Infinite processes in the real world

A long time ago, in ancient Greece, one of the philosophers asked a simple yet very important question - is matter infinitely divisible? He of course formulated the question in a much more intuitive way: what will happen if you take a stick and break it in half, than take one of the halves and break it in half again and so on. Thinking about this problem, he concluded that at some point we will not be able to continue breaking the stick. According to him, after a finite amount of time we will reach an indivisible component of matter. He named this indivisible component "atom".
As with any new idea, there were those who believed in it and those who concluded that this idea is wrong. Likely for both sides, there were no means to actually check it so they could argue as much as they wanted.

Even though we are much more advanced today we still don't know the answer to this problem. Ironically we have discovered particles which we named atoms only to find out that they can be split apart as well only a few years later. Although, to be really precise, we need to remember that the problem can be formulated as the "atom" being the basic component of a specific type of mater. In other words, one possible understanding of the problem is that it asks to find a "part" that if divided further looses the recognizable properties of the object we started with. If we formulate the problem in this way, then there are indeed such "atoms" - molecules.

At this point you are probably wondering what is this about and how is it connected to infinity. To understand this lets look on a somewhat famous paradox - the Thomson lamp. Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of all these progressively smaller times is exactly two minutes.
So, in the end, is the lamp on or off?

It turns out that there is no clear answer to this problem. While we know the state of the lamp at any time during the process, we cannot tell what is the state at the end. Now lets return to our original problem. Lets suppose for a second that "atoms" don't exist. With this in mind we can take the being from the lamp paradox and instead of it toggling the switch we will make it break sticks in half. Since there are no atoms, the process doesn't end before two minutes pass. But what do we have after two minutes?

In this case it is rather simple to look on the problem mathematically. Lets substitute the stick for the line [0,1]. The whole process can be described then as just a limit of [0,2^(-n)] when n goes to infinity. The limit is a single point, so that would mean that we will get a "particle" with size and mass equaling zero. However, that would suggest that the matter is build from particles with zero mass, and this is a rather bizarre conclusion.
The only possible result we can get from this line of thought is that if such a being actually exists then there are "atoms". However, if there is no such being then we cannot say anything.

While I would like to finish this post with at least a partial solution to the problems I presented, there is no solution as far as I know. There is, however, a funny "solution" to the Thomson lamp paradox. Lets assign numbers to the states of the lamp - 1 and 0. If we do this then the state of the lamp after n steps is: 1-1+1-1+...+(-1)^n.
Therefore, if we take the limit when n goes to infinity, we will get the state of the lamp after two minutes. So lets see what the limit is.

A=1-1+1-1+1-....

1-A=1-1+1-1+1-....=A
2A=1
A=0.5

As you can see, after two minutes the lamp is half on. :)

Gabriel's Horn

I didn't write anything about paradoxes for a long time, so here is a little something. Lets look on the volume we get by rotating the graph of the function y=1/x, for x>1. The object we get is called Gabriel's Horn. It is easy enough to show that its volume is finite and in is equal to pi. If we cut the horn in any finite point a we will get that the volume is exactly:If we now take the limit when a goes to infinity we will get that the volume is indeed pi. However the surface area is infinite. For any finite a we will get that it is exactly:


But the limit of this expression is infinity. Now that we know this , we can go on to the description of the paradox. Suppose that you want to paint the Horn with finite amount of paint. Obviously, it is not possible because the surface area is infinite. But you can fill it with a finite amount of paint. Lets now suppose that Horn is made from a transparent plastic. In this case, filling it with paint is the same thing as painting it.

As a result we get that it is both impossible and possible to paint the Horn with finite amount of paint. So which one is true? The solution is in fact rather simple. Firstly lets look on the graph of y=1/2x. Obviously this is again Gabriel's horn, but in a scaled down version. Lets put it inside the original while it is still filled with paint. In this way we painted it from the outside. How did we do it? The answer to this is in the distribution of paint. The thickness of paint is given by g=1/x-1/2x=1/2x. And this is the whole trick. We can paint even an infinite surface, the only thing we need to worry about is allowing the thickness of paint to approach zero in a way similar to this example (we need the integral of the paint distribution to be finite).

Indescribable numbers

This post is an attempt to explain what the term indescribable number means. Unfortunately, while this is a relatively well know term I frequently see it being misused. To understand it, we must firstly look on the proof that such numbers exist. It is a rather basic proof from set theory. What we need to do is to define two sets:

A={all the mathematical symbols and all the letters}
B={finite words in A}

Now, it is obvious that A is finite. B is not finite but it is only countably infinite (this is the smallest infinity). Therefore, B is smaller than the set of all numbers - R. Since all the possible descriptions are in B we conclude that there are numbers in R that cannot be described at all. Moreover, if you take away all the numbers that can be described the size of R will not change (this is a basic theorem in set theory). From this we can conclude that in fact most numbers are indescribable. This is a perfectly valid example of nonconstructive proof.

