Monday, April 30, 2012

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).

Sunday, April 15, 2012

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.

Monday, April 9, 2012

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!


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.

Friday, April 6, 2012

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.