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?

Positive Morality

Are we fundamentally cooperative or egoistic?

It would be easy to argue that everything that everyone does is based on pure egoism. I disagree with that viewpoint, but let me present it.

If Eve does something that benefit her and nobody else (or has a negative impact on others), we call the action egoistic. If Eve is kind to a friend in a selfless way, it is because she wants something in return. The notion of reciprocity. If she needs help at some later time, this friend will help her out (she assumes). It can actually be proven that reciprocity is the best strategy (tit for tat) in a 'game' supposed to model real life. Let us come up with a situation where no reciprocity is expected.

If you find a drunk man lying in the gutters, and help him get a taxi home, would you expect reciprocity? Let us assume no. But then your genes are. Through millennia of evolution your genes have found the perfect way for you to behave; group-orientation, giving, reciprocity, caring, emotions, family values, etc. But in the end all your behaviours are designed for one purpose only: You (or your genes at any rate).

Let us return to my viewpoint.

Survival of the fittest. In accordance with Darwin's theory of evolution, the strongest (those most fit to survive the world and bring offspring) survive (or they genes do anyway). This is easy to believe (note how a lot of people that are pro-Darwinistic consider the theory of gravity to be just a theory, but the theory of Evolution to be given as an Axiom of The World).

Few people would argue against that kindness and selflessness are important concepts that we use to model our world. So in everyday life these models are true in some sense. Some may argue that the more fundamental model of Evolution is more true (since it is in a sense more fine-grained; this is the typical Reductionist view), but I'd say we have another way to choose what to consider 'the most true'.

What you believe changes who you are. If you disagree, for example if you do not believe in free will and purpose, reading this has no value anyway (you're just doing it as a consequence of random or deterministic happenstance). The question is: Do you want to live in an egoistic world, where you can always just interpret everything as egoism? Or do you want to live in a kind world, and learn time and again that the world is not so kind? Or do you want to live somewhere in between, thinking the best of people, that is, the best that your experience permits?

Question anyone: If I give you a present (and conscious reciprocity is not in the picture), is there any experiment that would give a different result whether it was fundamentally an egoistic or a selfless act?

Sometimes the purpose can be one thing, and the important sub goal something else. This is how I view free will vs. Darwinism. Free will is the goal, but survivability is an important bi-product.

The reason I blogged about this today was essentially

http://www.youtube.com/watch?v=GcJxRqTs5nk

To sum up:

Return favours: Conscious egoism,

Empathy: Unconscious egoism, or something else?

Is giving away something with absolutely no future gain a "bad" Darwinian side effect of the powers of empathy, or is it a sign of our "true" nature?

Is this really a question for science or a question of some other kind?

Solving Conundrums 3

- Solutions to Solving Conundrums 1 (and 2)

The Second Conundrum

"Which way would you tell me to go if I were to ask?" (Then take that way)

If it's the one who always lie, he would tell you to go the wrong way, but as he is lying he has to lie to the question by telling you the right way.

The Fourth Conundrum

The solutions to this depends on what you mean by surprised. In practical terms, what the death-sentenced king does is to believe with all his mind that he will be killed tomorrow (he believes this every day). Tuesday morning he is not killed, but does he consider himself to be surprised?

There are three ways to define surprised (that I can think of). The first is "I am surprised whenever I am wrong". Using this interpretation, where is he wrong in his analysis? He must decide in advance on one day when he thinks he will be executed. If he decides 'I think I'll be executed on Tuesday', then he is surprised when he is executed on Friday (or, rather, when he is not executed on Tuesday).

The second interpretation is "I am surprised whenever I have a false negative, that is, when I predict that I will not be executed but I am." So if you have a false positive (you think you'll be executed but you're not) you don't count as surprised. Then the king is correct, his analysis is good, it is impossible to surprise him. (This goes against the way I told the story, but there exist different versions.)

