Interesting or true?

[A_Seagull]

Godel has shown us that ‘truth’ is a higher notion than ‘provability’ there are truths within a system that are not deducible from the axioms. ...

Isn't that 'some' systems?

Prime numbers are one example.

You can validate if something is prime, but you can’t seduce primes.

We can't calculate primes?

Nope. There is no simple formula for primes. In order to find primes one needs an algorithm to generate possibilities and evaluate them.

[TimeSeeker]

Isn't that 'some' systems?

No. ALL formal or informal systems. Including the one that's in your head which you call 'thinking'.

We can't calculate primes?

1. Define calculate in a way that is different to the way I define 'validate'.
2. No, we cannot. We can enumerate primes and we can validate primes.

If I told you to give me the 523452th, 8887682345124352th and 1983742897235234203862649866234597672364th prime number
you are gong to be busy for a very long time LOOKING for them, NOT calculating them!

[Arising_uk]

No. ALL formal or informal systems. Including the one that's in your head which you call 'thinking'. ...

Is Propositional Logic a formal system? If so I find it impossible to believe that there is a 'truth', i.e. a formula that exists that cannot be deduced from it's axioms. Can you give me an example of such a one?

1. Define calculate in a way that is different to the way I define 'validate'.
2. No, we cannot. We can enumerate primes and we can validate primes.

If I told you to give me the 523452th, 8887682345124352th and 1983742897235234203862649866234597672364th prime number
you are gong to be busy for a very long time LOOKING for them, NOT calculating them!

Ah! I understand, so counting is not calculating?

[Arising_uk]

Nope. There is no simple formula for primes. In order to find primes one needs an algorithm to generate possibilities and evaluate them.

Thank you, my Maths is terrible. Just to understand can you give me an example of a simple formula that produces a number with some unique property that doesn't need to be evaluated as having that property?

[TimeSeeker]

Ah! I understand, so counting is not calculating?

Well, both counting AND validating prime numbers are procedures. Or as computer scientists call them - algorithms.

The difference is that one algorithm is really simple.
1,2,3,4...

And the other isn't. The difference is computational cost. How much time/energy it will take you to get the answer.

https://en.wikipedia.org/wiki/Computati ... ity_theory

[Arising_uk]

post_id=379512 time=1539854248 user_id=16353]

Well, both counting AND validating prime numbers are procedures. Or as computer scientists call them - algorithms.

The difference is that one algorithm is really simple.
1,2,3,4...

And the other isn't. The difference is computational cost. How much time/energy it will take you to get the answer.

https://en.wikipedia.org/wiki/Computati ... ity_theory

Thank for your reply.

Even if it is costly this does not rule out that there is a way of calculating primes from axioms then?

[TimeSeeker]

Even if it is costly this does not rule out that there is a way of calculating primes from axioms then?

That violates the very definition of what a prime number is. The 'axioms' of a prime number would be its factors. And if it has factors - then it's not a prime

It relies on the very asymmetry of a process that is easy in one direction and very difficult in the other.

Quantum computation can address this by making it cheaper: https://en.wikipedia.org/wiki/Shor's_algorithm

[Arising_uk]

That violates the very definition of what a prime number is. The 'axioms' of a prime number would be its factors. And if it has factors - then it's not a prime ...

Pardon me for my ignorance of Maths once again but I'd have thought a Prime has two factors?

[TimeSeeker]

That violates the very definition of what a prime number is. The 'axioms' of a prime number would be its factors. And if it has factors - then it's not a prime ...

Pardon me for my ignorance of Maths once again but I'd have thought a Prime has two factors?

Well yes. 1 and itself. But every number divides by 1 and itself (except 0 and 1....). That's a given.

A prime has no OTHER factors, except 1 and itself.

[Arising_uk]

I guess what I'm trying to understand is what's the difference between a 'procedure", a set of logical axioms and an algorithm? As I'm pretty sure I could recreate the procedure for finding primes in Prolog and that would be a set of logical axioms?

[TimeSeeker]

I guess what I'm trying to understand is what's the difference between a 'procedure", a set of logical axioms and an algorithm? As I'm pretty sure I could recreate the procedure for finding primes in Prolog and that would be a set of logical axioms?

The distinction is technical rather than semantic, so let me switch gears. An axiom is something that cannot be doubted. You accept it or you reject it - for reasons that are your own. 1 + 1 = 2. Axiom. You could just as well accept an axiom which says 1+1 = 3. What you are accepting isn't the axiom as such, but the RULES which come married to the axiom. The RULES, which WHEN violated indicate that a contradiction has occurred.

And so if you do some calculation and you get a result that says 1+1 = 4, but the AXIOM says 1+1 = 3 then that is against the rules of the system.
That is all that a contradiction means: rule violation.

And the question of "what AXIOMATIC RULES should I accept?" is rather difficult to answer without appealing to a false authority

Lets take the black box as a fundamental building block or all systems we speak about:
https://en.wikipedia.org/wiki/Black_box

The generic model is: input->MAGIC->output

Even logic itself is a system so logic can be modeled as a black box.

You can think of logic as:

input->MAGIC->(True,False)

Or you can write a system which determines if a number is odd or even:

Input: N
Magic:
if reminder of (N/2) = 0 then output(even)
else output(odd)
Output: (even, odd)

or if the number is prime:

Input: N
Magic:
(some algorithm for testing prime numbers here)
Output: Prime, Not-prime

The point is that every one of those algorithms produces a BOOLEAN response.

True/False
Even/odd
Prime/Not-prime

And the statistical name for any magical box that classifies N inputs into 2 categories is a Binary classifier: https://en.wikipedia.org/wiki/Binary_classification

You can have a black box with 1 in put and 1 output (tautology)
You can have a black box with 1 in put and 2 outputs (ambiguity)
You can have a black box with M in puts and N inputs

The point is that the black box maps the inputs to a set of outputs.

Which is why it is commonly known as a transfer function: https://en.wikipedia.org/wiki/Transfer_function

[A_Seagull]

Nope. There is no simple formula for primes. In order to find primes one needs an algorithm to generate possibilities and evaluate them.

Thank you, my Maths is terrible. Just to understand can you give me an example of a simple formula that produces a number with some unique property that doesn't need to be evaluated as having that property?

2n+1 is an odd number for all integer n.

Incidentally there have been suggestions for possible formulae for primes along the lines of 2**(2n+1)-1 for n as an integer but they fail for high values of n.