Unfortunately, this simple and short proof (I didn't proof all I said, but it is all just basic theorems of set theory) does little to explain what an indescribable number is. Lets consider some examples. For the first example, look on the following set:

C={words in A shorter than one hundred letters}

It is obvious that C is finite. Therefore there is a maximum number described by C. The next integer number (lets call it Y) is thus "the first integer number that cannot be described in 100 letters". But this description is less then 70 letters long. And this means that this number should be in C. Obviously this is a paradox. We got a number that is both in C and not in C.

Here is another example of a similar problem. It is possible to proof that if you randomly choose a real number the probability of it being an indescribable number is 1. So lets randomly choose a number (since we are choosing only one number we don't need the choice axiom). Naturally we get an indescribable number. Well, lets call this number "indescribable number 1". In the case you didn't notice I just gave a description to an indescribable number.

What really is going on is just an indexing problem. It is important to understand that both B and C are just sets of index numbers. If we have such a set we can use it to index another (in our case the set R). Mathematically, description as it was used in those examples is just a function from B to R (or from C to R). But the way we apply the indexing is arbitrary - we can choose any function we want. Lets first look on the 100 letters case. When we defined the set C what we really defined is the pair (C, f). In this pair f is a description function that for any x in C returns a specific number in R (I suppose that all the words in C describe some number, but you can do without it). Since we cannot define Y without firstly defining (C,f) the word ""the first integer number that cannot be described in 100 letters" is assigned by f to some random number. Then when we got a description for Y, we basically created a new pair (C, g). In this pair g is a new function that agrees with f on all C except for one word - ""the first integer number that cannot be described in 100 letters". To this word it assigns Y. For us this may seem illogical, because we think about meaning of words. But in this proof meaning is not important - the words are just a way to index.

With this in mind, lets consider the second example. In this case the number we choose belongs to a set R\f(B). When we gave it a description all we did was to change f in such a way that now this number belongs to f(B). It is obviously not a problem to do such a change (if we wanted to change the function for an infinite amount of values it might have been a problem, but for one index it is always easy to do).

So, what is an indescribable number? After all, we just saw that it is possible to describe any random number. The answer to this is actually simple. Mathematically it is a number that was not indexed by the function f. In normal language it means that it is a number that wasn't described. Form a normal person point of view this is a weird definition, but mathematically it actually makes a lot of sense. The basic idea is that while we have the option to choose any function f, we can only choose one and we cannot change our choice to another function latter on (this is because we need our language to be consistent). Under this conditions it becomes obvious that both examples are just a misunderstanding of what an indescribable number is. You cannot take a number that is not described and give it a description, because the description is already in use and you can have only one number for one description .
But then, what is the point in saying that such numbers exit? It is after all obvious that there are numbers that are not described as of now. Unfortunately this is not what the theorem is about. The point of this theorem is not to say that there are numbers that we didn't describe. This theorem says that no matter what function f we use there are always numbers that are not in f(B). In other words, there are always will be numbers that we didn't describe even if we used all of B for the purpose of such description.

Dividing gold in a rational way

I finally got internet connection in my new apartment, I am really lucky it didn't take more time. Also, as I expected the semester in the university started on time despite the threats to not open it unless the government pays. I am doing 8 courses this semester, so I am really busy most of the week. Yesterday I was in the university from 8:00 to 20:00... Gladly today the lectures start at 16:00 so I can stay at home in the morning.

Now, about the title. One of the courses I am doing this semester is game theory. When I got the exercise for this course, one of the questions was about dividing gold between pirates (surprisingly, the other questions were rather difficult questions in analysis). What is interesting in this question is that the solution seems unbelievable, so I decided to post both the question and the solution on this blog:

The situation is as follows - five pirates got 50 coins of gold. They must find a way to divide those. The pirates all have different rank from 1 to five. They decide on how to divide the gold in a very simple manner: the pirate with the highest rank offers a way to divide the gold, and then the rest vote for or against his proposal. If the absolute majority is against he is killed and the process starts over with 4 pirates. There are three further assumptions. Firstly, the pirate who offers how to divide the gold wants to get as much gold as possible. Secondly, all the pirates are rational. Thirdly, if a pirate have no reason to vote for or against the proposal, he will vote against it.
A tip: go from the end to the beginning.