The third interpretation is a bit more complex. If it rains/not rains tomorrow, are you surprised? Not too much, because you don't have a strong prediction. But if a volcano erupts in your neighborhood, then you are surprised. If someone asked you beforehand you'd say that the chance of a volcano erupting so near you was almost 0, or just say 0. The interpretation is: "You are surprised whenever something occurs that you assigned less than 10% probability" (I chose 10% for convenience, you can substitute it with any number less than 50%). Every night the king can assign 50% probability to being executed the next morning. Then he is never surprised.

The Third Conundrum

When making statements in logic you can make almost any statement you can dream up. Almost. You are not allowed to define a logical variable by 'P:(not P)'. The easiest way to make sure your statements are definable is to only use logical variables that you have already defined, together with logical operations (and, or, not, etc.).

Let us define the logical variables:

P: not Q

Q: not P

This is impossible in the same way as

P: not Q

is impossible (you can't know if P is true or false, it can't be true or false, as you have not even defined Q).

If you'd rather think of this as a satisfiability question (see the last solution on this post) for 'P equivalent (not P)', then the answer is "no, there is no truth values satisfying this expression".

The First Conundrum

Using the clarified version:

If A) is correct then D) is also correct, and then A) is not correct as there are two correct solutions.

If B) is correct, then B) is false by its own statement.

If C) is correct, then C) is false by its own statement.

If D) is correct then A) is also correct and then D) is not correct.

So every answer gives a contradiction, hence none of them are correct. But isn't then alternative C) correct, since no of the alternatives are correct?

The resolution here comes from thinking about satisfiability. If you have a logical system you can give a number of variables A, B, C, ..., and a logical 'equation' (statement), for example

(A and B) or (C and (not A) and (not D) or (A equivalent B),

and ask the question "Is there a set of values for the variables that make the equation true?". The answer can be

"yes; A=true, B=false, ...", or

"no; there is no such choice of values".

Finally, the answer becomes "The question you posed has no correct answer". On this higher level, where you defined the question, you can say that there is no answer that can be correct (as all of them leads to contradictions).

The question cannot be satisfied. It is not so that 0 of the alternatives are correct on the level of the question. On the level above, it is so.

If you disagree, I would recommend a book on mathematical logic, and one on set theory (not naive set theory, but the serious kind), at least read the wikipedia articles

http://en.wikipedia.org/wiki/Satisfiability

http://en.wikipedia.org/wiki/Propositional_satisfiability

And to see how complicated stuff becomes when someone tells you everything, see

http://en.wikipedia.org/wiki/Russels_paradox

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

Solving Conundrums Part 2

- Hints (including clarifications) to Solving Conundrums Part 1

The First Conundrum

Restatement of the problem:

How many of the alternatives in this question is/are correct?

A) 1 of the alternatives

B) 2 of the alternatives

C) 0 of the alternatives

D) 1 of the alternatives

The Second Conundrum

This solution is supposed to only use a question referring twice to the brother who is alive. This one is connected to the third conundrum.

The Third Conundrum

This question is related to the post on Limitations of Logic

The Fourth Conundrum

The wikipedia-page on this (unexpected hanging paradox) is the worst I have seen. It is a lot simpler than that. Start with defining surprised.

If you think something (H) will happen (or not), there are four possible outcomes with respect to your information:

You said H would happen, and it does -> you're right

You said H would not happen, and it does not -> you're right

You said H would happen, but it does not happen -> False Positive

You said H would not happen, but it does happen -> False Negative

Are you surprised when you do a false negative, or are you surprised whenever you do either a false negative or a false positive?

Gambling All Of Mathematics

"A chess-master may gamble a piece, or even the entire game; but a mathematician writing an ad absurdum argument is gambling all of mathematics." - Couldn't find the source.

- This post can also be considered a hint to Solving Conundrums.

What is an ad absurdum argument? As always, you can see wikipedia, but let me take one of the most famous examples, and then give an explanation. Essentially the argument goes like this: Assume the opposite of what you want to show, arrive at a contradiction, conclude the opposite of what you assumed.

