Philosophy of Everything

Mathematics is an useful model for reality. However, most inventions that claims that use mathematics are only inspired by mathematics. Sometimes the mathematics was created after the fact. Encryption is 100% based on mathematics, most of the other technologies can be imagined without advanced mathematics - while arithmetic is always useful. Logic is just a tiny bit of mathematics that governs the binary gates used in computers. But nobody makes a complicated integral to decide how tight the electronical gates can be placed - this is based on experiments AFAIK. We still rely on trial and error for most of our inventions. And this is for a good reason, mathematics cannot provide the answer except for models that we already derived from experience. There are many non-linearities in real life that cannot be derived mathematically - take for example the friction of a body with air on very high speed. AFAIK we cannot mathematically calculate the portance of a certain airplane wing except by...

Russel's paradox is defining "the set of all sets that don't contain themselves". You can show that such construct leads to a contradiction. Read the wiki link for details. I started this article as answer to this article . There is another interesting paradox discussed by Bertrand Russell: "The smallest positive integer not definable in under sixty letters", actually better known as Berry's paradox  You can also prove that such number creates a contradiction, because you just defined it under sixty letters. For me, the above paradox shows that it is not enough to create a condition for something to exist with that condition - or a set of such elements. Such paradoxes argue that such objects defined by a predicate cannot exist sometimes.

In my 5'th grade, the new professor of Mathematics told us a puzzle: there is a jug with water staying in the desert, and a bird wants to drink some water. However, the jug have a tight neck, like the old Greek pottery for olive oil: it is like a sphere at the bottom, becoming tight while going up, only to enlarge a bit at the entry. At least this is how I imagine it. What is clear, the bird cannot reach the water because the head/beak are too big for the jug entry. How can the bird drink some water from the jug?

This article contains my personal reflections on irrational numbers, especially the informational property of their infinite strings of decimal - strings that often look close to random at a shallow look. It starts with an introduction of known facts, it continues with some less discussed properties and finishes with some philosophical considerations. Introduction Example of irrational numbers are numbers like √2 and ℼ (PI). Irrational numbers cannot be represented as a fraction like, 0.25 is represented by 25/100 or 1/4. Such numbers have an infinite number of decimals that are not repeating any groups we would take. Numbers like 0.(321) are not irrational, even if they have an infinite number of decimals, because they are repeating: 0.321 321 321 321 ... Such numbers with repeating decimal groups can be represented as a simple fraction: 0.321 = 321/999 an therefore are considered "rational". Rational numbers can be represented as fractions (like 234/789), and the...

Please check my informal introduction to "Kolmogorov complexity" if you are not familiar with the subject. In this article I will approach some less discussed ramifications of it and hopefully some new ideas: an example of a Kolmogorov compression that would require an arbitrary high amount of working memory before shrinking to the compressed string (conjecture). 1. Root cause why "Kolmogorov complexity" is uncomputable Kolmogororov complexity is uncomputable because Turing machines have a halting problem :) Let's start with the observation the the Halting problem of a Turing machine having a limited band is actually decidable, unlike the infinite band one. A "finite band Turing machine" is actually a finite state machine that will either stop without going through the same state, or will cycle infinitely. Having a limited number of possible states, it cannot run too much without repeating a previous state. As soon as the machine returns to...

I define intelligence as the capacity of one entity to create complex internal representations that can be used to predict about it's environment the likely outcome from a particular configuration or action, based on the regularities of the environment. Entity can be organic life form or (potentially) a natural or artificial inorganic intelligence. The "can be used" instead of "uses" also covers a case of "abstract only" entity that constructs purely theoretical abstract model (think mathematics) and does not use it for any adaptation purposes.

" P equals NP? "  is a million dollars unsolved problem . Introduction " P" is the class of problems where we have algorithms that solves the problem in polynomial time. " NP"  is the class of problems having solutions that can be proven in polynomial time once you "guess" the solution. This is analogous with the sorting problem: it's easier to check the correct sorting of a list than to actually sort a list. Of course, if you can fully solve the problem in polynomial time, you can also prove it in polynomial time, so P is included in NP. However, NP seems to contain some very hard problems that cannot be currently solved in polynomial time. It is believed that NP contains some problems that can never be possibly solved in P. I will not do a formal proof, more like an argument or a sketch. You might have more patience and skills to develop some of these ideas even if they might have flaws in the current form. You can take it a...

In a temperate musical octave , there are 12 possible sounds. However, in a certain melody, usually, only 7 of them are used. When you see, for example, "Symphony...in Do major", you should expect that it is based on the white piano keys only. When using others scales, like for "Symphony... in Re#", some of the white keys will be replaced by some of the black keys. Let's see the 12 keys in one piano octave:

We have this requirement: "Create the smallest program that will print the full text of the play Othello by William Shakespeare". I should warn you that the requirement hides a trick and you will hate the solution. The program does not need to run efficiently. Actually, it does not need to run in reasonable time. It just has to be very short. And to be proven to answer the requirement, eventually... No external random source is allowed and no input from user.

I will argue below that sorting of a big array, using only comparison, is inherently harder than solving the problem of choosing the best next move on a chess game. One could say that it is not fair to put in competition a big, however finite, problem like chess with an arbitrary big problem like sorting n elements. Fair enough; the interesting thing is that sorting becomes inherently harder than chess for an "n" that is way smaller than you would expect, like sorting 3000 elements. At least in theory... Sorting takes a minimum of n* log 2 (n) comparisons,  more exactly  log 2 (n!) This is about sorting "by comparison only" and can be proven in 2 step. You can skip to the "chess" part if you already know this. 1. Sorting can be done in  log 2 (n!) comparisons, that is just slightly less than n* log 2 (n). We are used to consider sorting an O (n* log 2 (n)) problem. Actually there are algorithms that can do the sorting with little less t...

“There are two types of people in the world, those who divide everybody into types and those who do not”. I am part of the second type ;) Is the text above true or false?

A friend told me once that we are living in an Universe full of unprovable truths. He was referring to the Gödel's incompleteness theorems . Without being very formal, this says that there are many mathematical propositions that might be true but we cannot prove them using a computer program or any predictive way of checking, no matter how huge amount of time we can spend. We can find the proof for some, by luck, but for most of them we cannot find either a proof or disproof. In fact this "unprovable" propositions are way more many than the provable propositions. The provable propositions are like the set of natural numbers "N" and the unprovable propositions are like the set of real numbers (R).