Is there an answer to any question?

Is there an answer to any question? Or to be more accurate – is there a deterministic answer to any question?

I was reading some things about prime numbers, and I found out that the number 232582657-1 is considered to be a prime. But in what sense is it a prime number? Is there any mathematical proof? Can I see the proof? Has it been checked with computers? No computers are completely reliable. Or maybe it was checked by humans? Very smart humans? Even the smartest humans are able to make mistakes, too.

So there is a probability that this number is prime. This probability may be very high, almost infinitely close to 1 maybe. But there is a probability that this number is not a prime, too.

Is there a definite answer at all, for any number, whether or not it is prime? It seems to me that it depends on the definition. Gödel proved that any theory of numbers cannot be proved to be inconsistent. It’s not complete. Not everything can be proved. So I guess there is a number for which there is no proof whether it is or is not a prime.

It makes sense to me. If we get exponentially close to infinity, it gets harder to prove whether a number is or is not a prime. My conclusion is that the uncertainty principle applies not only to physics, but to mathematics too. Or more generally speaking – it applies to anything, whether real or imaginary, whether exists in reality or not.

It occurs to me that there is an uncertainty in any definition, any question, any answer, any algorithm and any single step in computers – whether real computers or theoretical ones. The existence of positive integer numbers, such as the natural numbers, is just an assumption. It cannot be proved ad infinitum. It is taken for granted. It is an axiom. The closer we get to infinity, the harder it is to know whether two given numbers are equal. For large enough numbers, the answer whether n and n+1 are equal or not might not be deterministic. I’m not referring only to numbers we can represent with computers, but to theoretical numbers too. If there is no deterministic answer whether two numbers are equal, there is no deterministic answer whether they are prime, too.

So there is some uncertainty in every question. The uncertainty whether a given number is prime or not is just an example. But if there is uncertainty whether 232582657-1 is a prime, there is uncertainty whether 2 is a prime, too. It might be a very tiny uncertainty, it is easy for us to ignore, because improbable things don’t happen so often. We can define it to be an axiom. But there is no proof that there is no contradiction. The square root of 2 might be a natural number, too.

In what sense can it be “natural”? In the same sense it is “real”. Our definition of what is real and what’s natural is just intuition. It depends on our logic, from real life experience. We assume that any “real” number is either “natural” or not. But we know that a Turing machine can’t decide whether a given Turing machine defines a real number, and therefore it can’t decide whether it defines a natural number or not. It’s as hard as the halting problem. So why can we do that? Because we don’t put ourselves into the equation. We forget to check how our question is affecting the answer. Would the answer be different if we ask the question again? After we already received one answer? This is what the uncertainty principle is all about. Checking where a particle is affects the particle, looking at something affects it too. There is always uncertainty, sometimes small, sometimes big, and even the uncertainty itself cannot be calculated without uncertainty (ad infinitum).

I don’t think God plays dice. The uncertainty principle in itself is just an assumption, and there is some uncertainty in it, too. Determinism and randomness might not be two separate things, they might be identical. God can be one, and zero, and infinite and two and the number i at the same time. A cat can be both dead and alive, and also a human. Anything can exist and not exist at the same time. Reality is too complicated to put it into equations. Reality might be a very simple thing, too.

Consider this question: an infinite sequence of bits (0 or 1) is randomly generated. Can they all be 0? Can they all be 1? It can be proved by induction that they can be all 0, although the probability might be 0 too. Is the question whether they can be all 0 a deterministic question? And how do we define random? If the first 232582657-1 bits are 0, is it a random sequence? And how do we count so many bits, too?

The conclusion is that we can’t define determinism and randomness, not even in theory. We can make assumptions, the assumptions are true most of the time. But there is no way to know anything without uncertainty (even this sentence is itself an assumption). There is no way to tell whether a cat is dead or alive. Infinite order leads to infinite chaos, and infinite chaos leads to infinite order, too. Knowing everything and knowing nothing is the same thing. One is equal to zero, and they are both equal to God, too.