Example of an ad absurdum ("to the absurd") argument:

Are there a finite or infinite number of primes? (Primes are numbers p that cannot be factored into any other factors but p and 1. Primes per definition are > 1.)

Well, assume the statement Q to be true:

Q: 'There is a finite number of primes'

Then there are exactly n primes for some number n, and we can number these primes p_1, p_2, p_3 ... to p_n. Then we make a new number s by multiplying all these primes, and then adding 1 (s = p_1*p_2*...*p_n +1). This new number s will not be divisible by any of our primes, so the only possible factors of s are itself and 1. Hence there is at least n+1 primes. But this is impossible as n is the total number of primes.

We assumed Q and ended up with a contradiction, so Q must be wrong. Hence there is an infinite number of primes.

Example end.

So how does this work? I assume that we agree on what Logic is (the axioms and rules), as it would be far too cumbersome to write it out. Last time we talked about consistency of logic. This can be represented by the following axiom:

((P) and (not P)) equivalent to false

What we got in the proof was that the number of primes was exactly n and n+1, that is, exactly n and not n. What we got was equivalent to false by the above axiom. We showed that:

Q implies false

Now we want to use something called the rule of transposition:

(A implies B) is equivalent to ((not B) implies (not A))

In total we get

(not false) implies (not Q), which gives

true implies (not Q), which gives

not Q, which is

'There is an infinite number of primes'

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” - Sherlock Holmes

Have Mercy on my Implication

There is some sayings that irritates me. Like, "this is a quantum leap" (Norwegian: "Dette er et kvantesprang"). A quantum leap is the smallest possible leap in nature, in everyday life a quantum leap is the same as a continuous increase, as it is extremely small. That can be remedied, however, by thinking of the quantum leap as a leap in understanding. Physics at the scale of atoms were poorly understood until the concept of a quantum (the world is not continuous, but fundamentally quantized).

And then there are some that have no valid interpretation, some phrases that want me to knock someone's teeth out: "Yes, that implies truth."

It is common to confuse the following three words:

Correlation: Often when there is sun and rain we see a rainbow.

Causation: If you hit me on the head, I will feel pain.

Implication: If the moon is made of cheese, then I am eating chocolate right now.

(Since the moon is not made of cheese, the implication is true whether I am eating chocolate or not. Note also that moon and chocolate have nothing to do with each other.)

See Wikipedia's three pages for a more thorough explanation. Some may argue that implication is an abstract model for causation, but let us avoid philosophy right now.

Back to business: Instead of saying "X is true" some people say "X implies truth" to sound more wise. (In Norwegian: "X medfører riktighet"). This is completely bollocks! If you know that X implies truth, then you know nothing at all about X. ANYTHING implies truth! (Can you feel my frustration?) On the other hand, if someone were to say "Truth implies X", then you would know that X was true.

The Limitations of Logic

- Consistency, completeness, and Gödel's theorems.

- This is not really a hint to SolvingConundrums, but it is a prerequisite for understanding some of the solutions properly.

What is a logic? What is a mathematics? It is a set of axioms and some rules for deducing true statements.

Logic is the set of rules and axioms we have agreed to use. I will assume that we agree on these. (I refer to the mainstream choice, I know there are other candidates, like fuzzy logic, and I may come back to that later.)

A consistent system is one in which not both 'P' and 'not P' is true. In other words, a system where there is no statement P that is both true and false. Why is it so bad to have one such statement in our logic? Well, assume that we have one such statement, and call it P. Then for any other statement Z we get that P implies Z, because P is false. Then, since P is true, we can deduce that Z is true. The conclusion is that every statement in our logic is true (including 'not Z'). This is senseless – there is no difference between true and false anymore!

A statement-complete system is one in which every statement is either true or false, so that there is no unknowable thing. Having an incomplete system means that we can't know everything, even in principle, and that is bad. (When doing science we often think that there is just a matter of time and patience before we understand something.) This concept, I just made up, and it is a bad concept. What is truth, really, in a logical system? We still want to avoid epistemology (philosophy), so what then is truth? It is something that can be proved by using the axioms and logical rules.