The solution is that the first pirate (who has the highest rank) will get 48 coins and the pirates 3 and 5 will each get 1 coin. I don't want to post the full solution, if is obtained by repeating a few simple steps on the situation, so I will just write the beginning of the solution:
Suppose there is only one pirate, he then takes all the gold. If there are two pirates (number 4 and 5), no matter what way to divide the gold the forth will offer the fifth will be always against, because then the forth will die and he will get all the gold. If there are three (3, 4 and 5) then the fifth will be always against and the forth doesn't have a reason to vote for or against (we assume that his life is not important to him), so he will vote against unless he gets something - at least one coin. In this situation, if the third will give him one coin, the forth will vote for him, so the third can keep 49 coins.

Really bad logic

I found a site about logical paradoxes. I love such paradoxes, so I was really happy to stumble on such a site. However, while some of the paradoxes mentioned on that site are classical examples, the explanations were of extremely low quality. The reason I am writing this is that one of the paradoxes I read there has an interesting connection with set theory. I am not going to link to this site, but I will write the "paradox" here:

The paradox arises from considering the following two statements:
1. All ravens are black.
2. Anything that isn't black isn't a raven.
Obviously, both of the statements are true. Also, they clearly say the exact some thing. Therefore if I can find somethings that shows that the second one is true, it must follow that so is the first one. Lets suppose that I found a green parrot. It is green and it is not a raven, so it supports the second statement. So, because I found a green parrot I conclude that all the ravens are black.

The problem is that the line of thought shown in this "paradox" is not correct. I never heard before about this paradox, so it might be an invention of the owner of the site I find it on. Or, it is not explained properly there (who ever wrote it managed to make the barber paradox look weird..).

If we will look on this paradox from set theory point of view, we will see that statements say:
1. If a is in A, then a is in B.
2. If a is not in B, then a is not in A.
The paradox tries then to sell us the idea that: If c is in C and not in B then all a are in B. In this form the fact that this paradox is just a game of words becomes clear. While there are paradoxes, the only paradox in this is that some people (at least the owner of the site I found it on) actually see this as a paradox...

Quantum cat

A quantum cat. I am sure Schroedinger didn't think that his little experiments will become the reason for so many jokes..

Science is fun sometimes, isn't it?

Since in this case we see the cat, we are sure that he is in the box (and we know he is alive). And therefore this is not a paradox. However, it is always good to think about puzzles of quantum mechanics...

Relativity and quantum mechanics

In my previous post on this topic I have shown that while faster that light travel is impossible, it is possible for the electrons to move from one energy level to another in zero time. The distance it travels is like the distance between planets on our scale, however the time is zero.

The reason for such a result is very simple - relativity and quantum mechanics cannot be used together. It is not possible to apply relativity where one should use quantum mechanics. When we discussed individual electrons the relativity theory simply stopped working. The results that were correct for large scale become wrong on this scale.

But why is it so? There is after all a general agreement that a theory that works only under specific conditions should transform gradually to a different theory when the conditions it requires are changed. This part probably sounds a bit confusing, so here is a simple example:

Photo by wili_hybrid

A long time before Einstein, people noticed that for two system that are moving with a constant speed compared to each other, the system of coordinates has to be transformed when you move from one system to another. If, for example, you are on a train that moves with 50 kph east relative to the earth and you see someone who is sitting on the field outside, than from his system of coordinates you are moving with speed 50 kph east, but from your point of view he is moving with the same speed to the west. The transformation used to move from one system to another is the Galileo transformation. If your coordinates in one system are (x,y,z,t) than your coordinates in a system moving away from you with a constant speed v are (x',y',z',t'). If at the moment t=o the both observers where in the same place and the movement is only on the x axis we get that:

x'=x-vt
y'=y
z'=z
t'=t

However, according to relativity this is not correct when v is big enough. In relativity we use Laplace transformation instead of Galileo's. Under the same condition we will get:

y'=y
z'=z

The formulas look very different. Partially this is because the of units used. However, if we will go to the limit were c is significantly large than v (that is c is regarded as infinite), they will turn into the Galileo transformation. It is very easy to see - the only impostarnt part is to notice that the units need to be balanced, after this it is trivial. We can say therefore that relativity turns into classical mechanics when the speeds are low in comparison to the speed of light.

