Why do we dream?

February 20, 2009

Some people claim that our dreams are the manifestations of our subconscious, i.e. the brain’s way of telling us what we “really want”. However, I see no logical reasoning behind this claim. In reality, I think it mimics the same way of thinking that leads to religion, astrology, new-age “medicine”, and the likes ‚Äď human beings like to “personify” things. This claim is explained in Richard Dawkin’s book, the God Delusion.

When giving our dreams meaning, we personify our brain, giving it a separate entity from our own, believing that it somehow “guides” us in our life, bestowing on us its “mystic” wisdom. It is a comforting thought, since it provides us with hope that there is a bigger plan, or that we have a guiding force in our life, a guardian angel if you will, that watches over us and guides us. I see it as a romantic idea, wishful thinking. A human need.

However, scientific doctrine aims to remove humans from the equation, reaching conclusions which do not rely on the human observer. Therefore, if there is no logical explanation for this subconscious, which (or maybe I should say “who”?) tells us what we should do with our life, but only through dreams that we need to interpret using unscientific methods, i.e. intuition, I must look for an explanation which relies on known scientific axioms.

An axiom which immediately comes to mind is evolution.

evolution

I will not go into the whole evolution vs. intelligent design debate, since you can find plenty of websites which discuss it, and since frankly I could just as well debate evolution vs. the Flying Spaghetti Monster.

If you “believe” in evolution, and more importantly, if you understand it, you realize that each living organism on this planet looks the way it is today since, quite simply, all the other ways were not good enough. In other words – its current state is dictated by whichever changes that allowed it to survive better. This survival is not an inner mechanism or something that drives it. There is no “force” which makes evolution exist. Things which are best suited to survive do just that because they are best fitted to do so. Kind of recursive ūüôā

So let us go back to the title of this blog entry ‚Äď “Why do we dream?

dreaming_toc

I offer an evolutionary point of view, combined with computer science thinking.

The human brain is composed of connections. The more we think in a certain way, the more certain connections become stronger, reinforced. That is why astronauts undergo extensive underwater training before going on missions. It takes time for the human brain to adjust to new points of reference in space. Astronauts in microgravity usually lose their sense of direction and feel uncoordinated or clumsy. Because inner ear and muscular sensors seek terrestrial clues, astronauts must learn to rely on visual cues for balance and orientation. But even visual cues can be confusing ‚Äď astronauts in microgravity need to adjust to the fact that up and down don‚Äôt really matter in space like they do on Earth. They need to “force” their brain to think differently, and that takes time.

The same goes for human emotions. If for example you are a person who is always depressed, you will not be able to change overnight. Changing your way of thinking and behaving will take time, since you are “re-wiring” your brain (however, if you want a “quick fix”, you can always get a brain pacemaker transplant – http://en.wikipedia.org/wiki/Brain_pacemaker). The more you think differently, the more these connections become stronger, and the other become weaker. It is a very elegant code if you think about it from a programming point of view. It makes itself more efficient and streamlined, according to the relevant needs. Human emotions might not relate to evolution so clearly, but if we consider other brain functions such as moving about, breathing, or recognizing a lion in the bushes – it is vital for our survival that we carry out these actions successfully and as quickly as possible. Our brain has evolved in a manner which makes sure that whatever we do the most ‚Äď i.e. whatever we need to do to survive more, we do as efficiently as possible. Assuming of course that whatever it is that we do the most is beneficial for our survival ‚Äď perhaps not so true in the 21st century (for instance, I don’t think that reading this article improves your chances for survival, although I’ll be flattered if you think so), but most definitely true in most of our evolution, which took place in the wild, in much more harsh conditions.

lion31

[peek-a-boo]

So the connections in our brain constantly grow stronger in various ways, according to what we do and how we think. It has been known for quite some time that the augmentation of these connections mostly happens when we sleep, as corroborated by a recent study (http://www.eurekalert.org/pub_releases/2008-01/uow-sbc011808.php). This is known as plasticity (http://en.wikipedia.org/wiki/Neuroplasticity). It also makes sense ‚Äď as every person with a computer knows – it isn’t very smart to install/uninstall computer programs while they are running.

