Changes

As you all probably noticed I didn't post anything for a long time. The main reason for this is that I was overloaded with homework. Also, the things I wanted to do started to pile up. The unread count in google reader went to about 800, sent pages on SU are now at over 100 etc. I also have unanswered comments that piled up..
In the following days I hope to clean this all up, and continue to post more regularly. Also in order to prevent this pile up from happening in the future, I decided to take some measures. Those that are relevant to this blog are written below:

1. Removing the skribit widget - while I got some votes and even one suggestion from it, it is not active enough to be really useful. Update: I will of course firstly write about all the topics that are currently in the skribit widget.
2. Closing the comments for antonymous users - I don't get too much comments, but anonymous comments are somewhat annoying when you have two or more such comments on one post.
3. Closing the comments on my kiba dock posts - there are lots of comments there wich ask for help, and I don't really have the time to help people with computer problems over the internet.

Now, if you have an opinion about this changes you are welcomed to voice it. As of now none of the changes are made, and if enough people will comment saying that they want things to remain as they are now, I will reconsider the changes.

The future

Found on the web - Pizza 2015:

Operator: "Thank you for calling Pizza Hut. May I have your..."

Customer: "Hi, I'd like to order."

Operator: "May I have your NIDN first, sir?"

Customer: "My National ID Number, yeah, hold on, eh, it's 6102049998-45-54610."

Operator: "Thank you, Mr. Sheehan. I see you live at 1752 Meadowland Drive, and the phone number's 494-2399. Your office number over at Lincoln Insurance is 745-2302 and your cell number's 266-2566. Which number are you calling from, sir?"

Customer: "Huh? I'm at home. Where d'ya get all this information?"

Operator: "We're wired into the system, sir."

Customer: (Sighs) "Oh, well, I'd like to order a couple of your All-Meat Special pizzas..."

Operator: "I don't think that's a good idea, sir."

Customer: "Whaddya mean?"

Operator: "Sir, your medical records indicate that you've got very high blood pressure and extremely high cholesterol. Your National Health Care provider won't allow such an unhealthy choice."

Customer: "Dang. What do you recommend, then?"

Operator: "You might try our low-fat Soybean Yoghurt Pizza. I'm sure you'll like it."

Customer: "What makes you think I'd like something like that?"

Operator: "Well, you checked out 'Gourmet Soybean Recipes' from your local library last week, sir. That's why I made the suggestion."

Customer: "All right, all right. Give me two family-sized ones, then. What's the damage?"

Operator: "That should be plenty for you, your wife and your four kids, sir. The 'damage,' as you put it, heh, heh, comes to $49.99."

Customer: "Lemme give you my credit card number."

Operator: "I'm sorry sir, but I'm afraid you'll have to pay in cash. Your credit card balance is over its limit."

Customer: "I'll run over to the ATM and get some cash before your driver gets here."

Operator: "That won't work either, sir. Your checking account's overdrawn."

Customer: "Never mind. Just send the pizzas. I'll have the cash ready. How long will it take?

Operator: "We're running a little behind, sir. It'll be about 45 minutes, sir. If you're in a hurry you might want to pick 'em up while you're out getting the cash, but carrying pizzas on a motorcycle can be a little awkward."

Customer: "How the heck do you know I'm riding a bike?"

Operator: "It says here you're in arrears on your car payments, so your car got repo'ed. But your Harley's paid up, so I just assumed that you'd be using it."

Customer: "@#%/$@&?#!"

Operator: "I'd advise watching your language, sir. You've already got a July 2006 conviction for cussing out a cop."

Customer: (Speechless)

Operator: "Will there be anything else, sir?"

Customer: "No, nothing. Oh, yeah, don't forget the two free liters of Coke your ad says I get with the pizzas."

Operator: "I'm sorry sir, but our ad's exclusionary clause prevents us from offering free soda to diabetics."

Tautology and theories

What is a tautology? Simply put it is a statement that is always true. More specifically, it is a statement that is true because of its structure. Usually such a statement is not informative. The easiest way to explain this is by example, so lets look on the following statements:

1. All apples are round.
2. All apples are either round or not round.

I have never seen an apple that wasn't round, so the first statement is a correct one. However it is not a tautology. It is perfectly possible for a square shaped apple to exist, you just need to make it grow inside a box. Therefore this statement is not always true.
The second statement is always true, and therefore a tautology, but it doesn't say anything. That is, if all we know about apples is that an apple is either round or not round we don't know anything about apples.
It doesn't mean that tautologies are useless, they have both use and importance in certain cases.

Lets look on tautologies in a more formalistic way. In the example above I assumed we have an object "apple" and a property "round". However, this is not necessary. All we need to write this example is two symbols: P, Q. If we rewrite the example using this symbols we get:
1. P->Q
2. P->(Q(or)(notQ))

I don't want to explain the notation, if you don't know it you are welcome to use wikipedia.
This notation is far better that the previous one. With this notation we are free to chose the meaning of the symbols. For example we can suppose that the meaning of P is x=7, and the meaning of Q is x+4=0. In this case 1 is usually false (but not always) but 2 is true.
From this we see that the second statement is completely independent from reality. It doesn't matter if we use it to tell something about apples or about mathematics, it will always remain true. This is the true meaning of being a tautology.

Sometime ago I read a post that claimed that all theories are tautologies. From the above it should be obvious that this is wrong. But there is a certain moment that causes this confusion. A theory is basically a collection of statements (theorems) that can be logically concluded from a certain set of prepositions. Those prepositions are divided in two groups - tautologies and axioms. (We don't really need to include tautologies, but because they are always true they get included automatically). Axioms are not tautologies. They are just "rules" that we choose by ourselves. In physics, those rules are based on reality and experiment, but this doesn't have to be the case. For example - the parallel postulate of Euclid. It seems natural to us to think that it is correct. But it is possible to built a geometry without it. Naturally in such a geometry all the statement that were derived from the parallel postulate are no longer correct (they might be correct, but not necessarily).

To sum up this discussion, a theory is correct as long as the axioms from which it was built are correct. However, if we try to apply it to a "world" in which the axioms are not true, the theory will not work. By the way, this is a major problem for physics and economics. They can build wonderful theory only to find out that the world we live in is different from their initial assumptions. In mathematics this is not a problem because there is no desire to describe our world, but to built a mathematical structure.
This also makes easy to explain the statement that a theorem once proved is true forever - a theorem is dependent only on its own conditions, so as long as they are satisfied the theorem will always be true.