In Logic/Mathematics we desire proofs. If something is true (say 'P'), then we want there to exist a string of logical arguments showing that 'P' is true. If something is true, but there is no way of knowing that it is true, that is bad. Our definition of truth in a logical system is that which can be proved within the system. A system where every statement (which can be constructed in the system) is either provably true, or provably false, is known as a complete system. Wikipedia calls this syntactically complete, and gives a nice reformulation: A system is complete if and only if "no unprovable axiom can be added to it as an axiom without introducing an inconsistency."

Let me be clear: We will only use one concept, so that truth and provability is the same thing in Logic. It is possible to make a distinction between the two, but I don't want to do that.

What Gödel tellsus is that we can never have a complete system (given that it includes basic number theory and some extra technicalities). If humanity some day decides on a logical system to use for all eternity, then either that system is inconsistent or incomplete according to Gödel's proof. In other words, since we require our system to be consistent, there will be statements that are neither true nor false in our system.

What do we call these statements that are neither true or false in our logical system? We call them independent or undecidable. Examples of these are the axiom of choice (independent in ZF-logic), and the continuum hypothesis (independent in ZFC-logic). These examples are (to mathematicians) interesting statements, so the problem that Gödel found is not some weird technicality, but something we have to deal with.

What can be done with an independent statement? We can choose to add it as an axiom, or we can choose to add its negation as an axiom. Logic does not care which we choose, it will still be consistent regardless of our choice! If we want to model observable reality, however, we might care about which is "true", as in which choice results in the best model of reality.

When we have added our axiom (or its negation), then our logic is a new and more powerful logic, where more statements are provable (also known as 'theorems'), and where more statements can be constructed. Again Gödel's proof works, and we can find new (and possibly interesting) statements that are independent in this new logical system. And so on, and so on.

Here, those who are familiar with mathematics might want to try to cheat, and add an infinite sequence of axioms, each based on a previous level's known independent statements. Even this (and generalizations of this) will not work at all. The logic you end up with is either incomplete or inconsistent.

Gödels general examples that always work in a logical system T (not true nor false):

(A Gödel number is a proof translated to a number in number theory.)

"There is no Gödel number to this statement using the logical rules of system T"

(i.e.: There is no proof of this statement.)

"The logical system T is complete"

Solving Conundrums Part 1

[I was a little unsatisfied with my last post, the second part of Q.M. logic, so to the next theme I will take a different approach; starting with a problem to solve, and, within two weeks, give hints, and within four weeks, the solutions.]

I have a conundrum for you (that word tastes like soft thunder rolling over the horizon on a warm summer day). Well, I have several. All in the form of seemingly innocent questions, that soon become quite frustrating logical puzzles. I promise that I will solve all of them in a very concrete way; I am, after all, a mathematician (I do not, however, promise that you will like the solutions, as I am not a politician).

The First Conundrum

You sit down to have your exam in [logic-something-course], and get the following multiple-choice question:

"If you were to answer this question randomly what is the probability that you would be correct?

A) 25%

B) 50%

C) 0%

D) 25%

"

What is the correct answer?

The Second Conundrum

Suppose you are at a crossroads, and there are two paths, one will lead to riches, and the other to death. In the old days there were two brothers; one who always speak the truth and another who always lie. They were, of course, identical twins. The solution was to ask both of them, which road would your brother tell me to go to get riches, and then go the opposite way.

Sadly, one of the brothers was killed by an angry customer. As they were twins, noone knows who died and who still lives on. The only thing you know about the person in front of you is that he always either speaks the truth, or he always lies. What do you ask him? Will you get rich?

The Third Conundrum

Is the following statement true?

"This sentence is false."

What about the two next sentences, are any of them true?

"The next sentence is false.

The previous sentence is true."