So why do we dream? Well, my hypothesis is that our dreams are a “system-check” carried out by our brain. When you go to sleep, your brain shifts synapses in your brain, making some connections stronger, some weaker and perhaps even creating new connections or completely severing old ones. After making each change ‚Äď it is a good idea to run a system-check, don’t you think? That is also why our dreams seem so real. As far as we are concerned, the messages in our brain when we dream are identical to the ones we get when we are awake ‚Äď or else it wouldn’t be a very good systems-check, now would it? Also, that is why dreams relate to memories ‚Äď that is the information the brain has available to use for its systems-check. Furthermore ‚Äď dreams usually relate to recent events, since those are usually the areas which get modified.

But the most important part in my hypothesis is why I think my argument is true. After all, I can find lots of explanations for why we sleep, why is mine more logical than the rest? Well, I suggest that this systems-check is a direct result of evolution.

Consider that our brain evolved, and we started making more and more connections. Obviously, these connections were augmented in our sleep, since if you started making changes in your brain while you are awake and running away from a cheetah ‚Äď well, let’s just say your survival chances were not very good.

But some individuals also started having dreams. Those dreams were the systems-check carried out by their brain, making sure that all of these new connections were OK.

So those who had dreams one-upped their fellow tribe-members evolutionary-wise. They had less chance of suffering the consequences of “faulty-wiring”. And we all know how one bug in a program can wreak havoc…

blue_screen_of_death1


No rest for the wicked

August 10, 2008

I am currently SWAMPED under the workload from the university (finals period), so I’ll continue writing new posts as soon as I finish. Should be around early-October.


On the Abundance of Information

June 29, 2008

If I lived 50 years ago, I think I would have been very frustrated (although I probably wouldn’t be, since I wouldn’t know any better. So allow me revise my previous statement: if you were to send me to live 50 years in¬†the past, I would have been very frustrated. Wait, should I say “would have been” or “would be”? Grammar is always so difficult when you time travel ūüôā )

The reason I would be so frustrated is the lack of available information.¬†We live in an age,¬†that if unless¬†you are¬†looking for something very confidential, you are able to find the information you need and fast. How does the old saying go? If you looked for something on Google and came up with no results, then it means you have a very specific fetish ūüėČ

However, this plethora of data¬†has its own inherit problems, the greatest one being able to determine which information is relevant and correct. A lot of people claim that this new abundance of information is actually a “wolf in sheep’s clothing”, saying that you can’t really trust anything you read, and nothing is reliable.

To which I reply: “baloney!”

Lets review the evolution of information:

1) Information is not free and it is¬†controlled (i.e. the writers and editors¬†of encyclopedias, newspapers, etc.)¬†by a small group of people,¬†and the only way to attain this¬†information is to be born into royalty or other form of high class –¬†except for newspapers, which¬†is usually a means¬†for the small group to convey their idea of the truth to the masses¬†(pretty much the way things were up until the mid 20th century).

2) Information is free, monitored by a bigger group of people (i.e. gives room to different opinions), but attaining it is cumbersome and not easily available to everyone (think 1950’s-1980’s).

3) Information is free, monitored by everyone and for everyone, which results in a great deal of unreliable information.

I still prefer the third option. Allow me to explain.

The way I see it, the third option is the best because it puts the power in the hands of everyone. Sure, back in the 1950’s the information you received was considered more reliable, but your choices were limited. I prefer that you let me decide what seems reliable and what doesn’t, instead of giving me only one source of information, which you decided to be true.

Yes, there’s¬†a lot¬†of stupid people out there saying plenty of stupid things, passing them off as intelligent data. But it doesn’t mean there isn’t also some useful information out there. For instance, right now I’m a CS (Computer Sciences) student in Tel-Aviv University. Wikipedia and Google are my best friends. If I could add them to my facebook profile,¬†I would¬†:-).¬†Whenever I come across any new mathematical/computer-related term, I simply look it up, see the definition, or read an answer from a forum addressing¬†what I need to find out (e.g.: I need to convert a Double expression¬†to String. This is a relatively simple thing to do in Java, but when you are starting out, you still don’t have all of the syntax down. A simple search of: “java double to string” in Google will yield the proper method)¬†and carry on.¬† Why, even if I’m in the middle of a lecture, and the teacher mentions something which I don’t understand, I simply use my iPhone to look up the relevant information, get my bearings, and carry on with the lesson. Try to imagine the same situation 10 years ago, when the internet was just starting to grow. Now try to imagine it 50 years ago :-).