However, this doesn't happen with quantum mechanics. It is divided from relativity by a scale barrier, and when this barrier is approached the two theories start to contradict each other. A lot of work have been done to solve this problem. The main approach is to try to unify all the basic forces. Those forces are - Electricity, Magnetism, Gravitation, Strong and Weak. The first two are already unified for a lot of time. The weak force also can be unified with them. I also heard that the strong force was unified with the weak force, but I don't know the details. Gravitation is a problem however. For the other forces particle carriers where found - but not for gravitation. In fact, the question what gravitation really is, is still without answer. It is a mystery waiting to be solved...

A bit of trivia - It is a surprising fact that Einstein contributed a lot to both of these theories, but while he helped quantum mechanics to take roots he wasn't happy with the result. He was the one who proposed the duality of the photon, and he was the one who helped to promote the understanding that all particles have this duality.

I ended my previous post with a question - Was Einstein wrong? The answer should be clear, but I will say it anyway. He wasn't wrong. It is simply that by going to this very small scale I left the domain of relativity and there the rules are different.

You probably noticed that this post raises a very interesting question. Since faster than light travel is equivalent to time travel as I have shown, does the fact that the electron can move such great distance (on his scale) in zero time means time travel is possible? Nope. There is no time travel in this case. The reason for this is simple, but it requires getting used to. The electron belongs to the "quantum world". We can think about this world as being separated from our world by a "shield". This shield is called The Heisenberg Uncertainty principal. What it says is very simple - the uncertainty in the location and energy are always bigger than some constant number. It means that we simply cannot see too well what is going on in this "quantum world".
In fact the way I used to show that faster than light travel is equivalent to time travel depends on accurate measuring of distance and time. Since we cannot do this with the electrons, even this general result just doesn't apply.

By the way, if you want to be remembered forever in the history of science, finding a way to unify electricity and gravitation will surely achieve this goal...

Faster than light

I finished my post about time travel paradoxes with a promise to write about one very specific example in which faster than light travel (which is equivalent to time travel) is possible. I am going to write it in a slightly weird way - firstly I intend to explain why this particular example is not possible, and then to show that it happens nevertheless.


We all know how an atom looks like - it is just a little ball (nucleus) surrounded by a cloud of electrons. Most of the atom is empty space. The atom has a very interesting property: the electrons that orbit the nucleus can be only on a finite number of orbits = energy levels. I am not going to talk about why it is so, but what is important to note is that the distance between any two energy levels is very large. In fact, the atom can be viewed as a solar system in miniature - the distances between the orbits are comparable to the distance between planets. The electrons decide on which energy level to be according to the energy they have. The more energy an electron has the higher (further from the nucleus) it will be..

Thee is a simple way to cause an electron to jump from orbit to orbit - we just need to hit it with a photon. If the energy the photon has is the energy needed for the electron to move to a higher orbit, it will absorb the photon and move to this higher orbit. After discussing this we can go to the main question I will deal with in this post:

How much time does it take for the electron to move from one energy level to another?

Answer: Zero.

To explain this, lets firstly look on why this time must be more than zero. In the following paragraph I will develop a mathematical formula - if you don't understand what I am writing just skip it and go to the next paragraph.

Lets look on a regular piece of wire. If we will connect it to a battery (if you want this battery to be of any use lately don't do it) a current will flow in it. How much current? The usual way to define current is to look on the area of a cut of the wire = A. The current than is the amount of charge that passes in infinitesimal time=dt. To define it precisely we need first to define the density of charge flow. The density of charge flow (J) is the number of free electrons in the wire multiplied by their speed and by their charge: J=nvq. Note - J is a vector. The current is than defined as:

I=

dA is an area vector of an infinitesimal part of the cut area A. If, as it is with regular wire J=const we get that I is simply: I=JA. The formula with dA is good for any object in which a current can flow. Because of this we can use it in a slightly different way. Lets think about a general bounded 3d object. This formula is good for it - but instead of saying what the current is, it tells what is the rate of change in the charge of this object.
What the formula says exactly? It says that the rate of change of the charge is an integral on the charge flow density. Thus, the charge (electrons) that left this object is exactly the charge that passed the surface of this object - the integral is basically a flow throw the surface area of our object. This means that electrons cannot just jump outside - they must cross the surface somewhere. This conclusion seems obvious, but it isn't. Without this formula, having only the law that the total charge is conserved, we can say that if the electron jumped outside the total charge in the space is the same - so it is possible. But this simple formula says no. However, it doesn't say that it is impossible for the electrons to jump inside our object. To show that this is also impossible we need another formula, so lets develop it.