The Fourth Conundrum

Once upon a time, there was a proud king. His throne was usurped by a maniac, and the king was to be executed. The maniac said: "I will execute you this week, on Tuesday, Wednesday, Thursday or Friday. I will come and get you early in the morning, and it will surprise you!" The proud king then answered: "Well, fool, you cannot kill me on Friday, because then I will know it Thursday evening, so it will not be a surprise! Since you are unable to kill me come Friday, on Wednesday evening I will know it if you plan to kill me on Thursday, hence you cannot kill me on Thursday. Now, only Tuesday and Wednesday remains. So if I am not executed on Tuesday, I will know that you plan to kill me on Wednesday. Hence only Tuesday remains. But I know this, so there are no possible day when you can kill me!"

The maniac thought about this for a while, then answered "We will see". On Wednesday, the proud king was, to his surprise, executed (he had, after all, predicted that he would not be executed). Where was the flaw in his logic?

The Cat That Killed De Morgan

According to google (number of hits), it's supposed to be "the cat who killed", does anyone know for sure?

• Part two of Quantum Mechanical Logic. (Part one here).

In logic (classical/ordinary logic) we have something known as De Morgan's laws:

Let P be the statement 'It's raining outside', or any logical statement

Let Q be the statement 'The sun is shining', or any other logical statement

Then saying 'not(P and Q)' is the same as saying '(not P) or (not Q)', in words:

'It can't be both raining outside and sunny' is the same as

'It isn't raining outside, or It isn't sunny'

(yes, none of the sentences are necessarily true, but they are equivalent)

(in logic, 'A or B' means 'either A, or B, or both')

Does this work in Quantum Mechanics? No. Remember our perfectly grey cat. If you ask the cat

P: Are you black

Q: Are you grey

The statement 'not(P and Q)' [not both black and grey] is true. In the Quantum Mechanical sense, you cannot observe that it's grey and black at the same time (that is not a legal outcome). Hence (grey and black) is false, and 'not(grey and black)' is true.

The statement '(not P) or (not Q)' [not black or not grey] is true in 50% of the experiments. The part (not grey) is never true, as it starts out with being perfectly grey, and you cannot then observe it to be not grey. The statement 'not black', however, is true 50% of the time as it is a 50% chance that when we measure it to be 'not black' the cat will spontaneously become white. In total '(not P) or (not Q)' is true 50% of the time, and false the other 50% of the time.

So in Quantum Mechanical logic De Morgan's laws are not valid, the two statements are not equivalent. But in logic we require this law, so what to do? Well, even though I have called it Quantum Mechanical logic, it isn't really logic, but something else that has a strong similarity to ordinary logic; and it has some differences as we just saw.

To be fair to any mathematically inclined readers I want to add a comment. The "logical" 'or'-statement in Q.M. is usually taken to be a join of subspaces (the least subspace containing both the subspaces of Q and P), instead of a 'or' between two experimental outcomes, so that in our example the most natural thing to say is that (not black) or (not grey) is the linear span of the two, namely the whole 2-dimensional black/white subspace. Observing whether it is in this subspace would give a 'yes' with a 100% probability. It was, after all, grey. This seems to make my point moot, but alas, even with this more refined notion of "logical 'or'" you can find contradictions to De Morgan's laws (where meet and complement of subspaces is not the same as complement and join), see for example this page.

The Color Of A Cat

I'm late for my two week-appointment with my blog, so here's something special.

- How logic in Quantum Mechanics differs from the 'real world'.

Today I want to tell you one of the big secrets of Quantum Mechanics, using a parallel with a cat in it. By the time you have read this page (a couple of times) ordinary Quantum Mechanics will hopefully be clear, if not, don't hesitate to ask. Schrødinger's cat is another well known parallel with a cat, but it is about something else (in Q.M.).

We all use probability in our daily life, it's a handy tool. There is 1/6 chance of a die landing on a 6, there is 1/2 chance of a slice of bread landing with the peanut-butter-side down on the floor. Here, common sense dictates several wrong claims, like getting a 6 two times on a row (on a die) makes a third 6 less probable. Forgetting those fallacies, we all think that having enough information removes the probability. If I know the exact speed(s), air currents and form of the die and the table, I could (in theory) predict exactly on which side it would land.