Since the information I’m¬†looking for¬†is mostly¬†mathematical, the degree of “stupidity” I encounter is minimal. Problems start appearing when you are looking for information other than mathematical definitions. Let us say I was looking for a biography on Albert Einstein. Well, the first place I’d probably go to¬†is Wikipedia. And it’s not a bad place to start. It’ll definitely be¬†Google’s first¬†result.¬†People badmouth Wikipedia all the time, but personally I think it is a wonderful source of information. I wouldn’t use it for scientific research (since it is not 100% accurate), but it is a highly reliable and relatively very accurate source of information. The problem is that once you start looking for information such as biographies, history, art, society, etc., the reliabilty of the information decreases. And not because people have got their facts wrong (which is also a problem, but usually not in the websites I’m talking about), the problem is people are confusing facts with opinions.

When I look up the definition of what is a “Kernel” in linear algebra, there isn’t too much room for opinions. It’s a pure mathematical definition, and the only difference (barring any mistakes in the definition itself) between the various sources of information could be in the manner in which the term is explained – which is very useful, since people understand things in different ways (now compare that with the 1950’s, where you had only one textbook, and if you didn’t happen to think like the person who wrote it, your studies just became a whole lot more difficult). However, if I look up a controversial issue, like¬†for example:¬†“Jerusalem”, the information will not be accurate. I don’t mean that I’ll necessarily see false information, but rather that specific information will be ommitted or mentioned, according to the writer’s point of view. And people consider that a major issue. But now comes my next point – how is that different from the professor writing the article for Britannica? Doesn’t he also have his own opinions, prejudices and view of the world? His only advantage is that he is an expert on the subject. But does that mean he’ll be objective?

This raises the great question of “what is the truth?”, but I think I’ll overextend myself trying to tackle this subject. Maybe another time.

To sum matters up – I believe that the more information, the better. Smart search engines like Google and its ilk, as well as good ol’ common sense,¬†will help us seperate the relevant information from random ramblings of nitwits.


Procrastinators – unite tomorrow!

June 19, 2008

Yes, I haven’t written in a while. I have something in the works, but once again, the post-lecturers’-strike-university rears its ugly head, consuming every minute of my free time.

So in the meantime, enjoy this fascinating article about how the universe might not only be described by math, but made up of it as well!


The Prettier Side of Math

May 31, 2008

I’m pretty picky about the kind of math I like. Calculus and Linear Algebra? Blech. Discrete Mathematics and Data Structures? Yummy!

My problem with the former is that I believe this kind of math lives up to the famous quote by Charles Darwin, the father of the Theory of Evolution (see previous post): “A mathematician is a blind man in a dark room looking for a black cat which isn’t there.”

At least for me, when I try to solve such problems,¬†when I do manage to¬†solve them, I don’t get a feeling of satisfaction at the end. Either the exercise seems like Greek to me, and then I have to stumble in the dark until I find a solution (just trying different methods), or I just¬†follow a set order of procedures, which makes me feel like an overpriced computer program (hey, you don’t need to feed, dress¬†and house a computer program. Unless you consider electricity as food, an operating system¬†as clothes and¬†a case as a¬†house ūüôā ). Calculus sometimes has its finer moments, especially when you need to prove something fundamental, but most of the time I don’t care too much for it. Obviously, this isn’t a universal truth about math, and is purely a result of the way my mind works, and how¬†I solve problems.

However,¬†if we turn to¬†subjects like Discrete Mathematics, I¬†look at¬†questions as puzzles. Obviously it isn’t all fun and games, and¬†when you’re starting out¬†you have to do a lot of technical work to get everything down, but once you get past that¬†to the interesting¬†stuff – it gets a whole lot¬†nicer :-).

Before we dive into the math, consider the following riddle: You wake up in a room. You look around and all you see is a wall covered with numbers.¬†It’s a list of¬†499 numbers, all made up of 500 digits. Above the list you see the following message: “You have exactly¬†10 minutes to write down a new¬†number which is not already¬†on the list. If you write down a number which is already on the list, you will be trapped forever (punch and pie will be served).” What do you do?