What is Q? It is the total charge of our 3d object. Therefore it can be expressed as:

In this integral rho is the charge density (it is not J). If we will put this expression into the previous formula, and also use the Gauss theorem to change the right integral in our formula from JdA to divJdV we will get:

The both integrals are dV, so we get that:

It is usually written in this form:

The plus sign is a convention - it doesn't really matter, because the positive and negative flow direction is defined according to what is easier to work with.
What does this new formula says? It says one important thing - the change in the density of charge in an infinitesimal is equal to minus the divergence of J. This is exactly what we were looking for - now even the electrons inside our 3d object cannot jump. They all must flow passing all the midpoints.
Therefore it is impossible for the electron to jump from one energy level to another in zero time.

But is this correct? No. The math I used is correct, but there is one single thing that is left out - quantum mechanics. The formulas I developed don't take it into account. They are still correct, but they cannot be used on very small scales.

It is now time to explain what happens with out electron. When the electron is being hit by a photon, it consumes the photon and gains its energy. This energy is exactly the energy needed for it to jump to the next energy level (otherwise the photon will not be consumed). The collusion between the photon and the electrons happens in one instance - it is not a process. What this means is that the electron gets energy that corresponds to the next energy level. Therefore it must be in a higher energy level. It cannot just stay. If there would be no photons, than the electron would consume the light energy gradually and would "rise" to the next level. But the energy is transfered in a collusion.
Mathematically, the potential energy is space must be continues. Because of this it is not possible for the electron to have a higher energy while being in a place that corresponds to a lower energy. The reverse is also true - when the electron descends to a lower energy level it releases a photon and jumps down - in zero time.

So, was Einstein wrong in saying that faster than light travel is impossible? Were I wrong when I said in the previous posts that faster than light communication (or travel, it doesn't matter) is equivalent to time travel? This I will answer in the next post.

Time travel paradoxes

I recently wrote a post about why faster than light communication is impossible. I finished this post showing that such communication would be equivalent to time travel, and it is logical to assume that time travel is impossible.
However, is this indeed true? Time travel is often talked about in science fiction, but is there a reason to think that it is possible?

Photo by jonrawlinson

Firstly we need to distinguish between two types of time travel - to the past and to the future. We all know that it is possible to go to the future. We all do it, it is called living. It is also possible to travel to the future with a "faster speed" than normal life. All you need for this is to accelerate yourself to a high enough speed, and according to relativity you will travel to the future - when you will return to the place you started your travel you will see that your watch is late. How late depends on your speed, it might be a second or 1000 years. This effect is very real, we even have to take it into consideration when we communicate with satellites.

Traveling back to time is a different story. Unlike traveling to the future it was never done. So the only thing we can do is to discuss different theories. There are three main theories about traveling to the past:
1. Time travel is not possible - there are two versions: either there is no way or it will destroy the universe in the process.
2. Time travel is possible but the past cannot be changed.
3. Time travel is possible, but it will destroy part of the universe.

Lets look on all of the three theories:

1. Time travel is not possible:
There are two versions of this theory. The first one comes from a literal understanding of time travel. According to it in order to travel back one second in time, you need somehow to return the whole universe to the exact some state it was one second ago. It means that you need to place every single particle exactly at the same place. However this is clearly an impossible task.
The second version of this theory is based on time travel paradoxes. The paradox I will talk about here is called the grandfather paradox: Lets suppose that time travel is possible. Lets also suppose that someone (Mr X) traveled back in time. While he was in the past he 9accidently) killed his grandfather. In doing so he preventing himself from being born in the first place. But if he wasn't born how could he travel back in time?
There is also a more general version of this paradox - by traveling back in time you change the world past, so in the very moment you will get to the past the world you come from (the future) will no longer exist. And therefore, you never traveled back in time.
The conclusion from this is that if time travel is possible, you will destroy the whole universe by traveling back.

2. The past cannot be changed:
This theory is an answer to the question arising from the previous one. In the grandfather paradox, we assumed that it was possible for Mr X to kill his grandfather and to prevent his own birth. But what if this is not true? What if there is a law that not allows people to influence the past? In this particular example, we can suppose that Mr X will be stopped by police just in the right moment, or it would turn out that the men he killed wasn't his grandfather at all. For the more general version of this paradox, we can assume that either the results of the activities of time travelers just slowly vanish so they don't affect the future in any way, or that there is a fixed time line in which time travel appears together will all other things and nothing can be changed.
This theory allows for time travel and solves the paradoxes I presented, but there is a problem with it. Lets do the following thought experiment: Suppose you have a time machine, and a laser that shoots a bit of light into the time machine. The machine send the light back in time, so it goes out of it on the opposite side and two minutes before the laser was fired. On the wall after the time machine there is a detector that when hit by the laser been will send a signal to put a barrier between the laser and the time machine. It looks like this:Now if the time machine works, the laser will prevent itself from firing, but this would mean that the detector didn't close the barrier so the laser worked - and this is a contradiction. Note that it doesn't matter how the time machine works, and to how long ago the light pulse is sent.
If, as the theory says, we will assume that somehow "it all worked" it follows that something is broken - if for example the detector is broken, no paradox will be created. However, this is a very simple system. The only thing that is likely to always malfunction is the time machine itself. This means that the time machine doesn't work, and therefore time travel is impossible.