So the classical world ('real world', 'everyday world') probabilities are really hidden variables. Stuff we don't know. Probability is in the map and not in theterritory.

How does this differ from Q.M.? Let us give a parallel.

Say you have a perfectly gray cat. Perfect in the sense that it is exactly halfway between white and black on your gray-scale. If you ask 'is the cat gray?' what happens? The answer is 'yes', and, of course, the cat doesn't care. If you ask 'is the cat black' what happens? You get the answer 'sort of' or 'halfway black', and, again, the cat doesn't care.

Let us assume this cat is an electron, and color is some property of that electron. The cat is still perfectly gray. If you ask 'is the cat gray?' what happens? Well, the answer is 'yes' and the cat doesn't care. It's the same, so no surprises yet! If you ask 'is the cat black?', two things can happen:

1. Answer: 'yes, black' and the cat instantly changes color to black.
2. Answer: 'no, not black' and the cat becomes non-black, which, in this case (starting with a grey cat) would actually give you a white cat.

Poor cat. But which of the answers do you get? If you had 1000 such cats and asked them all, you would get answer 1) about 500 times, and answer 2) about 500 times, so we say that the probability of getting 1) is 1/2 and same for 2).

To digress, what Schrødinger's cat is about (if I understand it correctly), is whether this is actual probability. Are there any hidden variables determining which of the cats come out black, or is there an inherent True Probability in Nature? 'God does not throw dice' -Einstein. If anyone cares, I believe Einstein to be wrong about this, and that these experimental outcomes are determined by probability. I also believe the Schrødinger's cat experiment to be a bad argument, as the cat would measure whether it was alive or dead. You don't have to be a person to do an 'experiment', and not a cat either; any two molecules on a collision course will do an experiment to see whether they collide or not.

What is special about Q.M. logic? Grey can be a 'superpositon' of white and black. How do we model this? There is a certain thing in mathematics called a Hilbert space, where colors are unit vectors (or subspaces), and a vector [1,1] can be viewed as a superposition of [1,0] and [0,1].

Why? Well, experiments show that... But why? This borders on philosophy. From a scientific point of view, this is our best model – it works (there's a friggin' flag on the Moon and a rover on Mars).

Answers to odd numbered exercises

(This post will discuss the difference between a good and a bad scientific understanding/education.)

Why is there, in most math books (and physics, chemistry etc.), only solutions for some of the exercises? The last chapter is often "Answers to odd numbered exercises", but why not give answers to all of the exercises?

It could be laziness, but if you ask those who write the books they answer "the students learn better". Students, on the other hand, often complain, "how can we know that we are doing things right, without all the solutions?" Well, in mathematics, half the point is being certain that you are right. Even though this is close to the point I want to make, it's not exactly it, so let us hear a story.

"Once upon a time, there was a teacher who cared for a group of physics students. One day she called them into her class, and showed them a wide, square plate of metal, next to a hot radiator. The students each put their hand on the plate, and found the side next to the radiator cool, and the distant side warm. And the teacher said, write down your guess why this happens. Some students guessed convection of air currents, and others guessed strange patterns of metals in the plate, and not one put down 'This seems to me impossible', and the answer was that before the students entered the room, the teacher turned the plate around. "

(Taken from this page who cites Verhagen 2001.)

I see this all around me when people are trying to find a 'scientific' explanation for the world. The physics students in this story did a 'political argument', they wrote their bottom line first. If we write the conclusion first, it does not matter what kind of arguments we use to support it. When we write the conclusion, it's either correct or false – whatever arguments we write down after having decided does not influence the conclusion. You can give the best arguments for why the earth is flat, and how you can fall of the edge, but it doesn't change the world.

The kind of 'political thinking' where you choose your 'truth' first, and your arguments second is very common, and works fairly well when putting pressure on other people and on the society. But if you are faced with a difficult problem where there is a well defined answer, your arguments are supposed to help you find the correct solution.

