„Wovon man nicht sprechen kann, darüber muss man schweigen“
but if the thing can be interacted with, it can usually be mapped and defined and then named. But there will always be things which do not have names, at least in mathematics -- think of the real numbers.
I suppose if the number of nameable things is countable (because humans can only enumerate, and it is humans who name), then it trivially follows that some real numbers are unnameable. Which proves the existence of such entities.
The argument is something like the set of possible thoughts about physical objects is larger than the physical objects themselves. It feels similar in flavor to the idea of unameable objects.
There's something special about a name. The name of the God of the Bible is special. Christians are to call upon _the name_ of the Lord. We pray, hallowed by _thy name_. It is somehow denotes the summary essence of the thing being named, even if it doesn't give specific details of its characteristics.
If we assume the real numbers exist, then perhaps the paradox resolves because there are uncountably many reals and only countably many nameable things, but then perhaps the paradox does not resolve because we assume ZFC is true and we can well order the reals, hence name the first unnameable real.
The reals can be ordered, just use x < y. I think you mean that if ZFC is true, we could enumerate unnameable reals (choose one with the axiom of choice, remove it, choose another one, etc.), but you could not enumerate them all. But it is true that you could get a "first" unnameable real.
That ordering is not a well-ordering, which is what the GP specified. A well ordering requires that every non-empty subset has a smallest element. That's not true for the reals ordered by x < y: for example, the set of all reals > 0 has no smallest element.
No one has explicitly shown that the reals can be well ordered, but it's a consequence of the axiom of choice that every set can be well-ordered. So in ZFC there must be a well ordering of the reals, even though no one has found one. Issues like this are why not all mathematicians accept the axiom of choice.
Not a mathematician, so this question may be a bit thick. I see the problem with the set of reals > 0, but is it perhaps that in this case > 0 is the problem and for sets specified as >= 0 it's fine because 0 is a nameable real and the smallest element. Obviously you can't just exclude certain expressions arbitrarily though, so I don't know how you could justify that mathematically.
A well ordering on a set is a total order such that all subsets have a minimum element with respect to this order. The standard ordering of the reals is not a well ordering, but the axiom of choice is equivalent to the statement that all sets possess a well-ordering. A well-order of the reals would probably look pretty chaotic though.
But then by describing it you are committing it to a set of conditions this unnameable thing satisfies.
But then if you go beyond a narrow interpretation of that paradox and accept that naming and describing are both accomplishing the same fundamental thing, that being committing a thing to a condition (like a name) or set of conditions (like a description), you do run into the same problem.
Hmm, interesting. Now back to this E2E testing stuff I've been avoiding.
In mathematics, there are infinitely many "computable" numbers. That is, numbers which can be describe using any mathematics available. Then there are far more "non computable" numbers, which can't be described by anything finite.
Names are like variables in a function. you can name variables anything you want from a human understanding point of view (final cause), but the compiler doesnt care about that. The compiler only cares about the efficient cause of that variable in the sense of what it represents (stack/heap etc).
You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers.
Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?
Sounds akin to the complexity inherent in cellular automata. We know via the rules how to mutate successive generations, but backwards propagation, algorithmic simplification, etc may exist but not traceable from any given ruleset.
but if the thing can be interacted with, it can usually be mapped and defined and then named. But there will always be things which do not have names, at least in mathematics -- think of the real numbers.
This reminds me of a YouTube video I watched this week titled "A counting argument for why mind comes before matter": https://youtu.be/AtduNjJV-6E?is=nBZ9ztZsyeoVhCj1
The argument is something like the set of possible thoughts about physical objects is larger than the physical objects themselves. It feels similar in flavor to the idea of unameable objects.
There's something special about a name. The name of the God of the Bible is special. Christians are to call upon _the name_ of the Lord. We pray, hallowed by _thy name_. It is somehow denotes the summary essence of the thing being named, even if it doesn't give specific details of its characteristics.
There are some that unnameable with my mathematical understanding, but that's not saying much.
That ordering is not a well-ordering, which is what the GP specified. A well ordering requires that every non-empty subset has a smallest element. That's not true for the reals ordered by x < y: for example, the set of all reals > 0 has no smallest element.
No one has explicitly shown that the reals can be well ordered, but it's a consequence of the axiom of choice that every set can be well-ordered. So in ZFC there must be a well ordering of the reals, even though no one has found one. Issues like this are why not all mathematicians accept the axiom of choice.
But then by describing it you are committing it to a set of conditions this unnameable thing satisfies.
But then if you go beyond a narrow interpretation of that paradox and accept that naming and describing are both accomplishing the same fundamental thing, that being committing a thing to a condition (like a name) or set of conditions (like a description), you do run into the same problem.
Hmm, interesting. Now back to this E2E testing stuff I've been avoiding.
I think there is some analogy to be made here.
1) let x be a thing
2) I name x "Jeff"
3) all things are nameable (from 1 and 2)
another way to put this is that it's natural to take the paradox as a reductio.
Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?
Where is it a proof that there are many unnamed things ? I could only see it as an argument that there are no "unnameable" things.
This is the stuff of magic and folklore, and neatly resolved by Ursula K. LeGuin in _A Wizard of Earthsea_.