[deepsun]

What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?

What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?

What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?

I just making counter-examples to Jules Richard paradox formulation, showing that it's not properly formalized, but rather expressed in vague language, so cannot be considered strict mathematics.

[E6300]

> What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?

Would you mind producing such a phrase?

> What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?

> What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?

You may as well ask whether someone understanding a theorem has any bearing on its truth value. A well-formed sentence in a formal language has one and only one meaning regardless of whether a specific person understands it. That's the whole point of using a formal language. If meaning is actually somehow bound to its interpretation then communication of even the simplest of mathematics is totally impossible.

[deepsun]

> Would you mind producing such a phrase? Sure: "Current quarter of the Moon". Or "How Giants scored last season". Or "One, if P=NP, zero otherwise".

Well, I see your point, if we limit our phrases to only formal language, then you're right.

[E6300]

> "Current quarter of the Moon". Or "How Giants scored last season"

These sentences describe functions, not numbers.

> "One, if P=NP, zero otherwise".

This value is a constant. It doesn't change through time. We simply don't know what it is.

[deepsun]

> These sentences describe functions, not numbers.

Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2. Even it form suggests that we apply function "square root" to number 2. Less obvious example: π. It's also a relation between diameter and circumference.

If we go even deeper, trying to understand what, for example, number "2" means, we'll come up to their definition through functions (or classes) of equivalence of sets cardinalities (cardinal numbers), or order (ordinal numbers), where sets are defined using ZFC, for example.

I met this discrepancy while naively trying to program mathematical logic in college. In mathematics you just "take" a number, while in programming you "create" or "instantiate" a number, and it has a ton of consequences.

>> "One, if P=NP, zero otherwise".

> This value is a constant. It doesn't change through time. We simply don't know what it is.

Not necessarily a constant, as it's not proven to be a constant yet. As an example of how it may turn out to be non-constant, check out Continuum hypothesis proof (https://en.wikipedia.org/wiki/Continuum_hypothesis) : solution is independent of our axiom set. In other words -- we can state that ℵ1=c, or we can say ℵ1!=c, and both statements will not contradict anything else. We'll just get two different models, with one more axiom each.

[E6300]

> Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2

√ is a function. √2 is the result of applying √ to 2, which is a number. Since "how Giants scored last season" is a function, it can be applied to a value such as "2017-04-17", therefore "how Giants scored last season, and today is 2017-04-17" would be a number.

I don't follow your P=NP argument. Either P=NP or P!=NP. They can't both be simultaneously true. I also don't follow how the continuum hypothesis is relevant. = is not comparing the cardinalities of P and NP, it's asking whether they're the same set (i.e. all elements of P are in NP and vice versa). Again, either they're the same set or they're not. Even if there was a cardinality between that of N and that of R, I don't see how that would change how we compare sets for identity.