3. Local destruction:
This one is the attempt to unite the previous two theories. Basically it says that time travel is possible but because of the paradoxes described above, it will destroy the universe. However, the universe is a very large thing. So only a small part of it will be effected. Time travel will create a "wave of destruction" which will move over some finite distance, destroying everything. As it moves it will slow down and become less distractive, so after some finite distance it will just stop. Beyond this distance (this is, beyond a sphere with the time machine in the center), the universe will remain as it was. Inside this sphere however nothing will exist - a singularity will be formed.
From this is should be obvious that this theory also doesn't allow for time travel - because it can be used only as a weapon, and it is not possible to return or to do anything.

Conclusion:
Time travel is not possible, and therefore faster than light communication is also impossible.
But - not always. In the next post about this topic, I will write about some very specific situations in which faster than light travel and time travel are possible.

Why faster than light communication is impossible

Lets consider the following situation - You are 4 light years away from the Earth, on a distant planet X. While you there aliens land on that planet and capture you. They make you tell them from where you came, and tell you that they will go and destroy Earth. You manage to escape, and get to your spaceship. You cannot stop them, and they don't want to look for you - they just go to Earth.
Lets suppose that:
1. Their spaceship can fly at a speed lower than light, but faster than yours spaceship.
2. If the people on Earth are informed in advance about these aliens coming, they will likely be able to defend themselves.
3. You have a device that allows you to send message to Earth that will travel there faster than light.
4. You use the device and send the message in the exact same moment that the aliens leave that planet.

Now, after we have all the information, lets see what will happen.
From the view point of out story hero, he send the message and now goes home.
From the view point of Earth the message is received, and they prepare to fight the aliens.
But what the aliens see?

Lets denote the starting point (the planet X) and the starting time (sending of the signal) by (0,0) respectively, and the end point (Earth) and end time (signal received) by (x,t).
The speed of the alien spaceship is v<1 c="1)" style="text-align: center;">x'=

t'=

For point (0,0) we get (0,0), which is not surprising. For point (x,t) we get ( ,).
Firstly lets suppose that t=0. This means that the message went to Earth in zero time. We will get: (). The minus sign means that this event happened before t=o. Thus the signal was received before it was sent. This is clearly impossible - unless a time travel is involved.

Now, what happens if the message is not instant but is still faster than light? In this case t=x/(1+h) - time is distance divided by speed - where h is a positive number (remember I am working with c=1).
Lets look what is the condition for t'>0, for any v (we must find condition for any v because we don't know what is the speed of the alien spaceship):

>0

We can divide:

>0

And finally we get:

1>v+vh

Rearrange:

1-v>vh

If we will choose v=1-<1>

>(1-)h

If we will now take the limit when epsilon approaches zero, we will get that h=o. This means that it is not possible to send messages faster than light, because otherwise there are always be an observer for whom the order of the events changes - which means time travel.
Note that the way the signal is send is unimportant. The only thing I used is a spaceship which is totally unrelated to the way the device works.

Strictly speaking, this doesn't prove that it is impossible to comminicate faster than light - but the only way to do this involves time travel. Therefore it is logical to assume that it is simply impossible for anything to travel faster than light, including information.

Update: Read part two of this post - Time travel paradoxes.

Where the movement occurs?

Continuing the discussion on Zeno paradoxes, lets examine the moving arrow paradox. In the case you missed the two previous posts about Zeno, you can read about another two of his paradox here and here.

In this paradox the main problem is to explain when things move. It seems a strange thing to ask, but it is hard to give any answer to this one. We don't usually see any problem with our perception of time and space. But what will happen if we will start to analyze the world around us from a logical position? The results are often surprising.

Photo by Odalaigh

The paradox arises from the following line of thought: Lets take an arrow that is flying, and look on it in an infinitesimal time. More precisely, lets look on it in a point in time. In such a short time "period" (not even a period in fact, but a mere point in time), the arrow clearly doesn't move at all. The time is made from such points. Now, since the arrow doesn't move in anyone of these points, how can it be that it moves at all? It must remain at rest - and thus time and motion are illusions.

