espresso and a huge number

– (((((number of letters in wikipedia)!)!)!)!)!

– computation time, in steps, for a turing machine to compute a tsp on all atoms of universe

– one over the probability that a meteor hits my car and bounces over your car

– 10 snowflakes fall on your hand: one over the probability they are all equals

Actually i am testing the following game, one player against an another one, wins who names the biggest number. Right now the rules are the following:

– natural and finite number

– definition in english language commonly accepted by both player that effectively describes a number bigger than the previous

– the same operation or concept of number can’t be used again (exception: number you said concept is always allowed).

So a possible running of the game is:

P1: 1 (so simply naming a number is not more allowed)

P2: e^10 (exponentiation is not more allowed)

P1: the number you said plus 1 (no more plus)

P2: (((((number of letters in wikipedia)!)!)!)!)! (no more factorization and number of letters)

P1: Number you said in decimal transcription but in base 12

P2: Time in second it takes to count a number of objects as the number you said

Advertisements

[reader be aware: this is a post for getting attention from the web, you may get these information elsewhere and probably also better written]

There are a lot of proof about Cauchy-Schwarz inequality, there is also an entire book on it.

This is a proof without many advantages with respect to others, rather being very mechanic, however it is the only way someone reached my blog 🙂

Cauchy-Schwarz inequality states that:

\langle x,y\rangle \leq \left\|x \right\| \left\|y \right\|

Let’ s see that this inequality comes shortly from axioms of a vector space and from definition of scalar product

\langle\frac{x}{\left\|x \right\|}-\frac{y}{\left\|y \right\|}, \frac{x}{\left\|x \right\|}-\frac{y}{\left\|y\right\|} \rangle \geq 0

\langle\frac{x}{\left\|x \right\|},\frac{x}{\left\|x \right\|}\rangle + \langle\frac{y}{\left\|y \right\|},\frac{y}{\left\|y \right\|}\rangle -2\langle\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}\rangle \geq 0

1+1 -2\langle\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}\rangle \geq 0

and inequality follows.

The quality of this proof is that it comes out from the axioms of a vector space, unfortunately it does not bring an intuitive insight about this fundamental tool for analysis.

A paradox is a paradox?

September 14, 2007

As a computer scientist i cannot delete the previous ‘Hello World!’ post! But actually this is my first post.

I am starting this blog by explaining its title. I can remember when i attended high school that our philosophy teacher proposed us different sort of paradoxes and that no one convinced me. For example one was about Achille and the turtle , it says that if the turtle have some metres of advantage Achilles (also know as “swift-footed“) cannot catch it, the reason is that before reaching it he has to cover a half of the distance, but then another half and so on, while the turtle is still moving.

At the time we were proposed this solution:

\sum_{i=1}^{\infty} \frac{1}{2^{i}} =1,

but i was not convinced because it looked like a trick (and does not work if we say that Achilles should cover a third of the distance and so on). Right now i think at it in a very different way, no tricks, no analysis, it is simply a bug in our way of thinking; unfortunately this notion of bug is an act of faith, cannot believe it?

Let’s look to another famous paradox, a slightly modified version of the Liar paradox:

“this sentence is false”.

If it is true it is false, but if it is false it is true, ok, we cannot say it is true or it is false. Let’s say it is a paradox, that is, it has no meaning. This bring us to the very unlucky sentence ” ‘ “this sentence is false” has no meaning’ is true”, but then it is also true that ” ‘ “this sentence is false” has meaning’ is false”. If something is false it has a meaning and so the sentence can be rewritten ” ‘ “this sentence is false” has meaning’ has meaning” and so we get a contradiction with the fact it had no meaning (proof of Karl Popper).

We could solve the problem by saying ” ‘ “this sentence is false” has no meaning’ has no meaning” is true… ops I did it again! It looks like this is an unsolvable problem, but, oh my god! Can i say that it is?.

Actually someone, Kurt Godel, saved us. He said that it is NORMAL. He proved that every logical formal system has a statements that are not provable inside the system itself. In particular he introduces two statements: “this statement is not provable” and “this formal system is consistent”. The proof is technical indeed, but logic is technique!

Godel and other logicians in Europe are not the only ones who faced the matter, this stuff has been studied also on the opposite side of the world in Zen Buddhism, they are called zen koans, and their science is explained here.

[EDIT 28/09]: I am realizing that this post is a bit mystical, while the interested reader would like a technical detailed reason for which we can not say that ‘this sentence is false’ is a paradox. Saying that is a paradox is not a solution at all because introduces a new dichotomy. When we deal with this sentence we are in this situation:

true or false?

We define trueness as being where we pretend to be, and as falseness as not being as we pretend to be. For example the sentence ‘The word ‘short’ is short’ pretends to speak about reality, and it is real, while the sentence ‘The word ‘long’ is long’ pretends to speak about reality, but effectively it is not! So we say that the first is true and the second is false. In the case of ‘This sentence is false’ we have a boiling potato, or something more similar to the china syndrom, that says ‘I can’t stay here’ by the definition of falseness, everywhere we try to put it. Even if we define a new set:

true_false_no_meaningg.jpg

The problem is that the sentence is always in a wrong place and dividing the world, by default, is buggy.

Hello world!

September 11, 2007

Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!