Neo & Oracle

Oracle makes predictions about the outcome of a set of events just like . Neo on the other hand tries contradict the prediction after he knows what Oracle has predicted about him. Therefore, he behaves just like .

Foresight & Freewill are two mutually constructive meta- processes (i.e. a process that deals with processes), that evolve over time to become… God?

But why would God make a new version of itself? Is it the begnning and the end?

Economic history of various regions for the last 2000 years

What share of the World GDP did each region produce?

As you can see the above graph, India started of producing 33% of the total world production. That is, there was indeed a time when India was a economic superpower. But over 2000 years the Indian sub continent has gradually produced lesser and lesser of the what the whole world produced.

Nature of Freewill

Upon deep meditation about the most holy mysteries of The Proof of Turing’s Halting Problem, I have come to the conclusion that as defined in the one of the previous posts is the nature of freewill. And is the nature of foresight.

Those two are in a perpetual pissing contest with each other. Always trying to one up the other.

A new that can handle a wider set of inputs than its previous version is always being formulated in our minds as a response to experiences we have (i.e. using some kindaf Machine Learning equivalent). But after a new is formulated an equivalent is formulated automatically. This isn’t a big deal; all that has to be done is to contradict .

So freewill can be created in a mechanical fashion once foresight is created using a learning process.

Defending Reality

Who would have thought that reality would need defenders?

I am a bit pessimistic now. I think sooner or later, creationism will be taught in pre- university education all around the world. People are just waiting for US & UK to make the first step for mankind, backwards.

Creationists & IDists are everywhere. Just recently, I read a blog by a creationist in NUS, my university. He was studying life sciences (like me) and had better grades than I do, because the system cannot grade people based on their acceptance of reality.

As modern economy gets more and more based on technology, results that where derived using the scientific method will become mere rules of thumb.

I can easily see signs of that happening in my classes. Microbiology protocols have become rules of thumb. Nobody wants to know how they found it to be true. Examinations grade people who are better at manipulations using rules of thumb.

Teach the controversy is a noble principle except of course one must ask, why evolution? And why high school education?

My First Post On American Politics

I am not American, so don’t take me seriously. I don’t know enough about what is really happening in that country.

Given the ease with which people “can” rip apart potential VP Sarah Palin and given that the mainstream media pretends not to know anything bad about Sarah Palin, while the Internet is flooding with totally absurd things she did, makes it look as though something is not right.

It is scarily similar to a scenario John Grisham narrated in his book The Appeal. In the story, the bad guys set up a Obviously A. Villain type candidate against the good guy. He was meant to loose from the beginning. He had an unfavorable background, and he deliberately got himself into situations where he could be ripped apart.

I think the Republicans already know they are not going to make it this time. They are going to make it easy for the opposite side. They are getting ready for the period of actual governance by the Democrats, during which they will give them a bad image. So that they can come back after 4 years.

Republicans are going to let the people have the celebrity they want as a president. And laugh at every mistake Obama makes.

Frequently Recieved Complaints

These are the frequently received complaints against the proof of non-existence of a universal discussed in the previous few posts:

• You bastard! You set up into a situation it cannot handle: Yes. All I did was point out one situation cannot be correct about. There may be other more embarrasing scenarios won’t work on. Therefore, an ideal is impossible.
• What if I write an that looks into to see if contains self contradicting predicaments?: And do what? Return “undecidable” or “not a number”? If bites the bullet and says it can’t predict if the will halt or not, then it just confirms the proof that it is impossible to have a perfect that can predict whether any algorithm can halt or not.
• If there can be no way to predict whether “any” algorithm will halt or not, how do we predict things about the output of algorithms: We bite the bullet and play the undecidable card if we can’t decide the nature of output of the algorithm.
• Can’t I make arguments similar to the one against the existence of to any algorithm that makes a boolean decision about the output of an algorithm? Yes. All algorithms that makes a boolean decision about the nature of output, or classifies the output of all algorithms to one of 2 arbitrary sets based on an arbitrary criteria, can potentially be made to contradict itself by making a analogue that creates the opposite effect.

This means there is no way to predict anything about the output of all processes, because there will always be an undecidable scenario. There is however the enginneer’s solution i.e. make something that is correct almost all of the time.

What does all this mean?

• There are no short cuts to the future (a.k.a output) than to actually go there. Sometimes there is no future to go to. Things get so unpredictable. Heisenberg’s Uncertainty Principle is not just the property of the physics of this universe. Things will always be uncertain.
• Not all truths can be proved to be right or wrong with a fixed set of axioms. Somethings are accidentally true. Somethings are accidentally false. You may want to consider accepting those accidentally true things as axioms if it keeps becoming so all the time you verify it. And in case you were wondering: No. Modern religions are not just internally inconsistent, they do not even correspond with reality as we know it. BTW Any fool can make an infinite number of internally consistent world views e.g. String Theory.

Conclusion: Religions do not fall into same category as things that may be accidentally true like Goldbach’s Conjecture. Because as so far as we can verify Goldbach’s Conjecture hasn’t failed its prediction. On the other hand religions fail even the first few simple tests of correspondence with reality.