The explanation to this is very simple. In order for this paradox to work time must be infinitely divisible. But this is not true. There exists a basic unit, called Planck time. I already wrote about this in the first post about Zeno paradoxes, so I am not going to repeat myself here. What is interesting about this answer however is that it shows that Zeno paradoxes touch the very structure of our world. As such they are valuable, because they show us were we live.

The most interesting property of these paradoxes is that they show how easily we can deduct amazing facts about our world from pure logic. Zeno lived a long time ago and yet managed to create paradoxes that were only fully resolved scientifically by showing that our world is discrete. This was done only very recently. I believe that this is a great lesson both about logic and about science.

Counting the points

In the previous post, I wrote about an ancient paradox. The author of this paradox, Zeno of Elea, presented 7 paradoxes. In this post I will focus on another of his paradoxes.

This paradox arises from the following situation. Suppose you need to go to a shop to buy some milk. You live very near to the shop so you decide to take a walk. Lets suppose the shop is 100 meters away.

Photo by lime*monkey

If you read my previous post, you probably already guessed that Zeno claimed that it is impossible for you to get to the shop...

Where is the problem? Lets look on it in the following way - in order to get to the shop you need firstly to pass half of the distance to the shop. But even before this you have to pass a quarter of the distance. And before this an eighth of the distance... And so on.
It turns out that in order to get to the shop you need to pass an infinite number of such mid points. Zeno reasoned that this is impossible, because you would need an infinite amount of time for this. Therefore you cannot move at all. No matter how small the distance you move we can divide it in the some way and get an infinite number of mid points again. Conclusion - movement is only an illusion.

There is a simple solution to this paradox. Firstly, if we will sum all the time intervals the sum will be a finite number. This is the accepted solution to the paradox.
But this is not a full solution. In order to sum the time intervals we need to know two things - distance and speed. We know the distance, it is given in the problem. But we don't know the speed. We can decide that the man in the problem has a speed of ten meters per minute. If we would do so, it is trivial to find how much time it would take him to get to the shop:

And for all the mid points we would get the following infinite series:

5+2.5+1.25+ .... =

As you can see the sum of all the intervals is indeed a finite number. But can we assume that speed exists at all? The whole point of the paradox is to show that motion is only an illusion. By assuming that something called speed exists we assume that the paradox is wrong from the beginning.

What is interesting in this paradox is that it doesn't describe our reality as we know it. There is a hidden assumption in the paradox that the distance can always be divided. But this is not true from a physical prospective. Our world is discrete - there is a basic unit of length called Planck length. Below this length distance is simply meaningless. So in our world we have only a finite number of points between us and the milk shop, so there is no problem to get there in a finite time.

Zeno's paradoxes

Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic, and Bertrand Russell credited him with having laid the foundations of modern logic. He is best known for his paradoxes.

In this post I want to focus on one of his paradoxes. The paradox arises from the following situation:
A Greek athlete named Achilles is approached by a turtle. The turtle wants to compete against him in a 100 meter run.

Photo by lander2006

Since Achiles is the fastest runner, he decides to let the turtle start 20 meters ahead. The turtle speed is 1 m/s while Achiles can run at 10 m/s.
Now who do you think is going to win in the race? It seems that the logical answer is that Achiles will of course win. But is it indeed so?
Two seconds after the beginning of the race Achiles run 20 meters, bur he still needs to run 2 meters to reach the turtle. 1/5 of a second later, he runs 22 meters but the turtle is still ahead of him - by 1/5 of a meter. We can continue this way to infinity - Achiles will always need to pass some extra distance until he reaches the turtle.

It is clear that Achiles will never reach the turtle. No matter how much time will pass the turtle will always be slightly ahead of him. Moreover since there is no final step in this series, Achiles need to run for infinite time. This is not what happens in the real world, but what is the explanations to this?
One proposed solution is that as the distance between Achiles and the turtle approaches zero so it the time needed to pass this distance, and the sum of these time intervals is finite. Lets write it mathematically:
The intervals are:

2, 0.2, 0.02, 0.002 ......

This is a simple geometric series, and the sum is:

This is approximately 2.3 seconds. And this is of course a finite time.

However this is only a partial solution of the paradox. The reason why Zeno published this paradoxes was to defend the claims of his teacher who claimed that space, time and motion don't exist. In this paradox he tried to show that if we assume that space and time exist, what we get is that Achiles cannot outrun the turtle. And since this is not what happens in the word around us time and space don't exist - they are merely illusions.
This line of thought is probably exactly the opposite of the goal of this blog - as I see it, the goal of Math Pages is to show that math and the world around us are logical. Zeno however tries to claim that what we see is an illusion. Lets therefore take a look on another version of this paradox, a version that clearly shows the connection between this paradox and the claim that there is no time.