When solving a math exercise, would you write down the answer (42) at the bottom of the page, and then try to give sufficient arguments and 'good' calculations resulting in 42? Then you are learning how to get 42, not how to find the correct answer.

The power in science is being surprised whenever something implausible happens. If you can explain everything equally well, then you truly know nothing.

A rose by any other name

A few weeks ago I posted the following status to facebook:

"A shovel, by any other name, would still shovel dirt. A rose, on the other hand, would it still smell as sweet?"

This was the end result of one hour of deliberation, and it has significant philosophical depth. Apparently, facebook is not the place for something like that, so let me explain to you what it means. (I meant to do this two weeks ago, but you know...)

First one has to associate to it the well known saying by Shakespeare (said by Juliet in 'Romeo and Juliet', which is a good enough read, and written in funny English (by the way, has anyone noticed the similarities between Shakespeare-talk, and Yoda in Star Wars?)):

"What's in a name? That which we call a rose
By any other name would smell as sweet."

Modern research would answer: "Yeah, no, not really". Words, by their sound, and by their relation to other words (associations, connotations), does carry quite a bit of 'subconscious' prejudice.

How can this be? Studies show how the expensiveness of wine makes you like it more. So that if you don't know the price, most wines are equal (or even more expensive wines do poorer), but if you know that a wine is expensive, then you like it more. Now, you are probably thinking that the subjects reported to like it more, so that we can only conclude that the price affects how much we think we should like it. But no, alas, it also affects the amount of pleasant your brain generates. So the conscious price-information is taken into account when your brain decides how much it likes the wine on a subconscious level!

For the actual research article follow this link.

This should explain the second sentence of my facebook status, but what is the deal with the shovel?

Well, even if you are told that the shovel was expensive (maybe it's lined by gold or something) what happens? If it breaks, or is unable to contain enough dirt, then whatever it's called and how it's priced does not matter at all. Perhaps you like the expensive gold-shovel more, but the shovel that is best at shovelling dirt is the 'best shovel'.

To clarify, there is a distinction between two different values here. On one side it is the beauty, or the artistic value of a rose; it is summer and happiness, joy and love. On the other side it is the usefulness or practical value of the shovel. Even though it shovels dirt (a word with negative connotations) it is important to us. And this practical value would not be changed by renaming it.

As any other pair of concepts these are seldom seen apart. More often than not, the two values are entwined in any given object; there is a combination of artistic value and usefulness. But ideas, I think, are more powerful when we are able to distinguish between them.

Talent or no talent?

Today I saw several episodes of "Hjernevask" (Brainwash), a Norwegian TV-series on the debate nature vs. nurture, and the heavy political pressure towards the nurture side. I wanted to do a short discussion on Talent, whether it exists and what we can do about it.

When I was younger I did not believe in Talent. I thought everyone was a blank slate, and that everyone had the capacity to do anything. When the time came to choose which high-school I wanted to go to, I had to choose between studying science and music. Ironically I spent a lot of time doing research to figure out what was the best choice for me.

I talked with several people to find out who were the most satisfied with their job/career. Those who studied science/engineering/economics had the jobs they wanted, even though they did not consider themselves to be especially talented at their respective fields. Most of them hadn't even been passionately obsessed by their subject. When I talked to those who studied music I found that most of them had not, I repeat, had not, gotten the job they wanted. Many of them considered themselves talented in music, and most of them enjoyed it and obsessed over it, it was their work and their hobby. So they had more than average talent and had even spent a lot more effort. What went wrong?

Most people agree that music is something you can be talented in. If there's a shadow of a doubt I recommend [somecountry]'s got talent, like thisawe-inspiring-incredible 11 years old. So to do well in music you have to be (exceptionally lucky or) talented and obsessed (in my vocabulary obsessed is a positive word).

Luckily I went with science (my music teacher actually told me that I was good enough to study music, but if I could find something else I was equally good at, it would probably be a better choice).