(defun haltp (a i) …

Lets just say I have a way to decide whether a process will halt on a given input a.k.a as discussed in the previous post. Keep in mind that our dream function gives correct results for all possible inputs. Let us draw a table enumerating the outputs of for all possible algorithms operating on all possible inputs. Contrary to the definition of in the previous post let us use return values 1 and 0 to denote whether the input algorithm halts or not respectively.

I hope you do realize that every algorithm is a number. If you don’t believe me open up your executable files in hex editors. The compilers/assemblers convert your program into one huge number. All inputs are numbers to because they are binary in your computer.

So think of this table as similar. We enumerate all possible algorithms as a number on the left column. We enumerate all possible inputs as a number on the top row. Since our lovely “Will it halt?” procedure can decide whether any algorithm will halt for any any and all inputs, we are going to see whether such a procedure can exist in a Turing Machine.

If I need to show that such a perfect halting function which works in all cases does not exist, all I need to show is that there exists one situation when our hypothetical halt function fails.

I am going to define one such situation. In fact, I am going to define a function/process/algorithm called such that our hypothetically perfect function cannot decide whether will halt or not. Ready?

Here is how algorithm is defined: It finds whether the th algorithm halts on input and creates the opposite effect, i.e. becomes undefined (it makes itself undefined by looping forever) if returns 1 on the th algorithm with input as input. On the other hand, becomes halts by returning 0 if returns a 0 on the th algorithm and input as input.

Surely there is no function that behaves (halt/not halt) like in the enumeration of all possible algorithms for which make a correct prediction about whether it will halt or not. Because had there been one such function then the cell in the table must be equal to cell in the table, but we constructed in such a way that cell differs from whenever th cell is 1. So cannot be added to set of all algorithms that can act on to produce the correct results. But we defined as something that is capable of deciding whether any function will halt or not. And we found functions about which such decision cannot be made. Thus, it is impossible to have a perfect function which can decide whether any algorithm will halt or not.

This last paragraph we made contains an argument that remarkably similar to the Cantor’s diagonalization argument.

Confused? Did not understand the argument? Here is an alternate explanation which can be easily seen as equivalent to the above argument:

is defined as:

The different piecewise definitions are the different interpretations of . They are equivalent. returns 1 if algorithm halts on input , and returns 0 if it does not halt. What does is to see what would do and create the opposite effect.

The contradiction happens if we try to predict whether halts with input . Because if says halts, then runs forever and ends up contradicting what predicted would do. Similarly, if says runs forever, then halts and ends up contradicting what predicted would do.

Thus cannot act on and give a correct output. Thus we contradict ourselves because we had wished our to be ideal. i.e. capable of deciding whether any algorithm will halt or not. Thus a perfect cannot exist.

Feels like Heisenberg’s uncertainity, doesn’t it? More about the philosophical implications later.

Will it halt?

This is perhaps the most important thing we should like to know about any process. OK I agree it doesn’t seem like an important thing to know. What’s the big deal? So lets just say, I have it with me: I have an algorithm to determine whether an algorithm will halt or not and it is always right. I call it . It takes any algorithm as it’s first input and an input string , algorithm will work on, as its second input. It i.e. returns “will halt” or “will not halt” as its output.

Imagine what I could do with this ! I can prove the Goldbach’s conjecture and get the Field’s Medal with it. More importantly, I can prove everything and win all the field’s medals forever. ROFL!

OK here is how I will do it. But first, Goldbach’s conjecture:

Every even integer greater than 2 can be written as the sum of two primes.

Nobody has proved it as such. Distributed supercomputers have verified this to be true until . But hey! I have ! So I write a program that loops over all possible even integers greater than and halts whenever it finds an even number that cannot be written as sum of two primes. Now the best part is I don’t even have to run till the end of time. All I have to do is run . If it returns “will not halt”, then Goldbach’s conjecture is true or else it is false! What more evidence do we need than to know all even numbers greater than two can be written as sum of two primes? LOL!

Similarly, I could use to prove everything and anything!

This statement is false

Consider the above statement. If it were true, then what it says is the case; but what it says is that it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.

Thus the truth value of the statement is undecidable. More embarrassingly, there is no method/process/algorithm/procedure to determine whether or not such statement are true or false.

This is called the liar’s paradox. I used it in the previous post. We couldn’t decide whether or not the new binary number we created was in the set of all infinitely long binary numbers. Since we could construct such a number but not count it, we were forced to accept that there are more infinitely long binary numbers than there are natural numbers.

(To be continued…)

Cantor’s Diagonalization Argument

I need to say something, but to understand that you need to know this.

Let us enumerate all possible infinitely long lists of binary numbers, and try to count them using natural numbers on the left:

Let us make a new infinitely long binary number like this: Take the th bit of the th binary number, flip it, and write it down as the th bit of our new binary number. In other words lets the take the complement of the binary number represented by the diagonal of the above list.

Let us for a second, assume that this new binary number is  the th number in the list of all binary numbers given above. But Hey! If that were the case, then the th bit of that th number in the list would be equal to the th bit of the new number we have created. But that is not so. Because we flipped the th bit of the th number to get the new number we created.

So we have no choice but to accept that this new number is not in the list of all infinitely long binary numbers we tried to count using natural numbers. But did we not define the list as the list of all possible infinitely long binary numbers? Yet the new number can’t be in the list because we made it that way.

This is the essence of proof that there are actually more infinite sequences of ones and zeros than there are natural numbers.

Neat, huh?

(To be continued…)