In this version, Achiles runs after the turtle and counts the instances in which he reached the previous position of the turtle. Thus when he reaches 20 m this is one, when he reaches 22 meters this is 2 and so on. Lets suppose he outruns the turtle. To what number he counted?
As in the previous version, there are infinitely many such points so it turns out that he had to count to infinity. But this is impossible.
As far as I know there is no mathematical answer to this problem. It doesn't help that the time intervals become smaller, because he still needs to accomplish an infinite task in a finite time period.

There are two possible solution to this paradox. Firstly it depends on the fact that time is divisible to infinitely small parts. This is wrong however. It turns out that there are exists a basic unit of time - it is called Planck time. It is approximately 5.4*10^-44 seconds.
The problem with this answer is that this time is the smallest unit of time because physics begin to "break" on shorter time intervals.
The second solution is the Holographic universe. It is a new and weird theory that claims that our universe is in fact one particle that is simply viewed from different points of view. If so, then space is a sort of illusion and the paradox proves it's point. I don't think that I believe in this theory, but since it offers a logical explanation and not merely claims that space is an illusion, it is acceptable for me. By acceptable I mean of course acceptable as a theory and not as a fact.

There are two other Zeno paradoxes about which I want to write, but they will have to wait for the next post :)

Explanation of a Geometry Paradox

Last month I wrote about an interesting geometry trick, that resulted in a rather strange picture. In the post I explained why it happened, but it wasn't completely clear to some of the readers. So I decided to explain it again in a more complete way.

The problem is the following picture:

In the previous post I just wrote that the slope was different. For me such explanation seemed complete so I just continued ot talk about importance of proofs. In this post I will try to explain what it means that the slope is different.
First of all, it should be obvious that the area of both triangles should be equal, because they are built from the some blocks. This means that the missing segment area is still in the picture, but in a different place. Lets look on the following illustration:



In this picture the D dot is moved slightly to create a change in the slope. It is easy to see that the second triangle (on the right) is identical to the first (on the left) but it has another triangle added to it. This is exactly was was done in the original picture. The tiles were rearranged in such a way that the line connecting the points A and C is no longer straight. If the distance that the dot D has moved is small enough we don't see this, and the triangles seem identical.
Lets calculate how much the dot needs to move in order to account for the missing segment area.
In our picture the size of the triangle is: AB=5, BC=13. Thus according to Pythagoras:

The missing area is a 1x1 square. If the area of the ADC triangle is also equal to 1, we get the following equation:

And therefore h (the height of the ADC triangle) is: h=1/7 of a square. I rounded the figures a bit but the result is close enough. The result seems large, but it is not. It is of course possible to see, but it is not immediately visible.

The only thing that remains, is to show that indeed in the original picture the slope is different, and that the rearrangement of the pieces creates an extra triangle on top of the original one. This is very simple to do. All you need is to download the image to your computer and open it in an image editing program that supports layers and transparency. Now, cut the bottom triangle and put it in a different layer. Place it above the top triangle and play with transparency. Or you can watch the video below:

The program used to edit the picture is Gimp. It is a free program, capable of replacing Photoshop. The only problem with it is that it is not very intuitive.

I hope this post answered the question. If not, let me know in the comments.

Geometry trick

As I already wrote once, I like strange theories and paradoxes, but I believe that there is always some underlaying logic. For example look at this image:
Do you see the reason for this? There is nothing wrong with this picture. Moreover this is exactly what you will get if you will make your own triangle.

The explanation is very simple. The area of all of the smaller triangles is equal, in both pictures, to the original one. But if you will look carefully you will see that that the slope, of the second figure, is different.
Update: This wasn't clear to some of the readers, so I wrote a more detailed explanation.

Such things are the reason why in math everything must be proven. We cannot just trust our intuition. It was relatively easy to spot the reason in this case, but it is not always easy. Geometry we can at least visualize, and even make pictures. When the question is totally abstract it is much more difficult to understand it - especially if our intuition is wrong about it.

A few days ago I overheard two students talking about their linear algebra exam. There was a question in the exam to find a basis for some vector space. The student found it, but didn't prove it is indeed a basis. He didn't get any points for the question naturally. And he was angry about it...
But without proof there is nothing. Even if the answer is correct, without proof it is worthless. Such approach is the only way in which paradoxes can be solved and mistakes can be avoided