First, a basic introduction¬†to set theory (a mathematical branch). I promise – there will be¬†no equations, and everything will be explained in the most intuitive way possible. Let’s call all the Natural numbers N. “What are¬†natural numbers” you ask? Well, simply put, its numbers you can count with your fingers, i.e. 1, 2, 3, 67, 985431, 12341414, etc. You may need to have a lot of fingers, but you get the point – whole numbers, starting from¬†1 (sometimes zero is considered a natural number, and sometimes it isn’t. For¬†our purposes,¬†it isn’t) and up to infinity (infinity… another very interesting topic which I might tackle¬†in a future post).

Okay, so we know what¬†natural numbers are. We called them N. Now we need to define another group of numbers: Real numbers. We’ll call them… you guessed it:¬†R. What are the real numbers? every number possible. That’s right. Think of a number. It’s real. 1.543? It’s real. 980? Also real. -97234.123401213? Real as they come. Pi? trying to be tricky, huh? (Pi = 3.1415… is an irrational number which describers the ratio between a circle’s diameter¬†and its circumference, but I assume that if you got this far and this kind of stuff actually interests you, you’re probably pretty upset at me for even explaining this. Well, too bad for you. I’m thinking of the poor souls out there¬†who have lived life so far without knowing what pi is :-)) Well, foiled again. It’s also real.

So we defined N and R. Both groups of numbers are infinite in size, but “intuitively” it seems¬†that there are “more” real numbers than there are natural numbers, right? Well, you are correct in assuming that, but how do you prove it? As the famous phrase goes: “It’s not what you know, it’s what you can prove”.

Which brings us to the main subject of this post: Cantor’s diagonal argument.¬†Cantor’s diagonal argument proves just that. Before I explain Cantor’s argument I have to explain one more term: a Surjective function. Suppose I have a group of numbers on one side. And I have a group of bunnies on the other. Now, I want to assign a number to each bunny (hey, I gotta keep track of my bunnies, right?). So I draw a line between each number and each bunny. If there are more numbers than bunnies, i.e. I have some numbers which have no bunny to point to, then the function of numbers-bunnies is called surjective. It means that one group is “bigger” than the other. It’s quite easy to grasp when considering finite groups, but¬†becomes much harder when the groups are infinite.

In order to prove that the group¬†‘BIG’¬†has more members in it than the group ‘SMALL’, we need to prove that there is a surjective function from ‘BIG’ to ‘SMALL’, but we also have to prove that there is no surjective function from ‘SMALL’ to ‘BIG’ (if there is, then both groups are the same size).

We wanted to show that there are more real numbers than there are natural numbers. Finding a function from R to N which makes sure each natural number is “covered” is quite easy. Lets just look at the small segmant of real numbers from 0 to 1. We then define a function (a function doesn’t necessarily¬†have to be y=5x+7. A function can just describe some sort of relation between two groups of numbers. Think of it more like a computer program which has an input and an output. For instance, we can define a function that returns¬†0 if¬†it receives an odd number, and returns 1 if it receives an even number), which takes each real number between 0 and 1, for example 0.12345, and returns the corresponding natural number which is made up of the numbers after the decimal point, only in reverse,¬†i.e.¬†54321 (why do we flip the numbers? well, 0.12345 is actually 0.1234500000…, which might be hard to transform into a natural number. What natural number¬†should we transform it to? 12345000000…? (how many zeros should the natural number have?) So we just look where the zeros start, and flip it from there). Each natural number will be covered, so the function from R to N is surjective. However, creating such a function from N to R isn’t quite as simple, and is actually impossible. We will prove that ūüôā

We won’t prove that the natural numbers can’t “cover” the real numbers. We will prove¬†that¬†they can’t even cover the real numbers between 0 and 1!¬† (and of course that will be enough to claim that it can’t cover all of the real numbers)

Lets assume that there is such a surjective function from N to R. We will call¬†this function¬†f (reminder: f will receive as input any natural number, and will return a real number between 0 and 1).¬†Whenever f receives a natural number it returns it with a 0. at the beginning.¬†So, 7354 becomes 0.7354000… and so on.¬†But I don’t like all those zeros. So what we are going to do is employ an important rule in math which I will not explain here which states that 0.9999… = 1.¬†Therefore, f(7354) will equal 0.735399999…

Now, suppose we got all of f’s values and wrote them on a grid. What we will do now is define a new number: x.¬†This number¬†will be “built” in the following manner: we will go over the diagonal of the grid, and each digit we see, we simply increase it by 1 (9 becomes 0) and add the new digit to x, like so:

 Cantor\'s diagonal argument

 (2 becomes 3; 1 becomes 2; 6 becomes 7 and so on and so forth)

Now comes the tricky bit, which is the whole point behind this proof: x is a real number which is not covered by the function f, and therefore f is not a surjective function, which means that there are more real numbers than there are natural ones.