At the university I found that it was possible to have a talent for science. What happened was that I started to work vigorously, and my talent for mathematics came to the fore. Some of the things I have learned in one semester of hard work would take the average student at least a year (I guess). I tell you this not to show off, but to point out how extreme a contribution a talent can be.

People I talk with often agree that one can have a talent for music, but that it's impossible to have a talent for more 'normal' things, like studying calculus (undergraduate mathematics). Or some say that talent is pure nurture (that it's something you get from the environment, like teaching and parenting) as opposed to nature (the genes).

Taking the last point first, look at 'hjernevask' (For English subtitles follow theinstructions below the video.), or any of the research, or look at the youtube video I mentioned about the 11 years old girl Anna Graceman. I know several hard working singers in their twenties who don't have half the voice she has. When I was young my parents actually told me not to sing too loud. Tell me what Anna's parents have done, so that we can have more singers of her calibre. Frankly it's ridiculous to suggest that this has been caused by some random events in her environment, and that theory has no explanatory (or predicting) value whatsoever.

The other counterargument was that one can have a talent for music or sports, but not for studying science or philosophy, nor a talent for human interaction. This is true in some sense. Firstly, these fields require more talent to be successful in. There is a limit to how many football players and violinists we need in the world, and at least for the time being it seems there are a lot more candidates than jobs. Secondly, a talent for music and sports is a lot easier to see and measure, while a talent for philosophy would be hard to spot.

But why would you assume that one can be talented in a range of mental activities, but suddenly you draw a line between being talented at painting and being talented at understanding abstract concepts? It seems contrived to me, but I may have an explanation for why we (especially politicians) sometimes do this. This I will talk about now.

Before we continue let me agree that talent is not a yes/no question. You can have a little talent (the most common form), slightly more talent, a lot, etcetera. You can even have anti-talent in some sense. For some people anti-talent is a taboo, and for others a convenient excuse.

Why is it bad for you to tell someone they are not talented at, say, chemistry? Because they will become worse at it, it's sometimes a self-fulfilling prophecy. And sometimes you just don't know. Perhaps they are poor at it for some other reason; they don't work hard, they don't know whatever they should have learned before, they don't have the motor skills to do the experiments, they don't have a sufficiently good memory to remember all the different names.

Why is it bad for you NOT to tell someone they are not talented at chemistry? If someone spends obscene amounts of time at it, and gives all their effort, and still cannot manage, how do you think it feels when you tell them: "You don't work hard enough, give more effort."? And how do you think it will work out when they go for a university degree in chemistry? It might turn around, but then, it might not. In any case one should spend time looking for some other talent.

Why do you tell people that they can do anything they want? The main problem is that people are ridiculously happy in their comfort zone. If you tell people: "You probably won't be able to do that", then they don't even try! You have to say over and over "you can do anything" just to get them thinking slightly outside the box. But then, maybe there is a time when we ought to give a little guidance: "Have you tested your talent in anything else?".

The last strong objective that I see (while sitting here in my comfy chair this evening) is that your array of talents is highly connected to your self-esteem. Is the world fair? Is everyone good at something? Perhaps the world is unfairly kind, and gives some people exceptional talent, but at least it's not so unfair that there is someone out there with only poor talents? Right? And not one is born unable to use their right arm, right? But is this the basis on which we judge our fellow humans? Have we sunk so low that the only thing we care about is how good you are at doing [whatever it is that you do]? Doesn't trying hard and doing the best you can under the circumstances count anymore? Doesn't how much you care, your selflessness, and your humanity count? Why do we connect the worthiness of a person to their array of talents? And if we don't, why do people believe we do?

So what's the truth? Should we say, there is talent, or not? Do you make an educational system that assumes everyone is equal, or not?

I only know two things for certain. Even though it might not be wise to always communicate it, Talent is an important concept. The second thing is that we should have an educational system that searches for the talents in every child, so that after ten years of education, everyone can write down a list of things they are good at.

Addendum: The extent of this talent might be a bit easier to appreciate.