks are independent (you cannot wait and see what he chooses).

How is the prisoner's dilemma soled by two egoistical entities? If you and Joe are egoistic you will always choose to fight. Does this make any sense when both could have more food if both of you cooperated? Yes, as the choice is independent. No matter which choice Joe picks, it's always better for you to fight than cooperate. In that way outcome 2) is recognized as a stable Nash equilibrium (anyone seen the film "A beautiful mind"?).

Before we go on to iterate this problem, I want to take a small digression. How can we model the solution of a single prisoner's dilemma? Well, we can have utilitarianism (everyone's welfare is important) instead of egoism, but that was not our assumptions. One common solution is "kin selection". If Joe is your brother, then his survival will bring your genes (statistically about half of them) down the line. So his survival is half as important as yours: He getting 4 food has value 2 to your genes, but you getting 1 food each has value 1.5. This makes cooperating always better for your genes. (We just assumed probability of survival is linearly dependent on amount of food.) But what about your friend, they are not of your blood, can you still have kin selection? Before modern transportation it was highly likely that your children would have kids with the children of one of your friends some day. This might (with a slightly different setting, and some more assumptions) make cooperating with your friends a good strategy. From now on we assume that the two '

'prisoners' are complete strangers with nothing in common.

What its iterated prisoner's dilemma? Well, you go hunting with Joe every week for a few years. Now, every week you make a new choice, independent of Joe, whether you should fight or cooperate. But now you are allowed to remember everything that has happened up to this point. Now you are allowed to choose "Because Joe did X last time, I will do X now", which we called the tit for tat strategy. NB: you always start by cooperating. What is special about this strategy? It is essentially unbeatable. Let us disregard small deviations (like tit for tat with forgiveness or extra randomness). What does unbeatable mean? If you have a population of 100 people, each with their own strategy; some of them with the tit for tat strategy, some of them with completely different strategies then no one will get more food than you if everyone plays the iterated prisoners dilemma against everyone else (and you are using tit for tat). (This also relies on the assumption that the world is not dominated by a big number of tit for tat haters whose strategy is discovering the 'tit for tat strategy users' and killing them.)

Now the Theory of Evolution concludes that after many generations, everyone will have something close to the tit for tat strategy. Yes, this is dependent on even more assumptions. Maybe we should model this on the computer one of these days?