Ok, back up. Why is x not covered by f? After all, one could just say that it is made up of a series of numbers, so a natural number must have created it. But all of the real¬†numbers we created with f, using natural numbers as the source, were listed in the grid. And x is different by one digit from every number we created. Remember that we travel along the diagonal, meaning we make sure that it is different from every number in the grid by at least 1 digit – that’s the beauty of Cantor’s diagonal argument. No matter which numbers we choose to input in f, we can always create an x which is different from all of them. QED.

Now¬†lets return to the riddle this post started with. If you stayed on for this long, and¬†understood Cantor’s diagonal argument, you should be able to escape from the¬†room. If not – well, they did say that punch and¬†pie¬†will be served ūüôā¬†


Not writer’s block, more like a time block

May 29, 2008

Haven’t been able to find the time between work, “miluim” (army reserves) and university to write anything new, but I have some ideas, so don’t despair ūüôā

My next article will probably cover Cantor’s function, which is a very elegant way of proving that there are “more” real numbers (R)¬†than natural numbers (N), as well as define what exactly constitutes a random number (or more precisely – random sequence). If there’s time, I think I’ll also discuss Berry’s Paradox, another interesting way of looking at math and logic.


The misconception of science, or “what happens when a duck quacks in an opera house?”

May 25, 2008

I am troubled by the manner in which the majority of people assume that the things they hear or read are correct, without subjecting them to any sort of scientific method.

What do I mean exactly? well, one prime example is the story of the non-echoing quacking duck.

For those of you who are unfamiliar with this “hypothesis”, according to all of the wonderfully time-consuming and¬†misinforming PowerPoint presentations we receive in our inbox from people who we consider to be our friends, a duck’s quack does not echo. Now, this statement has all the makings of a psuedo-scientific fact. It involves, hold on to your hats – “sound waves”. As we all know, sound waves are a very “scientific” subject, yet a relatively simple concept to grasp, since we’ve all seen how a pond reacts to a stone thrown inside it. In addition,¬†this “theory” refers to a specific species in the wild kingdom. Since there are so many animals out there, surely one of them interacts with sound waves in this unusual manner.¬†For some reason,¬†when people receive facts that seem to be scientific, something in their brain tells them to automatically treat this new information as correct.

However, when we examine this¬†statement a little closer, using real¬†scientific methods, we see the absurdity of this so called “scientific fact”. Yes, physically, it is possible to create a sound wave which does not echo. You don’t even have to study physics to understand this, you just need to¬†have a basic understanding of the concept, and some good healthy logic. However, it is quite a leap of faith to believe that all the ducks in the world (every last one of the little quacking bastards), in all echo-creating conditions, have the ability to produce the exact sound (e.g. exact wavelength and frequency) that will not echo.

However, I do not wish to dwell specifically on the matter of the quacking duck, but more on the concept of how people tend to agree with what they read/hear without taking a moment to consider what they are being told, and examining this new information with their own brain for a change.

I am not saying that we should question everything. That seems like quite a tedious way to go about life. However, I believe that every person should strive to develop the ability to pick up on psuedo-scientific jibberish such as non-quacking ducks.

After all, most of mankind’s greatest scientific achievements came as a result of someone, somewhere, saying: “this can’t be right…”: Charles Darwin‘s Theory of Evolution (“Do we have to accept the story of how we came to be as told by a group of non-sceintific religious zealots?”), Albert Einstein‘s Special and General Theories of Relativity (“Who¬†decided that time, mass and space¬†have to be constant in all systems?”), and of course, Nicolaus CopernicusDe revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), which I perceive¬†to be¬†the catalyst of modern science as we know it, since Nicolaus’ question was¬†not only¬†humble in nature (“Why must we believe that man is the apex of creation?”), but¬†also led to a less romantic and more objective weltanschauung, which is key when one is attempting to analyze the world in a scientific manner.

But perhaps the problem is not in the manner in which people think. Perhaps the problem is thinking at all. Sadly, I tend to believe that the problem lies in the latter.