3 ms·

So long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.

lisper 7y ago
Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.
- dchyrdvh 7y ago
  We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.
  - lisper 7y ago
    Proofs must be finite so there can only be countably many of them.
    - dchyrdvh 7y ago
      The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.
      - lisper 7y ago
        > The same word can have infinitely many meanings. But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.
        dchyrdvh 7y ago
        What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.
        lisper 7y ago
        > What's the definition of "set"? It is a function from objects onto booleans. > what's the definition of the implication symbol The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions. Next question?
        dchyrdvh 7y ago
        Well, try to define a function without the recursion to sets.
        lisper 7y ago
        Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different. But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.

lisper 7y ago

Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.

dchyrdvh 7y ago
We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.
- lisper 7y ago
  Proofs must be finite so there can only be countably many of them.
  - dchyrdvh 7y ago
    The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.
    - lisper 7y ago
      > The same word can have infinitely many meanings. But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.
      - dchyrdvh 7y ago
        What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.
        lisper 7y ago
        > What's the definition of "set"? It is a function from objects onto booleans. > what's the definition of the implication symbol The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions. Next question?
        dchyrdvh 7y ago
        Well, try to define a function without the recursion to sets.
        lisper 7y ago
        Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different. But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.